The all in represents 2 times the pot. Assuming you raise and the preflop action is break even, or your hand range compares fairly favorably when called, you only need a 2/3rds or 67% chance of opponent folding for shoving with any two to be profitable even if you were drawing dead to any calling hand. It also represents a stack size that is still significant to most players.
A few things to keep in mind about this.
1)If either you just call or opponent calls in the blinds, the equation will be slightly different.
2)Looking at the probability you need opponent to fold for a total bluff to work is the "exploitative shoving".
Equilibrium shoving is a little different.
Equilibrium shoving depends largely on hand range vs hand range but basically opponent has to adjust calling range and you your shoving range until neither can make more by changing it...
This is quite difficult to solve for, but we can approximate for it
We make the assumption first that preflop decision is break even which may not be the case. If you are raising with too many hands and getting raised often enough it will not be. If you are putting in 3 big blinds to win 2.25, you have to win the pot right there 57.143% of the time and get raised off of it the remainder or get called in which case your equity is based upon post flop decisions only. That probably is too high. Conversely if you only raise 2.25% you only have to steal successfully 50% of the time.
So let's first assume you raise from the button with any two and the big blind defends with 42.9% of hands or 22+,A2+,K8+,QT+,JT,T8+,98,87,K2s+,Q6s+,J6s+,T6s+,95s+,85s+,74s+,64s+,54s. You also are assuming he never raises or if he does raise, he randomizes what hands he raises with so when you see a flop he always equally has the same probability of having EACH specific hand which isn't true. You are also assuming that opponents who folded fold randomly rather than placing more value on cards with aces in at least one of the cards. Otherwise, the probability the opponent holds an ace is higher, and the probability of a given flop changes.
Just for information purposes, equity PREFLOP with any two is still 43.06% if you are able to see your cards until the river. According to pro poker tools (assuming I am running the notation correctly).
propoker tools online simulator tool here.
(correction: I used the wrong notation on the suits, I should be using 6x5x for 64 suited, 64s means 6 of any kind, 4 of spades. So it should be 42.49% equity)
Now with a J82 flop? What is the equity?
It drops to 37.3%. That assumption is not quite right because that only tells us if both players check to the river what percentage of the time they win the pot. If one moves all in the other can adjust and call with strong hands and fold the weakest. This is where the calculation gets complicated.
Say for example you shove with any two on the flop. What if opponent calls with JJ, 88 and 22 only? First you have to calculate the equity when he calls, next you have to calculate the equity when he folds.
So you're saying there's a chance!
Equity when opponent calls is 6.33%. On rare occasion you will have draws to a flush or straight or back door trips or a 2 outer to make a higher set...
What I like to do is say out of 100 pushes, how many times my opponent calls. Out of all those calls, how many times does my opponent win, how much do I win and then figure it out.
This is tricky. Preflop the odds of being dealt a particular exact hand is 2/1326. Why 1326? The first card dealt is out of 52, the next is 51, order doesn't matter. Being dealt 8 of spades first than 7 of spades is the same as being dealt 7 of spades first, then 8. So (52*51)/2 is 1326 unique combinations considering suits as each unique. But there are 4 suits so if you are talking about 87 suited and not 8 of spades, 7 of spades, the odds are 4/1326. There are 6 combinations or ways you can make JJ so the probability is really 6/1326 for any pair. There are 9 ways you can make two unsuited cards.
In reality we get to look at our own hand so IF we don't have a jack there we can say the actual odds of opponent holding a specific hand changes. However, we are raising with any two and not going to even look, so we will ignore the slight changes in equity by adjusting for our hand.
However, we do know the flop. A J82 have been dealt. We KNOW there are only 49 unknown cards remaining in the deck. We also know there are only 3 of each matching card and then 2 after the first one is dealt.
So for JJ, our opponent can be dealt 3 cards and then 2 remaining but order doesn't matter. (3*2)/2=3. This is divided by (49*48)/2 to give us the odds our opponent has exactly JJ. To determine the odds JJ, 88 or 22, then we can multiply this by 3.
