1)Defend such that opponent is indifferent to bluffing. Meaning they cannot bluff profitably with any two and you have the best hand range possible without allowing them to bluff profitably with any two. Defend=call OR raise.
2)Bet/raise such that you have equity to call a bet plus an equal amount of bluffs. Meaning that if opponent defended properly by calling and it was checked to the river you would profit more than if you folded.
3) Tournaments mean opponents shouldn't defend as much and when they do we need more equity to have value (plus some bluffs)
Defend range of 2x pot bet:
1) 33.33%
2) 18.15%
3) 12.50%
4) 9.53%
5) 7.70%
6) 6.46%
7) 5.56%
8) 4.89%
Defend range for a 1x pot bet:
1) 50%
2) 29.29%
3) 20.63%
4) 15.91%
5) 12.95%
6) 10.91%
7) 9.43%
8) 7.41%
Defend range for half pot bet:
1) 66.66%
2) 42.27%
3) 30.66%
4) 24.02%
5) 19.73%
6) 16.73%
7) 14.53%
8) 12.83%
Equity range for 2x pot bet with 1 opponent:
1) 85%
2) 44.3%
3) 26.1%
4) 14.3%
5) 13.8%
6) 9.4%
7) 8.4%
8) 7.6%
Equity range for 1x pot bet with 1 opponent:
1) 100%
2) 100%
3) 100%
4)
5) 90%
6)
7) 80%
8)
tournament odds:
Opponents don't need to force opponents to break even in tournaments since they are willing to give up chips to fold their way up the money.
So if we risk 3 to gain 1.5 our opponents can fold more often since we are really risking some multiple of 3 to gain 1.5.
.80 tournament odds:
Opponents fold 2x pot 61.53845% of the time
Opponents fold 1x pot 44.444444% of the time
Opponents fold half pot 28.57142% of the time
Opponents fold 1/3rd pot 21.05% of the time
vs this range we need the following equity when called:
2x pot: 32%
1x pot: 26.67%
half pot: 20% equity
1/3 pot: 16% equity
1
Opponents fold 2x pot 66.66667% of the time
Opponents fold 1x pot 50% of the time
Opponents fold half pot 33.33333% of the time
Opponents fold 1/3rd pot 25% of the time
vs calling range (100%- above range) we need the following equity when called:
2x pot: 40%
1x pot: 33.333%%
half pot: 25% equity
1/3 pot: 20% equity
1.2
Opponents fold 2x pot 70.58823% of the time
Opponents fold 1x pot 54.54545% of the time
Opponents fold half pot 37.5% of the time
Opponents fold 1/3rd pot 28.57142% of the time
vs calling range (100%- above range) we need the following equity when called:
2x pot: 48%
1x pot: 40%
half pot: 30% equity
1/3 pot: 24% equity
1.4
Opponents fold 2x pot 73.68421% of the time
Opponents fold 1x pot 58.333333% of the time
Opponents fold half pot 41.17647% of the time
Opponents fold 1/3rd pot 31.81818% of the time
vs calling range (100%- above range) we need the following equity when called:
2x pot: 56%
1x pot: 46.6667%
half pot: 35% equity
1/3 pot: 28% equity
1.6
Opponents fold 2x pot 76.19047% of the time
Opponents fold 1x pot 61.53846% of the time
Opponents fold half pot 44.444444% of the time
Opponents fold 1/3rd pot 34.78260% of the time
vs calling range (100%- above range) we need the following equity when called:
2x pot: 64%
1x pot: 53.3333%
half pot: 40% equity
1/3 pot: 32% equity
Preflop defense .80 tournament odds 2x pot
1)38.46%
2)21.55%
3)14.94%
4)11.43%
5)9.25%
6)7.77%
7)6.70%
8)5.89%
Preflop defense .80 tournament odds 1x pot
1)55.56%
2)33.33%
3)23.69%
4)18.35%
5)14.97%
6)12.64%
7)10.94%
8)9.64%
Preflop defense .80 tournament odds 1/2 pot
1)71.42858%
2)46.55%
3)34.14%
4)26.89%
5)22.16%
6)18.84%
7)16.39%
8)14.50%
Preflop defense 1 tournament odds 2X pot
1)33.33%
2)18.15%
3)12.50%
4)9.53%
5)7.7%
6)6.46%
7)5.56%
8)4.89%
value vs this range:
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s
3: 44+,A9o+,KQo,A7s+,KTs+,QTs+,JTs
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs
6: 88+, AK, AQs+
7: 99+, AK, AQs+
8: TT+, AK, AQs+
value plus bluffs:
1: Any 2 (minus some bad hands)(85%)
2:22+,A2s+,K2s+,Q2s+,J2s+,T6s+,96s+,85s+,74s+,63s+,52s+,42s+,32s,A7o+,K9o+,Q9o+,J9o+,T7o+,97o+,86o+,75o+,64o+,54o,43o,32o (52%)
3:22+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,75s+,64s+,53s+,42s+,32s,A9o+,KTo+,QTo+,J9o+,T9o,98o
(29.4%)
4:22+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,75s+,64s+,53s+,42s+,32s,AJo+,KJo+,QJo,JTo (23.2%)
5: 44+,A9s+,K9s+,Q9s+,J8s+,T8s+,97s+,86s+,75s+,64s+,53s+,43s,32s,AQo+,KQo (16.4%)
6: 55+,A2s+,KJs+,QTs+,J9s+,T8s+,97s+,86s+,76s,65s,54s,43s,AKo (13.9%) - sometimes fold the weaker hands
7: 66+,ATs+,KQs,QJs,JTs,T8s+,98s,87s,76s,65s,54s,43s,32s,AKo (9.8%) - sometimes fold weaker hands
8: 77+,AQs+,A5s,KQs,QJs,JTs,T9s,98s,87s,76s,65s,54s,AKo (8.1%) (sometimes fold weaker hands)
Preflop defense 1 tournament odds 1x pot
1)50%
2)29.29%
3)20.63%
4)15.91%
5)12.95%
6)10.91%
7)9.43%
8)8.30%
equity to have value: 33%
1:22+,A2o+,k2o+,q2+,j2o+,T3o+,95o+,85o+,75o+,64o+,53o+ (85.5%)
2:22+,A2o+,K2o+,
3:22+,A2o+,K2o+,Q8o+,J8o+,T9o+,98o,87o,76o,A2s+,K2s+,Q2s+,J4s+,T6s+,95s+,84s+,74s+,63s+,52s+,43s (55.4%)
4:22+,A2o+,K8o+,QTo+,A2s+,K2s+,Q4s+,J8s+,T7s+96s+,85s+,74s+,63s+,53s+,43s (38.8%)
5:22+,A2o+,KTo+,QTo+,A2s+,K2s+,Q8s+,J8s+,T7s+,96s+,85s+,75s+,64s+,53s+,43s (35.1%)
6:22+,A2o+,KTo+,QJo,A2s+,K2s+,Q8s+,J8s+,T8s+,97s+,86s,76s,65s,54s (32.1%)
7:22+,A2o+,Kto+,A2s+,K4s+,Qts+,J9s+,T8s+,97s+,87s,76s,65s,54s (29.4%)
8:22+,ATo+,KTo+,A2s+,K5s+,QTs+,JTs,T9s,98s
value+bluffs
1:100%
2:100%
3:100%
4:77.6%
5:70.2%
6:64.2%
7:58.8%
8:39.6%
Preflop defense 1 tournament odds 1/2 pot
1)66.67%
2)42.26%
3)30.66%
4)24.02%
5)19.73%
6)16.73%
7)14.53%
8)12.83%
equity needed vs range: 25%
1.2
Opponents fold 2x pot 70.58823% of the time
Opponents fold 1x pot 54.54545% of the time
Opponents fold half pot 37.5% of the time
Preflop defense 1.2 tournament odds 2x pot bet
1)29.41177%
2)15.98%
3)10.96%
4)8.34%
5)6.73%
6)5.64%
7)4.85%
8)4.26%
tournament odds adjusted equity needed:48%
1:55+,A8o+,KJo+,A6s+,KTs+,QJs
2:66+,AJo+,ATs+
3:99+,AQo+,AJs+
4:TT+,AKo+, AQs
5:TT+,AKo+,AKs
6:AKs,JJ+
7:AKs,JJ+
8:JJ+
value+bluffs
1:22+,A2s+,K8s+,Q9s+,J8s+,T7s+,96s+,85s+,75s+,64s+,53s+,42s+,32s,A7o+,KTo+,QTo+,JTo,T9o (30.9%)
2:33+,A7s+,A5s,K9s+,Q9s+,JTs,T9s,98s,87s,76s,65s,54s,43s,AJo+,KQo (16%)
3:77+,A9s+,KTs+,QTs+,JTs,AJo+,KQo (10.6%)
4:77+,ATs+,KJs+,QJs,AQo+ (7.6%)
5:66+,A9s+,KQs,QJs,AKo (7.1%)
6:99+,AQs+,AKo (4.2%)
7:99+,AQs+,AKo (4.2%)
8:TT+,AQs+,AKo (3.8%)
Preflop defense 1.2 tournament odds 1x pot bet
1)45.454545%
2)26.15%
3)18.30%
4)14.06%
5)11.42%
6)9.61%
7)8.30%
8)7.30%
tournament odds adjusted equity needed:40%
1:22+,A2o+,K2o+,Q7o+,J8o+,t9o,A2s+,K2s+,Q2s+,J6s+,T7s+,97s+,87s,76s (48.4%)
2:22+,A2o+,K9o+,QTo+,A2s+,K3s+,Q9s+,JTs+ (30%)
3:22+,A2o+,KTo+,A2s+,K9s+,QJs (24.6%)
4:22+,A9o+,Kjo+,A2s+,KTs+ (16.7%)
5:22+,ATo+,KQo+,A9s+,KJs+ (12.5%)
6:33+,AJo+,ATs+,KQs (9.7%)
7:66+,AQo+,AJs+,KQs (7.1%)
8:88+,Aqo+,Ajs+ (5.9%)
value+bluffs
1:96.8%
2:60%
3:49.2%
4:33.4%
5:25%
6:19.4%
7:14.2%
8:11.8%
Preflop defense 1.2 tournament odds half pot bet
1)62.5%
2)38.76%
3)27.89%
4)21.75%
5)17.81%
6)15.08%
7)13.07%
8)11.54%
tournament odds adjusted equity needed:30%
Preflop defense 1.4 tournament odds 2x pot
1)26.31579
2)14.00%
3)9.68%
4)7.35%
5)5.92%
6)4.96%
7)4.27%
8)3.75%
tournament odds adjusted equity needed:56%
value
1:99+,AJ+,ATs+ (6.6%)
2:JJ+,AKo,AQs+ (3.3%)
3:JJ+,AKo,AKs (3%
4:QQ+,[AKs] (1.4%/1.7%)
5:QQ+ (1.4%)
6:QQ+ (1.4%)
7:QQ+ (1.4%)
8:QQ+ (1.4%)
value+bluffs
1:13.2%
2:6.6%
3:6%
4:2.8%/3.4%
5:2.8%
6:2.8%
7:2.8%
8:2.8%
Preflop defense 1.4 tournament odds 1x pot
1)41.6666667
2)23.62%
3)16.45%
4)12.61%
5)10.22%
6)8.59%
7)7.41%
8)6.52%
value
1:22+,A2o+,K9o+,QJo,A2s+,K7s+,Q9s+,JTs
2:55+,A9o+,KJo+,A5s,A7s+,KTs
3:77+,ATo+,KQo,A9s+,KQs
4:88+,AJo+,AJs+
5:99+,AQo+,AJs+
6:TTo+,AQo+,AQs+
7:TT+,AK,AQs+
8:TT+,AKo,AKs
value+bluffs
1:55.8%
2:27.2%
3:20%
4:13.6%
5:10.8%
6:9.4%
7:7.6%
8:7%
tournament odds adjusted equity needed:46.67%
Preflop defense 1.4 tournament odds half pot
1)58.82o353
2)35.83%
3)25.60%
4)19.89%
5)16.26%
6)13.75%
7)11.91%
8)10.50%
tournament odds adjusted equity needed:35%
J72 flop
Preflop defense 1.6 tournament odds 2x pot
1)23.80953
2)12.71%
3)8.67%
4)6.57%
5)5.29%
6)4.43%
7)3.81%
8)3.34%
tournament odds adjusted equity needed:64%
-
tournament odds adjusted equity needed:53.3%
Preflop defense 1.6 tournament odds 1x pot
1)38.46154
2)21.55%
3)14.94%
4)11.43%
5)9.25%
6)7.77%
7)6.70%
8)5.89%
tournament odds adjusted equity needed:53.3%
1:77+,A9o+,A8s+,KJs+
2:
3:
4:TT+,AKo+,AQs+
5:
6:
7:
8:JJ+
1:21.2%
2:
3:
4:7.6%
5:
6:
7:
8:3.6%
Preflop defense 1.6 tournament odds half pot
1)55.56%
2)33.33%
3)23.69%
4)18.35%
5)14.97%
6)12.64%
7)10.94%
8)9.64%
tournament odds adjusted equity needed:40%
Equity range for pot bet:
Equity range for half pot bet:
Flop equity vs
15%
with 60%
on J72
half pot bet
Aq+, AT, 22+,
23.7% with value
45% including bluffs
---------
Tournament odds divided by bubble factor equals your tournament odds.
pot odds/bubble factor=tournament odds
Bubble factor of 1.6
pot odds of 1.3125
Tournament odds of 1.3125/1.6=.8203125
Thursday, May 31, 2018
How wrong is ICM in Poker Tournaments?
Everyone knows ICM makes certain assumptions that are probably not true.
How wrong is it?
Very.
There's 2 things you need to know about poker tournaments.
1 is how fast you need to earn money to avoid all ins.
2 is how opponents would play if they adjusted a varient of GTO for the tournament odds.
Here is a variant of GTO. It assumes that it isn't worth even defending often enough to force opponent to break even as gaining chips isn't important as avoiding risk according to ICM. It also isn't worth raising very often either
For instance, rather than forcing opponent to break even in chips on bluffs, opponents should choose to force opponent to break even on reward relative to risk. The problem with this logic is risk is not linear and skill advantages can be gained as players adapt to ICM which assumes players have no skill advantage. Survival is important but risking a small percentage of your stack to gain a chip advantage or prevent opponent from getting one is worth a tradeoff. However,, risking a medium or large amount perhaps is not. ICM doesn't account for the fact that the chips you gain are only worth less than the chips you risk if the level of risk threatens your probability of survival proportionally. It does not. Many other assumptions are very far off. This is a modified GTO as described.
Preflop defense 1.4 tournament odds 2x pot
1)26.31579
2)14.00%
3)9.68%
4)7.35%
5)5.92%
6)4.96%
7)4.27%
8)3.75%
tournament odds adjusted equity needed:56%
value
1:99+,AJ+,ATs+ (6.6%)
2:JJ+,AKo,AQs+ (3.3%)
3:JJ+,AKo,AKs (3%
4:QQ+ (1.4%%)
5:QQ+ (1.4%)
6:QQ+ (1.4%)
7:QQ+ (1.4%)
8:QQ+ (1.4%)
bluff+value:
1:13.2%
2:6.6%
3:6%
4:2.8%
5:2.8%
6:2.8%
7:2.8%
8:2.8%
ICM assumes equal skill, but if everyone according to ICM they would no longer be of equal skill but instead of very different skill. If no one can call an under the gun raise of 2 times whatever is in the middle without the top 3.75% of hands, the raiser will succeed 73.66%. In fact, that is the expected success rate of a steal attempt from any position. A player who raises EVERY time it folds to him will may risk 3 to win 1.5 and that 1.5 may be worth substantially less according to ICM, such that according to ICM his opponents are forcing him to break even, but according to actual chips he wins about 2.84 big blinds per rotation. When the antes get involved if you risk 4.4 to win 2.2 you will win 4.16 big blinds per rotation on average. It may not always fold to you, but given the very tight hand ranges based upon the above, it will a LOT.If we assume a steady skill level equivalent, that is enough to win the WSOP main event without ever dipping below 60 big blinds. In other words, you will win close to 100% of the time if opponent allowed such easy stealing, especially when you add to the fact some ability to continue on good flops in small pots. That's how bad the assumption of "equal skill" and "risk applied evenly regardless of what percentage of chips you risk."
Conversely, even if you have aces and you risk it all again and again and opponents call you, your chances of elimination approach 100% as the field gets infinitely large.
So the ICM is disastrously bad.
Stealing every hand and avoiding big pots altogether isn't realistic, and with faster blind structures you need to gain chips much faster to avoid dipping below say 20 big blinds where stealing becomes increasingly exploitable to resteal shoves and even steal attempts risk chip volatility that can limit you to all in or fold. So some level of risk may have to be taken but if you can avoid it, how aggressive can you be with good hands?
How much equity do you need for 4bets, 3bets, and 4x raises and minraising at various blind levels?
To understand at what point raising actually produces NEGATIVE long term return we must determine kelly criterion breaking points.
That amount is 2x the kelly. In some situations, your kelly % is so high already that 2x kelly would put you at 100%. In those instances, you actually could go broke risking 2x the kelly but you could risk very close to 100% without a negative long term return or risk of (effective) ruin. Unfortunately that assumes chips are infinitely divisible which isn't the case. But given we are only risking a single buy in, we can still say it probably isn't a mistake to risk 2x the kelly or less especially since the action and speed of blinds may force everyone to take risks eventually. At some phase of the tournament you will trade risk of ruin for probability of higher finishes.
You may still call an all in risking more than 2x kelly, but understand ideally you wouldn't ever exceed 1x kelly. If you had 50% chance of a 1.16 to 1 payout, or 50% equity, you could risk up to 14.28% of your chipstack before it turned against you. That means a 9 big blind bet would not be acceptable at under 63 big blinds without more equity or some chance of opponent folding preflop.
http://www.albionresearch.com/kelly/
It's possible to create a more advanced kelly criterion having a full decision tree of possible outcomes and their reward as a percentage of our starting stack with a probability of that occuring but that is more advanced than I care to do at the time.
I want you to notice how limited your options get as you get shorter stacked. That creates a bizarre scenario whereby in a field of players that obey this, someone who takes risk and correctly anticipates everyone tightening up and is able to accumulate a large stack himself or herself, will get such an advantage that it may be worth some risk of ruin to at least some percentage of the time have enough chips to maintain a positive expected return over an extended period of time.
The nature of tournaments is very very unusual in this regard because at the same time, passing up positive EV is correct more so than ICM suggests due to opportunity costs.
However, passing up any positive EV scenario that is also less than 2x the kelly and probably 1x the kelly and definitely some fraction of the kelly is probably a big mistake.
So Tournament poker is both solvable and unsolvable. We can solve ICM (stupid) we can even solve a theoretical risk % in certain situations based upon assumptions on our opponents, but if our opponents react a certain way there is an infinite series of counter strategies and always different incentives for different players that in turn create different strategies and constantly adapting players based upon how others adapt.
Minraising and folding to reraises becomes acceptable down to 5 big blinds as long as you have a 60% chance of about a 1:1 payment, although you ideally want to only risk half that which means 10 big blinds. If you only have a 55% chance of success you can minraise fold at 10 big blinds, although you ideally would seek to do so down to 20 big blinds. If you could limp in and have it checked to the river, you could do so down to 5 big blinds if you were 55% to win vs one other caller but ideally 10 big blinds. With 3 big blinds you could limp in and check it down all the way down to 2.5 big blinds but ideally 5 if you had 60% chance of winning.
Of course there are always other options and different ways to try to play.
It's tricky to suggest a 3bet strategy without reworking a lot of the numbers.
But if we wanted to raise over a few limpers and risk 5.2, we could do it down to 25 big blinds with a 55% chance of success or more and down to 13 big blinds with 60% chance of success. But ideally we down to 50 big blinds with a 55% chance of success and 25 big blinds with a 60% chance of success.
3bets could be worked out if the goal was to win preflop, but we would need a very high success rate realistically speaking. In most cases 3bets are met with calls, however 3bet plus Cbet works a fairly high percentage of the time but takes a lot more risk, risks potential 4bets and you still will get called or raised on the flop. When you are called on the flop you may still win but a large percentage of the time it will require you to call at least one other bet. This makes looking at risk management strategies very difficult.
But it can be done.
If we look at a 9BB 3bet plus a 9.75 Cbet and we think our opponent will fold preflop 40% of the time, will raise us off our hand 10% of the time, then fold on the flop 20% of the time, raise us on the flop 10% of the time, will call and make one more bet on the river which we win 8% of the time (and lose 2% of the time), and bet on the river forces us to fold 5% of the time whereas 5% of the time we check it down and win and 2% of the time we lose another big bet... we can actually determine that risking 25% of our chipstack on this series of plays (considering 18.75 big blinds as a full bet)
This means we can make this play ideally with 75 big blinds but still probably fairely safely with 37.5 big blinds. If we instead 3bet 7.5 and risk 16.5 total then we can try it down to 66 big blinds and really down to 33 big blinds. In reality the chances of success should go up slightly as our opponents also get shorter stacked and have more to lose but if i change the assumptions about how often anything happens on any street it changes the results.
If you get creative you can probably find ways to accumulate chips without big risks. You may prefer to limp raise 3bet 12 big blinds with a large percentage of your limp in hands when you have a big stack or face opponents who have a small stack to discourage raising over your limp ins so that you can limp in frequently later on. Limping in creates really small multiway pots and few people know how to play multiway correctly because it's too nuanced for most people to really know how strong of hand they need. You may find ways to keep betting really small.
ALthough your bets after the flop are very correlated to bets and hand ranges before the flop, one way to approach postflop is first start preflop with profitable steal attempts and if you are called then you can treat each decision separately. FOr instance, if you have risked 5% of your chip stack preflop, assume those chips are gone and make decisions based upon how much you have in front of you. So if you had 50 big blinds and now raised 2.5 big blinds and got called and you now have 47.5 big blinds left, if you want to place a pot sized bet of say 7 big blinds that represents nearly 15% of your stack so you need a more than 55% and less than 60% chance of success ideally, but 55% and even less than 55% is acceptable, particularly if opponent calls a and may check additional streets rather than raises. If you risk less than the pot you can do so with a lower chance of success and still avoid unnecessary risk of ruin. You may allow more bad beats by betting less or checking but you also invite bluffs and manage risks.
You may choose to play a little bit more like a limit player in tournament in terms of bet sizing. Keep the bet small on every street. The problem is you may get raised and you may have to play a big pot or you may have to take a stand or change your strategy. There is a reason Helmuth and Negreanu have so much tournament success. They one way or another do unconventional things to manage risks. People say minraise folding is horrible and exploitable but as long as they can find weak enough opponents and have strong enough hands it's fine. They say limping under 12 big blinds is silly and yet Negreanu will limp down to 7 big blinds looking for a spot postflop to get his money in or take down the pot on a semibluff. In some environments he may be able to check to the river and get paid or even minbet on the river and get paid off with a small pair or jam when he hits and get paid off as well. THe important thing is he doesn't subscribe to risk his tournament life just because it is plus EV. and even as a short stack they still find a way to preserve and even grow their chipstack. THe blinds may rise faster than they can grow their stack, but if they grow their stack grow their stack and either double up plus a nice premium of antes and blinds or go bust They will have grown their stack to ssuch a larger amount a high percentage of the time that they don't need to double up as often as everyone else. True, some players would prefer doubling up or going home or taking down enough blinds while risking their life multiple times and when they do win they'll have a big stack of chips, but repeated all in risks eventually have an enormous proobability of a bustout and unless they can convert chipstacks into wins a very high percentage of the time it isn't necessarily better. If Negreanu or Helmuth double up they can continue to accumulate chips and if they happen to double up quickly enough twice in a row they will immediately accelerate their rate of chip accumulation.
