Waiting for Aces

If you are waiting for aces, what are the chances you get them with X hands left? What about waiting for QQ or better?

The probability of getting AA in a hand is 1/221, so the probability of not getting it is 220/221, and the probability of not getting it in x hands is (220/221)^x. This means that if we can solve for the probability of NOT getting it in X hands we can determine that the remaining percentage we DID get it at least once over X hands.

For example (220/221)⁶⁰=~.761771. Probability of getting aces is then 1-.761771=~.238229

 What about Aces OR kings.
(2/221) or (119/221)⁶⁰=!0.579576102. 1-0.579576102=~.420424.


Action is information. You could run some math to figure out what the odds of one of your opponents getting aces, but it would probably not be as relevant as the odds BASED upon how they act. You could easily define their rasing range on hands they would play in that manner based upon history and be reasonably confident and more accurate...

Define a hand range as 10%, then 3bet range as 4%, and of the top 4% of hands, aces is .4524% of that, which makes it about an 11.3% chance that an opponent who reraised you with that kind of range has aces. It could be higher or lower. Realistically you aren't betting probabilities but probability ranges. You might statistically depending on the sampling size be able to say that you can be 90% confident opponent has between a 8.3% and 14.3% of aces. You could say that about ANY specific pocket pair, so the chances that you are ahead with QQ in that spot is probably about 77.4% Although the chances opponent has AK is 1/82.875=1.2% which gives you about a 30% chance of being in a race, and combine that with a 22.6% of being way behind and suddenly QQ doesn't look great... Especially if you have to put in another bet that opponent may just fold if he has less than KK. Folding actually in this spot may not be a terrible option if you have faith in opponent's 3bet range. However, calling and avoiding flops like AKx, Axx, while trying to keep pot smalls on flops like JTA, and possibly shoving with very few king high flops (since opponent will have to respect YOUR range which would rarely shove there with less than a king), and some AAx and KKx flops and trying your best to get opponent to try to bluff and get all the money in on Queen high flop or lower.

But if the shove is all in preflop, you probably would want to call unless you feel table is weak and can pick away at small pots and pick your spots a little better in a tournament.

Take the time to construct a table which uses the math to determine the probability. Look at the range in which you'll expect a hand in that range 50% of the time as a good range to plan on as an all in hand, but adjust it as you become more and more short stack as you approach fewer and fewer hands left. Be willing to raise all in with 1.5x that hand range since you don't always get action (I actually go for 2x the range I can wait for as I don't want to wait until I am short stacked if I can help it... Exception is when I am deep enough in tournament and want to be even more patient and wait on hands which are less than 50% likely to come because they way I can fold my way up the money and THEN develop the tight image and make up for it with extreme aggression once blinds go higher and fewer people with fewer hands are in.).

You can develop an entire strategy based upon how many hands you can wait for, provided you take the blind structure into account.

Letting opponents exploit you

Now a lot of people are focused on "equilibrium" poker now days. I admit, it is very interesting and can be very profitable. However, I think there is a lot to be said about an exploitable strategy.Many people in turn play an exploitable strategy, but THEN in certain spots try to balance their ranges. But in reality there are plenty of spots where there is a certain "quid pro quo". If your opponents are letting you rob them blind in one spot, rather than too aggressively try to play back at them in another spot, just let them exploit you in one spot, and exploit them in another. Ths is especially true in tournaments. Sometimes it's okay to let them get away with crazy ridiculous all in bluffs if they were to move in with every hand, but you trust your read that they wont and even if they do, there is an adjustment period and a counter exploitation.

For example, what if you could raise the blinds with a min bet, and do this every time and steal a pot with a pot size risk more than half the time everytime? You could coast your way into the tournament provided you don't go all in. Granted, opponents will adapt, but you can as well. This is to illustrate that even folding aces would be correct if in theory you could indefinitely steal your way to first place without ever getting knocked out of the tournament or ever being at risk.

Realistically this assumption is terribly misinformed and I would not suggest folding aces.... However, folding Jacks or queens? that's another story.

Most people who play lose expect to be able to get paid off and they expect to get played back at. And when they get ace-x or top pair they can't lay it down because they anticipate players will play back at them with air. I at times struggle with BOTH being extremely exploitative YET also making big laydowns during that style... Such a style is mathematically incorrect... Theoretically if you raised 75% of your hands and folded 5/6 times to a reraise a player could reraise 10% of the time and be profitable and just fold to your 4bet all ins almost every time as well. But what happens if you switch strategy without them knowing it and go from raising 75% of the time and 4betting 1/6 (or 4beting with the top 12.5% of hands) to raising 40% of the time and 4betting 1/2 of the time? You'd be 4bet stealing a lot (20% of all hands and half the hands you raise). But by the time they realize it you have a ton of chips and they've been conditioned to fold and assume you finally got a hand. They probably won't notice whether you are raising 75% of all hands or 50% of all hands because you could just be getting dealt 27off and 29off several hands in a row. Then you could be dealt AJ+,666+ and be moving in. If opponen tis 3betting you very light and assuming you are moving in with 77+ and AJ+ they have to wait for JJ+ or AK to be able to call. Maybe TT or AQ, I haven't calculated the equity and it depends on how much dead money is in the pot. But if you are 4betting with 20% of hands (55+, A2s+, AT+, K7s+, QTs, JTs, KT+, Q9s, J9s, T8s+, 98s, QJ) you have a very good chance of getting them to fold way too much. And even when they call you, you still may happen to have a very strong hand.  Alternatively, you could tighten way up and take more of a passive approach to see more flops if you recognize there are greater exploitation opportunities on the flop.

Losing a huge portion of your stack or busting out of the tournament isn't a big deal if you can win often enough to compensate for your losses as well. And if you are far enough away from money, or at a key stage where massive chip accumulation can significantly improve your chances of finishing deep, there is no problem at all with taking this strategy for a few massive steals and then moving back to exploitative/exploitable again.


Even though a strategy may be theoretically "mathematically incorrect" you have to challenge the assumptions made because it may still be correct in real life. This is BECAUSE math often requires certain "flawed assumptions" to simplify. The flawed assumption is that players adapt perfectly to whatever style you have. Realistically if they THINK you are playing loose, you can switch to tight for quite some time before they realize you have switched styles. So you can be quite "exploitable" for quite some time and switch styles to the opposite.


But in reality that is sometimes EXACTLY what you should do, and when I DO, I have a great chance of doing very well, provided I don't get impatient and regret that decision rather than just focus on the game.

Strategy in Focus: (fundamentally sound:post flop scenario analysis)

In the post, 4 poker fundamental strategies, I gave a few preflop strategies. We are going to put one of the strategies in focus and talk about how we will use the information to construct a postflop baseline.
There are 117,600 possible flops, so we must "generalize". What we want to do is look at cards 2 ,3,4,5 as if they are one single card, cards 6,7,8,9 as if they are one card, T,J,Q,K as one and ace as one. We then can get down to 20 unpaired flops, plus 16 paired flops, plus 4 flops of 3 of a kind for a total of 40 unique situations.