Ahh... but there is a problem, it isn't a "random" hand opponent has, we KNOW information and can adjust to this.
We know our opponent plays a certain range (well we assume, but for now we will assume that we know exactly how opponent plays.)
Pairs in a random set of cards has a different probability of existing than in a fixed range of cards.
In real life since our opponent just called we might assume that he doesn't have AA or KK or that those two hands are LESS likely while QQ and AK are also LESS likely. But that complicates things yet again.
So let's just assume there is an equal probability of opponent holding any one of the ranges. A better way to approach this is to determine given our opponent's hand range, what are the odds he flops a set of any kind. Since he will play 22 through AA with even probability the chance ANY of his pairs flop a set works in this case.
The odds of flopping a set given a pocket pair are calculated by looking at odds of NOT flopping a set on each hand, then multiplying them and subtracting by one. So there are 2 cards in the deck out of 50 remaining that hit a set. 48 do NOT hit a set, then assuming you miss the set there are 47 out of 49 that do not. Then 46 out of 48. Multiply these together to determine total odds that NO set is hit on the flop. The remainder or 1 minus that amount is the odds a set or quads IS hit. You get 11.7551%... GIVEN a pair is held.
What's the chance opponent holds a pair?
In a random list of hands it is simply the odds of a single pair times 13. However, it isn't "random" list of hands, but a known range. There are a total of =13*6=78 combinations of hands that make a pair. We must look at the combination of ALL hands in opponent's holdings and then the percentage of them that are pairs.
So if he has a pair he can flop a set 11.7% of the time. Of his total hand range this means in a given flop he has a (78/566)*.117 or about a 1.62% chance of having a set.
So our opponent clearly will be folding way too much if 98.38% of the time he will fold the flop.
Just to show you the math so you understand the process,
98.38% of the time you win 8.25 big blinds.
1.62% of the time you are called.
Of the 1.62% of the time you are called, you win 24.75 big blinds 6.33% of the time.
Of the 1.62% of the time you are called, you lose 16.50 big blinds 93.67% of the time.
Total expectation of the move?
(.9838*8.25)+(.00102546*24.75)+(.01517454*-16.50)=net profit of 7.89 big blinds for the all in move if opponent only calls you with a set.
Question, against an opponent likes this, at what point do we have too large of a stack to move in for any profit?
About 574.25 times the big blind or nearly 70 times the pot!
However, most opponents only play say 77+ and AQ and maybe some others so pairs are a larger representation of their hands if you are up against a tighter opponent preflop.
Nevertheless, there is a clear lesson. First, you cannot wait to flop a set against opponents unless you have some way to keep up with the blinds in the meantime. Second, you better be very selective about what hands you play if you do not have a clear plan, or be the hyperaggresser rather than forced to make a difficult decision. You clearly can afford to fold when OTHER people push all in if you have more than your ample time to return the favor.
To find "equilibrium" we must see if we can improve our opponent's decision by calling more or less often. Intuitively we don't need to even test playing less often after the flop since we know that given the preflop range our opponent will have to call more.
Okay, so in a J82 flop what if our opponent also calls with any pair?
Now starting with ANY pocket pair as well as in this case any jack, any 8 and any 2 will call. BUT when they do we will still have a chance of winning. In fact, we improve to a 37.8% chance of winning when called. Awesome.
However, there are also 278 hands out of 566 that opponent can call with! That means he folds only 566-278=288/566=50.88% of the time.
50.88% of the time you win 8.25 big blinds.
49.11% of the time you are called.
Of the 49.11% of the time you are called, you win 24.75 big blinds 37.8% of the time.
Of the 49.11% of the time you are called, you lose 16.50 big blinds 62.2% of the time.
Total expectation of the move?
(.5088*8.25)+(.185636*24.75)+(.305464*-16.50)=net profit of 3.752 big blinds for the all in move if opponent only calls you with a set.
We improved our opponent's chances, but clearly pushing with any two is still favorable.
Question: Against this opponent at what point do we have too large of stack relative to the blinds to push?