<br />
Kelly based bets vs 100% calling station or calls vs 100% betting station.
2.2+2.2+2.2=
6.6 flop
13.2 turn
26.4 river
52.8 after river betting.
2.2+3.3+6.6+13.2=25.3 risk
27.5 reward.
Vs 100% calling station or if you call down 100% aggro
Full Kelly
Pot preflop half pot postflop
1.087 to 1
54%
11.68% 216BBs /108
57%
17.4% 145BBs /72.5
60%
23.2% 109BBs/54.5
65%
32.8% 77/38.5
70%
42.4% 60/30
80%
61.6% 41/20.5
-------
2/3 streets or float plus check to induce bluffs or one stab at the pot with equal mixture value+bluffs.
2.2+3.3+6.6=12.1 risk
14.3 reward
1.1818181 to 1
52%
11.38% 106/53
54%
15.08% 80/40
57%
20.62 58.6/29.3
60%
26.15% 46/23
65%
35.38% 34/17
70%
44.62% 27/13.5
80%
63.08% 19.2/9.6
---------
1/3 streets check dry flop bet turn check river or bet flop and give up.
2.2+3.3=5.5 risk
7.4 reward
1.4 to 1
48%
10.86% 50.6/25.3
52%
17.71% 31/15.5
54%
21.14% 26/13
57%
26.29% 21/10.5
60%
31.43% 17.5/8.75
65%
40% 13.75/6.875
70%
48.57% 11.3/5.7
80%
65.71% 8.4/4.2
<br />
<br />
Repeat for pot sized bets and other decisions. Build tournament strategy that you can test based upon structure va others such as trying to double up to preserve utility vs trying to preserve life maximally vs some balance.
<br />
How wrong is it?
Very.
There's 2 things you need to know about poker tournaments.
1 is how fast you need to earn money to avoid all ins.
2 is how opponents would play if they adjusted a varient of GTO for the tournament odds.
Here is a variant of GTO. It assumes that it isn't worth even defending often enough to force opponent to break even as gaining chips isn't important as avoiding risk according to ICM. It also isn't worth raising very often either
For instance, rather than forcing opponent to break even in chips on bluffs, opponents should choose to force opponent to break even on reward relative to risk. The problem with this logic is risk is not linear and skill advantages can be gained as players adapt to ICM which assumes players have no skill advantage. Survival is important but risking a small percentage of your stack to gain a chip advantage or prevent opponent from getting one is worth a tradeoff. However,, risking a medium or large amount perhaps is not. ICM doesn't account for the fact that the chips you gain are only worth less than the chips you risk if the level of risk threatens your probability of survival proportionally. It does not. Many other assumptions are very far off. This is a modified GTO as described.
Preflop defense 1.4 tournament odds 2x pot
1)26.31579
2)14.00%
3)9.68%
4)7.35%
5)5.92%
6)4.96%
7)4.27%
8)3.75%
tournament odds adjusted equity needed:56%
value
1:99+,AJ+,ATs+ (6.6%)
2:JJ+,AKo,AQs+ (3.3%)
3:JJ+,AKo,AKs (3%
4:QQ+ (1.4%%)
5:QQ+ (1.4%)
6:QQ+ (1.4%)
7:QQ+ (1.4%)
8:QQ+ (1.4%)
bluff+value:
1:13.2%
2:6.6%
3:6%
4:2.8%
5:2.8%
6:2.8%
7:2.8%
8:2.8%
ICM assumes equal skill, but if everyone according to ICM they would no longer be of equal skill but instead of very different skill. If no one can call an under the gun raise of 2 times whatever is in the middle without the top 3.75% of hands, the raiser will succeed 73.66%. In fact, that is the expected success rate of a steal attempt from any position. A player who raises EVERY time it folds to him will may risk 3 to win 1.5 and that 1.5 may be worth substantially less according to ICM, such that according to ICM his opponents are forcing him to break even, but according to actual chips he wins about 2.84 big blinds per rotation. When the antes get involved if you risk 4.4 to win 2.2 you will win 4.16 big blinds per rotation on average. It may not always fold to you, but given the very tight hand ranges based upon the above, it will a LOT.If we assume a steady skill level equivalent, that is enough to win the WSOP main event without ever dipping below 60 big blinds. In other words, you will win close to 100% of the time if opponent allowed such easy stealing, especially when you add to the fact some ability to continue on good flops in small pots. That's how bad the assumption of "equal skill" and "risk applied evenly regardless of what percentage of chips you risk."
Conversely, even if you have aces and you risk it all again and again and opponents call you, your chances of elimination approach 100% as the field gets infinitely large.
So the ICM is disastrously bad.
Stealing every hand and avoiding big pots altogether isn't realistic, and with faster blind structures you need to gain chips much faster to avoid dipping below say 20 big blinds where stealing becomes increasingly exploitable to resteal shoves and even steal attempts risk chip volatility that can limit you to all in or fold. So some level of risk may have to be taken but if you can avoid it, how aggressive can you be with good hands?
How much equity do you need for 4bets, 3bets, and 4x raises and minraising at various blind levels?
To understand at what point raising actually produces NEGATIVE long term return we must determine kelly criterion breaking points.
That amount is 2x the kelly. In some situations, your kelly % is so high already that 2x kelly would put you at 100%. In those instances, you actually could go broke risking 2x the kelly but you could risk very close to 100% without a negative long term return or risk of (effective) ruin. Unfortunately that assumes chips are infinitely divisible which isn't the case. But given we are only risking a single buy in, we can still say it probably isn't a mistake to risk 2x the kelly or less especially since the action and speed of blinds may force everyone to take risks eventually. At some phase of the tournament you will trade risk of ruin for probability of higher finishes.
You may still call an all in risking more than 2x kelly, but understand ideally you wouldn't ever exceed 1x kelly. If you had 50% chance of a 1.16 to 1 payout, or 50% equity, you could risk up to 14.28% of your chipstack before it turned against you. That means a 9 big blind bet would not be acceptable at under 63 big blinds without more equity or some chance of opponent folding preflop.
http://www.albionresearch.com/kelly/
It's possible to create a more advanced kelly criterion having a full decision tree of possible outcomes and their reward as a percentage of our starting stack with a probability of that occuring but that is more advanced than I care to do at the time.
I want you to notice how limited your options get as you get shorter stacked. That creates a bizarre scenario whereby in a field of players that obey this, someone who takes risk and correctly anticipates everyone tightening up and is able to accumulate a large stack himself or herself, will get such an advantage that it may be worth some risk of ruin to at least some percentage of the time have enough chips to maintain a positive expected return over an extended period of time.
The nature of tournaments is very very unusual in this regard because at the same time, passing up positive EV is correct more so than ICM suggests due to opportunity costs.
However, passing up any positive EV scenario that is also less than 2x the kelly and probably 1x the kelly and definitely some fraction of the kelly is probably a big mistake.
So Tournament poker is both solvable and unsolvable. We can solve ICM (stupid) we can even solve a theoretical risk % in certain situations based upon assumptions on our opponents, but if our opponents react a certain way there is an infinite series of counter strategies and always different incentives for different players that in turn create different strategies and constantly adapting players based upon how others adapt.
Minraising and folding to reraises becomes acceptable down to 5 big blinds as long as you have a 60% chance of about a 1:1 payment, although you ideally want to only risk half that which means 10 big blinds. If you only have a 55% chance of success you can minraise fold at 10 big blinds, although you ideally would seek to do so down to 20 big blinds. If you could limp in and have it checked to the river, you could do so down to 5 big blinds if you were 55% to win vs one other caller but ideally 10 big blinds. With 3 big blinds you could limp in and check it down all the way down to 2.5 big blinds but ideally 5 if you had 60% chance of winning.
Of course there are always other options and different ways to try to play.
It's tricky to suggest a 3bet strategy without reworking a lot of the numbers.
But if we wanted to raise over a few limpers and risk 5.2, we could do it down to 25 big blinds with a 55% chance of success or more and down to 13 big blinds with 60% chance of success. But ideally we down to 50 big blinds with a 55% chance of success and 25 big blinds with a 60% chance of success.
3bets could be worked out if the goal was to win preflop, but we would need a very high success rate realistically speaking. In most cases 3bets are met with calls, however 3bet plus Cbet works a fairly high percentage of the time but takes a lot more risk, risks potential 4bets and you still will get called or raised on the flop. When you are called on the flop you may still win but a large percentage of the time it will require you to call at least one other bet. This makes looking at risk management strategies very difficult.
But it can be done.
If we look at a 9BB 3bet plus a 9.75 Cbet and we think our opponent will fold preflop 40% of the time, will raise us off our hand 10% of the time, then fold on the flop 20% of the time, raise us on the flop 10% of the time, will call and make one more bet on the river which we win 8% of the time (and lose 2% of the time), and bet on the river forces us to fold 5% of the time whereas 5% of the time we check it down and win and 2% of the time we lose another big bet... we can actually determine that risking 25% of our chipstack on this series of plays (considering 18.75 big blinds as a full bet)
This means we can make this play ideally with 75 big blinds but still probably fairely safely with 37.5 big blinds. If we instead 3bet 7.5 and risk 16.5 total then we can try it down to 66 big blinds and really down to 33 big blinds. In reality the chances of success should go up slightly as our opponents also get shorter stacked and have more to lose but if i change the assumptions about how often anything happens on any street it changes the results.
If you get creative you can probably find ways to accumulate chips without big risks. You may prefer to limp raise 3bet 12 big blinds with a large percentage of your limp in hands when you have a big stack or face opponents who have a small stack to discourage raising over your limp ins so that you can limp in frequently later on. Limping in creates really small multiway pots and few people know how to play multiway correctly because it's too nuanced for most people to really know how strong of hand they need. You may find ways to keep betting really small.
ALthough your bets after the flop are very correlated to bets and hand ranges before the flop, one way to approach postflop is first start preflop with profitable steal attempts and if you are called then you can treat each decision separately. FOr instance, if you have risked 5% of your chip stack preflop, assume those chips are gone and make decisions based upon how much you have in front of you. So if you had 50 big blinds and now raised 2.5 big blinds and got called and you now have 47.5 big blinds left, if you want to place a pot sized bet of say 7 big blinds that represents nearly 15% of your stack so you need a more than 55% and less than 60% chance of success ideally, but 55% and even less than 55% is acceptable, particularly if opponent calls a and may check additional streets rather than raises. If you risk less than the pot you can do so with a lower chance of success and still avoid unnecessary risk of ruin. You may allow more bad beats by betting less or checking but you also invite bluffs and manage risks.
You may choose to play a little bit more like a limit player in tournament in terms of bet sizing. Keep the bet small on every street. The problem is you may get raised and you may have to play a big pot or you may have to take a stand or change your strategy. There is a reason Helmuth and Negreanu have so much tournament success. They one way or another do unconventional things to manage risks. People say minraise folding is horrible and exploitable but as long as they can find weak enough opponents and have strong enough hands it's fine. They say limping under 12 big blinds is silly and yet Negreanu will limp down to 7 big blinds looking for a spot postflop to get his money in or take down the pot on a semibluff. In some environments he may be able to check to the river and get paid or even minbet on the river and get paid off with a small pair or jam when he hits and get paid off as well. THe important thing is he doesn't subscribe to risk his tournament life just because it is plus EV. and even as a short stack they still find a way to preserve and even grow their chipstack. THe blinds may rise faster than they can grow their stack, but if they grow their stack grow their stack and either double up plus a nice premium of antes and blinds or go bust They will have grown their stack to ssuch a larger amount a high percentage of the time that they don't need to double up as often as everyone else. True, some players would prefer doubling up or going home or taking down enough blinds while risking their life multiple times and when they do win they'll have a big stack of chips, but repeated all in risks eventually have an enormous proobability of a bustout and unless they can convert chipstacks into wins a very high percentage of the time it isn't necessarily better. If Negreanu or Helmuth double up they can continue to accumulate chips and if they happen to double up quickly enough twice in a row they will immediately accelerate their rate of chip accumulation.
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Kelly based bets vs 100% calling station or calls vs 100% betting station.
2.2+2.2+2.2=
6.6 flop
13.2 turn
26.4 river
52.8 after river betting.
2.2+3.3+6.6+13.2=25.3 risk
27.5 reward.
Vs 100% calling station or if you call down 100% aggro
Full Kelly
Pot preflop half pot postflop
1.087 to 1
54%
11.68% 216BBs /108
57%
17.4% 145BBs /72.5
60%
23.2% 109BBs/54.5
65%
32.8% 77/38.5
70%
42.4% 60/30
80%
61.6% 41/20.5
-------
2/3 streets or float plus check to induce bluffs or one stab at the pot with equal mixture value+bluffs.
2.2+3.3+6.6=12.1 risk
14.3 reward
1.1818181 to 1
52%
11.38% 106/53
54%
15.08% 80/40
57%
20.62 58.6/29.3
60%
26.15% 46/23
65%
35.38% 34/17
70%
44.62% 27/13.5
80%
63.08% 19.2/9.6
---------
1/3 streets check dry flop bet turn check river or bet flop and give up.
2.2+3.3=5.5 risk
7.4 reward
1.4 to 1
48%
10.86% 50.6/25.3
52%
17.71% 31/15.5
54%
21.14% 26/13
57%
26.29% 21/10.5
60%
31.43% 17.5/8.75
65%
40% 13.75/6.875
70%
48.57% 11.3/5.7
80%
65.71% 8.4/4.2
<br />
<br />
Repeat for pot sized bets and other decisions. Build tournament strategy that you can test based upon structure va others such as trying to double up to preserve utility vs trying to preserve life maximally vs some balance.
<br />
Saturday, May 19, 2018
Poker Formula
So Let's wrap up everything we know about both preflop and postflop play to construct a mathimatically sound strategy.
Preflop:
We want to raise preflop with hands that have value plus some ratio of value to bluffs.
So what has value?
Keep in mind our opponents should collectively raise of call often enough to force us to break even on our bluffs. That means that if we are raising 3 big blinds (or twice the amount of the small blind plus big blind) our opponents can call us such that they fold 2/3rds of the time. In other words, when called against optimal opponents, a hand will only have value when it has equity over a hand that given the information available has a 66.67% chance of being best. we also will mix in some bluffs.
That puts the equivalent of this range of hands
1 opponent left to act (everyone else folded) 33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
TO determine "value" and "Bluff to value" we need to determine which hands have enough equity given the blinds in the pot to break even if checked to the river. In reality implied odds and reverse implied odds play a role but we will revisit and adjust later.
So with 3x big blind and someone not in the blinds calling and both the blinds folding we risk 3 when called and lost, but win 4.5. We only need to win 40% of the time or 40% equity over the hand ranges above to have VALUE. Value meaning we extract money from the decision to raise if we are called. Bluff means a hand where we raise with even though it isn't in the "value" range.
In reality a lot of the hands like 76s, 65s, 54s, 43s, 32s I would rather have than hands like K3o that may have more equity but have worse reverse implied odds, very few semi bluffing opportunities and very few possible hands where you can ever bet large. Hands with lower equity may have more REAL value due to implied odds vs hands that mathematically using this simple metric are stronger due to reverse implied odds.
number of opponents | hands
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s
3: 44+, A9+,KQ, A7s+,KTs+,QTs+,JTs
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs
6: 88+, AK, AQs+
7: 99+, AK, AQs+
8: TT+, AK, AQs+
In reality the blinds should defend more often but at a lower cost which means the odds will be different and the value will be different, but we will have position on them and the additional probability of having a stronger hand will allow for stronger hands when we show down..
Flop Play.
The strategy after the flop depends on the flop and the bet size. There's more of a tendency to standardize preflop than there is postflop. We could also work out strategies preflop to bet larger with even better hands. Note that we are making assumptions that opponents play you "optimally". I put "optimally" in quotes because in reality the preflop decisions should be pretty radically adjusted due to postflop play and mostly due to implied odds and reverse implied odds. If a hand cannot be bet with confidence even when you pair your hand and it doesn't produce good draws that get paid off when they hit there isn't nearly as much value in it as the equity demonstrates. It may do well in limit games when betting is small after the flop or in games that check a lot post flop or have players making bad decisions like large bets.
The first rendition will be pot sized flop bets. With a pot sized flop bet you must call often enough to force your opponents to break even on bluffs. The bet frequency we will make such that you have enough equity when called to break even or better, plus we will add bluffs. But when we call we only want to call such that we have equity to do so plus whatever is needed to force opponent to break even on bluffs (preventing opponent from bluffing every single hand profitably). Ideally these will converge, if not we will have to go back to the drawing board on preflop hand selections or create lines in which we call the flop to bluff the turn or bluff raise the flop which is another set of equiations we won't cover in this post.
We technically should determine this for approximately every flop. We will approximate based upon understanding the highest card as well as determining how coordination (straight and straight draw possibilities and increased two pair probabilities) and flush potential (potential flush and flush draws) and pairing impacts a given high card flop and generalizing this understanding to all flops.
In total there will be 7 unique flops plus variations for each flop. Ace, King, Queen, Jack, Ten, 9 or 8, and 7 or less. Note I have combined the 9 high and 8 high flops. I will average the decisions together. 7 or less I will just determine what to do with a 7 high flop and assume the flops below should be treated the same.
We are assuming an opponents hand ranges are the following:
1 opponent left to act (everyone else folded) 33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
We are assuming we took the lead in betting preflop and everyone in the blinds and everyone else folded except for a caller not in the blinds. What hands have equity to call a pot sized bet (you could make one with all of these hands and perhaps more but there's probably more value in sometimes checking with both strength and weakness or betting pot with big hands and bluffs and checking with medium strength hands)
ace high flop/
1 opponent preflop. For instance, autofolders in the blinds you are in the cutoff. Any paired flop, any pocket pair, any gut shot draw with backdoor flush draw 67 or higher, Any King high with both cards higher than the other 2. (king 6 vs A25 flop)
Sometimes queen jack but the conditions are tricky
Queen ten suited with backdoor flush draw
You will be playing 43% of hands preflop.
On an A26 flop, you will have equity to continue when facing a pot sized bet with about 30% of all preflop hands which is about 70% of your opening range before we even introduce bluff raises or calls to bluff future streets or balance range of future decisions. In other words you can call a pot sized bet about 70% of the time given these assumptions, and that also means you can continuation bet with this percentage of hands plus bluffs. I won't pick out all of the bluffs to make just yet. For now our preflop hand range is very compatible with ace high flops and we could probably play more hands with one opponent if this is representative of how other flops will play. However, much the disclaimer with preflop play, play on the flop also will have reverse implied odds or implied odds.
As of now you don't have to worry about widening call range to make it impossible for opponent to exploit you by bluffing, you can solely choose to call with value and call more than enough. Smaller bet sizes will give you even better odds to call with even weaker hands. You could call a half sized bet with t8 with a backdoor flush draw for comparative purposes and jack 7 off suit.
2 omitted for time purposes
3 99+, any paired flop (even 27 on A29 flop), any open ended draw or double gutter (45o on A27 flop or QJ on an AT8 flop.). With 3 opponents left to act preflop, you will play
44+, A9+,KQ, A7s+,KTs+,QTs+,JTs or 13.4% of hands preflop. You will have a hand 9.4% of ace high flops not even including double gutters and pairing a higher card (like KT on an AT2 flop). About 70% of flops you will have a hand that can stand up to a pot sized bet again. You will be able to bet all of these hands plus bet hands for bluffs as well if you are the one doing the betting if you choose.
4 omitted for time purposes
5 omitted for time purposes
6 TT+, any open ended or double gutter. Any paired flop card around ten or higher or low pair with gutshot.
WIth 6 opponents left you will play 88+, AK, AQs+ 4.7% of hands and thus have 3.8% of hands that can call or you can call pot sized bets about 80% of the time. When you get into smaller percentages of hand ranges of opponent the specific hands chosen matter. I choose a mixture that only sometimes plays certain hands and lower pairs and fewer ace hands. If more ace hands are included then even pocket jacks aren't really good enough to call.
7 omitted for time purposes
8 qq+, a paired jack with Jack ten or higher, queen jack with double gutter and back door flush draw, open ended straight draw.
Remember, I didn't add in the "bluffs" preflop. You can work them in as long as you don't increase them such that you have to fold more than 50% of the time to a pot sized bet.
Although I omitted some hands, you can guess that not a lot will change and that 4 and 5 will be somewhere between 3 and 6 and 7 will be somewhere between 6 and 8. With 7 players left to act, I would imagine it's something like JJ+, open ended, a paired flop card of jack or higher and maybe some tens.
Although this may suggest playing more preflop hands is possible, particularly those that can flop big draws or hit unbeatable hands by the river, let's not jump to conclusions based on one flop. Also, let's remind ourselves that although we technically only need to call a minimum of half the time to a pot sized bet (or 33% of the time to a half pot bet), we will sometimes face the blinds defending and the preflop equity will be less and so having a comfortable margin of strength may be a good thing or even necessary.
Also, the first step to understanding quality poker is understanding "optimal" play or how to play vs perfect opponents (well a very simplified variation of it, anyways). The next step is understanding what to do when opponents are NOT playing optimal in order to extract more chips.
The FREQUENCY is the important thing to understand. Opponents should be able to call often enough to force you to break even on your bluffs. If they do not call this often, you can literally bluff every time and you may even want to pass up betting for value as well in favor of inducing bluffs from your opponent or bluffing on a future street. If they call MORE often you should reduce your bluffing frequency to semibluffs or eliminate it. The technical goal of bluffs should theoretically be to bluff frequently enough to break even on bluffs in order to force your opponent to call widely so as to extract more value from your value bets.
If opponents play MORE hands preflop than suggested more hands are going to have value and the amount of value it has vs opponents range is going to increase. Since you will have the ability to bet for value more, you can bluff more frequently overall without changing your bluff to value ratio. Opponents that play too many hands preflop have to have a flaw in their game somewhere. Either they make up for playing too many hands by calling all the way to the river too often, fold too much on one street vs the others, or play all streets too loosely or too tightly, or they overbet good hands and try to compensate for loose play by getting paid off when they hit or they bluff too often. Good players still generally play too many hands, they just are good at picking spots to bluff, spots to call to bluff later, spots to get paid off and avoid trouble and make up for their "suboptimal" play by exploiting bad opponents.
A good player with good information on opponents actually SHOULD play more hands than "optimal" because they will correctly recognize where the flaws opponents have, or will isolate a loose player even before they really know where the postflop flaws are.
For instance if a loose player plays 40% of all hands when they should play 10%, every hand in the top 20% of hands probably has value over this player if hands are checked to the river. So if you assumed an optimal opponent (raising 10%) you should only maybe 5% of hands plus some bluffs plus some hands that are behind but with position, equity, implied odds, etc you have a little bit of value... but you have value with about the top 20% of all hands plus you can add in some bluffs and calls with enough equity. You can dramatically widen your range and even raise to isolate yourself to play with weak hand ranges and then from there sort of play as if you started with a preflop hand that was strong enough to open up with a raise and create value against opponents that make mistakes. Once you spot how this opponent makes mistakes after the flop you may theoretically be best off playing close to 100% of all hands with this opponent because either opponents will fold too often in which case you don't have to have any equity to be able to bluff profitably, or they will not be able to fold even to very large bets and you can control the size of the pot and bet larger to the degree by which you have an edge over your opponents calling range. In practice even bad opponents make some adjustments, plus there are a lot of other players involved as well.