Now as we broke up preflop raising strategies into 8 possible raising range per strategy, we can get down to 320 situations we need to solve. Rather than solve all 8 raising strategies, we can solve about half of them and get a close enough idea. This still is 160 possible situations to solve per strategy. PLUS evaluating flush draw flops. How in the world do I expect to do all of this? By getting to work immediately. What use will this even have? By having all 40 strategies laid out on a spreadsheet, and the corresponding strategies given a set of hand ranges of that particular strategy and how you handle the flop, you will have an incredibly solid framework for correct poker decisions, shaving off hundreds of hours of practice, plus recency bias and results oriented biases among other biases, and developing a very good idea backed by math of what decisions to make and how to analyze them.

So the strategy in focus for the time being is this raising strategy. The most common spot will be in a pot that we raise the standard 3x big blind. It really won't matter as long as we aren't figuring all in strategies or complex 3bet on the flop strategies.

UTG: 77+,A7s+,A5s,AT+,K9s+,QTs,JTs,KQ
UTG+1: 66+,A7s+,A5s,A4s, AT+,K9s+,QTs,JTs,KJ+,Q9s,J9s,T9s
UTG+2: 55+,A3s+, AT+,K9s+,QTs,JTs,KJ+,Q9s,J9s,T9s,QJ
lojack: 55+,A2s+, AT+,K7s+,QTs,JTs,KT+,Q9s,J9s,T8s+,98s,QJ
hijack: 44+,A2s+,A9+,K4s+,Q7s+,J8s+,KT+,T7s+,97s+,87s,76s,QT+,JT,
cutoff: 33+,A2s+,A8+,K3s+,Q6s+,J7s+,K9+,T7s+,97s+,86s+,75s+,64s+,54s,Q9+,J9+,T9,
Button: 22+,A2s+,A7+,K2s+,Q3s+,J4s+,K7+,T6s+,96s+,86s+,74s+,63s+,53s+,Q8+,J8+,T6+,97+,85+,75+,
SB/BB: All cards

We will start with UTG: 77+,A7s+,A5s,AT+,K9s+,QTs,JTs,KQ.

Now we want to simplify these hands as well to be notated according to the simplification system we set up. But we also want to know the PERCENTAGE of all of our starting hands that contain such a combination.

The notation is L=low,M=mid,F=facecard,A=ace and  the various 2 card combinations.
Taking away the significance of order (LM is the same as ML), you have 15 combinations.
Then you have 4 additional hands:LP=low pair, MP=midpair, FP=facecard pair, AP=Pair of aces for a total of 19.

In this case, we have 3 mid pairs, 4 facecard pairs, 1 pair of aces, and a slew of other 2 card combinations. We must be aware of the probability that we have a particular specific hand so when a particular flop comes , we can evaluate the approximate probability of our RANGE of hands pairing.

Although we only play 12.67% of hands OF the hands we play, we get the following breakdown GIVEN we are in the hand with a single raise:
LA    2.38%
MF    2.38%
PA    3.57%
MA    7.14%
PM    10.71%
PF    14.29%
FF    21.43%
FA    38.10%
total=100%

Now if for example we analyze a flop with 3 low cards we know 2.38% of hands have at least one low card in them. We also can look at the best 50% of hands on a LLL flop.
PA,LA(made straight OR pair with straight draw and overcard),PF,PM. That gives us over 30% of our cards. That's actually ABOUT right when facing a TWICE the pot sized bet since opponent will get 1 to 2 on his bets, and we have to call about 1/3rd of the time to keep him honest. The best hands after that are then ace and a face card. If we used all of them it would give us 69% of the flops which is too much (but about right for a half pot sized bet). If we just use AK,AQ(including suited) we have 50% of the flops. From a game theory perspective without considering "outs" and potential shove equilibriums on future streets, we now can specifically say that given a pot sized bet on 3 low card flops with this starting hand range, we should call with: 77+,AQ+,A5s. Note that A7s on a 345,346,456,563,346 flop will also be worth calling or semibluffing occasionally, but most likely not on the other flops.
If we are first to act we want to BET half the time and check half the time. However, we still need to disguise our hands.50% of the hands we bet with SHOULD be in this range, and 50% of the hands we check with should be in this range. We also will bluff 50% of the time with hands NOT in this range, and check 50% with hands we will likely fold.  IF we are faced with a bet after checking, we will raise with HALF of the hands we want to keep, and call with about half. You want to mix in raises with the weak end and strong end of the range along with the calls so as to not be predictable. If you need a way to randomize it if the time ends in an odd number, choose to check raise, if it's an even number choose to check call. On the turn you might use the second hand to randomize.

Now you have LLL flop which is one of 40 situations figured out for one of 8 possible opening hand ranges. You also should note a notable flaw in this plan is that given a low flop only 30% of the time you have a pair. Given a HIGH card flop there is an informational advantage that opponents have as they know you are much more likely to have paired. For now we won't worry about this, BUT the way you can eliminate giving this edge away is randomly playing some low suited connectors every few rotations and folding some of your other marginal hands a bit more. But this actually gives up a hand strength edge to reduce the informational edge the opponent has over you. Pick your poison.

Now you want to repeat this for this particular hand range for all 40 situations. Then pick the UTG+2 (skip UTG+1) and repeat, then pick UTG+3(lowjack) and repeat, then pick the cutoff and repeat, then look at the situation using all cards and repeat.

Hey, winning poker is not easy, and hard work pays off. One way to ensure your poker success is to manage your bankroll well and outwork everyone to ensure you maintain your sustainable and substantial edge at least for your given stakes.

You also may be surprised what having this information as a reference can do for you sometimes. You can put opponent on a range, check your spreadsheet table and decide his range and determine what you need to have to call him approximately.

Do NOT leave decisions to emotion, but ground them in mathematics.

Hey, I could continue this until I completed the entire range of 40 potential flops, but you don't expect me to do all the work for you do you?

Some Notes on Flop Texture

Super bluffing:to play a hand correctly, one must be very aware of the flops ad the likelihood anyone else has hit and properly evaluate the hand strength.

Recognize how good the hand is and how likely it is to have alternatives.
On KQT board there is AJ,J9,KK,QQ,TT,KQ,KT,QT,KJ,QJ,TJ,AK,AQ,AT,K9,Q9,T9,99, then any pocket pair plus any J and any 9 and any ace. All of these hands are very likely potential holdings. On a K23 board there is only KK,22,33,K2,K3,23,AK,any pocket pair,A2,A3,KX,45, and only a few are likely holdings. As such, a hand like a single pair or even ace high is actually much stronger... Even QJ is surprisingly high on hand strength. Unless someone has a K, a bluff can be very profitable. If you size bluffs according to how likely opponent is to hit given his range it will be much better.
Put a KK2 board and only KK,22,K2,Kx and a pocket pair are relevant. Unlike AX, many of the KX hands aren't in the players hand range. As such ace high is often the best hand at this stage. Knowing this, you can get tricky and basically slow play a bluff to extract more chips before you make a larger bluff. At lower level you can just bet half the pot and give up if you are faced with resistance.

Book examples.