Answer: About with 47.81 big blinds into 8.25 pot or about 5.8 times the pot!
However, that only compares it against the option of folding. Perhaps the better question would be to compare it to checking it down and winning a portion of the 8.25 big blinds. Since checking it down would earn 37.3% of the pot anyways, it probably against most opponent isn't necessary to move all in. If you have position you also have an informational advantage.
37.3 times out of 100 we win 8.25, 63.7 times we lose nothing.
We still expect to yield 3.07725 big blinds just checking it down.
So while you can say it's 'profitable" to shove all in, there is a large risk if you have an extended bankroll or are risking your tournament lifeas you not only will make money from checking it down but you will at times make significantly MORE money while avoiding pointless risks that create so much volatility you'd go broke or bust out of a tournament even with a large edge.
So let's revise.
Question: If our opponent calls with a set to a shove only, at what point is checking it down the better option?
Answer: At about the point where a shove is 356.7 big blinds or about 43.24 times the pot.
With a skill advantage you may be able to play in smaller pots and win more than 3 big blinds, or if you are out of position you or have a skill disadvantage you may win less.
Question: If our opponent calls with any pair to a shove, at what point is checking it down the better option?
Answer:At about the point where the shove is 22.13 big blinds or larger into 8.25 pot or about 2.7 times the pot. We should start thinking about playing this long ball when we have less than 25 big blinds.
You can see that you can make a huge mistake by shoving if you have deep stacks and your opponent correctly calls you. You also can see that from your opponent's perspective, we have been able to substantially improve his decision by calling with more hands.
It's worth thinking about not shoving as well as you may have an exploitative edge greater than your equity of checking it down, OR opponent may be able to make it too pricey to call when you are behind to actually see the full equity. If your opponent is very passive, just check it down UNLESS you have a hand that has a greater than 50% chance of winning. This will add equity. Of course so will getting the better hand to fold by making small bets. But the idea of adding too many layers to the game is very tricky to accurately quantify. That doesn't mean you can't gain a larger edge from doing so or being more "intuitive" about it, just that you won't have a lot of confidence on decisions that are already less than certain. And the lower the confidence, the more unnecessary risk of making decisions that actually lose chips in the long run or make less than other decisions would.
Preferably, position should have an advantage that helps one to gain equity but there are many vulnerable hands worth playing.
Additionally, there are options such as shoving in with hands that have little to no equity and "playing poker" with those that don't. This forces our opponent to loosen up even more on his all in calls while tightening up on his calling of small bets than if you just randomized it. However, it puts us in much less favorable position when we are called.
Alternatively, if you only shove in with the top 10% strongest hands and the top 6% weakest hands, or top 20% strongest hands and top 12% weakest hands you likely still have the best hand a good percentage of the time when opponent calls, and it still allow you to still bluff a high percentage of the time, while still gaining some equity from intermediate strength hands and draws that you show down to the river.
Keep in mind that if we just CALL the flop we probably NEED to yield at least 3 big blinds to risk calling otherwise we lose money on our calls. This is pretty interesting since we earn about 3 by checking down and earn about 3 by a long ball strategy at the listed amount so if our opponent bets with 43% of hands we can call with any two at those levels. From the big blind we can probably call even with deeper stacks or with a slightly tighter opponent.
Question: Against this opponent, who auto continuation bets on 100% of flops, how frequently can we call with the intention of check raising all in? How deep of stacks are too deep for this?
If our opponent auto continuation bets from the big blind, we can anticipate how large the bet will be and check raise shove with deeper stacks than that as well....
So for example say pot will be 8.25 big blinds minus 1 or 7.25 since the big blind is priced in. So our opponent may bet say 4 big blinds on the flop. That puts 11.25 into the pot and we can shove at a MINIMUM with 2.7 times that behind or 30.4 plus the initial 2 big blinds that we called preflop means around 33 big blinds you can go to long ball check raise defending aggressive button and cutoff raisers that have a range of 43% of hands with any two cards.