Turn play
Turn play PROBABLY doesn't change a lot. In theory you should assume some optimal range of hands of opponent, make sure the call frequency is enough to force him to break even and be able to shed a lot of hands that don't improve, or even weak hands to additional bets. But in practice opponents think their hand is good and keep betting it or slow down and play to catch bluffs. With an ace high flop the strength of hands is unlikely to change much aside from the gutshots, double gutters and backdoor draws. On 7 high flops, the turn can go from a 7 high board to an ace high board which totally changes the nature of the hand. I think mostly you can treat the turn like a 4 card flop and look at the highest card. Some straight and flush draw possibilities may open up on the turn or the board may pair and we have yet to see how that impacts decisions but aside from that ace high flops probably don't change a whole lot to the turn. TO the river the draws that opened up either hit or they don't but if there are only backdoor draws and you initiated the bet of the pot sized bet, and they call that shouldn't be a problem most of the time (some players may call with 67 of spades on a 7A2 board with a single spade and improve to a pair with a flush draw or on a A78 board they could pick up an open ended straight draw or even a 15 hour straight and flush draw, but these are not too common.
I'm speculating as I have yet to do the work, but what I do know for sure is a flop with low cards has a lot more ways it can change. A 7 high has about 28 cards in the deck that change the top pair card plus 9 cards that pair the board and probably a few that set up straight and flush draws too that are more likely to be of the open ended or double gutter since with only 6 cards below 7 if you have 4 of them you are bound to create some kind of connected cards or paired board.
River play
With all of the cards out and betting in prior streets the pot will have gotten inflated, hands are much more well defined and you can make large all in bets with some ratio of pure bluff to max value if you'd like. The math becomes much more clear as to how frequently to bluff vs betting for value, particularly in situations in which you either have a monster or nothing.
Bluff to value ratio:
Optimal bluff to value ratio depends on the bet size proportional to the pot. Here is the formula:
X= (B-Bz-Pz)/(By+Py-Bz-PZ)
y= opponent's equity vs value bets
z= opponent's equity vs our bluffs
B= our bet size
P= our bet size + pot
x= frequency with which our bets should consist of value
1-x= frequency with which our bets should contain bluffs.
So if pot has 1 pot in it and we bet 9 with either the nuts or a total bluff...
x= [9-9(1)-10(1)]/[9(0)+10(0)-9(1)-10(1)]
x=.526
x-1=.47
value=.526
bluff=.473
If we know on the turn that we will hit the value hand 18% of the time how often must we carry a hand that will end up being a bluff? (.18/.526)-.18= 16.2%.
If you really want to do close to optimal poker you should start with possible rivers and work backwards and then determine which hand range with the optimal frequencies assuming a bet on every street (or whatever will get you to the expected bet size on the river) will produce those ranges from the turn to the river, then the flop to the turn and then what preflop percentage.
You'd probably only really need to look at about 28 rivers and work backwards but you could get more accurate doing more. 3 variations of 7. Actually 4 if you want to include pairs and 5 if you want to include combination of both flush and straight combinations. 3 card flush, 4 card flush, some sort of connectedness of varying degrees, pairs, etc.
A more complicated and more accurate solution would include varying degrees of betting that USUALLY bets large in way ahead/way behind flops and small in draw heavy dynamic flops (where board is certain to change strength of hand with lots of draws) with the optimal bluff to value bet ratio associated with that bet size and some percentage flipping the bet size on both sides. These are "decoy" bets much in the way bluffs force opponents to widen the hand range. Decoy bets make your bet patterns harder to read and force your opponent to widen the calling range vs large bets even though it's usually an optimal frequency of bluff to value with a really strong hand, sometimes it is instead a weaker hand or draw. I only say this for completeness.
Let's look at bluff to value with bluffs having 25% equity on average (as they will contain semibluffs and backdoor straightflush draws or bottom pair or overcards or something) and value bets having 70% equity on average.
9xpot:61.4% value to 38.6% bluff
2xpot: 77.78% value to 22.22% bluff
pot size: 92.6% value to 7.4% bluff
75% of the pot and lower: 100% value to 0% bluff.
The problem with this formula is it only considers one street and then checking to the river. In reality semibluffs can become value bets on future streets and pure bluffs can also bluff again.
Also, opponent's hands can call with the ability to bluff on certain turns and rivers, and it doesn't consider how to properly deal with and make check raises. Additionally, sometimes checking is best and there usually should be some kind of balance.
Balancing ranges:
There are certain lines of play when considering multiple streets that can either lead to "blended ranges" or "mixed ranges" or "polarized ranges". Polarized ranges are either really strong or really weak. betting polarized ranges forces opponent to call with the worst of it some percentage of the time in order to prevent you from being able to bluff. Mixed ranges that have draws, weak pairs and strong hands allow you have to determine which hand to bet thinly for value or play to catch bluffs on opponents and can beat ranges and also allow you to bet different sizes with some mixture of strong and weak or blended ranges so that you can polarize your range when betting large for instance. It can get pretty complicated solving for game theory optimal (GTO) strategies when you introduce this kind of concept. But there are certain principals you can understand from a lot of the complicated solutions.
1)You should be more willing to play drawing hands with slightly less equity on draw heavy boards and even backdoor straight and flush draws or backdoor ace high flush draws if they can make "nut" type of hands like flushes and straights (even if they aren't the absolute nuts you can play with bluffing frequencies that allow you to represent flushes when you don't have it and force your opponent to have to call with worse).
2)You can use busted straight draws to represent flush cards and backdoor flush draws to represent standard flush draws to balance your bluffs and value bets on the turn or river.
3)If your hand range is more clearly definable you can mix in some percentage of weak and strong hands by varying your actions. For instance, if you only check medium strength hands that may be okay and even sometimes may be optimal, but if you do it all the time in all situations the predictability loses something in real situations even if in practice it's GTO. Sometimes that's okay, but in practice opponents usually aren't playing that close to GTO and instead they are trying to spot things they notice. So if you occasionally check with really strong hands or bet bluff catching hands on the turn and/or river and maybe even check raise instead and get other medium strength hands that would catch bluffs to fold at worse it isn't a big mistake and it may actually extract value in the long run from opponents misinterpretting you as a weak player.
Exploiting
Exploiting opponents that play with the wrong frequencies is pretty straight forward. If opponents don't bluff often enough you can fold stronger hands. If they bluff too much you can bluff catch weaker hands. If they start out with weaker hands your value range widens and your bluff range could widen if they play better after the flop. If opponent's don't call often enough you can bluff 100% of the time and play more hands preflop to be in situations where you can bluff more often. If they call too often you shouldn't ever bluff unless it's to build up a bigger pot to bluff on a future street where they fold too often. 3betting to exploit opponents is more complicated since you have to introduce the possibility that opponent will 4bet and if they call the pot may introduce more jam all in situations on the turn or river. 3 betting can be done preflop to build up a bigger pot against a suboptimal opponent, it can isolate yourself with players that play too many hands and thus you have a range of hands equity edge against. It can also build up pots when opponents fold too often to prevent you from bluffing profitably on later streets. Or it can build up pots while bluffing for opponents that call with too many weak hands even to large bets, fold too many draws and backdoor draws (that they can/should bluff or semibluff with later) and that can both define your opponent's strength and set up a big overbet jam
3betting SO far I haven't really done anything about 3betting preflop or on the flop. 3betting can be a polarized range of really weak with some amount of bluffs. You could 3bet some draws with the intention of giving up if you don't improve and possibly buying free cards or building the pot to jam by the river.
Facing 3bets
A much more simplified play that's more unexploitable than GTO is to let opponent bluff often enough such that you can 3bet and make back your loses. So if opponent bets the pot you can fold 2/3rds of the time and then 4bet on the 3rd and either make everything back or be faced with a 4 bet in which you can then fold often enough such that you can 5bet jam to make everything back. Stack sizes matter because if a 4bet is a jam, you can't afford to wait. You are basically allowing your opponent to win some share of the blinds and antes but preventing him from winning your stack as an average. If you want to neutralize opponent and prevent him from being able to bluff profitably you should call OR raise half the time. You could call at the ratio to prevent opponent from bluffing. You can call anytime you have equity to call plus you can add in some calls to bluff future streets. A more GTO solution would have more complicated % of calls and raises with a mixture of hands that sometimes call and sometimes raise to balance your ranges and allow for proper bluff to value bet ratios on future streets on certain cards.
https://forumserver.twoplustwo.com/15/poker-theory/question-developing-proper-bluff-value-ratios-1623904/
https://www.pokerstrategy.com/news/content/Do-you-have-the-right-bluff-to-value-ratio-_98179/
https://forumserver.twoplustwo.com/32/beginners-questions/how-calculate-bluff-value-ratio-jam-1513678/
http://www.runitonce.com/chatter/gto-simplified/
https://www.pokerstrategy.com/video/30133/
https://www.pokerstrategy.com/video/40552/
http://www.pokerplayer365.com/poker-strategy/advanced-strategy-how-to-use-the-value-to-bluff-ratio-to-improve-your-river-ev/
https://www.pokervip.com/strategy-articles/texas-hold-em-no-limit-intermediate/gto-aggression-in-poker
https://www.youtube.com/watch?v=j7xSHC90_Og
https://ua-video.com/sxoEWZY5hG4-ev-aggregation-and-marginal-range-building.html
https://ocw.mit.edu/courses/sloan-school-of-management/15-s50-poker-theory-and-analytics-january-iap-2015/lecture-notes/MIT15_S50IAP15_L7_GameTheor.pdf
http://ktiml.mff.cuni.cz/~bartak/ui_seminar/talks/2017ZS/SuryaChembrolu_DeepStack.pdf
https://www.reddit.com/r/poker/comments/2mhsgq/gto_is_so_much_more_than_unexploitable/
okay, king high flop.
There range:
1 opponent left to act (everyone else folded) 33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
our range:
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s
3: 44+, A9+,KQ, A7s+,KTs+,QTs+,JTs
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs
6: 88+, AK, AQs+
7: 99+, AK, AQs+
8: TT+, AK, AQs+
I am assuming they bet with 100% of their range. In practice this will not be the case, but I'm simultaneously trying to approximate how often you could bet with and still have equity vs an opponent that calls the maximum. Just because you could bet doesn't mean you always should and you probably should check some of the hands with the equity to call a pot sized bet or make one and bet some with less equity than this as well.
1: ANy open ended, any pair, any pocket pair, <b>Ace 6 or higher usually</b> (if it's KQ9 flop or something with multiple high, connected cards than no, if it's K78 with backdoor draw or K72 with 2 overcards to bottom pair and minimal chance of opponent hitting then yes), medium gutshots with backdoor flush draws and overcards to the non King cards. Bottom gutshot with outs to double gutshots or open ended like JT on K96 board as a queen makes a straight an 8 provides an open ended and a 7 provides a double gutshot giving not only 4 outs to hit straight, but 8 houts to improve from 4 outs to 8 outs plus 6 outs to at least get a pair and possibly improve from there. Technically when offsuited this JT is just below calling range where as a JT with a backdoor flush draw is just above. However I would certainly still play it due to implied odds and future equity plus you need some hands to call with to bluff later, or you can bluff raise or bluff on future streets yourself to balance stronger hands you call with and gain equity back later. Queen high with gutshot. Ace high backdoor flush draws are usually good. The lower ace hands need a little something else like a backdoor wheel draw (A4s with a 2 or 3 or 5) and/or lower cards.
[A hand like Q6s with a backdoor flush draw actually has enough equity for a HALF pot bet just for comparative purposes.]
notes: We can see that hands like QJ and QT become strong enough to call with a gutshot while weak ace high hands usually also can call. Unlike ace high flops where king high flops couldn't call unless they had 2 cards higher than the others, ace high flops can call without that condition because of the ability for an ace to beat the best hand and an ace hits our opponents a little harder than a king. The lower cards can get counterfeitted by runner runner higher 2 pair on the board. I imagine that as the flop cards get lower hands with 2 overcards or just one overcard plus posibility of having the high card vs non pair hands will increase in value
2:ommitted
3:ommitted
4:ommitted
5:any pair, any pocket pair, ace king, any open ended, AT with gutshot and backdoor flush draw
6:ommitted
7:ommitted
8:ommitted
Jack high flop:
1:Any pair, any pocket pair, any open ended draw, any ace, KTo+, K9 or higher with backdoor flush draw and 2 overs to the other two cards. King 8 with backdoor flush draw and connected card like a 7 and when the other card is like a 2 or something low. gutshot with backdoor flush draw and over cards to either bottom 2 pair like 98clubs on J72 flop with a club or 67clubs with J53 flop.
2:ommitted
3: any pair, usually any pocket pair, any open ended, AT, KQ gutshot, KTs+ gutshot, QT+gutshot with backdoor flush draw, T9 gutshot backdoor draw, 98 gutshot with backdoor flush and backdoor double gutshot/open ended (98 on J65) as a T or Q brings 8 outs to straight.
4:ommitted
5: Any pair, usually any pocket pair* (JT flops need stronger hands for weakest pairs) Any open ended, Ace King Offsuit, Ace Queen with back door flush draw.
6:ommitted
7:ommitted
8:Any Pair, any pocket pair*(JT flop need stronger pairs), any open ended, Ace King, KQ with gutshot, backdoor flush draw and low card on board like a 2.
1,3,5
1)43.3% of all hands start preflop. Hits a little over 28.5% of the time. About 65.6% of hands preflop.
There's about a 45% chance we flop something with suited connectors. Something being pair or better, flush draw or open ended straight draw. With 1 gap connectors some of the straight draws that aren't also flush draws might go down. With 2 gap they go down more. There are also some gutshots plus backdoor draws that in certain circumstances are okay. I'm going to say I can add bluffs preflop up until the point where I can no longer stand calling a pot sized bet vs an opponent that plays as described. Since there is a 40-45% chance that any suited hand I add will be enough to continue I can add 2-2.5 times more hands than it might appear.
We have 43.3% of hands that we start with preflop and 28.5% hit. We can bluff such that we get this down to half or 21.5%. But as we increase the preflop hands, we also will increase that number. We could play nearly all hands except maybe a few 82o-84o,72-73o,62o but I don't think we want to bluff with more than an equal amount of hands that represent value. Basically I will play every hand but T2o-T5o,92o-95o,82o-84o,72o,73o,62o. In a real situation heads up hands with 40%-49% equity preflop won't have value so unless there are antes involved with fewer value hands you'd also probably want fewer bluff hands but It's not totally unreasonable to have more than twice the amount of bluff hands and the post equity percentages do support playing nearly every hand as opposed to 87% of hands.
3)14.3% hands preflop we start with and about 11.3% hit. So nearly 80% of preflop selection. We can widen this to twice that hand selection and still have an estimated 58% chance of "hitting" the flop hard enough to call a pot sized bet with equity to continue.
5)77+, AQ+, ATs+, KQs(6.9%). we hit the flop a little more than 5.1% of the time (open ended straight draws and flush draws not included).
Playable range if opponent raises 3bbs with this many left to act. THis is also assumed GTO calling/raising range vs your approximate optimal range:
1 opponent left to act (everyone else folded) 33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
Opening value range (Has 40% equity or more vs the above ranges):
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s (44.3%)
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s (26.1%)
3: 44+, A9+,KQ, A7s+,KTs+,QTs+,JTs (14.3%)
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs (6.9%)
6: 88+, AK, AQs+ (4.7%)
7: 99+, AK, AQs+
8: TT+, AK, AQs+
Bluffs should be added. I will put this here to approximate:
1: Value:22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
Bluffs: Any remaining 2 except for offsuited T2-T5,92-95,82-84,72-74 and you can sometimes fold others if you want.
3: Value:44+, A9+,KQ, A7s+,KTs+,QTs+,JTs
bluffs:[22+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,75s+,64s+,53s+,42s+,32s,KTo+,QTo+,J9o+,T9o,98o]
5:Value: 77+, AQ+, ATs+, KQs
Bluffs:[66-44,A9s,KJs-KTs,Q9s+,J8s+,T8s+,97s+,86s+,75s+,65s,54s,43s,32s,KQo]
It looks like we can mostly add bluffs such that we raise twice as often without any problems as far as I can tell
total hands (approximation)
1: Any 2 (minus some bad hands)(85%)
2:22+,A2s+,K2s+,Q2s+,J2s+,T6s+,96s+,85s+,74s+,63s+,52s+,42s+,32s,A7o+,K9o+,Q9o+,J9o+,T7o+,97o+,86o+,75o+,64o+,54o,43o,32o (52%)
3: 22+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,75s+,64s+,53s+,42s+,32s,A9o+,KTo+,QTo+,J9o+,T9o,98o
(29.4%)
4:22+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,75s+,64s+,53s+,42s+,32s,AJo+,KJo+,QJo,JTo (23.2%)
5: 44+,A9s+,K9s+,Q9s+,J8s+,T8s+,97s+,86s+,75s+,64s+,53s+,43s,32s,AQo+,KQo (16.4%)
6: 55+,A2s+,KJs+,QTs+,J9s+,T8s+,97s+,86s+,76s,65s,54s,43s,AKo (13.9%) - sometimes fold the weaker hands
7: 66+,ATs+,KQs,QJs,JTs,T8s+,98s,87s,76s,65s,54s,43s,32s,AKo (9.8%) - sometimes fold weaker hands
8: 77+,AQs+,A5s,KQs,QJs,JTs,T9s,98s,87s,76s,65s,54s,AKo (8.1%) (sometimes fold weaker hands)
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I will do more work later. There is software available to make more significant strides as opposed to me making a ton of small assumptions all of which that add up. I think once I work to a few rivers I can work backwards from ace high river boards and king high and be more precise about how assumptions are made. The order they come in matters as it effects which hands continue but there are few poker players perfect enough for us to really have to do the super tough precise work of actually mapping out GTO play from preflop to the river. When it becomes possible to only sometimes take certain actions and only sometimes play certain hands and when one range of hands changes the decisions vs a different range of hands and when there are adjustments and counter adjustments it becomes really difficult to get EXACT GTO play, but there's something called epsilon equilibrium which lets us know how far away from GTO we are.
If we have 100 big blinds say a lot of play is around 10% or 10 big blinds from equilibrium and we can get to 5, I think that's a profit of 5 big blinds per 100 (it actually depends which runouts you are solving for and how common they are and ideally you can find a few really bad players, but assuming this is on average and you can enter these situations fairly frequently). We can probably get under 5 on specific situations and when we generalize and apply a one size fits all strategy we will sometimes be slightly off but we can still earn a pretty high rate of profit by studying and generally getting a feel for how equilibrium play behaves. FInding a good method to sometimes make certain decisions and randomize it proportional to some statistic would be important. For instance, if you are supposed to bet 1/4th the time with KQs on a certain say ace high board when you have a flush draw, you can choose to only bet heart flush draws and generalize when the other cards change as long. The massive amounts of profit come from exploitative play. When an opponent makes certain types of mistakes and you get good at spotting where you can very liberally exploit this opponent. If an opponent folds too often on the turn, you can enter even with zero equity hands (they don't exist but assume they do) and raise and build up a pot and bet the turn. Since you sometimes will have a hand and sometimes have equity even when called it will be insanely profitable until that player adjusts. Making sure that opponent doesn't have reason to adjust is also important so sometimes passing up the opportunity to go after him and finding ways to represent a tighter range of hands is a good idea(such as playing tighter than optimal when that player is not in the hand and sometimes folding when he is so it doesn't look like you are exploiting him). People have a bias to see patterns where they don't exist and ignore patterns that aren't recent enough. So if you play every other hand some exploitable player is in the pot except when another player makes a huge raise to kick you out plus occasionaly play 2 in a row when you have a strong enough hand and occasionally pass up th eweakest of hands just so you aren't predictably playing every other it will still be incredibly profitable when you also add equilibrium play everywhere else.
The simplest easy to understand exploitative play is to identify points of major weakness related to frequency. Preflop is the easiest to spot as you will see showdowns and see what hand player turns over and you will also see them in too many hands preflop (or not enough but that is much more rare). Focus on one player at a time starting with the player that enters the pot the most often. Too loose preflop creates problems that can't be masked without making another mistake once the community cards come out.
Frequency errors will be seen if opponent folds 7 of the last 10 half potsized bets. They should be calling closer to 7 out of 10 than rather than folding 7 out of 10. If opponent calls 8 out of 10 or something on top of starting with too weak of hand you can take them to value town. Opponents should roughly call the half sized pot bet more than 67% of the time and pot sized bet just more than 50% of the time. (or raise) and 2x pot bet more than 33% of the time. If opponents are below these frequencies you can bluff at will and sometimes check stronger hands than most bluff catchers since opponents will fold if you bet more often. If they are significantly above this frequency, you can probably remove your bluffs and value bet with weaker hands that you normally might use to bluff catch.
There are other more nuanced weaknesses such as offering too big of implied odds, or calling too liberally on draws, or calling with hands that just barely have the odds to call but have terrible reverse implied odds, or not having the right balance of range so as to make their specific hand too predictable. Some of those aren't obvious and aren't even going to be spotted in the normal amount of play since you don't know if opponent just has a run of certain cards or if he's playing too high of frequency with specific situations in a certain way.
In reality we probably have our opponent have a slightly more defined hand range since an optimal opponent should be 3betting some percentage of the time with strong hands as well as weak (and maybe rarely some medium hands) and calling generally with medium strength hands (and perhaps rarely with both strength and weakness).
However, this work at least gets us part of the way there in understanding how one would determine how to play closer to equilibrium and with the right software and working back from the river backwards you can probably make additional fine tuned adjustments to improve a strategy.
I also didn't do much work vs the blinds so it isn't totally going to model GTO play but again we want to provide a foundation for solid play without a lot of over the top work.
Summary:
Based upon the number of opponents we should raise for value with the following hands, plus bluffs:
number of opponents | hands
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s
3: 44+, A9+,KQ, A7s+,KTs+,QTs+,JTs
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs
6: 88+, AK, AQs+
7: 99+, AK, AQs+
8: TT+, AK, AQs+
Our opponents should each call (or raise) about this often given our raise point and assuming autofolders in the blinds and before adjusting for positional advantage:
[if we raised with] 1 opponent left to act (everyone else folded) [then remaining opponent(s) should call or raise this often]33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
With ace high flops vs one opponent we should be willing to call or bet the pot with any pair, any gutshot, any backdoor straight and flush draw, any king high with both cards higher than the other 2, sometimes queen jack and queen ten with backdoor flush draw (and backdoor broadway straight).
As number of opponents before acting and thus strength of hand increases, we should approach QQ+, a paired jack with JT, QJ with double gutshot and backdoor flush draw and open ended draws.