In the book "Kill Everyone" the benefit with various flops for shoving 4 times the pot are given. Assuming a particular raising range by the initial raiser and the range of any two cards by the player who calls and shoves the flop, how loose must you call? This info is provided in the book with 3 example flops giving creditability to a selectively hyper aggressive strategy after the flop. If shoving all in with any two for 4 times the pot calls for such a loose calling range, just imagine if you start with a tighter range of hands and are even a tad selective about your spots to have big draws how opponent will have to call. And what if instead you are the initial raiser who had the opportunity to steal the blinds first and also attempt to pick up pots with bets on the flop first and can recognize how the turn changes things. This provides the possibility for some real aggressive pushing and the opportunity to avoid pre flop all in coin flips.

The super bluffer style:
The superbluffer will rarely move in without "outs" But he will know when to take small stabs at pots and when to just back off. He certainly will open up his range significantly on some flops in some spots to include hands without "outs"... he will also shove with his monsters in many spots as well. The superbluffer will at times play "small ball" and then only push on river near an equilibrium solution or not at all. At times this player will let opponent do the bluffing and snap off calls. But often they will flat call or minraise certain flops with potential to bluff small on the turn or with a push all in if the right turn card comes. The superbluffer must back down if faced with another bet or the wrong card, but may just check the turn if the back door draw is there at times so he can induce another bet and shove if he hits.  The key is knowing the type of board and the player.


Various Flops:
AA2-AA5, these flops are basically the same with only a minor difference being the straight draw possibilities. However, hands like 23,24,25,34,43 aren't in very many players holdings. Perhaps 45s may be. In other words, the player either hit the ace or not. Since there are only 2 aces in the deck the odds are slim, however you can be pretty confident that if an ace was dealt, it was played. As such these flops are different than most paired flops. But still very similar. Do not try to risk it all on a draw semibluff because you could very  well be drawing dead, or facing redraw outs from opponent. Either bluff small with nothing or not at all. If you have an ace you might slow play it for a street if opponent is likely to bluff. An okay play is the minraise on the flop with a fold to any resistance beyond that, followed by a very small (40% of pot) bluff on the turn if they check to you and otherwise giving up the pot.
Treat AA6-AA9 about the same.
AAT-AAK these flops are much more likely to hit someone than any other paired flop. You have straight draws and 2 pair with full house possibilities as well as anyone with an ace. Even though paired boards may be good to bluff these are not. If you have an ace rarely should you slow play unless you have a boat. The only reason to check is to control pot size and induce a bluff but that should only be done on one street and very rarely.NEVER try to bluff these flops.
KKx,QQx,JJx. These should be looked at the same. Some people may play J9 or Q9 but not K9 but they will play K2s but not Q2S or J2s. So it is about as likely to hit them. Provided there are no straight draw opportunities or pair with back door straight possibilities, this is one of the ideal flops to bluff at. Given a hand AK the chances of hitting 3 of a kind, fullhouse or quads are about 1.45%. However, given this flop and opponent's hand range, the odds go up since you know they are starting in the hand with a face card more often than not... As such you can deduce that KKx is more likely than 772 to have hit them. Estimate a bit more than 2% chance. Then leave a little margin of error for guys who may have overpaired or 2pair with inability to fold or even crazy call with ace high. You thus may want to be willing to commit up to 35 times what you risked pre flop towards an aggressive bluff.  1/35=2.8%. You can see how money this is, but be aware, other savvy poker players will be doing the same. Know your opponent and make that big last all in on a regular if you think they will fire. Out of position:
check-call followed by check raise push, against opponents that will double barrel continuation bet, or just a check raise push.
In position a flat call or minraise followed by reraise push. Or a reraise push, or just a medium (half pot) bet followed by a push.

KK,QQ,JJ+ face card.
The face cards get much more dangerous. Particularly if that face card is an ace. Even a 9 comes into play with straight draws but a lower connected card is the danger. KKQ,JJT,QQJ as these make the mode straight possibilities and open enders players with ace high and a broadway straight draw will correctly be bolder at trying to snap off bluffs since if they are wrong they may have outs. If they have AK on QQK flop they have full house redraw PLUS back door straight draws plus are less likely to give up the pot. These flops are great semi bluffing flops if you pick up a draw as well but only against the right opponent with right stack size.

22-99x no face card
These boards are good to take small bluffs at but only shove against hyper aggressive players that continuation bet all the time and get into large pots or in more desperate situations.

22-99 plus face card
These boards you should treated in virtually the same way as a face card plus two low cards. EXCEPT it is slightly worse since you will be virtually drawing dead enough of the time. Even though opponent knows that, they are less likely to give you credit, so be more selective.

Q27,J27,K27,Q26,etc. in this case, this is a "dry flop" meaning not many if any drawing possibilities, and only one real face card that can help. Excellent bluffing opportunity, but I want to have at least an over card, ideally two because opponent could have AJ on 27J and my A T is no good. I actually like slow playing top pair here, and playing my bluffs similarly in that I will let another card peal off and maybe just flat call a bet. Types of hands that should set up potential bluffs are those with lots of outs to a draw, such as a 56 with a 4 or 7 on board a 3 and/or 8 works too but not as well. This way if the draw comes you can make a very aggressive semibluff all in if need be
the introduction of a 9 and/or two connected cards such as J78,Q78 make the flop a bit more interesting. OR if stacks are deep and opponent is likely to pay you off when you hit, you can occasionally take a free card or cheap card and just call the turn and then go for big over push when you hit to not be predictable. I might also push if I pick up a pair in that sits having 5 outs on the turn to two pair or trips. Only 10% chance of hitting my hand, but better than nothing, AND I also will be pushing with top pair, sets, two pair,etc taking that exact same line. Typically however, you will brick out on back door draw and instead if a scare card doesn't come, you can just proceed with a small stab at the pot as if you were slow playing top pair or better. Only when you pick up that back door draw are you occasionally willing to push. Putting in a minraise should be a rare, but powerful technique to represent an even bigger monster.

Qx9,Jx9,Kx9,J98,Q97,K98, the introduction of a 9 and/or two cards such as J78,Q78 make the flop a bit more interesting. Now there are straight possibilities or straight draw possibilities. With some kind of hand if you can you may want to shut out straight draws. I go to showdown a bit more often by checking the dangerous flops in position or a small bet, or leading small to keep pot small and check at least on one of the 3 streets. The risk of letting him draw out can sometimes be made up for by disguising your hand as a draw yourself and getting opponent to bluff. Since I play big hands this way, I play my draws this way most of the time. You can't play draws strong and made hands weak or vice-versa without someone picking up on it. But you could go with a 60/40 split. Of the two, I would rather play draws strong than pairs strong and pairs weak more than draws and only a set or two pair or made straight equally as strong.

Coordinated boards
456,567,789,QJT,etc
These boards are a nightmare. Even if you have a straight or straight draw you could be drawing dead. Even if you have a straight you could be outdrawn by a set or two pair or even 9T on a 789 board can hit a jack to spit or beat you or a QJ could hit a T for a higher straight, or you could be up against a nut straight. This board is the worst. Then there are the flush draws on top of it that compound the problem. You have to have a big piece of the flop and/or a chance to draw out to continue and even then I hate it.)
Even if you hit aces you might fold to this flop since you are either way behind to drawing dead, or barely ahead with opponent drawing live.
It is an okay semibluff in board but only against a really good opponent.
I hate these boards as you cannot really bluff, you cannot really stay with decent hands, and it's a nightmare if you have a made hand. Just consider these boards poor and keep the pot as small as you can.
Introducing a single gap doesn't help much as on a KQT board someone has JT or QJ or KJ or KQ or AK or KK or any two pair combination.