I would actually go with that move with deeper stacks than that because:
1)Opponent probably won't call with underpair to the board. Maybe with AK if they think you are making a move, which stands to reason why you need to anticipate that your opponents will try to adapt and you may need to make adjustments
2)The out of position disadvantage plus autocontinuation bet opponent means checking it down is probably not the best way to play and probably will gain less than 3 big blinds by trying to just play normal poker or small ball poker out of position.
3)It's a good practice to defend your blind aggressively. Even though you have zero equity in just calling, you need to come up with other ways to defend so it isn't just reraise or fold. Doing so will give you more flops to opponents who are afraid to raise you, and will give you more blind vs blind confrontation as well as more walks to people that just fold around when you are in the big blind.
4)You can fold the weakest of hands on the flop as well as the weakest hands preflop and probably gain substantial equity on this move.
HOWEVER, you better be prepared to ADJUST based upon the flop and that means you will have to do the work to know how to play many other flops correctly.
Calling gives up 2 big blinds instead of 3 so at minimum you need to not have so many big blinds where shoving doesn't at least yield 2 big blinds since that is the equivalent of breaking even on all streets. In a standard pot this would be 31.1 big blinds with a shove of 3.77 times the pot. However, since you only have to call 2 big blinds since you already paid 1 and since the pot is smaller, and since our opponent has added money on the flop, we can just use the 3.77 multiple to the 11.25 big blinds or at about 42.4 big blinds.
With MORE blinds, I would rather just defend with a tighter range of hands than that and use the same move up to 50 or 60 big blinds. Alternatively you can call with the same amount and just be more selective about shoving and otherwise either lead out with a bet or call and lead out on the turn with less than a shove if opponent is weak enough. Or call and check it down with the stronger range of hands otherwise fold and push with your draws and monsters.
To just say this without calculation though is more speculative. It's important to actually put in the work to not lose money to opponents who have.
When you have a hand you could get tricky and try to call but there probably isn't a whole lot of added value unless you know the opponent will bet twice. In principal calling the flop and shoving the turn instead of the check raise doesn't change much except it may give an opponent with AK more reason to fold and also a chance to catch But it may give you a chance to catch as well. Fa
Both of these sets of numbers provides a range of thinking that is good enough to think about exploitative poker. Players generally are somewhere between calling with any pair and waiting for a set. However, some very loose online players may not give up ace king or an open ended straight draw particularly if they see you shove a lot. Also, our opponents can shove themselves when first to act to take that move away from us. Nevertheless, we are looking at equilibrium play so we can do one of two things.... show how loose our opponent has to be to defend against this strategy, or we can instead refine our opponent's preflop strategy to be much tighter. From the button we will gain some more preflop but not be able to shove as much post flop without giving up an edge.
Working on some of these numbers in either direction will allow us to both try to approximate equilibrium calling strategies and how large of stacks shoving all in is acceptable against a different preflop range of opponents on a particular flop as well as provide ranges for exploitative play against opponents that call too often or not often enough vs our behavior.
This can be compared with other strategies of trying to gain equity to really sharpen up out long ball game.
Of course, it is a LOT of work to keep doing this but you have such a massive edge when you learn these sorts of shoving all in strategies. You can even add in variables such as always betting flop and shoving turn knowing that opponents who just call will have difficult decision or have to trap you and if they have overcards they only have 3 outs to where they probably call, under pocket pair has 2 outs, overpair will call, top pair may call, mid pair or bottom pair has 5 outs and in some cases more (straight draws?) So you can work in turn shoving which is less common and will be a lot tougher for opponent to deal with, especially if you build up a pot and normally play small ball.
You need to have a good feel for opponents who call with AK and AQ on the shoves. So you need to do the work vs loose callers also. You also need to understand that as your stack sizes get bigger opponents tend to respect shoves as "less desperate" as well.
But I have given you a glimpse on how to actually do all the math to determine what shoving strategy works and what strategy is too aggressive, and have given you some food for thought on your expectations of just playing small ball as the alternative. If you are a better small ball player, you should use long ball shoving only when you are making say 4 or 5 big blinds from that move as a result of either having a tighter preflop hand selection OR being more selective on the flop.