As we approach a jack high board with 1 opponent we play any pair, any open ended draw, any ace high, KT+, K9 or higher with backdoor draw and 2 overs to bottom 2 cards, or K8 with a 7 and flush draw plus 2 overs to bottom two cards. WIth 5 opponents preflop on jack high board: Any pair, any pocket pair, Any open ended, Ace King Offsuit, Ace Queen with back door flush draw.
half pot sized bet is a different equation. What also needs to be done is we need to add in bluffing hands preflop which should have some big draw potential but equity just below what we can raise for value with. We can look at some of the decisions after the flop and frequency and work backwards to determine how many bluffs we can add maximum, but you still want MORE value preflop than bluffs.
okay, sorry for the stream of consciousness, no editing style to this but heyyyy that's the way it goes.
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edit:
I'm adding preflop hands such that my range is closer to 50% call range vs a pot sized bet. We can BLUFF raise and should and in response our opponent should widen his playing range.
To do this I want a more typical (non ace, non king) flop. I think a queen high flop is close enough
There's flaws in assuming opponent bets 100% of his range, he probably will bet strong hands plus mixture of weak hands and semibluffs and check back some medium strength hands and occasionally check some stronger hands, but I think it's reasonably close for most opponents. Just keep in mind true GTO play is probably a bit tighter, particularly for low pocket pairs and weak kickers and it may be just a bit looser for draws that have a chance to make the nuts as they can make up for lost equity later and be used for bluffing combinations.
We are going to make different assumptions on the turn although I required a minimum of a strong enough hand to have equity.
TURN
The calculations we made are intentionally slightly flawed for simplicity and flexibility. The reality is our opponent shouldn't always bet all of his range and we shouldn't always call just because we have equity to do so. We also shouldn't always fold if we don't have the equity either. Some folding of the weakest hands that play bad on future streets (like low pocket pairs, hands that may be outkicked, and hands that may be playing vs stronger draws), some raising or calling of draws too weak to continue and some calling or raising of hands that are strong could improve upon the strategy.
However, this allows the strategy to fairly reasonably represent both betting and calling decisions. We can either bet the pot when it's our turn to act with these hands as they would have the equity to call if bet plus betting gives us some equity when opponents fold... or we can call a pot sized bet with similar logic and only some occasional/minor adjustments.
Note HOW it's flawed when you try to adapt to the known/best guess information. IF we are to do anything different with this information it should be to be biased towards folding more than indicated when our opponent bets. We can make adjustments for whether or not we bet if our opponent checks or if we are first to act but those are less significant than whether or not to call.
IF opponent's play more hands than we assumed, and thus continues weaker than we assume when they bet, the profitability increases. If opponents frequency of betting the pot drops from 100% towards a more optimal amount, and thus the strength of the hand increases to better than we assume, the profitability decreases. IF we bet and our opponent folds some percentage of the time and when they choose to continue they call rather than raise our profitability increases. If there are difficult decisions on future streets, it favors continuing and betting or raising (or calling) with hands that can improve to nut hands or close to nut hands (flushes, straights, full houses) that are unlikely to be outkicked or beat when opponent calls or places a large bet and punishes hands that have little chance to improve and bets are likely to scare away hands that it is ahead of.
We made the flop to be fairly flexible in that there is equity to call if opponent bets all of his assumed range. His assumed range is probably a bit tighter than his actual range so even if opponent only bets SOME of his actual range it still may be a pretty good solution. This allows us to bet with that range knowing that even if our opponent calls 100% of the time it's still profitable. If opponent sometimes folds rather than calls, it's likely to be profitable too, however turn and river play must adapt to this information. If our opponent often checks rather than bets, or bets half the pot rather than full pot based upon hand strength, the hand strength of our opponent could go up enough to where we need to adjust and fold more often.
BUT with the turn play we will now assume that our opponent sheds 50% (or less) from his hands when opting to continue. If he sheds more hands then we will have some profitability from betting the pot with the hands in the 33%-49% equity (or less if we stray from the above as suggested and occasionally bluff with worse) on the flop. If he sheds fewer hands we will approach showdown with a superior holding than we assumed and have more equity when called.
This will more accurately reflect either an opponent who shows some discretion on when to bet the pot or an opponent who shows some discretion on when to call if WE make such a bet.
The trick is we now need to flip this around and determine if/when our opponent has the equity to continue vs OUR pot sized bet according to the hands we sort of approximated above.
For now I'm only going to use Jack high boards.
So if our opponents start with this range of hands based upon where we raised from:
1 opponent left to act (everyone else folded)33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
And on the jack high flop we bet pot size bet when we have a hand strong enough to call a pot sized bet and THEY then call only when they have a hand with enough equity to continue vs THAT range... NOW we can assume with THAT filtered range they bet the pot with 100% of all hands that remain and we can cut down our hand range.... we will do this again for the river and then use actual equilibrium calculators to understand the FULL scope of GTO play on the river and then we can adjust the turn and river cards to see how the strategy would change and we can adjust the holdings to see if we can improve upon it. We'd also have to see how adding a higher overcard such as a queen on the turn or river as well as straight and/or flush cards might effect strategy. We'd also potentially have to redo this process with a different card selection of one player to see if it improves or declines the EV of that player.
So Flop bet by u.
Opponent has 33% of starting hands.
22+,A2s+,K2s+,Q6s+,J7s+,T8s+,
Th7h,Ts7s,97s
+,9s6s,86s+,75s+,64s+,54s,4s3s,3s2s,A8o+,Ac7d,Ac7h,Ad7h,
KJo+,KcTd,KcTh,KcTs,KdTh,KdTs,KhTs,KsTh,Kc9s,Kd9s,Kh9s,QTo+,Qc9d,Qc9h,Qc9s,Qd9h,Qd9s,JTo,Jc9d,Jc9h,Jc9s,Jd9c,Jd9h,Jd9s,Jh9s,Js9c,T9o,9c8d,9c8h,9c8s,9d8h,
9d8s,9h8s
We bet Any pair or pocket pair, open ended straight draw, Any ace high, K9s with backdoor flush draw, K8 with backdoor straight draw and 7 (backdoor Jack to 7 straight draw), gutshot with backdoor flush draw plus two cards higher than the lowest card
So one opponent on a J72 flop we bet pot with
22+,A2s+,KJs,QJs,J7s+,Tc9c,Th9h,Ts9s,Tc8c,Th8h,Ts8s,T7s,9c8c,9h8h,9s8s,97s,87s,A8o+,KJo,QJo,J9o+
This doesn't include hands that we added for preflop bluffing.
[If we added those we'd be able to bet the pot with:
22+,A2s+,KJs,K9s-K8s,K2s,QJs,Q2s,J2s+,T7s+,T2s,97s+,92s,87s,82s,72s+,62s,52s,42s,32s,A2o+,K7o,K2o,Q7o,Q2o,J2o+,T7o,97o,87o,76o,32o]
about 46.5% of hands]
I added those.
Our opponent can call a pot sized bet on the flop with
AQo+,ATs+(with backdoor flush draw), any pocket pair, any pair, any open ended straight draw. T8s,T9s,98s with gutshot and backdoor flush draw plus backdoor double gutshots are barely good enough (it wasn't before I added the hands from preflop bluffing).
Being that they start out with 33.3% of hands this gives them about 16.7% of calls. This is just enough to prevent us from being able to bluff profitably with any two. Most likely they will find some more calls in there as there will be more coordinated flops that offer more straight draw possibilities.
I constructed a partial 33% of hands meaning that the opponent only played SOME of certain cards and as such he only can SOMETIMES call with certain hands. Never the less on a J72 flop we can proceed with equity calculation vs the following hands from the opponent:
22+,ATs+,A7s,A2s,KJs,K7s,K2s,Tc7c,Th7h,97s,76s,7d5d,7h5h,7s5s,4c2c,4s2s,3c2c,3s2s,ATo+,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h
We have bet pot with:
22+,A2s+,KJs,K9s,Kc8c,Kh8h,Ks8s,K2s,QJs,Q2s,J2s+,T7s+,T2s,97s+,92s,87s,82s,72s+,62s,52s,42s,32s,A2o+,K7o,K2o,Q7o,Q2o,J2o+,T7o,97o,87o,76o,32o
-------------------------------------------------------
I will use a turn card of the 8 of clubs. We now want to determine which hands in our range have 33.33% equity or more over our opponent's range.
We now need a pocket pair of 5s or an unpaired pair of 4s or higher (many of these won't be in our holdings). Even ace high flush draws with ace 9 of clubs aren't quite good enough while AT of clubs are. Ace king is now not strong enough for a pot sized bet (in fact it is not even strong enough for a half pot bet). open ended straight flush draws like t6 of clubs or 65 of clubs aren't strong enough purely on equity.
On this example flop we have 28.4% of preflop starting hands that we can continue with a pot sized bet on the turn... compared to 46.5% that we continued from preflop to the flop to see the turn bet. This is 61% of our hands, above the 50% needed. As such, we can add raises or pot bet bluffs on the flop with hands that have less equity than the ones we chose to continue with. For instance KQ with backdoor flush draw was NOT strong enough to call a pot size bet but we may wish to make one. Same thing with some king high backdoor flush draws and perhaps some hands with 3 cards to a straight draw and more gutshots. If we are looking at this as calling a bet on the flop then we can raise SOME of our stronger hands as well as weaker as long as we still have the hand strength to call a pot sized bet half the time (or more). The point I want to illustrate is that the flop action depends on what potential options we have and decisions we are forced to make on the turn.
22+,AJs,KJs,Kc8c,Kh8h,Ks8s,QJs,J2s+,T7s+,97s+,87s,72s+,AJo,A7o,K7o,Q7o,J2o+,T7o,97o,87o,76o
The decisions on the river will in turn possibly allow decisions that we don't currently have the equity to make. The bluff to value decisions made on the river may require we bring additional hands to the river like straightflush draws on the turn that lack the equity to call a pot sized bet for value... and the break even bluff value on the river may allow for profitable value on the river to compensate for lost value from calling without the equity to do so, and it may allow for hands that don't have enough equity on the turn to make up for it on the river when they hit and it may help hands that have the equity extract more value by calling without quite enough equity.
Decisions to the river depend a lot more on the exact card than previous streets. Draws either get there or don't. Weak hands either improve or they don't. Multiple overcards either hit or don't. Pairs become weaker in the presence of overcards, possible straights and flushes and connected boards that increase chances of two pair hands.
But now we need to determine if we bet with the range of hands
22+,AJs,KJs,Kc8c,Kh8h,Ks8s,QJs,J2s+,T7s+,97s+,87s,72s+,AJo,A7o,K7o,Q7o,J2o+,T7o,97o,87o,76o
on the turn after either caling or betting the pot on the flop, how do our opponents who first reacted to a pot bet on the flop by calling with 16.7% of hands on the flop (started with 33.3% preflop):
22+,ATs+,A7s,A2s,KJs,K7s,K2s,Tc7c,Th7h,97s,76s,7d5d,7h5h,7s5s,4c2c,4s2s,3c2c,3s2s,ATo+,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h
Well they won't be able to continue with a pair of 2s anymore and pocket 6s and lower is out. It may be pocket 7s and 8s are too weak as well and even 2 9s may be too weak but in the example board 7s and 8s make a set and 9s make a gutshot draw as well. If I adjust the 8 of clubs to make it a 6 of clubs the equity drops from 42% to 37% and that's without adjusting all of the 7 hands.
75s is out except for when it includes a flush draw.
Now have to respond to determine how our opponents will have the equity to continue from the turn to the river. THEN we can look at what happens for each possible river card given a pot sized bet in terms of which has the equity to call on the river.
Our opponent strengthens to the following hands:
77+,22,AJs,Ac7c,KJs,Kc7c,Tc7c,9c7c,7c6c,7c5c,AJo,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h
This is 9.2% of hands after calling a turn bet compared to 16.7% after the flop and 33.3% preflop.
Unlike from preflop to the flop where it was almost exactly perfect, 55% of hands that call the flop can call the turn. This can be lower which means our opponent can bluff raise the flop or call the flop to bluff future street(s) with a few more hands if they want.
1 opponent. initiate with 85% preflop, 43.3% flop, 28.4% turn. 17.3% River.
defend with: 33% preflop, 16.7% flop, 9.2% turn. 3.3% River (River doesn't have equity enough which means some bluff raises and/or some looser requirements on earlier street... OR else the runout is not representative and in other situations you will call closer to 4% or 5%... it's hard to say so far)
Both parties can afford to bluff raise or float more on the flop
If the river is a 7 of clubs pairing the board and offering flush possibilities, while counterfitting Jack 2 two pair. Jack 3 has 17.3% equity. Even jack ten is not strong enough to call vs opponent's range and JT only has 22.5% equity. Queen jack leaps to 36.4% and ace jack has 59.6%. What a difference a kicker makes.
The overall range after such a runout from the turn to the river shows us at 46.781% equity despite 28.4% of hands vs 9.2% and starting with 85% vs 33%. Other river cards will significantly effect equity. There's no way around the possible draw and pair and overcard and low card and the impact they have unlike in the first two streets.
So if our opponent bets his entire range, we can call with any flush, QJ or higher, QQ+, any two pair, ay 7, any full house, T9 for straight. In the actual board I did, there are no flushes that aren't also strong hands that would have called anyways as the jack of clubs and 8 of clubs and 7 of clubs are on board.
JJ+,88-77,22,AJs,KJs,QJs,J8s-J7s,T9s,T7s,97s,87s,72s+,AJo,A7o,K7o,Q7o,J8o-J7o,J2o,T7o,97o,87o,76o (17.3%)
If we bet this much, our opponent needs to call with
KK+, trips, probably any two pair, any straight, any flush, any full house. This will tighten his range to about 3.3%. Our opponent will for the first time NOT have enough hands on the river to prevent exploitation here which is sort of odd considering he started with fewer hands. However, it may be possible this river card is unusually strong for our range and as such it may not necessarily mean the defender needs to float the flop or turn or more hands preflop.
This isn't GTO, necessarily, but knowing that many top pair hands are not going to be strong enough by the river unless they improve, you may be better off playing to bluff catch earlier on with top and middle pair while bluffing with bottom pair and maybe some pocket pair and perhaps checking a street or two and perhaps bluff catching some draw heavy boards that miss with mid pocket pair after bluffing yourself. Again, you also should prefer the backdoor draws to straights and flushes a little more than you might think just looking at equity because although a hand like 43 of clubs on a jack 7 2 board might only have 20.7% equity with a backdoor flush draw and a few backdoor gutshot and backdoor open ended straight draw possibilities, when it hits the flush even when the board pairs it will have 93.33% equity vs the original 33.33% preflop range and 88.5% equity vs the opponent's turn calling range that sees the river after 2 potsize bets. vs the hands that opponent will call another pot size bet with it probably will have a lot of equity as well. If the 43 misses, you usually will know by the turn, although some runouts will pair the turn and miss the river, or produce a good draw on the turn then miss and some of the ones that hit will only produce a gutshot or just a flushdraw and not both. Nevertheless, calling with less equity with certain bluffing lines plus certain boards that hit that are not predictable and thus carry some good implied odds probably will be worth giving up some equity at least SOME percentage of the time.
Let's summarize and condense this a little bit.
Preflop with one other player and blinds on autofold, we should raise with 85% of hands if opponent defends with 33.33% of hands. Defending hand range may look something like this:
22+,A2s+,K4s+,Q6s+,J7s+,T7s+,98s,87s,76s,65s,54s,43s,A7o+,K9o+,Q9o+,J9o+,T9o
These are opponents rough range to define simply, but when calculating, some of the hands at the margins were sometimes folded and sometimes replaced with other drawing hands.
These are the "value hands" vs our opponent's assumed range:
22+,A2s+,K4s+,Q6s+,J7s+,T7s+,98s,87s,76s,65s,54s,43s,A7o+,K9o+,Q9o+,J9o+,T9o
everything else is "bluff hands" This has a bluff to value ratio of about 1:1... This means if our opponent raises with any two, half the time we have the best hand. But obviously opponent probably won't raise that much. If opponent raises we should call such that his bluffs break even or call 1/(Opponents ratio of raise to the pot) and must call or raise at least 1/(x+1) to prevent exploitation where Xis the number of times our bet we'd gain by raising if opponent folded. For instance, if we bet 3 opponent raise 9 and there's 1.5 big blinds in the pot dead from autofolding blinds, there would be 10.5big blinds. If we raised some amount times the 10.5, we would gain back about 3.5 times what we lost. So we can fold 3.5 times and on the 4.5 time we have to call, or we have to call or raise 22.22% to prevent our opponent being able to raise profitably with any two. But if opponent is betting 9 to win 4.5 you actually would need to call him 50% of the time to deny his profit. What gives? The opponent can profit off of what is already in the pot. Giving up more than 50% of the time (but less 77.78% of the time) than doesn't eat into what you put at risk if when you win you get back 3.5 times what you lost. The GTO play is to deny opponent profits. I believe It's unexplitable to play GTO and play 50% of the time but playing MORE than 50% or less than 22.22% is exploitable at that point.
If we know opponent plays fewer hands, our value range opening shrinks but our bluff range can expand. In fact if opponent folds more often it's profitable to raise with any two, and may be more profitable to limp in with strong hands if opponents make mistake after the flop.
If we know opponents play MORE hands, our value range expands, but our bluff range can decline.
If opponent makes more mistakes after the flop we can call with more hands too if our stack sizes are large enough because we either get better implied odds, profits from bluffing, perhaps both (if we can bluff small and value bet big) or an informational edge.
Assuming those hands go to the flop
J72 flop: ANy pair or pocket pair, OESD,Ace high, K9+bdfd, K8bdfd+bdsd,gsbdfd+ocb2
raise vs 1: 22+,A2+,K3+,Q8+,J9+,Q4s+,J7s+,T7s+,98s,87s //[Any two minus a few]
defend: 22+,ATs+,A7s,A2s,KJs,K7s,K2s,Tc7c,Th7h,97s,76s,7d5d,7h5h,7s5s,4c2c,4s2s,3c2c,3s2s,ATo+,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h (16.7%)
oesd=open ended straight draw
bdfd=back door flush draw
bdsd=back door straight draw (run of 3 consecutive cards only)
gs=gutshot
gsbdfd=gutshot with backdoor flush draw
+ocb2=plus overcards to bottom 2
fd=flush draw
bddg=back door double gutshot or backdoor open ended
TURN:J728 turn
We should bet pot with or call pot bet with:
22+,AJs,KJs,Kc8c,Kh8h,Ks8s,QJs,J2s+,T7s+,97s+,87s,72s+,AJo,A7o,K7o,Q7o,J2o+,T7o,97o,87o,76o
defend:
77+,22,AJs,Ac7c,KJs,Kc7c,Tc7c,9c7c,7c6c,7c5c,AJo,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h (9.2%)
River:
bet or call with:
any two pair, ace jack, sets, flushes, straights, high pocket pair (QQ+)
JJ+,88-77,22,AJs,KJs,QJs,J8s-J7s,T9s,T7s,97s,87s,72s+,AJo,A7o,K7o,Q7o,J8o-J7o,J2o,T7o,97o,87o,76o (17.3%)
defend:
KK+, trips, probably any two pair, any straight, any flush, any full house.
KK+,JJ,88-77,22,AcJc,Ac7c,KcJc,Kc7c,Tc7c,9c7c,7c6c,7c5c
This will tighten his range to about 3.3%. Our opponent will for the first time NOT have enough hands on the river to prevent exploitation here which is sort of odd considering he started with fewer hands. However, it may be possible this river card is unusually strong for our range and as such it may not necessarily mean the defender needs to float the flop or turn or more hands preflop.
The river you are better off polarizing your hand ranges such that you bet the strongest hands and weakest and check to catch bluffs on your medium strength hands. You may also wish to do what is needed to have more hands on the river that can be bluffed on the river. That is better than than turning holdings that have pretty good showdown value into bluffs Bluffs can be forced to fold when raised and that probably lose when called but may win when opponent checks often enough that it would be silly to bet these hands. But that is more complicated and we still aren't quite there in terms of strategy yet
Now we want to do all the calculating for UNDER the gun which is a MUCH MUCH stronger range preflop and the decisions postflop will tend to be much stronger although also less likely for certain two pair and trips hands to be contained.
8/ opponents:
preflop: 77+,AQs+,A5s,KQs,QJs,JTs,T9s,98s,87s,76s,65s,54s,AKo (8.1%)
defend: TT+,9d9h,9d9s,9h9s,8c8d,AQs+,AcJc,AhJh,AsJs,KQs,9c8c,9h8h,9s8s,7c6c,7h6h,7s6s,AKo (5.1%)
flop:J72
8:Any Pair, any pocket pair*(JT flop need stronger pairs), any open ended, Ace King, KQ with gutshot, backdoor flush draw and low card on board like a 2.
22+,22+,AKs,KcQc,QJs,JTs,87s,76s,AKo (6.1%)
defend: 99+, any pair 5+ probably, open ended.
TT+,9d9h,9d9s,9h9s,8c8d,AJs,76s (3.2%)
turn J728
raise/call:JJ+,88-77,22,QJs,JTs,87s,7c6c (4.1%) (Just barely exploitable, more hands earlier)
JJ+, sets, pair of jacks+ straights, mid pair with flush draw (almost) , ace king with flush draw and gutshot (almost)
defend:KK+,JJ,8c8d,7c7s,AcJc (1.60%)
KK+,sets, AJs with flush draw
river:J7287
raise/call:AA,JJ,88-77,22,87s,7c6c (2.6%)
aces, flushes, straights, trips, boats
defend:JJ,8c8d,7c7s,AcJc (.7%) (exploitable, need more hands earlier)
flushes, sets, full house (trips not enough)
We can see that it's mostly very premium hands that hit the range
Quarter pot bet
With a quarter pot bet if you are calling you only need 1/5 equity and to prevent your opponent from being able to bluff profitably with any two you need to call him with at least 80% of hands.
In other words 80% of all hands should have equity to continue and if not you will need to call and bluff or extract extra value on future streets such as with a flush draw that can bet big when it hits plus backdoor flushes and straight draws that can sometimes bluff with cards that miss opponent's range and typically hit yours so you bet as if the flush hit forcing opponent to bluff catch more frequently and thus getting paid more when you have a hand.
So let's start with the heads up range of hands and then the UTG 9 handed range of hands.
85% of hands vs 33% of hands
The flop it's ridiculous, basically a hand like T8 high on a J24 board is good enough. A 43 backdoor flush draw on a J72 board is good enough..
I would pretty much just call every flop even though some hands have 18% equity instead of 20% because the hands are going to narrow considerably on the turn I believe after backdoor draws are gone and cards that didn't improve remain unpaired. Also the player defending basically should call every hand. People probably don't, especially in tournaments and so betting really really small on the flop might be marginally profitable with any two cards. You might lose some equity by not betting larger with your good hands but you can bet a disproportionate amount on the river with a more optimal bluff to value frequency. Just an idea.
On the turn, you need to turn an open ended draw or a pair or have QT high or better. Actually QT is not quite strong enough without a gutshot but if we remove the possible flush draw it gets really close above 19%.
If you are defending with a 33% range vs 85% range queen 2 high is enough. High card gutshots like 98 on a J7 board or even 76 on a J438 are strong enough (19.34% with a draw to the nuts is close enough to 20% given another street of betting with a possible nut hand if a 5 comes)
So calling down really light becomes acceptable even if opponent occasionally doesn't bet with his weakest of hands and narrows down his range on each street.