Two broadway, one low card
KTX,KJX,AKX,AJX,AQX,ATX,QTX,QJX,etc
These hands are NOT for bluffing... Ever. Even semibluffs are bad. Continuation bets are the only acceptable bluff but even then trying to get a small to mid pair to fold isn't worth risking extra chips. Plus when you hit top two pair or top pair top kicker, you will have broadway straight draws so keep pot small and fold the all ins. Let opponent make the mistake of bluffing this type of board.


Ace is much more likely to have hit someone than any other broadway card... Adjust past hand mentions with an ace instead accordingly.


Suitedness:
Having 2 suits of the same kind shouldn't change much about the flop other than semibluff opportunities and potentially present future bluffing opportunities to represent the flush. The overall odds of a flush hitting are small. As a big bluffer, and even a small bluffer (Think Daniel Negreanu) suitedness is awesome. You can bluff big on a semibluff, you can represent a suit you don't have, pick up a back door draw and bluff the draw and value bet if you hit. You can pick up a straight or straight draw and then bluff big while representing hitting the flush with potential outs to a straight and/or flush. You can represent flushes you don't have, you can make small bet into 3 suits of the same kind on flop as if you either have it made or the draw. But don't give your opponents credit for a flush too often unless their playing style strikes you as honest. With 3 suited you can give him credit for a draw though. They are still a concern, but shouldn't change as much as most people might think.


Turn play
The turn is the spot where there is added flavor. You may pick up a double gutshot draw or open ended draw, a straight flush draw or either a straight or a flush draw of some kind. You also will want to chase out that type of action. Yet sometimes not doing so is fishy enough that you actually can bet big as if you missed your draw and fool opponent. If you pair your draw such that you conceivably could have made a straight it is a great spot to represent it and shove with outs to it and outs to two pair or trips as well.
I.E. 46 on a K57 board and 4 or 6 turn.
But you don't always shove just to do it if chip stack sizes are too large in relationship to the pot.

Stack sizes
When it comes to making decisions, stack sizes are crucial. For example, if the pot has 100 and your stack size is 10,000 an all in bet would most likely rarely ever be very smart. You are risking far too much to get far too little. Theoretically your opponent could call with only the very best possible hand on the flop if is was your strategy and still make a ton of chips in the long run. We will discuss stack sizes later, but for now know that it is crucial towards a solid fundamental "all in" strategy.

Watch a pro
Phil Ivey and Hoyt Corkins are both good at the longball all in strategies. watch them and pay attention to the amount in the pot if possible and their stack sizes as well as their opponent's stack sizes.

4 Poker Fundamental Strategies

Preflop :
Strategy 1) Raise 2-2.5big blinds: play the top 1/ (number of players left to act-1) hands, rarely 3bet and if you do only do a min 3bet with a wider range of hands. Avoid 4 betting. Flat call raises and limp behind other raisers with a wider hand range. Start with worst hand and make up for it with smart, tactical play especially after the flop.
Strategy 2) Equilibrium strategy. Raise 2.5-3.5big blinds: play the top 1/(number of players left to act+0) hands, 3bet with the top 50% of opponent's range(assume they raise the same percentage as you do). 3bet size should be 3 times opponent's bet. 4bet with 1/3rd to 1/4th of your initial raising range.
Strategy 3) Raise 3.5-4.5big blind:Play the top 1/(numbers of players left to act+1) hands. You will come in with best hand more than 50% of the time. You also want to 3bet with a hand that is best more than 50% of the time (maybe 60%). 3bet size should be 4 times opponent's bet. 4bet should occur 1/4 times you raise.
Strategy 4) Raise 5-5.5 big blinds:Play extremely tight. top 1/(number of players left to act +2)


Link to preflop hand ranges.
Google docs poker spreadsheet.
spreadsheet embed on page below

Laying The Foundation For A Conditional Spreadsheet

In this spreadsheet we will need to approximate a typical set of hand ranges that each person folds if folded to. We are only interested in the hand strengths given multiple folds, and we will not worry about a raise since that just represents a particular hand strength and fodls after that may not mean as much. While someone might open the pot with A3s they probably will not call a raise with it and probably will even fold ace jack so the fact that people fold after a raise isn't very actionable data to me.

On the other hand, every fold with no one forcing the action before they have acted has to be taken as information, even if the assumption is slightly inaccurate, it is more accurate than assuming they fold at random.

As such, I want to estimate a tight/solid gameplan where the players only open when they are likely to have the best hand. the formula will be 1/# of players in the pot. Although technically we could have the raiser simply match the opponent's range, we will instead include the player. That means with 9 players total, first to act will open with 1/9=11.11% of hands. With 8 they open with the top 1/8 hands and so on. Unfortunately hand strength is somewhat subjective or at least not easily ascertainable. In other words, some hands are strong if the action ends at the flop, other hands are strong to the river. Some hands are strong in multiway action, others are much stronger heads up. Some hands are strong given a certain hand range folds, others are weaker given those same hand ranges folding.

But any assumption is better than "random folds".
So rather than get into a complex strategy of occasionally playing a particular hand and having a certain probability of playing each particular card, I will keep it simple.
I will use the hands ranged by EV according to actual games.

http://www.tightpoker.com/poker_hands.html

Here is a "solid tight preflop strategy" based upon the above. Note that it is perfectly okay to have a range that includes losing hands as it better disguises your other hands and ultimately you have a range of "lower end" hands that are "break even" which should improve the action you get with better hands and make your style less easy to exploit. Additionally, the results of the hands in the chart via the aforementioned link does not say why some of those hands are losing. Some of them are because amateurs play them wrong and lose a lot, skewing the average. Of course the experts could also skew some of the worst hands positively as well. Nevertheless, I doubt when this study was done people paid attention to game theory and position as well as they should. With that being said we have the following strategy.

Top 1/9 hands: 77+, AJ+, KQ, A8s+, K9s+, QTs+, JTs
Top 1/8 hands: 77+, AT+, KQ, A7s+, A5s, K9s+, QTs+, JTs
Top 1/7 hands: 66+, AT+, KJ+, A7s+, A5s, A4s, K9s+, Q9s+, J9s+, T9s
Top 1/6 hands: 55+, AT+, KT+, QJ, A3s+, K9s+, Q9s+, J9s+, T9s
Top 1/5 hands: 55+, AT+, KT+, QJ, A2s+, K7s+, Q9s+, J9s+, T8s+, 98s
Top 1/4 hands: 44+, A9+, KT+, QT+, JT, A2s+, K4s+, Q7s+, J8s+, T7s+, 97s+, 87s, 76s
Top 1/3 hands: 33+, A8+, K9+, Q9+, J9+, T9, A2s+, K3s+, Q6s+, J7s+, T7s+, 97s+, 86s+, 75s+, 64s+, 54s
Top 1/2 hands: 22+, A7+, K7+, Q8+, J8+, T6+, 97+, 86+, 75+, A2s+, K2s+, Q3s+, J4s+, T6s+, 96s+, 86s+, 74s+, 63s+, 53s, 43s

This strategy will lay the foundation for determining the probability of each card being left in the deck and as a result it can allow more accurate understanding of what opponent is likely to have... and thus, using a monte carlos simulation using poker software come up with a weighted average of how the hand range fares vs an average hand.