If position is worth 1 big blind when you have equity of 3, we can put that in reverse and say that in position you will not want the long ball strategy. However, you also can say the check means weakness more often than the bet does against certain opponents so that is worth something.
In position you might therefore not want to shove in favor of keeping the pot small and "playing poker" unless you are under 18 big blinds on the flop or about under 20 big blinds preflop. This is because you can trap opponent, your opponent has to bet or risk you not betting when he is strong and you are weak, and when you are strong and he is stronger he'he risks committing more chips. You can wait until the last move on the river to shove depending on opponent's actions as well with the right proportion of bluffs to actually having a hand to be very deadly as well.
In tournaments if you have an edge, you can place a bit more premium on a survival due to opportunity cost of getting profitable situations in the future if you instead play tighter, cautious and more patiently. However, you should also consider that there is a premium on getting more chips since it allows you to play more pots, playing more speculative hands and allowing you to have enough chips to stay up with the blinds in the future... so it's a two edge sword that depends on how much better you are with a deep stack and how top heavy the payouts are, and what a particular image means for the future if you do double up and survive at some point. I wish I had a way to weight the benefit of having more cips vs the benefit of the opportunity of waiting. It is easy for me to calculate what the odds of getting a better hand are or come up with a max patience strategy, but I don't know how to turn that into actionable data yet and measure it against "utility" just yet.
However, nevertheless, I would repeat this exercise with the loosest opponents who call with ace high, solve for equilibrium in them, and also repeat with different preflop ranges. I will also suggest repeating this with different flops. There are probably under 100 general "flop types" if you just look at high, medium and low cards plus an ace and suitedness adds several more types but you can keep the "flop type" fairly simply if you'd like. Certainly a jack and a queen might be significantly different vs some opponents who play different ranges of hands so you may want to just see how much it differs for a few, but once you have that in mind, you can kind of get a "feel" for how much to adjust it by so you don't really need to work out every single card. The 2-s through 7 don't really matter much at all but the ace makes a big difference for a lot of opponents.
You can also think about "moves" like check calling the flop and check raise shoving the turn against the double barrel better by anticipating bet sizes and stack sizes and doing this same excersize but with a 4 card flop and think about defending your blind a lot sooner in a tournament when you have a much larger stack size. You also will want to make adjustments for the 2x and 2.x preflop raisers and 4x so you have a feel for stack sizes and how to adjust as well.
What I've done is I've looked at shoving with ANY draw on the flop, any set, any overpair and breaking even on everything else and consciously adjusting my preflop hand range so I can approximately break even on the shoves (assuming preflop PLUS pairs played to river is profitable) I don't still have that data but basically you look at odds of flopping set vs odds of flopping the draws you push with, and you balance them out to the right ratio depending on your equity of someone calling and the proportions of hand you play and how deep the stacks are. This is somewhat complicated but basically if opponent will call you half of the time with a pair, you want to have enough equity in your shoves to break even. That way if opponent DOESN'T call with pairs and better flush draws or straight draws you win enough on those hands to profit. Plus you can small ball any pair and gutshot draw based upon opponent a little bit, and occasionally you can small ball your sets and draws and shove the turn.
Well, I'm not making this blog to do ALL of the work for you, so hopefully that provides enough information and a framework for you to figure it out from there.
FYI, If you can learn to get the right balance against the right opponents in the right situations, it is MASSIVELY profitable since opponents have to give up a large pot or risk making a very bad call. It is very hard to know what range of hands your opponent is shoving with unless you play with him a LOT and even then what if he switches his style? How would you know if he is shoving with a much tigher range?
The way I like to use this information is by going into that "mode" every rare once and awhile so that it is maximally effective, and then going back to small ball or a much tighter all in strategy.... Especially if I can show a massive bluff or semibluff a time or two when stacks are deep, or show the goods when stacks are short and then proceed to do mostly the opposite.
Next up we will look at pushing with any two vs a much tighter opponent, then how tightening up our range changes this information.
No comments:
Post a Comment