77+,AQs+,A5s,KQs,QJs,JTs,T9s,98s,87s,76s,65s,54s,AKo
vs 4.89%
Almost the whole range is good except no backdoor flush draw, no gutshot, no pair. Even A5 high is good if you have backdoor flush draw. KQ is no good unless you also have backdoor flush draw or gutshot.
I'm really not sure this exercise is worth doing with 1/4th the pot bets because your call frequency is so high. Just throw out the obvious hands that have no chance and even if you call with any two cards it is never that bad of a decision if opponent is betting widely enough. On the turn you have to at least have some potential and on the river I'm sure you have to have at least a pair of some kind. I don't think we need to get into the details to define if the worst of pairs is okay if everyone narrows down their range. I could technically see there being a problem from the turn to river as if you call with every two on the flop and then on the turn you fold hands that have no draw no pair you probably shed more than 20% of your hand range. So you probably will need to float with some hands to bluff the river or bluff raise the turn every once and awhile. Perhaps this is a lesson that double barreling with small bets or even triple barreling (missed draws when you also have a lower missed draw) can be profitable just because opponents probably don't call the turn or river often enough with a small bet even when they know what you're doing.
Half pot is more where modern poker is played so you could be very, very thorough about different situations and even range from 40% to 60% since people sometimes bet above or below the half pot mark. The better you are at finding which hands have equity and which hand ranges have enough hands to not be bluffed with any two but not many more, the better off you will be. I think certain "all in" situations also need to be studied
-------------------------
I could go through and do a lot more work and I do want to come back to half pot play but first I have to cover the antes and I also want to cover 3 handed poker and 5 handed poker where I think there is a lot of room for error and exploitation.
There are different SIZES of antes. This should not be compared on an absolute basis but relative to the small blind. In some instances the small blind is 6 times the ante and in others it is 4 or even 3. Usually it is 5. But then there are the number of opponents at the table. 9 handed with all 9 players seated with the ante 1/3rd the small blind such as in 25 ante with 75/150 blinds there will be 3 small blinds or 1.5 big blind plus 1.5 big blinds for 3 big blinds in the pot. If you bet only 2 big blinds with a min raise and your opponents defend such that you break even on your bluffs, your opponents have to collectively call 60% of the time.
If you bet 3 big blinds and it's 5 handed with antes 1/6th the size of the small blind, then there is about 2.3333 big blinds and so if you risk about that much opponents have to defend about 50% of the time. If you bet 3 big blinds your opponents have to defend about 40% of the time. I believe bets tend to be on the smaller side so opponents probably will defend closer to 50% of the time in most situations.
As such, when calling out of the blinds and assuming the blinds fold you need about 33% equity vs the range that defends to have the equity to play for value.
1 vs 50% defender:
I want to start with 50% We want 33% equity over the top 50% of hands. 27 offsuit has 29.6% equity. 74 offsuit has 32.81% equity 94 has 31.963% equity T2 has 32.6% equity.
SO I'm not saying 100% of hands have value to raise with, but pretty dang close, You can bluff so play 100% of hands easily.
3) You have value with more like 75% of all hands or maybe 60% and can play all 100% when you include the bluffs
5)22+,A2o+,K6o+,Q9o+,J9o+,T9o+,98o,A2s+,K2s+,Q4s+,J7s+,T6s+,95s+,85s+,75s+,64s+,53s+
You have value with 45.1% of hands and can safely play 90% if you include bluffs.
7)22+,A2o+,K8o+,Q9o+,J9o+,T9o,A2s+,K2s+,Q4s+,J6s+,T6s+,96s+,86s+,76s,65s
About 40% of hands have value, so you can play about 80% hands if you include bluffs.
One thing I sort of failed to do was adjust the defend percentage. In reality the defend percentage probably should be a lot wider, particularly with antes because although we can play to deny equity from opponent, we actually have enough equity to call with weaker hands vs opponent's implied value range, much less his bluff range. In theory that means we should widen our range and opponent can widen his value range and narrow his bluffing range... but it involves a lot of moving parts to determine what the ideal raise and defend percentage is so I prefer keeping it here rather than continue to upgrade the strategy. If we happen to defend too widely against a much tighter opponent we could be in a very poor situation after the flop. While we are much less likely to enter these situations vs any given opponent we have to also respect that there is a table full of raisers and the odds that at least one opponent has a strong hand should bias the range given a raise to much stronger of a hand. In reality there should be a lot more calling and multiway action with squeeze plays and 3bets since everyone should adjust their assumptions about opponent so given you assume opponent raises with 80% of all hands 40% of all hands have value and although some medium strength hands you are better off calling, some bluffs plus the stronger hands gives you a pretty wide range of hands to 3bet. And as a result opponent can then say you raise with the top 40% of hands and then some of the strongest of the top 20% plus some bluffs can 4bet. Then you can 5bet with the premium of the top 10% plus bluffs. It's a little bit ridiculous to think about the profitability and just how loose and aggressive you can be with the antes.
There are a lot of reasons NOT to get nearly as crazy as suggested because
1)Tournament odds/ICM model. If you acquire 100% of the chips you do not win 100% of the prizepool, but probably closer to 20%. So you should be much tigher, particularly with decisions that are for large percentage of your chips and risk ruin, instead favoring a passive or tighter strategy.
2)It isn't enough to look at what the range of the raiser is, but before you 3bet, you also have to look at the probability that any one of the remaining opponents have a hand they can 4bet you with (in your upper half of the range) and the probability that at least someone started with it.
If there are 9 opponents and 8 people fold, they did not fold pocket aces and pocket kings and the probably are more likely to have folded a 2 through 9 plus a random card than an ace or king plus a random card. As such, you have to tighten your hand range considerably more when you open for a raise or 3bet someone who opened after several folds.
3)This is largely theoretical as very few opponents will widen their range enough to consider doing anything close!
4)CALLING is better if you have a post flop edge and your opponent is not perfect
5)Risk management! - There's something called the kelly criterion, although it's impractical in a tournament to apply the kelly betting strategy and never go broke, the long term rate of growth actually declines as you risk a greater percentage of your chips past a certain point.
Since all bets after the flop tend to be proportionally large and thus your hands need to get proportionally stronger, it's a very wise and strategic move in a way almost ZERO percentage of poker players truely understand to be far more passive or to radically reduce the bet size. Negreanu and Helmuth both seem to understand this intuitively to some degree.
6)Future opportunity - If players are so radically far away from optimal play once the antes kick in there should be a ridiculous amount of future opportunity to extract profit and keep up with the escalating blinds and antes as long as you don't get too short stacked. Try to exchange variance and profitability for survivability and consistency. There certainly may be certain points where it's worth taking the worst of it in order to have the chips to steamroll, but that's probably sometime before the antes kick in or before the small blind goes from 1/6th to 1/3rd the small blind or before the bubble or before the final few tables when you also have about 30 big blinds. Under 20 and you are kind of left playing for resteal shoves and double ups or instead you can limp in. Negreanu will limp in even with only 9 big blinds. Helmuth will minraise fold down to 7 big blinds and minraise call a shove after that. Most players would say that's absurd citing profitability, Expected value, tournament odds, and whatever reason, but I'll take them in a tournament over some keyboard warrior.
I wouldn't get quite as crazy as suggested because I'm pretty sure with more opponents
Let's with antes look at how often to call however. Remember, if opponent oppens with 50% of hands we have equity with close to any two. We can figure this out without reworking the hands. So.
Equity to defend vs value range (I'm not including bluffs)
1)100% of hands
2)100% of hands
3)90% of hands
4)90% of hands
5) I assume 85-90% of hands
6) I assume 85-90% of hands
7) 80% of hands 22+,A2o+,K2o+,Q2o+,J3o+,T6o+,96o+,86o+,75o+,64o+,54o, any suited
8) I assume 75% of hands or so.
You can see it's ridiculous. There's almost no incentive for anyone to fold preflop.
Multiway action gets a little bit different as well. Basically if you already have a call in front of you and you assume the blinds fold, you have to call 2 big blinds to potentially win let's say 6.4 not including your 2. This means you need 23.80% equity vs opponents hand ranges, plus strong enough hand to withstand a squeeze often enough to resequeeze collectively or call to prevent a player from being able to squeeze with any two... I would definately assume a tighter range here, looking for something like 40% raiser plus 80% defender.
call after a raise and a call with assumed 80% range and 40% range:
A2o+, K2o+,Q4o+,J7o+,T6o+,96o+,86o+,75o+,65o+,54o+,all suited. About 73% of hands.
Granted, we are assuming the small and big blind players give up the blinds and they probably won't all of the time. That gives us a little bit less to play for OR at least more opponents and we require a little stronger hand. BUT we also will have position over the blinds which especially matters when our decision is on the button and there is no one left to act.
In practice, implied odds and playability after the flop and being able to draw to premium hands matters a lot more, especially multiway. You'd much rather be drawing to straights and flushes than trying to play for low pairs or even top pair top kicker vs lower pairs and draws and straights and flush draws. I would certainly give up the possibility that I am giving up some value to avoid squeezes, avoid trouble hands and play hands that have a little stronger edge anyways...
We can probably mix in bluffs and call more to bluff later or
Bubble odds.
Some smart people using the ICM model have figured that since winning 100% of the chips doesn't win 100% of the prize pool that the chips you win are worth less than the chips you stand to lose. They have refined this further by assuming everyone has equal skill and determining just how much the chips you gain are worth relative to the chips you lose at any given situation. In the book called "Kill Everyone" (sequel to kill phil) they go over a lot of situations to illustrate how a tournament changes over time in it's tournament odds. On the bubble your tournament odds rise to around 1.6 in a lot of tournaments. Some are as high as 1.8. Sit N Gos are 2.1 on the bubble I believe. Super sattelites can go up to ridiculous amounts since you're only waiting for the first person to bust and there's no need to risk anything.
At 1.60 this means you are risking 1.6 times what you are getting on a 1:1 bet. Or perhaps it means you are getting 1/1.6 or 62.5% what you risk. In other words, if you are putting up 2 big blinds to win 2.2 big blinds, you are really putting up 3.2 to win 2.2. If that's the case, you need 37% equity. Or is it putting up 2.2 to win 1.375 and you need 38.1% equity instead of about 33%. I don't remember exactly but the point is, the bubble factors force all of the above calculations to be a bit tighter, requiring a bit more equity. I think that's important to keep in mind, however... WIN WIN WIN WIN WIN WIN WIN.
We will move ahead and you can adjust accordingly.
Let's look at flop play when you have the antes involved. In reality there will be a lot more 3 handed and 4 handed play, particularly blind defends. So I'm going to start with something different and that's approximate 3 handed play. Hand range values change when 3 handed
What if we assume the big blind calls at a discount whenever he has halfway decent equity and let's say small blind folds and someone else calls not in the blinds.
If there is 2.2 big blinds or so preflop then we put in 2 more plus opponent not in the blinds calling 2, plus the big blind calls for 1. So we now need about 27.78% equity vs 2 other hands. The big blind we will mark as the top 70% of hands. The other caller will still defend such that we can't profit with any two if the blinds were on autofold.
So the minimum we will start with is when there are 3 other opponents left and the one in the blind calls such to prevent us from breaking even on bluffs.
If we are risking 1 to make 1 we stick with the 20.63% range of hands. If they call with more as they certainly could, it just expands our value range into some of the bluffs and increases the amount of value the value range has so I don't see a big flaw in calculating it this way other than to say you probably could expand the raising range, especially if you are going to be heads up vs the blind(s) a lot of the time with position and the initiative.
You have equity/value to play
22+,A3o+,KTo+,QTo+,A2s+,K4s+,Q8s+,J8s+,T8s+,98s,87s,76s,65s
This is a little over 30% of hands. We can add in bluffs so that we play around 60% of hands. 30% is quite a bit lower than 75% having value as calculated heads up when the blinds fold, but it's still quite a bit when we add in bluffs.
If the big blind is the only caller, we are risking 2 to win 3.2 when called so we need 38% equity when called vs blind calling range. Our opponent only needs to call 1 to win 4.2 so he only needs like 19% equity. basically close to any two. We have 38% equity vs the top 70% of hands with close to any two ourselves 22+,A2o+,K2o+,Q2o+,J3o+,T6o+,96o+,86o+,76o+,A2s+,K2s+,Q2s+,J2s+,T2s+,93s+,84s+,74s+,64s+,54s
or 71.9% of hands.
So do we play 100% of hands or not? I'd say no, but we can play around 60% of hands as that gives us bluffs plus value in multiway pots and if we only get to play against the blinds we pretty much have a pure value range against a properly loose defender. That may not quite be 70% of all hands
Let's go to the flop.
Our range:
22+,A2s+,K2s+,Q2s+,J2s+,T5s+,95s+,85s+,74s+,63s+,53s+,42s+,32s,A2o+,K7o+,Q7o+,J7o+,T7o+,97o+,86o+,75o+,65o,54o,43o
Caller on button:
20.63%
Caller in blinds:
70%
Flop J56 with 2 hearts. This will more properly represent multiway flop as there will be draws.
Let's look at some weird situations. We want to make a 1/4 sized bet called by 2 opponents, a half pot bet called by 2 opponents or a pot size bet called by one opponent.
1/4 pot bet.
When we make a 1/4 pot bet it may get some raises sometimes but it may also allow for a cheap draw sometimes. Since we are risking 1/4th to win 4/4ths in the pot plus 2/4ths from our opponents or 6/4ths we only need to win 1/7 times assuming future decisions break even. They probably won't call with 100% of their range, but what we gain in equity from folds we probably lose from stronger hands. So it equals out. Winning 1/7 means even 14.28% equity is enough vs opponent's calling range.
This means. any pair, any pocket pair,
A2o+,K4o+,Q7o+,any gutshot 98, 23 (gutshot) or open ended(34o,78o), any flush draw.
K2s+ backdoor flushdraw,
Q3s+ backdoor flush draw.
T8s+ backdoor flush draw
98s backdoor flush draw(even on flops like J34h where we don't have a gut shot.)
87s backdoor flushdraw.
If we started with the top 60% of hands, that's
22+,A2s+,K4s+,Kc3c,Kh3h,Kc2c,Kh2h,Q7s+,Qc6c,Qh6h,Qc5c,Qh5h,Qc4c,Qh4h,Qh3h,Qh2h,J2s+,Tc9c,Th9h,Tc8c,Th8h,Tc7c,Th7h,T6s-T5s,95s+,85s+,74s+,63s+,53s+,43s,32s,A2o+,K7o+,Q7o+,J7o+,97o+,86o+,75o+,65o,54o,43o,32o
or 57% of the preflop hands. This is a LOT!!
If you can control the bet size by a bet when opponents just call, whereas a check will induce a larger bet a huge percentage of the time. do it. I even like doing it with strong hands because I think I can make good folds to big bets on the turn, read the board well and make good decisions and keep both opponents in. 2 1/4th pot bets are the same as 1 1/2 pot bet with 1 caller and loosely betting with your good hands will protect your range if opponent raises, so I don't mind betting small with strength too. Although I still prefer big pot big hand, small pot small hand more often if possible.
Now let's look at a half pot bet size with 2 callers.....
With a half pot bet we have 2 callers to make 2 times the pot and we only have to call half .5/2.5=.20.
We only need 20% equity.
any pair, any pocket pair, A7o+,KQo, any gutshot (98,23), or open ended (34,78), any flush draw.
A6 backdoor flush draw
That is 41.9% hands out of 60% or about 70% of our preflop range.
22+,A5s+,Ah4h,Ah3h,Ah2h,KcQc,KhQh,KJs,KhTh,Kh9h,Kh8h,Kh7h,K6s-K5s,Kh4h,Kh3h,Kh2h,QJs,QhTh,Qh9h,Qh8h,Qh7h,Q6s-Q5s,Qh4h,Qh3h,Qh2h,J9s-J2s,Th9h,Th8h,Th7h,T6s-T5s,95s+,85s+,74s+,63s+,53s+,43s,32s,A6o+,KQo,K7o,Q7o,J7o+,97o+,86o+,75o+,65o,43o,32o
Now collectively between us and one of the other opponents the defenders have to prevent the better from being able to profit with any two. So if our opponent bets half the pot we collectively need to call him 66.67% to make him indifferent to bluffing or not. We'll easily do that here. We could probably be a lot tighter, and perhaps we should be since opponents can narrow their ranges. But we do have flexibility to call with any of these hands to bluff on later streets or fold some of the weaker hands as well if it's necessary to carry more or fewer hands onto the turn or river.
I would personally want to fold low pocket pairs and also any high card hand without at least two overcards or some kind of backdoor draw... like I would fold A7-A9 I might still call AT sometimes and I probably call AQ and AK. But if I were on the button with this range or knew my opponents would not raise, also would call with more backdoor draws like any ace of clubs or better and any sort of backdoor straight draw with 2 clubs like K7 or K4 of clubs or even T9 of clubs.
If we assume our opponents narrow their range significantly before they determine it's worth betting then we'd have to produce some kind of logic to arrive at their betting frequency and then look at equity vs those kind of hands and determine if we have enough. I'd say both the player calling and placing the bet probably would narrow their betting range somewhat and have a stronger hand.
We can probably throw out some of the high cards, low pocket pair. I'd be tempted to toss out the top pair weak kicker hands even though it has lots of equity (45%) vs the full range of our opponents, as they narrow their range if it doesn't improve we are going to probably be in trouble in any big pots where we see betting on the turn and river. I'd almost rather have 56 or 78 for 3 to a straight and a pair because those have not only 5 full outs to improve to 2 pair but also 8 outs to bring an open ended straight draw and 8 outs to bring a gutshot straight draw on the turn. Then I have a lot of runner runner full houses, runner runner straights and 2 chances at 2 outs to trips and 1 out and 1 out to runner runner quads. these hands are all very very likely to win and 3 outs to make 2 pair certainly will be strong. Plus I will want to have some possibility of bluffing as if I made the heart flush draw on the river with some of these hands.
How about a pot sized bet on the flop with 1 opponent and 2?
With one opponent it's tricky because if we place the bet and only one of them calls, it's generally the stronger of the two so we can't simply look at the equity of 1 or the other, we have to narrow them down. If opponent bets the flop and we call we can't just look at the opponent's range he came into the pot with because he bet the pot knowing he had 2 opponents which means his range has to be stronger if he's optimal
Let's just for now stick with the range and look for equity of 25% vs 2 opponents.
Any flush draw, any pair, AQo+, high gutshot with a heart or suited with backdoor draw, open ended. We're still going to have a lot of hands that can continue vs assumed range.
OKAY!
Preflop:
We want to raise preflop with hands that have value plus some ratio of value to bluffs.
So what has value?
Keep in mind our opponents should collectively raise of call often enough to force us to break even on our bluffs. That means that if we are raising 3 big blinds (or twice the amount of the small blind plus big blind) our opponents can call us such that they fold 2/3rds of the time. In other words, when called against optimal opponents, a hand will only have value when it has equity over a hand that given the information available has a 66.67% chance of being best. we also will mix in some bluffs.
That puts the equivalent of this range of hands
1 opponent left to act (everyone else folded) 33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
TO determine "value" and "Bluff to value" we need to determine which hands have enough equity given the blinds in the pot to break even if checked to the river. In reality implied odds and reverse implied odds play a role but we will revisit and adjust later.
So with 3x big blind and someone not in the blinds calling and both the blinds folding we risk 3 when called and lost, but win 4.5. We only need to win 40% of the time or 40% equity over the hand ranges above to have VALUE. Value meaning we extract money from the decision to raise if we are called. Bluff means a hand where we raise with even though it isn't in the "value" range.
In reality a lot of the hands like 76s, 65s, 54s, 43s, 32s I would rather have than hands like K3o that may have more equity but have worse reverse implied odds, very few semi bluffing opportunities and very few possible hands where you can ever bet large. Hands with lower equity may have more REAL value due to implied odds vs hands that mathematically using this simple metric are stronger due to reverse implied odds.
number of opponents | hands
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s
3: 44+, A9+,KQ, A7s+,KTs+,QTs+,JTs
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs
6: 88+, AK, AQs+
7: 99+, AK, AQs+
8: TT+, AK, AQs+
In reality the blinds should defend more often but at a lower cost which means the odds will be different and the value will be different, but we will have position on them and the additional probability of having a stronger hand will allow for stronger hands when we show down..
Flop Play.
The strategy after the flop depends on the flop and the bet size. There's more of a tendency to standardize preflop than there is postflop. We could also work out strategies preflop to bet larger with even better hands. Note that we are making assumptions that opponents play you "optimally". I put "optimally" in quotes because in reality the preflop decisions should be pretty radically adjusted due to postflop play and mostly due to implied odds and reverse implied odds. If a hand cannot be bet with confidence even when you pair your hand and it doesn't produce good draws that get paid off when they hit there isn't nearly as much value in it as the equity demonstrates. It may do well in limit games when betting is small after the flop or in games that check a lot post flop or have players making bad decisions like large bets.
The first rendition will be pot sized flop bets. With a pot sized flop bet you must call often enough to force your opponents to break even on bluffs. The bet frequency we will make such that you have enough equity when called to break even or better, plus we will add bluffs. But when we call we only want to call such that we have equity to do so plus whatever is needed to force opponent to break even on bluffs (preventing opponent from bluffing every single hand profitably). Ideally these will converge, if not we will have to go back to the drawing board on preflop hand selections or create lines in which we call the flop to bluff the turn or bluff raise the flop which is another set of equiations we won't cover in this post.
We technically should determine this for approximately every flop. We will approximate based upon understanding the highest card as well as determining how coordination (straight and straight draw possibilities and increased two pair probabilities) and flush potential (potential flush and flush draws) and pairing impacts a given high card flop and generalizing this understanding to all flops.
In total there will be 7 unique flops plus variations for each flop. Ace, King, Queen, Jack, Ten, 9 or 8, and 7 or less. Note I have combined the 9 high and 8 high flops. I will average the decisions together. 7 or less I will just determine what to do with a 7 high flop and assume the flops below should be treated the same.
We are assuming an opponents hand ranges are the following:
1 opponent left to act (everyone else folded) 33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
We are assuming we took the lead in betting preflop and everyone in the blinds and everyone else folded except for a caller not in the blinds. What hands have equity to call a pot sized bet (you could make one with all of these hands and perhaps more but there's probably more value in sometimes checking with both strength and weakness or betting pot with big hands and bluffs and checking with medium strength hands)
ace high flop/
1 opponent preflop. For instance, autofolders in the blinds you are in the cutoff. Any paired flop, any pocket pair, any gut shot draw with backdoor flush draw 67 or higher, Any King high with both cards higher than the other 2. (king 6 vs A25 flop)
Sometimes queen jack but the conditions are tricky
Queen ten suited with backdoor flush draw
You will be playing 43% of hands preflop.
On an A26 flop, you will have equity to continue when facing a pot sized bet with about 30% of all preflop hands which is about 70% of your opening range before we even introduce bluff raises or calls to bluff future streets or balance range of future decisions. In other words you can call a pot sized bet about 70% of the time given these assumptions, and that also means you can continuation bet with this percentage of hands plus bluffs. I won't pick out all of the bluffs to make just yet. For now our preflop hand range is very compatible with ace high flops and we could probably play more hands with one opponent if this is representative of how other flops will play. However, much the disclaimer with preflop play, play on the flop also will have reverse implied odds or implied odds.