Spreadsheet Of Conditional Odds Introduction

I continue in the process to find the formula of poker. We have laid the entire formula out, as
Ca=C+(p*B)+((1-p)*E). We then defined many of the variables which are very conditional based upon responses on path to get the ultimate formula.

Now we actually need some sort of way of approximating an "equilibrium" style of sorts while also providing the ability to evaluate hands based upon how opponents have responded before us. This has never truly been done before in multi-way pots. Approximated equilibrium of the past have failed to take advantage of information provided when opponent's ahead of you fold a particular estimated hand range. This will change the odds of you running into an ace, but also strengthen hands like AK as more people fold and the liklihood increases of:
1)The opponents left to act having a weaker ace

2)The opponent you face having lower cards which are more likely to have been folded
3)An Ace or King being more likely to come on the flop as it is more likely than not that they were not folded.
4)A lower likelihood of opponent hitting a set or pair with lower cards.
5)A lower likelihood of opponent pairing or making a draw with unpaired cards than usual and thus an increased chance that ace high holds up if opponent doesn't have a pocket pair already.

6)A better semibluff opportunity with greater chance of cards that can help make a straight as well as greater chances of pairing the ace or king than from early position or with simply "random" folds.

In fact, you may actually elect to slow play AK in very late position after many fold, to keep an ace with a bad kicker in, to under represent your hand and potential semibluff opportunities... OR you may choose to try to build up a larger pot than usual and hope to put in a semibluff into a larger pot knowing your odds are better that it works with all the previous folds than in normal spots. Also, for the same reasons, you may actually elect to fold small and even medium pocket pair if the steal attempt fails and opponent is only likely to do that with overcards to your pair. You also may be MORE likely to draw and call or limp behind after multiway action because higher cards is more likely which means a lower flop is coming.

We want to incorporate this more advanced information of at least some approximation of what a fold represents and how the "variable change" effects our chances of winning. We also want to leave the formula open to be able to deviate from it given more known information as it develops.

By approximating the equilibrium, we will then be able to actually given any number of potential conditions, plug all the data into the formula as needed for now.

There is a high degree of difficulty of estimating both the probability that all opponents ahead of you fold, and how previous opponents folding changes the effective probability they have a hand, and hand strength due to variable change and conditional probabilities.

I gotta be honest, I am going to probably struggle with this formula and maybe even work backwards a bit at times as I will possibly run into dead ends.

So you will have to bear with me...

Formula of Poker 2

In the last post Constructing the Formula of Poker we established the formula of poker
Ca=C+(p*B)+((1-p)*E)

Where

Ca=Chipstack AFTER result of hand

C=Current chip stack
p=probability steal attempt wins

x=probability it doesn't win blinds

B=chips steal attempts gain if successful.
We will more clearly define E.

E=equity gained/lost if opponent responds=(Pr*Er)+((1-Pr)*Ec)

Where

Pr=Probability opponent raises, given he responds

Er=equity gained/lost if he reraises (3bets)

(1-Pr)=probability opponent calls, if he doesn't fold

Ec=Equity gained/lost if he calls.

We can thus notate it from
Ca=C+(p*B)+((1-p)*E)
and substitude E for  (Pr*Er)+((1-Pr)*Ec) to get
Ca=C+(p*B)+{(1-p)*[(Pr*Er)+((1-Pr)*Ec)]}

The problem is, we aren't done yet. Afterall the equity gained/lost if he doesn't fold is still contingent upon a lot of various options before and after the flop.

The equity gained/lost if opponent raises has it's own formula. The equity gained/lost if opponent just calls has it's own formula.


Equity gained lost if he raises equals to (-amount risked*percentage of the time you fold)+(equity you gain when you call*probability you call)+(Equity you gain when you raise*probability you 4bet raise).
Percentage of the time you fold equals (1-(probability you call+probability you raise))
Therefore
Er=(-Ar*Pf)+(Ec3*Pc3)+(E4*p4)
OR
Er=(-AR*(1-(Pc3+p4)))+(Ec3*Pc3)+(E4*p4)

Where
Er=Equity gained/lost if he 3bets
Ar=Amount risked in raise
Pf=Percentage you fold GIVEN this line of play
Ec3=Equity you gain when you call a 3bet GIVEN this line of play
Pc3=percentage of the time you call a 3bet GIVEN this line of play
E4=Equity you gain when you 4bet
p4=percentage of the time you 4bet.

Make a mental note that we still have to calculate the Ec in the formula
Ca=C+(p*B)+{(1-p)*[(Pr*Er)+((1-Pr)*Ec)]}
Right now we will finish "Er" and the various formulas that need to be derived from other variables to truly complete the formula.

At this point the "raise" will be assumed to be an "all in" so we can put an end to what otherwise would be an indefinite formula since we have not limited by chipstack or number of additional raises over the top of the previous raise. We have finally just about finished one possible line of play that goes raise, 3bet followed by you pushing all in. We only need to determine the equity gained/lost if you push which will be contingent upon whether opponent calls or fold and finally if he calls what percentage of the hands that you win. Also, the amount of chips in the pot is won if he folds. At some point you may be able to condense the formula and relate it to terms of a particular hand range.

However, in doing so we created several more possible outcomes for the way the hand could be played out. In some cases, our opponents 3bet will be all in. In that case, the probability of a "raise" will be zero and the equity if you call will have no future value other than the equity between hands.

Equity gained from all in 4bet= (probability opponent folds*chips in the pot minus the chips you have put in)+(Probability opponent calls*Probability your hand wins*Chips in pot minus your own)
E4=(Pf4*(Cp-Cu))+(Pc4*Pw*(Cp-Cu))
probability opponent folds can be expressed as
(1-Pc4)
Therefore
E4=((1-Pc4)*(Cp-Cu))+(Pc4*Pw*(Cp-Cu))
Finally we have a basic possible "end" (of many) to the equation aside from having to plug in the amount of chips we have and somehow having a way of determining or estimating the percentage of hands our opponent our opponent will call when facing an all in.

Now that we have come to an "end" we can go back and plug this equation in by substituting E4 for ((1-Pc4)*(Cp-Cu))+(Pc4*Pw*(Cp-Cu)) to previous parts and then go and begin to complete the remaining unfinished parts of the problem.
So we were previous left with
Ca=C+(p*B)+{(1-p)*[(Pr*Er)+((1-Pr)*Ec)]}
Where
Er=(-AR*(1-(Pc3+p4)))+(Ec3*Pc3)+(E4*p4)
AND
E4=((1-Pc4)*(Cp-Cu))+(Pc4*Pw*(Cp-Cu))
OR
Ca=C+(p*B)+{(1-p)*[(Pr*Er)+((1-Pr)*Ec)]}
Where

Er=(-AR*(1-(Pc3+p4)))+(Ec3*Pc3)+([((1-Pc4)*(Cp-Cu))+(Pc4*Pw*(Cp-Cu))]*p4) 
and we can eliminate E4 from the equation.