As of now you don't have to worry about widening call range to make it impossible for opponent to exploit you by bluffing, you can solely choose to call with value and call more than enough. Smaller bet sizes will give you even better odds to call with even weaker hands. You could call a half sized bet with t8 with a backdoor flush draw for comparative purposes and jack 7 off suit.
2 omitted for time purposes
3 99+, any paired flop (even 27 on A29 flop), any open ended draw or double gutter (45o on A27 flop or QJ on an AT8 flop.). With 3 opponents left to act preflop, you will play
44+, A9+,KQ, A7s+,KTs+,QTs+,JTs or 13.4% of hands preflop. You will have a hand 9.4% of ace high flops not even including double gutters and pairing a higher card (like KT on an AT2 flop). About 70% of flops you will have a hand that can stand up to a pot sized bet again. You will be able to bet all of these hands plus bet hands for bluffs as well if you are the one doing the betting if you choose.
4 omitted for time purposes
5 omitted for time purposes
6 TT+, any open ended or double gutter. Any paired flop card around ten or higher or low pair with gutshot.
WIth 6 opponents left you will play 88+, AK, AQs+ 4.7% of hands and thus have 3.8% of hands that can call or you can call pot sized bets about 80% of the time. When you get into smaller percentages of hand ranges of opponent the specific hands chosen matter. I choose a mixture that only sometimes plays certain hands and lower pairs and fewer ace hands. If more ace hands are included then even pocket jacks aren't really good enough to call.
7 omitted for time purposes
8 qq+, a paired jack with Jack ten or higher, queen jack with double gutter and back door flush draw, open ended straight draw.
Remember, I didn't add in the "bluffs" preflop. You can work them in as long as you don't increase them such that you have to fold more than 50% of the time to a pot sized bet.
Although I omitted some hands, you can guess that not a lot will change and that 4 and 5 will be somewhere between 3 and 6 and 7 will be somewhere between 6 and 8. With 7 players left to act, I would imagine it's something like JJ+, open ended, a paired flop card of jack or higher and maybe some tens.
Although this may suggest playing more preflop hands is possible, particularly those that can flop big draws or hit unbeatable hands by the river, let's not jump to conclusions based on one flop. Also, let's remind ourselves that although we technically only need to call a minimum of half the time to a pot sized bet (or 33% of the time to a half pot bet), we will sometimes face the blinds defending and the preflop equity will be less and so having a comfortable margin of strength may be a good thing or even necessary.
Also, the first step to understanding quality poker is understanding "optimal" play or how to play vs perfect opponents (well a very simplified variation of it, anyways). The next step is understanding what to do when opponents are NOT playing optimal in order to extract more chips.
The FREQUENCY is the important thing to understand. Opponents should be able to call often enough to force you to break even on your bluffs. If they do not call this often, you can literally bluff every time and you may even want to pass up betting for value as well in favor of inducing bluffs from your opponent or bluffing on a future street. If they call MORE often you should reduce your bluffing frequency to semibluffs or eliminate it. The technical goal of bluffs should theoretically be to bluff frequently enough to break even on bluffs in order to force your opponent to call widely so as to extract more value from your value bets.
If opponents play MORE hands preflop than suggested more hands are going to have value and the amount of value it has vs opponents range is going to increase. Since you will have the ability to bet for value more, you can bluff more frequently overall without changing your bluff to value ratio. Opponents that play too many hands preflop have to have a flaw in their game somewhere. Either they make up for playing too many hands by calling all the way to the river too often, fold too much on one street vs the others, or play all streets too loosely or too tightly, or they overbet good hands and try to compensate for loose play by getting paid off when they hit or they bluff too often. Good players still generally play too many hands, they just are good at picking spots to bluff, spots to call to bluff later, spots to get paid off and avoid trouble and make up for their "suboptimal" play by exploiting bad opponents.
A good player with good information on opponents actually SHOULD play more hands than "optimal" because they will correctly recognize where the flaws opponents have, or will isolate a loose player even before they really know where the postflop flaws are.
For instance if a loose player plays 40% of all hands when they should play 10%, every hand in the top 20% of hands probably has value over this player if hands are checked to the river. So if you assumed an optimal opponent (raising 10%) you should only maybe 5% of hands plus some bluffs plus some hands that are behind but with position, equity, implied odds, etc you have a little bit of value... but you have value with about the top 20% of all hands plus you can add in some bluffs and calls with enough equity. You can dramatically widen your range and even raise to isolate yourself to play with weak hand ranges and then from there sort of play as if you started with a preflop hand that was strong enough to open up with a raise and create value against opponents that make mistakes. Once you spot how this opponent makes mistakes after the flop you may theoretically be best off playing close to 100% of all hands with this opponent because either opponents will fold too often in which case you don't have to have any equity to be able to bluff profitably, or they will not be able to fold even to very large bets and you can control the size of the pot and bet larger to the degree by which you have an edge over your opponents calling range. In practice even bad opponents make some adjustments, plus there are a lot of other players involved as well.
Turn play
Turn play PROBABLY doesn't change a lot. In theory you should assume some optimal range of hands of opponent, make sure the call frequency is enough to force him to break even and be able to shed a lot of hands that don't improve, or even weak hands to additional bets. But in practice opponents think their hand is good and keep betting it or slow down and play to catch bluffs. With an ace high flop the strength of hands is unlikely to change much aside from the gutshots, double gutters and backdoor draws. On 7 high flops, the turn can go from a 7 high board to an ace high board which totally changes the nature of the hand. I think mostly you can treat the turn like a 4 card flop and look at the highest card. Some straight and flush draw possibilities may open up on the turn or the board may pair and we have yet to see how that impacts decisions but aside from that ace high flops probably don't change a whole lot to the turn. TO the river the draws that opened up either hit or they don't but if there are only backdoor draws and you initiated the bet of the pot sized bet, and they call that shouldn't be a problem most of the time (some players may call with 67 of spades on a 7A2 board with a single spade and improve to a pair with a flush draw or on a A78 board they could pick up an open ended straight draw or even a 15 hour straight and flush draw, but these are not too common.
I'm speculating as I have yet to do the work, but what I do know for sure is a flop with low cards has a lot more ways it can change. A 7 high has about 28 cards in the deck that change the top pair card plus 9 cards that pair the board and probably a few that set up straight and flush draws too that are more likely to be of the open ended or double gutter since with only 6 cards below 7 if you have 4 of them you are bound to create some kind of connected cards or paired board.
River play
With all of the cards out and betting in prior streets the pot will have gotten inflated, hands are much more well defined and you can make large all in bets with some ratio of pure bluff to max value if you'd like. The math becomes much more clear as to how frequently to bluff vs betting for value, particularly in situations in which you either have a monster or nothing.
Bluff to value ratio:
Optimal bluff to value ratio depends on the bet size proportional to the pot. Here is the formula:
X= (B-Bz-Pz)/(By+Py-Bz-PZ)
y= opponent's equity vs value bets
z= opponent's equity vs our bluffs
B= our bet size
P= our bet size + pot
x= frequency with which our bets should consist of value
1-x= frequency with which our bets should contain bluffs.
So if pot has 1 pot in it and we bet 9 with either the nuts or a total bluff...
x= [9-9(1)-10(1)]/[9(0)+10(0)-9(1)-10(1)]
x=.526
x-1=.47
value=.526
bluff=.473
If we know on the turn that we will hit the value hand 18% of the time how often must we carry a hand that will end up being a bluff? (.18/.526)-.18= 16.2%.
If you really want to do close to optimal poker you should start with possible rivers and work backwards and then determine which hand range with the optimal frequencies assuming a bet on every street (or whatever will get you to the expected bet size on the river) will produce those ranges from the turn to the river, then the flop to the turn and then what preflop percentage.
You'd probably only really need to look at about 28 rivers and work backwards but you could get more accurate doing more. 3 variations of 7. Actually 4 if you want to include pairs and 5 if you want to include combination of both flush and straight combinations. 3 card flush, 4 card flush, some sort of connectedness of varying degrees, pairs, etc.
A more complicated and more accurate solution would include varying degrees of betting that USUALLY bets large in way ahead/way behind flops and small in draw heavy dynamic flops (where board is certain to change strength of hand with lots of draws) with the optimal bluff to value bet ratio associated with that bet size and some percentage flipping the bet size on both sides. These are "decoy" bets much in the way bluffs force opponents to widen the hand range. Decoy bets make your bet patterns harder to read and force your opponent to widen the calling range vs large bets even though it's usually an optimal frequency of bluff to value with a really strong hand, sometimes it is instead a weaker hand or draw. I only say this for completeness.
Let's look at bluff to value with bluffs having 25% equity on average (as they will contain semibluffs and backdoor straightflush draws or bottom pair or overcards or something) and value bets having 70% equity on average.
9xpot:61.4% value to 38.6% bluff
2xpot: 77.78% value to 22.22% bluff
pot size: 92.6% value to 7.4% bluff
75% of the pot and lower: 100% value to 0% bluff.
The problem with this formula is it only considers one street and then checking to the river. In reality semibluffs can become value bets on future streets and pure bluffs can also bluff again.
Also, opponent's hands can call with the ability to bluff on certain turns and rivers, and it doesn't consider how to properly deal with and make check raises. Additionally, sometimes checking is best and there usually should be some kind of balance.
Balancing ranges:
There are certain lines of play when considering multiple streets that can either lead to "blended ranges" or "mixed ranges" or "polarized ranges". Polarized ranges are either really strong or really weak. betting polarized ranges forces opponent to call with the worst of it some percentage of the time in order to prevent you from being able to bluff. Mixed ranges that have draws, weak pairs and strong hands allow you have to determine which hand to bet thinly for value or play to catch bluffs on opponents and can beat ranges and also allow you to bet different sizes with some mixture of strong and weak or blended ranges so that you can polarize your range when betting large for instance. It can get pretty complicated solving for game theory optimal (GTO) strategies when you introduce this kind of concept. But there are certain principals you can understand from a lot of the complicated solutions.
1)You should be more willing to play drawing hands with slightly less equity on draw heavy boards and even backdoor straight and flush draws or backdoor ace high flush draws if they can make "nut" type of hands like flushes and straights (even if they aren't the absolute nuts you can play with bluffing frequencies that allow you to represent flushes when you don't have it and force your opponent to have to call with worse).
2)You can use busted straight draws to represent flush cards and backdoor flush draws to represent standard flush draws to balance your bluffs and value bets on the turn or river.
3)If your hand range is more clearly definable you can mix in some percentage of weak and strong hands by varying your actions. For instance, if you only check medium strength hands that may be okay and even sometimes may be optimal, but if you do it all the time in all situations the predictability loses something in real situations even if in practice it's GTO. Sometimes that's okay, but in practice opponents usually aren't playing that close to GTO and instead they are trying to spot things they notice. So if you occasionally check with really strong hands or bet bluff catching hands on the turn and/or river and maybe even check raise instead and get other medium strength hands that would catch bluffs to fold at worse it isn't a big mistake and it may actually extract value in the long run from opponents misinterpretting you as a weak player.
Exploiting
Exploiting opponents that play with the wrong frequencies is pretty straight forward. If opponents don't bluff often enough you can fold stronger hands. If they bluff too much you can bluff catch weaker hands. If they start out with weaker hands your value range widens and your bluff range could widen if they play better after the flop. If opponent's don't call often enough you can bluff 100% of the time and play more hands preflop to be in situations where you can bluff more often. If they call too often you shouldn't ever bluff unless it's to build up a bigger pot to bluff on a future street where they fold too often. 3betting to exploit opponents is more complicated since you have to introduce the possibility that opponent will 4bet and if they call the pot may introduce more jam all in situations on the turn or river. 3 betting can be done preflop to build up a bigger pot against a suboptimal opponent, it can isolate yourself with players that play too many hands and thus you have a range of hands equity edge against. It can also build up pots when opponents fold too often to prevent you from bluffing profitably on later streets. Or it can build up pots while bluffing for opponents that call with too many weak hands even to large bets, fold too many draws and backdoor draws (that they can/should bluff or semibluff with later) and that can both define your opponent's strength and set up a big overbet jam
3betting SO far I haven't really done anything about 3betting preflop or on the flop. 3betting can be a polarized range of really weak with some amount of bluffs. You could 3bet some draws with the intention of giving up if you don't improve and possibly buying free cards or building the pot to jam by the river.
Facing 3bets
A much more simplified play that's more unexploitable than GTO is to let opponent bluff often enough such that you can 3bet and make back your loses. So if opponent bets the pot you can fold 2/3rds of the time and then 4bet on the 3rd and either make everything back or be faced with a 4 bet in which you can then fold often enough such that you can 5bet jam to make everything back. Stack sizes matter because if a 4bet is a jam, you can't afford to wait. You are basically allowing your opponent to win some share of the blinds and antes but preventing him from winning your stack as an average. If you want to neutralize opponent and prevent him from being able to bluff profitably you should call OR raise half the time. You could call at the ratio to prevent opponent from bluffing. You can call anytime you have equity to call plus you can add in some calls to bluff future streets. A more GTO solution would have more complicated % of calls and raises with a mixture of hands that sometimes call and sometimes raise to balance your ranges and allow for proper bluff to value bet ratios on future streets on certain cards.
https://forumserver.twoplustwo.com/15/poker-theory/question-developing-proper-bluff-value-ratios-1623904/
https://www.pokerstrategy.com/news/content/Do-you-have-the-right-bluff-to-value-ratio-_98179/
https://forumserver.twoplustwo.com/32/beginners-questions/how-calculate-bluff-value-ratio-jam-1513678/
http://www.runitonce.com/chatter/gto-simplified/
https://www.pokerstrategy.com/video/30133/
https://www.pokerstrategy.com/video/40552/
http://www.pokerplayer365.com/poker-strategy/advanced-strategy-how-to-use-the-value-to-bluff-ratio-to-improve-your-river-ev/
https://www.pokervip.com/strategy-articles/texas-hold-em-no-limit-intermediate/gto-aggression-in-poker
https://www.youtube.com/watch?v=j7xSHC90_Og
https://ua-video.com/sxoEWZY5hG4-ev-aggregation-and-marginal-range-building.html
https://ocw.mit.edu/courses/sloan-school-of-management/15-s50-poker-theory-and-analytics-january-iap-2015/lecture-notes/MIT15_S50IAP15_L7_GameTheor.pdf
http://ktiml.mff.cuni.cz/~bartak/ui_seminar/talks/2017ZS/SuryaChembrolu_DeepStack.pdf
https://www.reddit.com/r/poker/comments/2mhsgq/gto_is_so_much_more_than_unexploitable/
okay, king high flop.
There range:
1 opponent left to act (everyone else folded) 33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
our range:
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s
3: 44+, A9+,KQ, A7s+,KTs+,QTs+,JTs
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs
6: 88+, AK, AQs+
7: 99+, AK, AQs+
8: TT+, AK, AQs+
I am assuming they bet with 100% of their range. In practice this will not be the case, but I'm simultaneously trying to approximate how often you could bet with and still have equity vs an opponent that calls the maximum. Just because you could bet doesn't mean you always should and you probably should check some of the hands with the equity to call a pot sized bet or make one and bet some with less equity than this as well.
1: ANy open ended, any pair, any pocket pair, <b>Ace 6 or higher usually</b> (if it's KQ9 flop or something with multiple high, connected cards than no, if it's K78 with backdoor draw or K72 with 2 overcards to bottom pair and minimal chance of opponent hitting then yes), medium gutshots with backdoor flush draws and overcards to the non King cards. Bottom gutshot with outs to double gutshots or open ended like JT on K96 board as a queen makes a straight an 8 provides an open ended and a 7 provides a double gutshot giving not only 4 outs to hit straight, but 8 houts to improve from 4 outs to 8 outs plus 6 outs to at least get a pair and possibly improve from there. Technically when offsuited this JT is just below calling range where as a JT with a backdoor flush draw is just above. However I would certainly still play it due to implied odds and future equity plus you need some hands to call with to bluff later, or you can bluff raise or bluff on future streets yourself to balance stronger hands you call with and gain equity back later. Queen high with gutshot. Ace high backdoor flush draws are usually good. The lower ace hands need a little something else like a backdoor wheel draw (A4s with a 2 or 3 or 5) and/or lower cards.
[A hand like Q6s with a backdoor flush draw actually has enough equity for a HALF pot bet just for comparative purposes.]
notes: We can see that hands like QJ and QT become strong enough to call with a gutshot while weak ace high hands usually also can call. Unlike ace high flops where king high flops couldn't call unless they had 2 cards higher than the others, ace high flops can call without that condition because of the ability for an ace to beat the best hand and an ace hits our opponents a little harder than a king. The lower cards can get counterfeitted by runner runner higher 2 pair on the board. I imagine that as the flop cards get lower hands with 2 overcards or just one overcard plus posibility of having the high card vs non pair hands will increase in value
2:ommitted
3:ommitted
4:ommitted
5:any pair, any pocket pair, ace king, any open ended, AT with gutshot and backdoor flush draw
6:ommitted
7:ommitted
8:ommitted
Jack high flop:
1:Any pair, any pocket pair, any open ended draw, any ace, KTo+, K9 or higher with backdoor flush draw and 2 overs to the other two cards. King 8 with backdoor flush draw and connected card like a 7 and when the other card is like a 2 or something low. gutshot with backdoor flush draw and over cards to either bottom 2 pair like 98clubs on J72 flop with a club or 67clubs with J53 flop.
2:ommitted
3: any pair, usually any pocket pair, any open ended, AT, KQ gutshot, KTs+ gutshot, QT+gutshot with backdoor flush draw, T9 gutshot backdoor draw, 98 gutshot with backdoor flush and backdoor double gutshot/open ended (98 on J65) as a T or Q brings 8 outs to straight.
4:ommitted
5: Any pair, usually any pocket pair* (JT flops need stronger hands for weakest pairs) Any open ended, Ace King Offsuit, Ace Queen with back door flush draw.
6:ommitted
7:ommitted
8:Any Pair, any pocket pair*(JT flop need stronger pairs), any open ended, Ace King, KQ with gutshot, backdoor flush draw and low card on board like a 2.
1,3,5
1)43.3% of all hands start preflop. Hits a little over 28.5% of the time. About 65.6% of hands preflop.
There's about a 45% chance we flop something with suited connectors. Something being pair or better, flush draw or open ended straight draw. With 1 gap connectors some of the straight draws that aren't also flush draws might go down. With 2 gap they go down more. There are also some gutshots plus backdoor draws that in certain circumstances are okay. I'm going to say I can add bluffs preflop up until the point where I can no longer stand calling a pot sized bet vs an opponent that plays as described. Since there is a 40-45% chance that any suited hand I add will be enough to continue I can add 2-2.5 times more hands than it might appear.
We have 43.3% of hands that we start with preflop and 28.5% hit. We can bluff such that we get this down to half or 21.5%. But as we increase the preflop hands, we also will increase that number. We could play nearly all hands except maybe a few 82o-84o,72-73o,62o but I don't think we want to bluff with more than an equal amount of hands that represent value. Basically I will play every hand but T2o-T5o,92o-95o,82o-84o,72o,73o,62o. In a real situation heads up hands with 40%-49% equity preflop won't have value so unless there are antes involved with fewer value hands you'd also probably want fewer bluff hands but It's not totally unreasonable to have more than twice the amount of bluff hands and the post equity percentages do support playing nearly every hand as opposed to 87% of hands.
3)14.3% hands preflop we start with and about 11.3% hit. So nearly 80% of preflop selection. We can widen this to twice that hand selection and still have an estimated 58% chance of "hitting" the flop hard enough to call a pot sized bet with equity to continue.
5)77+, AQ+, ATs+, KQs(6.9%). we hit the flop a little more than 5.1% of the time (open ended straight draws and flush draws not included).
Playable range if opponent raises 3bbs with this many left to act. THis is also assumed GTO calling/raising range vs your approximate optimal range:
1 opponent left to act (everyone else folded) 33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
Opening value range (Has 40% equity or more vs the above ranges):
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s (44.3%)
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s (26.1%)
3: 44+, A9+,KQ, A7s+,KTs+,QTs+,JTs (14.3%)
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs (6.9%)
6: 88+, AK, AQs+ (4.7%)
7: 99+, AK, AQs+
8: TT+, AK, AQs+
Bluffs should be added. I will put this here to approximate:
1: Value:22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
Bluffs: Any remaining 2 except for offsuited T2-T5,92-95,82-84,72-74 and you can sometimes fold others if you want.
3: Value:44+, A9+,KQ, A7s+,KTs+,QTs+,JTs
bluffs:[22+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,75s+,64s+,53s+,42s+,32s,KTo+,QTo+,J9o+,T9o,98o]
5:Value: 77+, AQ+, ATs+, KQs
Bluffs:[66-44,A9s,KJs-KTs,Q9s+,J8s+,T8s+,97s+,86s+,75s+,65s,54s,43s,32s,KQo]
It looks like we can mostly add bluffs such that we raise twice as often without any problems as far as I can tell
total hands (approximation)
1: Any 2 (minus some bad hands)(85%)
2:22+,A2s+,K2s+,Q2s+,J2s+,T6s+,96s+,85s+,74s+,63s+,52s+,42s+,32s,A7o+,K9o+,Q9o+,J9o+,T7o+,97o+,86o+,75o+,64o+,54o,43o,32o (52%)
3: 22+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,75s+,64s+,53s+,42s+,32s,A9o+,KTo+,QTo+,J9o+,T9o,98o
(29.4%)
4:22+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,75s+,64s+,53s+,42s+,32s,AJo+,KJo+,QJo,JTo (23.2%)
5: 44+,A9s+,K9s+,Q9s+,J8s+,T8s+,97s+,86s+,75s+,64s+,53s+,43s,32s,AQo+,KQo (16.4%)
6: 55+,A2s+,KJs+,QTs+,J9s+,T8s+,97s+,86s+,76s,65s,54s,43s,AKo (13.9%) - sometimes fold the weaker hands
7: 66+,ATs+,KQs,QJs,JTs,T8s+,98s,87s,76s,65s,54s,43s,32s,AKo (9.8%) - sometimes fold weaker hands
8: 77+,AQs+,A5s,KQs,QJs,JTs,T9s,98s,87s,76s,65s,54s,AKo (8.1%) (sometimes fold weaker hands)
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I will do more work later. There is software available to make more significant strides as opposed to me making a ton of small assumptions all of which that add up. I think once I work to a few rivers I can work backwards from ace high river boards and king high and be more precise about how assumptions are made. The order they come in matters as it effects which hands continue but there are few poker players perfect enough for us to really have to do the super tough precise work of actually mapping out GTO play from preflop to the river. When it becomes possible to only sometimes take certain actions and only sometimes play certain hands and when one range of hands changes the decisions vs a different range of hands and when there are adjustments and counter adjustments it becomes really difficult to get EXACT GTO play, but there's something called epsilon equilibrium which lets us know how far away from GTO we are.