Now before we integrate this into the original formula, we still must solve for Ec3 or equity you gain when you call a 3bet. This will start to get really tedious and complicated beyond this point as it will introduce FLOP play, several more variables and then from those variables, turn and river play.

This work will basically be the same as the "Ec" just with a slightly different starting pot size and likely much different hand ranges. As such the analysis and expectations of hand ranges will thus be different and the results will be different, but the equation will look very similar.

Ec and Ec3 BOTH are basically equal to equity post flop. We can express them as the same equation PLUS preflop pot size and express bets as a size in relationship to the pot size if we want.

Actually, we could go back to the initial equation and come up with bet size assumptions and express it in number of big blinds. But I think we are getting ahead of ourselves and still have a long road of plenty of math in front of us.

Also remember that we started this formula with the assumption that you always open up for a raise. This may actually not be ideal with certain stack sizes as the opportunity to limp raise all in as well as gain more implied odds to maximize your post flop edge may provide a larger outcome. At some point we still may want to introduce another formula and it's potential outcomes if we just limp in. I don't necessarily know if it's needed. Afterall you can just express the amount raised as the call amount with zero percentage chance that the big blind opponent folds and the equity gained/lost will just change.

Additionally this formula was is will leave open the possibility to play with different raise sizes if we can estimate how our opponent may differ in strategy given certain bet sizes.

For now I must save additional calculations for another post as I am beginning to grow tired and need to refresh my thoughts with a new blank sheet.

Constructing the formula of poker

"It depends" is one of the most common responses to most poker questions. Rather than sit there like an idiot and give a stereotypical answer like most do because they don't care or know enough to really analyze it themselves thoroughly enough to show why it depends and exactly "what" it depends on, instead I am going to begin the construction of the ultimate "formula" of poker. "solving" poker doesn't mean the game is over by any means. So much of the equation can be analyzed more in depth to come up with more accurate information. Intuition can use how the people are playing and how the table conditions have changed to anticipate hand ranges and thus probabilities. As the information is constantly changing, so must variables of the formula itself.

Poker is a very situational problem with conditional probabilities. IN other words an "IF-THEN" type of programming language of an equation. Or "IF" this happens "THEN" the odds are influenced in this manner and the optimal decision may change. IF you face this opponent who has adapted in such a way to have good odds of playing a particular style THEN they change to something else.

To understand the equation we need to account for multiple variables that potentially can connect with an alternative set of equation. Those wondering if you ever were going to use high school algebra and pre-calc, you got your answer! YES! Now you can see why the F(x) equations come in handy.

What your expectations of your chip stack are in a given hand give us the following equation assuming you plan to open up for a raise. We can go back and modify this later if you have a strategy of occaionally limping with the potential of limp-raising as well based upon the percentage of hands you limp and raise with and what informational advantages/disadvantages you give your opponents by doing so. Nevertheless, here is how we might begin to construct the formula:

Chip stack after result of hand=Current chip stack+(probability hand steals blinds* chips you win if you successfully steal)+(probability it doesnt*equity gained/lost if opponent responds).
Or
Ca=C+(p*B)+((x)*E)
Where
Ca=Chipstack AFTER result of hand
C=Current chip stack
p=probability steal attempt wins
x=probability it doesn't win blinds
B=chips steal attempts gain if successful.
E=equity gained/lost if opponent responds.

In this case, the probability the steal attempt doesn't is equal to (1-p) so we can substitute x for (1-p)

Ca=C+(p*B)+((1-p)*E)

Now what you would do is break down different parts of the equation.
In this case, the probability the steal attempt wins will be a very complicated formula in itself, and really is one of the reasons poker is not really entirely solvable, especially considering your opponent's range will change. Also even knowing information about those who folded ahead of you will provide "variable change" or information that changes the problem to have a different effective probability that the remaining opponents have a particular hand and what your equity is against that hand. Whether or not those left to act have a playable hand to a raise or a playing style/information about you or tournament  conditions even that will influence them to raise.

But basically you will need to take 1 minus the probability that each successive opponent will have a playable hand and use that information to determine the probability that one or more of them will either call or raise. We won't be substituting that for now, or considering the possibility of multiway pots just yet. Without getting into all those fancy calculations, you might just intuitively estimate probability of seeing action based upon experience for now.

The chipstack will be known, the chips the steal attempts gain if successful will probably be 1.5 blinds or around 2.5 blinds if there are antes.

That brings us to the "E" part of the equation or "equity gained/lost if opponent responds". This too is entirely opponent dependent. The opponents will dictate what percentage of the time they fold, and if they choose to play, what percentage of the time they play they will raise or just call. Given that they raise, the decision is then back on you to determine based upon your raise hand range, what your equity is and what your fold range and 4bet range or call range is, and then the equity gained from that decision may hinge partially again on what percentage the opponent will call a 4bet shove shove or call a 4bet.
To simplify the equation at some point you have to just figure an all in and use chip sizes and probability of winning if called to determine the results.

BUT I DIGRESS...
E=(Pr*Er)+((1-Pr)*Ec)

as said before E=probability gained/lost if opponent RESPONDS in some way (call/fold)
Pr=Probability opponent raises given he responds
Er=equity gained/lost if he reraises (3bets)
(1-Pr)=probability opponent calls if he doesn't fold
Ec=Equity gained/lost if he calls.

Now I will not yet continue this equation... but just know that the equity gained/lost if at some point there is a "call" will have it's own set of equation with potential actions and probabilities of certain actions on the flop and probabilities of opponents foldings, probability that you will have a hand in your range that you can bet and/or raise on the flop and same thing on the turn and/or river with potential all ins that end the hand along the way with potential drawing equity.

The complicated part is actually not the formula itself with all these huge amounts of contingencies that you keep having to come up with a formula and ultimately plug it into ONE formula, but it's determining the actual values and probabilities based upon the opponents.

With multiple opponents there contains different numbers and thus you actually would have to duplicate the formula for each opponent and weight a probability assigned to each particular opponent and the probability that THEY are the one in the hand when determining value before you raise.

The formula itself is complicated... sure.. but the NUMBERS within the formula depend upon playing style. With completely unknown opponents you can still determine a baseline solid strategy to use that closely enough approximates the information needed to at least have some informational edge. Then as you gain information, the formula itself can be more easily defined.

What will greatly help you find these values is if you have detailed stats on opponents by using some kind of "poker tracker" or hold'em manager" program that does a lot of the leg work for you.

Trying to account for all these subtle nuances is incredibly important, but still very difficult.

YET, what no one has told you yet... is that it CAN be done, and is not all that hard... just very, very time consuming and perhaps actually coming up with the most accurate numbers require very complex calculations that may require advanced knowledge of excel and/or programming skills.

But that's about all i can give you for now!