If we have 100 big blinds say a lot of play is around 10% or 10 big blinds from equilibrium and we can get to 5, I think that's a profit of 5 big blinds per 100 (it actually depends which runouts you are solving for and how common they are and ideally you can find a few really bad players, but assuming this is on average and you can enter these situations fairly frequently). We can probably get under 5 on specific situations and when we generalize and apply a one size fits all strategy we will sometimes be slightly off but we can still earn a pretty high rate of profit by studying and generally getting a feel for how equilibrium play behaves. FInding a good method to sometimes make certain decisions and randomize it proportional to some statistic would be important. For instance, if you are supposed to bet 1/4th the time with KQs on a certain say ace high board when you have a flush draw, you can choose to only bet heart flush draws and generalize when the other cards change as long. The massive amounts of profit come from exploitative play. When an opponent makes certain types of mistakes and you get good at spotting where you can very liberally exploit this opponent. If an opponent folds too often on the turn, you can enter even with zero equity hands (they don't exist but assume they do) and raise and build up a pot and bet the turn. Since you sometimes will have a hand and sometimes have equity even when called it will be insanely profitable until that player adjusts. Making sure that opponent doesn't have reason to adjust is also important so sometimes passing up the opportunity to go after him and finding ways to represent a tighter range of hands is a good idea(such as playing tighter than optimal when that player is not in the hand and sometimes folding when he is so it doesn't look like you are exploiting him). People have a bias to see patterns where they don't exist and ignore patterns that aren't recent enough. So if you play every other hand some exploitable player is in the pot except when another player makes a huge raise to kick you out plus occasionaly play 2 in a row when you have a strong enough hand and occasionally pass up th eweakest of hands just so you aren't predictably playing every other it will still be incredibly profitable when you also add equilibrium play everywhere else.
The simplest easy to understand exploitative play is to identify points of major weakness related to frequency. Preflop is the easiest to spot as you will see showdowns and see what hand player turns over and you will also see them in too many hands preflop (or not enough but that is much more rare). Focus on one player at a time starting with the player that enters the pot the most often. Too loose preflop creates problems that can't be masked without making another mistake once the community cards come out.
Frequency errors will be seen if opponent folds 7 of the last 10 half potsized bets. They should be calling closer to 7 out of 10 than rather than folding 7 out of 10. If opponent calls 8 out of 10 or something on top of starting with too weak of hand you can take them to value town. Opponents should roughly call the half sized pot bet more than 67% of the time and pot sized bet just more than 50% of the time. (or raise) and 2x pot bet more than 33% of the time. If opponents are below these frequencies you can bluff at will and sometimes check stronger hands than most bluff catchers since opponents will fold if you bet more often. If they are significantly above this frequency, you can probably remove your bluffs and value bet with weaker hands that you normally might use to bluff catch.
There are other more nuanced weaknesses such as offering too big of implied odds, or calling too liberally on draws, or calling with hands that just barely have the odds to call but have terrible reverse implied odds, or not having the right balance of range so as to make their specific hand too predictable. Some of those aren't obvious and aren't even going to be spotted in the normal amount of play since you don't know if opponent just has a run of certain cards or if he's playing too high of frequency with specific situations in a certain way.
In reality we probably have our opponent have a slightly more defined hand range since an optimal opponent should be 3betting some percentage of the time with strong hands as well as weak (and maybe rarely some medium hands) and calling generally with medium strength hands (and perhaps rarely with both strength and weakness).
However, this work at least gets us part of the way there in understanding how one would determine how to play closer to equilibrium and with the right software and working back from the river backwards you can probably make additional fine tuned adjustments to improve a strategy.
I also didn't do much work vs the blinds so it isn't totally going to model GTO play but again we want to provide a foundation for solid play without a lot of over the top work.
Summary:
Based upon the number of opponents we should raise for value with the following hands, plus bluffs:
number of opponents | hands
1: 22+, A2+, K3+, Q8+, J9+, Q4s+, J7s+, T7s+, 98s, 87s
2: 22+, A4+, KT+,QJ,A2s+, K8s+,Q9s+,J9s+,T8s+,98s
3: 44+, A9+,KQ, A7s+,KTs+,QTs+,JTs
4: 55+, AT+,KQ, A8s+, KTs+,QTs+,JTs
5: 77+, AQ+, ATs+, KQs
6: 88+, AK, AQs+
7: 99+, AK, AQs+
8: TT+, AK, AQs+
Our opponents should each call (or raise) about this often given our raise point and assuming autofolders in the blinds and before adjusting for positional advantage:
[if we raised with] 1 opponent left to act (everyone else folded) [then remaining opponent(s) should call or raise this often]33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
With ace high flops vs one opponent we should be willing to call or bet the pot with any pair, any gutshot, any backdoor straight and flush draw, any king high with both cards higher than the other 2, sometimes queen jack and queen ten with backdoor flush draw (and backdoor broadway straight).
As number of opponents before acting and thus strength of hand increases, we should approach QQ+, a paired jack with JT, QJ with double gutshot and backdoor flush draw and open ended draws.
As we approach a jack high board with 1 opponent we play any pair, any open ended draw, any ace high, KT+, K9 or higher with backdoor draw and 2 overs to bottom 2 cards, or K8 with a 7 and flush draw plus 2 overs to bottom two cards. WIth 5 opponents preflop on jack high board: Any pair, any pocket pair, Any open ended, Ace King Offsuit, Ace Queen with back door flush draw.
half pot sized bet is a different equation. What also needs to be done is we need to add in bluffing hands preflop which should have some big draw potential but equity just below what we can raise for value with. We can look at some of the decisions after the flop and frequency and work backwards to determine how many bluffs we can add maximum, but you still want MORE value preflop than bluffs.
okay, sorry for the stream of consciousness, no editing style to this but heyyyy that's the way it goes.
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edit:
I'm adding preflop hands such that my range is closer to 50% call range vs a pot sized bet. We can BLUFF raise and should and in response our opponent should widen his playing range.
To do this I want a more typical (non ace, non king) flop. I think a queen high flop is close enough
There's flaws in assuming opponent bets 100% of his range, he probably will bet strong hands plus mixture of weak hands and semibluffs and check back some medium strength hands and occasionally check some stronger hands, but I think it's reasonably close for most opponents. Just keep in mind true GTO play is probably a bit tighter, particularly for low pocket pairs and weak kickers and it may be just a bit looser for draws that have a chance to make the nuts as they can make up for lost equity later and be used for bluffing combinations.
We are going to make different assumptions on the turn although I required a minimum of a strong enough hand to have equity.
TURN
The calculations we made are intentionally slightly flawed for simplicity and flexibility. The reality is our opponent shouldn't always bet all of his range and we shouldn't always call just because we have equity to do so. We also shouldn't always fold if we don't have the equity either. Some folding of the weakest hands that play bad on future streets (like low pocket pairs, hands that may be outkicked, and hands that may be playing vs stronger draws), some raising or calling of draws too weak to continue and some calling or raising of hands that are strong could improve upon the strategy.
However, this allows the strategy to fairly reasonably represent both betting and calling decisions. We can either bet the pot when it's our turn to act with these hands as they would have the equity to call if bet plus betting gives us some equity when opponents fold... or we can call a pot sized bet with similar logic and only some occasional/minor adjustments.
Note HOW it's flawed when you try to adapt to the known/best guess information. IF we are to do anything different with this information it should be to be biased towards folding more than indicated when our opponent bets. We can make adjustments for whether or not we bet if our opponent checks or if we are first to act but those are less significant than whether or not to call.
IF opponent's play more hands than we assumed, and thus continues weaker than we assume when they bet, the profitability increases. If opponents frequency of betting the pot drops from 100% towards a more optimal amount, and thus the strength of the hand increases to better than we assume, the profitability decreases. IF we bet and our opponent folds some percentage of the time and when they choose to continue they call rather than raise our profitability increases. If there are difficult decisions on future streets, it favors continuing and betting or raising (or calling) with hands that can improve to nut hands or close to nut hands (flushes, straights, full houses) that are unlikely to be outkicked or beat when opponent calls or places a large bet and punishes hands that have little chance to improve and bets are likely to scare away hands that it is ahead of.
We made the flop to be fairly flexible in that there is equity to call if opponent bets all of his assumed range. His assumed range is probably a bit tighter than his actual range so even if opponent only bets SOME of his actual range it still may be a pretty good solution. This allows us to bet with that range knowing that even if our opponent calls 100% of the time it's still profitable. If opponent sometimes folds rather than calls, it's likely to be profitable too, however turn and river play must adapt to this information. If our opponent often checks rather than bets, or bets half the pot rather than full pot based upon hand strength, the hand strength of our opponent could go up enough to where we need to adjust and fold more often.
BUT with the turn play we will now assume that our opponent sheds 50% (or less) from his hands when opting to continue. If he sheds more hands then we will have some profitability from betting the pot with the hands in the 33%-49% equity (or less if we stray from the above as suggested and occasionally bluff with worse) on the flop. If he sheds fewer hands we will approach showdown with a superior holding than we assumed and have more equity when called.
This will more accurately reflect either an opponent who shows some discretion on when to bet the pot or an opponent who shows some discretion on when to call if WE make such a bet.
The trick is we now need to flip this around and determine if/when our opponent has the equity to continue vs OUR pot sized bet according to the hands we sort of approximated above.
For now I'm only going to use Jack high boards.
So if our opponents start with this range of hands based upon where we raised from:
1 opponent left to act (everyone else folded)33.00%
2 opponent left to act (everyone else folded) 18.15%
3 opponent left to act (everyone else folded) 12.50%
4 opponent left to act (everyone else folded) 9.53%
5 opponent left to act (everyone else folded) 7.70%
6 opponent left to act (everyone else folded) 6.46%
7 opponent left to act (everyone else folded) 5.56%
8 opponent left to act 4.89%
And on the jack high flop we bet pot size bet when we have a hand strong enough to call a pot sized bet and THEY then call only when they have a hand with enough equity to continue vs THAT range... NOW we can assume with THAT filtered range they bet the pot with 100% of all hands that remain and we can cut down our hand range.... we will do this again for the river and then use actual equilibrium calculators to understand the FULL scope of GTO play on the river and then we can adjust the turn and river cards to see how the strategy would change and we can adjust the holdings to see if we can improve upon it. We'd also have to see how adding a higher overcard such as a queen on the turn or river as well as straight and/or flush cards might effect strategy. We'd also potentially have to redo this process with a different card selection of one player to see if it improves or declines the EV of that player.
So Flop bet by u.
Opponent has 33% of starting hands.
22+,A2s+,K2s+,Q6s+,J7s+,T8s+,
Th7h,Ts7s,97s
+,9s6s,86s+,75s+,64s+,54s,4s3s,3s2s,A8o+,Ac7d,Ac7h,Ad7h,
KJo+,KcTd,KcTh,KcTs,KdTh,KdTs,KhTs,KsTh,Kc9s,Kd9s,Kh9s,QTo+,Qc9d,Qc9h,Qc9s,Qd9h,Qd9s,JTo,Jc9d,Jc9h,Jc9s,Jd9c,Jd9h,Jd9s,Jh9s,Js9c,T9o,9c8d,9c8h,9c8s,9d8h,
9d8s,9h8s
We bet Any pair or pocket pair, open ended straight draw, Any ace high, K9s with backdoor flush draw, K8 with backdoor straight draw and 7 (backdoor Jack to 7 straight draw), gutshot with backdoor flush draw plus two cards higher than the lowest card
So one opponent on a J72 flop we bet pot with
22+,A2s+,KJs,QJs,J7s+,Tc9c,Th9h,Ts9s,Tc8c,Th8h,Ts8s,T7s,9c8c,9h8h,9s8s,97s,87s,A8o+,KJo,QJo,J9o+
This doesn't include hands that we added for preflop bluffing.
[If we added those we'd be able to bet the pot with:
22+,A2s+,KJs,K9s-K8s,K2s,QJs,Q2s,J2s+,T7s+,T2s,97s+,92s,87s,82s,72s+,62s,52s,42s,32s,A2o+,K7o,K2o,Q7o,Q2o,J2o+,T7o,97o,87o,76o,32o]
about 46.5% of hands]
I added those.
Our opponent can call a pot sized bet on the flop with
AQo+,ATs+(with backdoor flush draw), any pocket pair, any pair, any open ended straight draw. T8s,T9s,98s with gutshot and backdoor flush draw plus backdoor double gutshots are barely good enough (it wasn't before I added the hands from preflop bluffing).
Being that they start out with 33.3% of hands this gives them about 16.7% of calls. This is just enough to prevent us from being able to bluff profitably with any two. Most likely they will find some more calls in there as there will be more coordinated flops that offer more straight draw possibilities.
I constructed a partial 33% of hands meaning that the opponent only played SOME of certain cards and as such he only can SOMETIMES call with certain hands. Never the less on a J72 flop we can proceed with equity calculation vs the following hands from the opponent:
22+,ATs+,A7s,A2s,KJs,K7s,K2s,Tc7c,Th7h,97s,76s,7d5d,7h5h,7s5s,4c2c,4s2s,3c2c,3s2s,ATo+,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h
We have bet pot with:
22+,A2s+,KJs,K9s,Kc8c,Kh8h,Ks8s,K2s,QJs,Q2s,J2s+,T7s+,T2s,97s+,92s,87s,82s,72s+,62s,52s,42s,32s,A2o+,K7o,K2o,Q7o,Q2o,J2o+,T7o,97o,87o,76o,32o
-------------------------------------------------------
I will use a turn card of the 8 of clubs. We now want to determine which hands in our range have 33.33% equity or more over our opponent's range.
We now need a pocket pair of 5s or an unpaired pair of 4s or higher (many of these won't be in our holdings). Even ace high flush draws with ace 9 of clubs aren't quite good enough while AT of clubs are. Ace king is now not strong enough for a pot sized bet (in fact it is not even strong enough for a half pot bet). open ended straight flush draws like t6 of clubs or 65 of clubs aren't strong enough purely on equity.
On this example flop we have 28.4% of preflop starting hands that we can continue with a pot sized bet on the turn... compared to 46.5% that we continued from preflop to the flop to see the turn bet. This is 61% of our hands, above the 50% needed. As such, we can add raises or pot bet bluffs on the flop with hands that have less equity than the ones we chose to continue with. For instance KQ with backdoor flush draw was NOT strong enough to call a pot size bet but we may wish to make one. Same thing with some king high backdoor flush draws and perhaps some hands with 3 cards to a straight draw and more gutshots. If we are looking at this as calling a bet on the flop then we can raise SOME of our stronger hands as well as weaker as long as we still have the hand strength to call a pot sized bet half the time (or more). The point I want to illustrate is that the flop action depends on what potential options we have and decisions we are forced to make on the turn.
22+,AJs,KJs,Kc8c,Kh8h,Ks8s,QJs,J2s+,T7s+,97s+,87s,72s+,AJo,A7o,K7o,Q7o,J2o+,T7o,97o,87o,76o
The decisions on the river will in turn possibly allow decisions that we don't currently have the equity to make. The bluff to value decisions made on the river may require we bring additional hands to the river like straightflush draws on the turn that lack the equity to call a pot sized bet for value... and the break even bluff value on the river may allow for profitable value on the river to compensate for lost value from calling without the equity to do so, and it may allow for hands that don't have enough equity on the turn to make up for it on the river when they hit and it may help hands that have the equity extract more value by calling without quite enough equity.
Decisions to the river depend a lot more on the exact card than previous streets. Draws either get there or don't. Weak hands either improve or they don't. Multiple overcards either hit or don't. Pairs become weaker in the presence of overcards, possible straights and flushes and connected boards that increase chances of two pair hands.
But now we need to determine if we bet with the range of hands
22+,AJs,KJs,Kc8c,Kh8h,Ks8s,QJs,J2s+,T7s+,97s+,87s,72s+,AJo,A7o,K7o,Q7o,J2o+,T7o,97o,87o,76o
on the turn after either caling or betting the pot on the flop, how do our opponents who first reacted to a pot bet on the flop by calling with 16.7% of hands on the flop (started with 33.3% preflop):
22+,ATs+,A7s,A2s,KJs,K7s,K2s,Tc7c,Th7h,97s,76s,7d5d,7h5h,7s5s,4c2c,4s2s,3c2c,3s2s,ATo+,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h
Well they won't be able to continue with a pair of 2s anymore and pocket 6s and lower is out. It may be pocket 7s and 8s are too weak as well and even 2 9s may be too weak but in the example board 7s and 8s make a set and 9s make a gutshot draw as well. If I adjust the 8 of clubs to make it a 6 of clubs the equity drops from 42% to 37% and that's without adjusting all of the 7 hands.
75s is out except for when it includes a flush draw.
Now have to respond to determine how our opponents will have the equity to continue from the turn to the river. THEN we can look at what happens for each possible river card given a pot sized bet in terms of which has the equity to call on the river.
Our opponent strengthens to the following hands:
77+,22,AJs,Ac7c,KJs,Kc7c,Tc7c,9c7c,7c6c,7c5c,AJo,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h
This is 9.2% of hands after calling a turn bet compared to 16.7% after the flop and 33.3% preflop.
Unlike from preflop to the flop where it was almost exactly perfect, 55% of hands that call the flop can call the turn. This can be lower which means our opponent can bluff raise the flop or call the flop to bluff future street(s) with a few more hands if they want.
1 opponent. initiate with 85% preflop, 43.3% flop, 28.4% turn. 17.3% River.
defend with: 33% preflop, 16.7% flop, 9.2% turn. 3.3% River (River doesn't have equity enough which means some bluff raises and/or some looser requirements on earlier street... OR else the runout is not representative and in other situations you will call closer to 4% or 5%... it's hard to say so far)
Both parties can afford to bluff raise or float more on the flop
If the river is a 7 of clubs pairing the board and offering flush possibilities, while counterfitting Jack 2 two pair. Jack 3 has 17.3% equity. Even jack ten is not strong enough to call vs opponent's range and JT only has 22.5% equity. Queen jack leaps to 36.4% and ace jack has 59.6%. What a difference a kicker makes.
The overall range after such a runout from the turn to the river shows us at 46.781% equity despite 28.4% of hands vs 9.2% and starting with 85% vs 33%. Other river cards will significantly effect equity. There's no way around the possible draw and pair and overcard and low card and the impact they have unlike in the first two streets.
So if our opponent bets his entire range, we can call with any flush, QJ or higher, QQ+, any two pair, ay 7, any full house, T9 for straight. In the actual board I did, there are no flushes that aren't also strong hands that would have called anyways as the jack of clubs and 8 of clubs and 7 of clubs are on board.
JJ+,88-77,22,AJs,KJs,QJs,J8s-J7s,T9s,T7s,97s,87s,72s+,AJo,A7o,K7o,Q7o,J8o-J7o,J2o,T7o,97o,87o,76o (17.3%)
If we bet this much, our opponent needs to call with
KK+, trips, probably any two pair, any straight, any flush, any full house. This will tighten his range to about 3.3%. Our opponent will for the first time NOT have enough hands on the river to prevent exploitation here which is sort of odd considering he started with fewer hands. However, it may be possible this river card is unusually strong for our range and as such it may not necessarily mean the defender needs to float the flop or turn or more hands preflop.
This isn't GTO, necessarily, but knowing that many top pair hands are not going to be strong enough by the river unless they improve, you may be better off playing to bluff catch earlier on with top and middle pair while bluffing with bottom pair and maybe some pocket pair and perhaps checking a street or two and perhaps bluff catching some draw heavy boards that miss with mid pocket pair after bluffing yourself. Again, you also should prefer the backdoor draws to straights and flushes a little more than you might think just looking at equity because although a hand like 43 of clubs on a jack 7 2 board might only have 20.7% equity with a backdoor flush draw and a few backdoor gutshot and backdoor open ended straight draw possibilities, when it hits the flush even when the board pairs it will have 93.33% equity vs the original 33.33% preflop range and 88.5% equity vs the opponent's turn calling range that sees the river after 2 potsize bets. vs the hands that opponent will call another pot size bet with it probably will have a lot of equity as well. If the 43 misses, you usually will know by the turn, although some runouts will pair the turn and miss the river, or produce a good draw on the turn then miss and some of the ones that hit will only produce a gutshot or just a flushdraw and not both. Nevertheless, calling with less equity with certain bluffing lines plus certain boards that hit that are not predictable and thus carry some good implied odds probably will be worth giving up some equity at least SOME percentage of the time.
Let's summarize and condense this a little bit.
Preflop with one other player and blinds on autofold, we should raise with 85% of hands if opponent defends with 33.33% of hands. Defending hand range may look something like this:
22+,A2s+,K4s+,Q6s+,J7s+,T7s+,98s,87s,76s,65s,54s,43s,A7o+,K9o+,Q9o+,J9o+,T9o
These are opponents rough range to define simply, but when calculating, some of the hands at the margins were sometimes folded and sometimes replaced with other drawing hands.
These are the "value hands" vs our opponent's assumed range:
22+,A2s+,K4s+,Q6s+,J7s+,T7s+,98s,87s,76s,65s,54s,43s,A7o+,K9o+,Q9o+,J9o+,T9o
everything else is "bluff hands" This has a bluff to value ratio of about 1:1... This means if our opponent raises with any two, half the time we have the best hand. But obviously opponent probably won't raise that much. If opponent raises we should call such that his bluffs break even or call 1/(Opponents ratio of raise to the pot) and must call or raise at least 1/(x+1) to prevent exploitation where Xis the number of times our bet we'd gain by raising if opponent folded. For instance, if we bet 3 opponent raise 9 and there's 1.5 big blinds in the pot dead from autofolding blinds, there would be 10.5big blinds. If we raised some amount times the 10.5, we would gain back about 3.5 times what we lost. So we can fold 3.5 times and on the 4.5 time we have to call, or we have to call or raise 22.22% to prevent our opponent being able to raise profitably with any two. But if opponent is betting 9 to win 4.5 you actually would need to call him 50% of the time to deny his profit. What gives? The opponent can profit off of what is already in the pot. Giving up more than 50% of the time (but less 77.78% of the time) than doesn't eat into what you put at risk if when you win you get back 3.5 times what you lost. The GTO play is to deny opponent profits. I believe It's unexplitable to play GTO and play 50% of the time but playing MORE than 50% or less than 22.22% is exploitable at that point.
If we know opponent plays fewer hands, our value range opening shrinks but our bluff range can expand. In fact if opponent folds more often it's profitable to raise with any two, and may be more profitable to limp in with strong hands if opponents make mistake after the flop.
If we know opponents play MORE hands, our value range expands, but our bluff range can decline.
If opponent makes more mistakes after the flop we can call with more hands too if our stack sizes are large enough because we either get better implied odds, profits from bluffing, perhaps both (if we can bluff small and value bet big) or an informational edge.
Assuming those hands go to the flop
J72 flop: ANy pair or pocket pair, OESD,Ace high, K9+bdfd, K8bdfd+bdsd,gsbdfd+ocb2
raise vs 1: 22+,A2+,K3+,Q8+,J9+,Q4s+,J7s+,T7s+,98s,87s //[Any two minus a few]
defend: 22+,ATs+,A7s,A2s,KJs,K7s,K2s,Tc7c,Th7h,97s,76s,7d5d,7h5h,7s5s,4c2c,4s2s,3c2c,3s2s,ATo+,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h (16.7%)
oesd=open ended straight draw
bdfd=back door flush draw
bdsd=back door straight draw (run of 3 consecutive cards only)
gs=gutshot
gsbdfd=gutshot with backdoor flush draw
+ocb2=plus overcards to bottom 2
fd=flush draw
bddg=back door double gutshot or backdoor open ended
TURN:J728 turn
We should bet pot with or call pot bet with:
22+,AJs,KJs,Kc8c,Kh8h,Ks8s,QJs,J2s+,T7s+,97s+,87s,72s+,AJo,A7o,K7o,Q7o,J2o+,T7o,97o,87o,76o
defend:
77+,22,AJs,Ac7c,KJs,Kc7c,Tc7c,9c7c,7c6c,7c5c,AJo,KcJd,KcJh,KcJs,KdJh,KdJs,KhJc,KhJd,KhJs,KsJc,KsJd,KsJh,QJo,JTo,Jc9d,Jc9h,Jd9h,Jd9s,Jh9c,Jh9s,Js9h (9.2%)
River:
bet or call with:
any two pair, ace jack, sets, flushes, straights, high pocket pair (QQ+)
JJ+,88-77,22,AJs,KJs,QJs,J8s-J7s,T9s,T7s,97s,87s,72s+,AJo,A7o,K7o,Q7o,J8o-J7o,J2o,T7o,97o,87o,76o (17.3%)
defend:
KK+, trips, probably any two pair, any straight, any flush, any full house.