Edit: This post has since been continued on the post Formula of Poker 2.
Text URL to copy/paste if you can't follow hyperlinks in your browser:
http://nutballpoker.blogspot.com/2013/10/formula-of-poker-2.html

Why Most poker players get things wrong

The fact that even an unknown opponent made even a single fold is actually very powerful information. If you know opponent's range of hands he plays, it's even more powerful. If you assume a more stable raising strategy of only coming in for a raise with the best 1/9 hands (1st to act 9 handed) and you rank them in a certain way you still have an assumption that can give you more information than not. For example, the probability that your opponent folds a 2 is higher than the probability that he folds an ace. So much so in fact that you can effectively say there are less than the full 4 set of twos in the deck. You instead might approximate there are 3.75 2's in the deck and 3.95 aces in the deck after a single fold. Add in consecutive folds and it is extremely powerful information.

As a result, after 8 people fold, the chances of the big blind having pocket aces is greater. At the same time, the hand strength of low suited connectors vs a random hand goes down while the hand of ace king goes way up. The chances of him having any ace are greater so a low ace suited is not as good. AT is probably about the same, ace jack is probably slightly stronger, and ace queen and ace king are much stronger.. That information is so much of an advantage that if you have ace king you go from being a dog in a standard heads up game to a favorite over even a mid pair. Better yet is there is more likely to either be an ace or king in your opponent's hand (in which case you have him dominated), and more likely that an ace or king comes on the flop (which also plays in your favor). This not only changes hand strength, but also what constitutes "equilibrium" strategy in poker for everything to pushing ranges preflop to postflop drawing strategies. This information can become a huge edge. You can more accurately determine what cards will come on the flop and even after the flop. Learn to compute this kind of stuff in a spreadsheet, and you will know things that others do not, and be able to adjust accordingly.

Unfortunately it is a lot of work. You can either assume hand ranges, or learn to do some kind of Monte Carlo simulation. I will try to come up with one an instructions how to do it. I have not yet done the work myself.

There are many ways to approach the same problem. I want to use a montecarlo simulation rather than trying to map out all permutations and calculate raw specific probabilities.
I want to assume specific hand ranges that the opponents will play even if this may be an erroneous assumption it still will be closer to the truth than assuming they fold randomly.

To do this I need to approach everything separately. For example, rather than just look at the hand ranges for everyone as the same, I must isolate it, calculate the probability that it will fold to a particular person and given that, determine what percentage of the time the person it has folded to has a particular card. Each card 2 through A will have to be looked at individually. For now suitedness doesn't matter since I am not looking at probability of exact cards, although down the road I may wish I had put all 2652 2 card combinations of deals in there.

In the future I will work towards constructing a spreadsheet that can more accurately make calculations.

Slow gear, really slow gear, and downright nitty.

Before the final table, if you have plenty of chips left to survive, you still want to think about entering the pot for a raise when you are more likely than not to have the best hand and a slightly larger range. The easy rule of thumb is count the number of players left to act plus yourself and divide ONE by that number. This is the hand range you play. For example, if you are on the button you should play the top 1/3 hands and look at them as most likely the best. Rather than try to aggressively exploit players like you do all tournament, in this spot, you merely seek to go over the top with again... the best hand. You are going to give credit to your opponent and fold enough to basically break even. In other words, if opponent bets 3 times your bet, you can fold your bet 3 times and on the 4th go all in and win everything back. In other words, take your original hand range and divide it by 4 for a standard 3x range. So if you play 33% of hands and the blind plays back at you, wait for the top 33/4=8.25% of hands to play.

I actually am very okay with being much more patient than that both in the all in front and in the initial raise front. The reason being you can coast your way to the final table, pick better spots, avoid all ins and still have a very good chance of making it. Many times in many parts of the tournament, I will approach a slow gear. Probably will enter this slow gear after I make the money. This slow gear is about understanding how long I could wait if I only were to bet with my best hands and developing a strategy around that.

In this case, you basically want to just determine even considering the rates that the blinds go up how many "true hands left" you have before you are blinded down to nothing. Really probably take away 10 of those hands since you don't really want it to come down to desperation where everyone calls you and the double up only puts you back to where you are now in terms of overall chips. This might be between 20 and 40 hands left. Say it's 30 for example. Then take 1/30 and that's your call all in range. You should be willing to limp in, raise, or reraise with half of that or 1/15. You might adjust slightly for position since you will have more information preflop after everyone else has act. You still have to realize this is a rule of thumb so if the tightest poker player in the world goes all in you may still fold, and if the loosest one does you may still fold. As things happen and your chips diminish or grow, your percentage must be adjusted to reflect recent data. Also, you may pick some spots to play the player rather than your cards and just try to steal every now and then, but at least with this baseline you know the maximum you can wait, and you can maybe push with twice the hand range you could wait for.

Please be aware that as the tournament gets short handed at 3 tables left and 2 tables left, the calculation on the amount of hands left changes quite a bit since you will pay a lot more blinds at each blind level and the cost per blind level for folding basically doubles. As such, there is an akward phase where you may want to come off the slow gear every now and then.

FINALLY the ULTIMATE slow gear that is downright nitty is the final table gear with a full table... If you have a healthy stack, this is where you make a lot of money from just sitting on the sideline, letting some players do a lot of work and get all their chips back later with a single pot or two at an elevated blind level.

So you will stick to pocket JJ or better, and maybe TT and AK in late position when not facing action. You might fold TT and JJ and AK when facing a raise, and QQ to a reraise. In reality I doubt most people will possess the ability to resist some marginally better hands than that, but it's a good perspective to start with planning. I believe this strategy was used by Helmuth to coach someone from the final table to a WSOP top 3 finish.

Now when it gets down to 5 players left or 3 you may come out to play and widen the range, but really what you are looking to do is double up with a monster chance of winning and THEN come out of your super tight mode and go back to bullying the table. Helmuth probably would only have you open the raising range slightly, but I think your tight image will REALLY pay off and allow you to grind away a very large chunk of chips with just a handful of blind steals at the elevated levels and maybe a decent sized pot.

You really make money as players bust themselves but that's still not the goal of this. The goal is to have basically a low varience strategy that allows you to just hang on. If your opponents are raising and such you might get a 3xBB steal plus 1.5BB plus about a big blind worth of antes for 5.5. Since you will be folding all but 1/35 hands you basically pay for over 3 rotation which will cost you just over 4.5 big blinds plus 3 rounds of antes. You win probably somewhere between 2.5big blinds and 5.5 big blinds with the opportunity for perhaps much more... EXCEPT the blinds after 30 hands will on average rise.

How much? that depends upon structure, but a 25% rise twice makes sense. This takes 2.5big blinds to 5.5 up to 4 big blinds to 8.50 with potential for more. Since you lose about 8 bit blinds this is a slightly unprofitable strategy if you don't get any more value from your hand than a steal. That is perfectly acceptable since everyone will reduce the size in big blinds of their stack and many resort to risking all their chips. Throw in a single additional steal after or just before the big hand, and you will add 4 big blinds to that total and you're better than break even. Considering the significantly tight image you will create by folding will give you a license to steal later, it is likely very worth the downside. Afterall if you have say 40 big blinds, that is way more extra room than you need anyways, so just keep a slow, patient, low varience strategy to start with.

This just is much better than the strategy of potentially going bust just to slightly increase the chance of winning.