KK+,JJ,88-77,22,AcJc,Ac7c,KcJc,Kc7c,Tc7c,9c7c,7c6c,7c5c
This will tighten his range to about 3.3%. Our opponent will for the first time NOT have enough hands on the river to prevent exploitation here which is sort of odd considering he started with fewer hands. However, it may be possible this river card is unusually strong for our range and as such it may not necessarily mean the defender needs to float the flop or turn or more hands preflop.
The river you are better off polarizing your hand ranges such that you bet the strongest hands and weakest and check to catch bluffs on your medium strength hands. You may also wish to do what is needed to have more hands on the river that can be bluffed on the river. That is better than than turning holdings that have pretty good showdown value into bluffs Bluffs can be forced to fold when raised and that probably lose when called but may win when opponent checks often enough that it would be silly to bet these hands. But that is more complicated and we still aren't quite there in terms of strategy yet
Now we want to do all the calculating for UNDER the gun which is a MUCH MUCH stronger range preflop and the decisions postflop will tend to be much stronger although also less likely for certain two pair and trips hands to be contained.
8/ opponents:
preflop: 77+,AQs+,A5s,KQs,QJs,JTs,T9s,98s,87s,76s,65s,54s,AKo (8.1%)
defend: TT+,9d9h,9d9s,9h9s,8c8d,AQs+,AcJc,AhJh,AsJs,KQs,9c8c,9h8h,9s8s,7c6c,7h6h,7s6s,AKo (5.1%)
flop:J72
8:Any Pair, any pocket pair*(JT flop need stronger pairs), any open ended, Ace King, KQ with gutshot, backdoor flush draw and low card on board like a 2.
22+,22+,AKs,KcQc,QJs,JTs,87s,76s,AKo (6.1%)
defend: 99+, any pair 5+ probably, open ended.
TT+,9d9h,9d9s,9h9s,8c8d,AJs,76s (3.2%)
turn J728
raise/call:JJ+,88-77,22,QJs,JTs,87s,7c6c (4.1%) (Just barely exploitable, more hands earlier)
JJ+, sets, pair of jacks+ straights, mid pair with flush draw (almost) , ace king with flush draw and gutshot (almost)
defend:KK+,JJ,8c8d,7c7s,AcJc (1.60%)
KK+,sets, AJs with flush draw
river:J7287
raise/call:AA,JJ,88-77,22,87s,7c6c (2.6%)
aces, flushes, straights, trips, boats
defend:JJ,8c8d,7c7s,AcJc (.7%) (exploitable, need more hands earlier)
flushes, sets, full house (trips not enough)
We can see that it's mostly very premium hands that hit the range
Quarter pot bet
With a quarter pot bet if you are calling you only need 1/5 equity and to prevent your opponent from being able to bluff profitably with any two you need to call him with at least 80% of hands.
In other words 80% of all hands should have equity to continue and if not you will need to call and bluff or extract extra value on future streets such as with a flush draw that can bet big when it hits plus backdoor flushes and straight draws that can sometimes bluff with cards that miss opponent's range and typically hit yours so you bet as if the flush hit forcing opponent to bluff catch more frequently and thus getting paid more when you have a hand.
So let's start with the heads up range of hands and then the UTG 9 handed range of hands.
85% of hands vs 33% of hands
The flop it's ridiculous, basically a hand like T8 high on a J24 board is good enough. A 43 backdoor flush draw on a J72 board is good enough..
I would pretty much just call every flop even though some hands have 18% equity instead of 20% because the hands are going to narrow considerably on the turn I believe after backdoor draws are gone and cards that didn't improve remain unpaired. Also the player defending basically should call every hand. People probably don't, especially in tournaments and so betting really really small on the flop might be marginally profitable with any two cards. You might lose some equity by not betting larger with your good hands but you can bet a disproportionate amount on the river with a more optimal bluff to value frequency. Just an idea.
On the turn, you need to turn an open ended draw or a pair or have QT high or better. Actually QT is not quite strong enough without a gutshot but if we remove the possible flush draw it gets really close above 19%.
If you are defending with a 33% range vs 85% range queen 2 high is enough. High card gutshots like 98 on a J7 board or even 76 on a J438 are strong enough (19.34% with a draw to the nuts is close enough to 20% given another street of betting with a possible nut hand if a 5 comes)
So calling down really light becomes acceptable even if opponent occasionally doesn't bet with his weakest of hands and narrows down his range on each street.
77+,AQs+,A5s,KQs,QJs,JTs,T9s,98s,87s,76s,65s,54s,AKo
vs 4.89%
Almost the whole range is good except no backdoor flush draw, no gutshot, no pair. Even A5 high is good if you have backdoor flush draw. KQ is no good unless you also have backdoor flush draw or gutshot.
I'm really not sure this exercise is worth doing with 1/4th the pot bets because your call frequency is so high. Just throw out the obvious hands that have no chance and even if you call with any two cards it is never that bad of a decision if opponent is betting widely enough. On the turn you have to at least have some potential and on the river I'm sure you have to have at least a pair of some kind. I don't think we need to get into the details to define if the worst of pairs is okay if everyone narrows down their range. I could technically see there being a problem from the turn to river as if you call with every two on the flop and then on the turn you fold hands that have no draw no pair you probably shed more than 20% of your hand range. So you probably will need to float with some hands to bluff the river or bluff raise the turn every once and awhile. Perhaps this is a lesson that double barreling with small bets or even triple barreling (missed draws when you also have a lower missed draw) can be profitable just because opponents probably don't call the turn or river often enough with a small bet even when they know what you're doing.
Half pot is more where modern poker is played so you could be very, very thorough about different situations and even range from 40% to 60% since people sometimes bet above or below the half pot mark. The better you are at finding which hands have equity and which hand ranges have enough hands to not be bluffed with any two but not many more, the better off you will be. I think certain "all in" situations also need to be studied
-------------------------
I could go through and do a lot more work and I do want to come back to half pot play but first I have to cover the antes and I also want to cover 3 handed poker and 5 handed poker where I think there is a lot of room for error and exploitation.
There are different SIZES of antes. This should not be compared on an absolute basis but relative to the small blind. In some instances the small blind is 6 times the ante and in others it is 4 or even 3. Usually it is 5. But then there are the number of opponents at the table. 9 handed with all 9 players seated with the ante 1/3rd the small blind such as in 25 ante with 75/150 blinds there will be 3 small blinds or 1.5 big blind plus 1.5 big blinds for 3 big blinds in the pot. If you bet only 2 big blinds with a min raise and your opponents defend such that you break even on your bluffs, your opponents have to collectively call 60% of the time.
If you bet 3 big blinds and it's 5 handed with antes 1/6th the size of the small blind, then there is about 2.3333 big blinds and so if you risk about that much opponents have to defend about 50% of the time. If you bet 3 big blinds your opponents have to defend about 40% of the time. I believe bets tend to be on the smaller side so opponents probably will defend closer to 50% of the time in most situations.
As such, when calling out of the blinds and assuming the blinds fold you need about 33% equity vs the range that defends to have the equity to play for value.
1 vs 50% defender:
I want to start with 50% We want 33% equity over the top 50% of hands. 27 offsuit has 29.6% equity. 74 offsuit has 32.81% equity 94 has 31.963% equity T2 has 32.6% equity.
SO I'm not saying 100% of hands have value to raise with, but pretty dang close, You can bluff so play 100% of hands easily.
3) You have value with more like 75% of all hands or maybe 60% and can play all 100% when you include the bluffs
5)22+,A2o+,K6o+,Q9o+,J9o+,T9o+,98o,A2s+,K2s+,Q4s+,J7s+,T6s+,95s+,85s+,75s+,64s+,53s+
You have value with 45.1% of hands and can safely play 90% if you include bluffs.
7)22+,A2o+,K8o+,Q9o+,J9o+,T9o,A2s+,K2s+,Q4s+,J6s+,T6s+,96s+,86s+,76s,65s
About 40% of hands have value, so you can play about 80% hands if you include bluffs.
One thing I sort of failed to do was adjust the defend percentage. In reality the defend percentage probably should be a lot wider, particularly with antes because although we can play to deny equity from opponent, we actually have enough equity to call with weaker hands vs opponent's implied value range, much less his bluff range. In theory that means we should widen our range and opponent can widen his value range and narrow his bluffing range... but it involves a lot of moving parts to determine what the ideal raise and defend percentage is so I prefer keeping it here rather than continue to upgrade the strategy. If we happen to defend too widely against a much tighter opponent we could be in a very poor situation after the flop. While we are much less likely to enter these situations vs any given opponent we have to also respect that there is a table full of raisers and the odds that at least one opponent has a strong hand should bias the range given a raise to much stronger of a hand. In reality there should be a lot more calling and multiway action with squeeze plays and 3bets since everyone should adjust their assumptions about opponent so given you assume opponent raises with 80% of all hands 40% of all hands have value and although some medium strength hands you are better off calling, some bluffs plus the stronger hands gives you a pretty wide range of hands to 3bet. And as a result opponent can then say you raise with the top 40% of hands and then some of the strongest of the top 20% plus some bluffs can 4bet. Then you can 5bet with the premium of the top 10% plus bluffs. It's a little bit ridiculous to think about the profitability and just how loose and aggressive you can be with the antes.
There are a lot of reasons NOT to get nearly as crazy as suggested because
1)Tournament odds/ICM model. If you acquire 100% of the chips you do not win 100% of the prizepool, but probably closer to 20%. So you should be much tigher, particularly with decisions that are for large percentage of your chips and risk ruin, instead favoring a passive or tighter strategy.
2)It isn't enough to look at what the range of the raiser is, but before you 3bet, you also have to look at the probability that any one of the remaining opponents have a hand they can 4bet you with (in your upper half of the range) and the probability that at least someone started with it.
If there are 9 opponents and 8 people fold, they did not fold pocket aces and pocket kings and the probably are more likely to have folded a 2 through 9 plus a random card than an ace or king plus a random card. As such, you have to tighten your hand range considerably more when you open for a raise or 3bet someone who opened after several folds.
3)This is largely theoretical as very few opponents will widen their range enough to consider doing anything close!
4)CALLING is better if you have a post flop edge and your opponent is not perfect
5)Risk management! - There's something called the kelly criterion, although it's impractical in a tournament to apply the kelly betting strategy and never go broke, the long term rate of growth actually declines as you risk a greater percentage of your chips past a certain point.
6)Future opportunity - If players are so radically far away from optimal play once the antes kick in there should be a ridiculous amount of future opportunity to extract profit and keep up with the escalating blinds and antes as long as you don't get too short stacked. Try to exchange variance and profitability for survivability and consistency. There certainly may be certain points where it's worth taking the worst of it in order to have the chips to steamroll, but that's probably sometime before the antes kick in or before the small blind goes from 1/6th to 1/3rd the small blind or before the bubble or before the final few tables when you also have about 30 big blinds. Under 20 and you are kind of left playing for resteal shoves and double ups or instead you can limp in. Negreanu will limp in even with only 9 big blinds. Helmuth will minraise fold down to 7 big blinds and minraise call a shove after that. Most players would say that's absurd citing profitability, Expected value, tournament odds, and whatever reason, but I'll take them in a tournament over some keyboard warrior.
I wouldn't get quite as crazy as suggested because I'm pretty sure with more opponents
Let's with antes look at how often to call however. Remember, if opponent oppens with 50% of hands we have equity with close to any two. We can figure this out without reworking the hands. So.
Equity to defend vs value range (I'm not including bluffs)
1)100% of hands
2)100% of hands
3)90% of hands
4)90% of hands
5) I assume 85-90% of hands
6) I assume 85-90% of hands
7) 80% of hands 22+,A2o+,K2o+,Q2o+,J3o+,T6o+,96o+,86o+,75o+,64o+,54o, any suited
8) I assume 75% of hands or so.
You can see it's ridiculous. There's almost no incentive for anyone to fold preflop.
Multiway action gets a little bit different as well. Basically if you already have a call in front of you and you assume the blinds fold, you have to call 2 big blinds to potentially win let's say 6.4 not including your 2. This means you need 23.80% equity vs opponents hand ranges, plus strong enough hand to withstand a squeeze often enough to resequeeze collectively or call to prevent a player from being able to squeeze with any two... I would definately assume a tighter range here, looking for something like 40% raiser plus 80% defender.
call after a raise and a call with assumed 80% range and 40% range:
A2o+, K2o+,Q4o+,J7o+,T6o+,96o+,86o+,75o+,65o+,54o+,all suited. About 73% of hands.
Granted, we are assuming the small and big blind players give up the blinds and they probably won't all of the time. That gives us a little bit less to play for OR at least more opponents and we require a little stronger hand. BUT we also will have position over the blinds which especially matters when our decision is on the button and there is no one left to act.
In practice, implied odds and playability after the flop and being able to draw to premium hands matters a lot more, especially multiway. You'd much rather be drawing to straights and flushes than trying to play for low pairs or even top pair top kicker vs lower pairs and draws and straights and flush draws. I would certainly give up the possibility that I am giving up some value to avoid squeezes, avoid trouble hands and play hands that have a little stronger edge anyways...
We can probably mix in bluffs and call more to bluff later or
Bubble odds.
Some smart people using the ICM model have figured that since winning 100% of the chips doesn't win 100% of the prize pool that the chips you win are worth less than the chips you stand to lose. They have refined this further by assuming everyone has equal skill and determining just how much the chips you gain are worth relative to the chips you lose at any given situation. In the book called "Kill Everyone" (sequel to kill phil) they go over a lot of situations to illustrate how a tournament changes over time in it's tournament odds. On the bubble your tournament odds rise to around 1.6 in a lot of tournaments. Some are as high as 1.8. Sit N Gos are 2.1 on the bubble I believe. Super sattelites can go up to ridiculous amounts since you're only waiting for the first person to bust and there's no need to risk anything.
At 1.60 this means you are risking 1.6 times what you are getting on a 1:1 bet. Or perhaps it means you are getting 1/1.6 or 62.5% what you risk. In other words, if you are putting up 2 big blinds to win 2.2 big blinds, you are really putting up 3.2 to win 2.2. If that's the case, you need 37% equity. Or is it putting up 2.2 to win 1.375 and you need 38.1% equity instead of about 33%. I don't remember exactly but the point is, the bubble factors force all of the above calculations to be a bit tighter, requiring a bit more equity. I think that's important to keep in mind, however... WIN WIN WIN WIN WIN WIN WIN.
We will move ahead and you can adjust accordingly.
Let's look at flop play when you have the antes involved. In reality there will be a lot more 3 handed and 4 handed play, particularly blind defends. So I'm going to start with something different and that's approximate 3 handed play. Hand range values change when 3 handed
What if we assume the big blind calls at a discount whenever he has halfway decent equity and let's say small blind folds and someone else calls not in the blinds.
If there is 2.2 big blinds or so preflop then we put in 2 more plus opponent not in the blinds calling 2, plus the big blind calls for 1. So we now need about 27.78% equity vs 2 other hands. The big blind we will mark as the top 70% of hands. The other caller will still defend such that we can't profit with any two if the blinds were on autofold.
So the minimum we will start with is when there are 3 other opponents left and the one in the blind calls such to prevent us from breaking even on bluffs.
If we are risking 1 to make 1 we stick with the 20.63% range of hands. If they call with more as they certainly could, it just expands our value range into some of the bluffs and increases the amount of value the value range has so I don't see a big flaw in calculating it this way other than to say you probably could expand the raising range, especially if you are going to be heads up vs the blind(s) a lot of the time with position and the initiative.
You have equity/value to play
22+,A3o+,KTo+,QTo+,A2s+,K4s+,Q8s+,J8s+,T8s+,98s,87s,76s,65s
This is a little over 30% of hands. We can add in bluffs so that we play around 60% of hands. 30% is quite a bit lower than 75% having value as calculated heads up when the blinds fold, but it's still quite a bit when we add in bluffs.
If the big blind is the only caller, we are risking 2 to win 3.2 when called so we need 38% equity when called vs blind calling range. Our opponent only needs to call 1 to win 4.2 so he only needs like 19% equity. basically close to any two. We have 38% equity vs the top 70% of hands with close to any two ourselves 22+,A2o+,K2o+,Q2o+,J3o+,T6o+,96o+,86o+,76o+,A2s+,K2s+,Q2s+,J2s+,T2s+,93s+,84s+,74s+,64s+,54s
or 71.9% of hands.
So do we play 100% of hands or not? I'd say no, but we can play around 60% of hands as that gives us bluffs plus value in multiway pots and if we only get to play against the blinds we pretty much have a pure value range against a properly loose defender. That may not quite be 70% of all hands
Let's go to the flop.
Our range:
22+,A2s+,K2s+,Q2s+,J2s+,T5s+,95s+,85s+,74s+,63s+,53s+,42s+,32s,A2o+,K7o+,Q7o+,J7o+,T7o+,97o+,86o+,75o+,65o,54o,43o
Caller on button:
20.63%
Caller in blinds:
70%
Flop J56 with 2 hearts. This will more properly represent multiway flop as there will be draws.
Let's look at some weird situations. We want to make a 1/4 sized bet called by 2 opponents, a half pot bet called by 2 opponents or a pot size bet called by one opponent.
1/4 pot bet.
When we make a 1/4 pot bet it may get some raises sometimes but it may also allow for a cheap draw sometimes. Since we are risking 1/4th to win 4/4ths in the pot plus 2/4ths from our opponents or 6/4ths we only need to win 1/7 times assuming future decisions break even. They probably won't call with 100% of their range, but what we gain in equity from folds we probably lose from stronger hands. So it equals out. Winning 1/7 means even 14.28% equity is enough vs opponent's calling range.
This means. any pair, any pocket pair,
A2o+,K4o+,Q7o+,any gutshot 98, 23 (gutshot) or open ended(34o,78o), any flush draw.
K2s+ backdoor flushdraw,
Q3s+ backdoor flush draw.
T8s+ backdoor flush draw
98s backdoor flush draw(even on flops like J34h where we don't have a gut shot.)
87s backdoor flushdraw.
If we started with the top 60% of hands, that's
22+,A2s+,K4s+,Kc3c,Kh3h,Kc2c,Kh2h,Q7s+,Qc6c,Qh6h,Qc5c,Qh5h,Qc4c,Qh4h,Qh3h,Qh2h,J2s+,Tc9c,Th9h,Tc8c,Th8h,Tc7c,Th7h,T6s-T5s,95s+,85s+,74s+,63s+,53s+,43s,32s,A2o+,K7o+,Q7o+,J7o+,97o+,86o+,75o+,65o,54o,43o,32o
or 57% of the preflop hands. This is a LOT!!
If you can control the bet size by a bet when opponents just call, whereas a check will induce a larger bet a huge percentage of the time. do it. I even like doing it with strong hands because I think I can make good folds to big bets on the turn, read the board well and make good decisions and keep both opponents in. 2 1/4th pot bets are the same as 1 1/2 pot bet with 1 caller and loosely betting with your good hands will protect your range if opponent raises, so I don't mind betting small with strength too. Although I still prefer big pot big hand, small pot small hand more often if possible.
Now let's look at a half pot bet size with 2 callers.....
With a half pot bet we have 2 callers to make 2 times the pot and we only have to call half .5/2.5=.20.
We only need 20% equity.
any pair, any pocket pair, A7o+,KQo, any gutshot (98,23), or open ended (34,78), any flush draw.
A6 backdoor flush draw
That is 41.9% hands out of 60% or about 70% of our preflop range.
22+,A5s+,Ah4h,Ah3h,Ah2h,KcQc,KhQh,KJs,KhTh,Kh9h,Kh8h,Kh7h,K6s-K5s,Kh4h,Kh3h,Kh2h,QJs,QhTh,Qh9h,Qh8h,Qh7h,Q6s-Q5s,Qh4h,Qh3h,Qh2h,J9s-J2s,Th9h,Th8h,Th7h,T6s-T5s,95s+,85s+,74s+,63s+,53s+,43s,32s,A6o+,KQo,K7o,Q7o,J7o+,97o+,86o+,75o+,65o,43o,32o
Now collectively between us and one of the other opponents the defenders have to prevent the better from being able to profit with any two. So if our opponent bets half the pot we collectively need to call him 66.67% to make him indifferent to bluffing or not. We'll easily do that here. We could probably be a lot tighter, and perhaps we should be since opponents can narrow their ranges. But we do have flexibility to call with any of these hands to bluff on later streets or fold some of the weaker hands as well if it's necessary to carry more or fewer hands onto the turn or river.
I would personally want to fold low pocket pairs and also any high card hand without at least two overcards or some kind of backdoor draw... like I would fold A7-A9 I might still call AT sometimes and I probably call AQ and AK. But if I were on the button with this range or knew my opponents would not raise, also would call with more backdoor draws like any ace of clubs or better and any sort of backdoor straight draw with 2 clubs like K7 or K4 of clubs or even T9 of clubs.
If we assume our opponents narrow their range significantly before they determine it's worth betting then we'd have to produce some kind of logic to arrive at their betting frequency and then look at equity vs those kind of hands and determine if we have enough. I'd say both the player calling and placing the bet probably would narrow their betting range somewhat and have a stronger hand.
We can probably throw out some of the high cards, low pocket pair. I'd be tempted to toss out the top pair weak kicker hands even though it has lots of equity (45%) vs the full range of our opponents, as they narrow their range if it doesn't improve we are going to probably be in trouble in any big pots where we see betting on the turn and river. I'd almost rather have 56 or 78 for 3 to a straight and a pair because those have not only 5 full outs to improve to 2 pair but also 8 outs to bring an open ended straight draw and 8 outs to bring a gutshot straight draw on the turn. Then I have a lot of runner runner full houses, runner runner straights and 2 chances at 2 outs to trips and 1 out and 1 out to runner runner quads. these hands are all very very likely to win and 3 outs to make 2 pair certainly will be strong. Plus I will want to have some possibility of bluffing as if I made the heart flush draw on the river with some of these hands.
How about a pot sized bet on the flop with 1 opponent and 2?
With one opponent it's tricky because if we place the bet and only one of them calls, it's generally the stronger of the two so we can't simply look at the equity of 1 or the other, we have to narrow them down. If opponent bets the flop and we call we can't just look at the opponent's range he came into the pot with because he bet the pot knowing he had 2 opponents which means his range has to be stronger if he's optimal
Let's just for now stick with the range and look for equity of 25% vs 2 opponents.
Any flush draw, any pair, AQo+, high gutshot with a heart or suited with backdoor draw, open ended. We're still going to have a lot of hands that can continue vs assumed range.
OKAY!
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