Ideally if you could you would have a strategy that just does a little better than break even at no risk of going broke, but with the blind structures for online tournaments that is probably being optomistic. Nevertheless, since the play will go from 10 or 9 handed at the final table to short handed in a hurry, and then you have to pay much more to see 10 hands of play, that is when you can after a few rotations of playing only marginally looser, start to really go for a few key major steals in large pots and drastically open up your range.

Realistically you will start with a strategy that may be worse than break even (or much, much better than break even if opponents slip up), then kick in one that is very exploitative throwing out plenty of small bets to win lots of small pots and then scaling back a bit again, but not nearly as much as before. If you get yourself too short stack you may get all in and called with very much the best of it, and walk away a big chip leader. If not, don't worry.

The goal may require you to really strike and make a steal attempt and follow it through or reraise steal all in with a completely terrible hand. You really have nothing to lose at this point so using your tight image to make an utterly insane move for a huge stack of chips is great.

What you want to do next is fun. Find a way to maintain chips for awhile, note the people you can really steal from. you may want to keep the short stacks alive so you can continue to steal from the medium stacks. As long as you are accumulating chips while they try to fold their way up to second, you at some point actually will take the lead. The chip leader may try to knock the short stacks out and lose. If not, don't worry, just keep chipping away. It won't take too many extra steals plus some good monster bluffs or just extracting a bit of value out of a decent hand before you are the chip leader. Even if you get down to 2 handed heads up, Just keep with the min raise on the button and avoid out of position and keep aggression on. Push if you have to.

Slow Down nutball - The "slow gear"

The fun thing about poker tournaments is you can look at it basically like an equity chart and consider bankroll management skills. EXCEPT you will be forced to be much more aggressive. You need a strategy that will get all the chips to win. OR you need a strategy that can maintain pace ahead of the blinds without violating money management principals... (or as little as possible).

Basically if you have done any work on money management strategy, whether that be for stocks, sports betting, options or anything else, you know that getting "too aggressive" can be a major flaw in one's "game"

The red and blue lines are both examples of what it might look like if you go "beyond" the maximally aggressive strategy. In fact, the red and blue line within a limited sample size could technically still look like that even if they were under betting the "optimal maximally aggressive" strategy since the assumption within money management calculations like the "kelly criterion" is that you maximize it over an "unlimited amount of bets".

Note:

If you bet half the amount you still get 3/4 of the long term return with HALF the volatility.


Theoretically if you could ALWAYS bet a fractional amount you would never go broke, but in poker tournaments the object is forced. BEYOND twice the kelly and the LONG term expectation is negative. HOWEVER, in tournaments there are people who are still successful doing this.

What you might see is a wide potential set of possible strategies. Some either get a lot of chips REALLY fast OR fizzle out. OR one where you cautiously accumulate chips and then either keep the same strategy and just try to squeak in the money and from there perhaps get aggressive... OR survive until the bubble and then get aggressive. The reality of the situation is it is NOT unlimited, but VERY fixed in both TIME (because of inevitable rising of the blinds) AND in number of chips (so a strategy should not be looked at as maximizing the RETURN BUT instead MAXIMIZING the probability of getting the best result in cash based upon finish. In some cases, you just can maximize the probability of getting first.

What that means is, at some point, you should probably slow down. There are plenty of people at the table who will want table respect as well as taking your reign as the table bully, especially once you are already in the money. Rather than fight them on it, give it up willfully before you lose a bunch of chips in the process. Remember, your goal is to get all the chips while maximizing payout, not get them as fast as possible.

The dark green strategy gets in trouble and the light green gains at it's expense. This is not necessarily because the strategy isn't more profitable, but because it is a higher risk, more volatile strategy that is unnecessary. The blue line is example of a smoother more cautious strategy that makes it and has a point where he takes more caution around the same time as well. The black strategy is a bit volatile but not crazy, yet he does not slow down and is forced as a short stack to make some more desperate decision. The green, the blue and black make the final table even though the green may have a more profitable strategy in terms of chips, he does not neccessarily in terms of tournaments. Actually the blue strategy is probably the best because somehow it manages to avoid major confrontation. I believe this is the reason Daniel Negreanu is one of the best because he will take lots of pots down, and lots of pots to the river and avoid any major decision. He has an "equity curve" that has a "low standard deviation" but high enough expectation and possibility to gain to stay ahead of the blinds. the dark red and gray strategy strugle to keep up with it, and the red strategy has a "boom or bust" strategy. The green does as well except he is slightly more cautious until the blinds start to go up and he manages even after a big loss to play patient and smart enough to hang on as long as he can until he gets desperate. He finally makes a move on one of the most aggressive players who fails to slow down (dark green) and uses his chips to vault his way to a win.

I hope this illustrates that "time" and "chips" both eventually dictate how far you get, how much you get paid, and who wins. If I extended this line 3 times as long, the green and black strategies would probably go broke more often, however, perhaps the blinds would also catch up to the blue strategy if not for a shift in style. Another factor is the speed of play and rate that blinds go up as well as the blind structure. Sometimes the blinds will go up 20%, other times they will go up 33%. Even some tournaments when antes kick in it increases the cost per rotation by as much as 50%. Sometimes the ante are always equal to 1/5th of the small blind, other times they are 1/3rd or 1/4th. All these minor distinctions will or should impact decision making slightly. In terms of this tournament the green strategy or black may still be best depending upon final table play.

Another determiner of which strategy is best is the payout. If you are in a large super satellite where the top 50 places score an entry into the next tournament, there is no distinction between 1st and 50th. As a result it actually is extremely costly to take on all the risks of going bust when a more cautious approach may work.
Bankroll management is hugely important for similar reasons and I would never advoate extending your bankroll and looking like any of these EXCEPT blue but in tournaments I would. However the at the table strategy may end up similar at times.

In MOST tournaments however, there will get to be a situation in which you already have double the second highest stack or a large chip lead, and/or a very comfortable amount of big blinds such as 50 late in the tournament. That is when you stop trying to steal blinds, and you wait and start building up a tight image. The "slow gear" at this stage and at the final table is a bit different than the one before it, but that's for the next post.

Why post flop equilibrium is different than the rule of thumb

I said previously that the "rule of thumb" is basically to balance half your decisions so that you continue half the time on each street and half the time you bet. So much of whether that decision is correct depends upon stack sizes.

The reason you actually should not do that is each bet grows progressively larger on each street so you actually could be a bit tighter since the result is a combination of all streets and if you are tighter you are more able to make it all back. On the other hand, if stack sizes are small enough and bets are large enough in proportion to them, that may become a reason to stay in with actually a wider range of hands, with either the potential to shove or chceck raise all in at some point depending upon stack sizes. If opponent makes the move and takes away your check raise, he will only be able to do it on a smaller percentage of hands.

IF you are out of position you actually can at times be looser given a small enough chip stack size relative to the pot. Ironically, being out of position is usually a key disadvantage that should actually require you to be much tighter than the 50% rule of thumb suggested, BUT NOT a disadvantage, if the stacks are large enough OR small enough. It seems unusual, but smaller stack sizes actually take away the advantage of position size not quite to the extent of large ones, but it allows the big oversized all in with a monster hand and it allows you to neutralize it by checking and making a decision on another street. Interestingly enough LARGE enough stack sizes hurt the out of position player less too but there is that akward range of putting about 12.5%-22.5% of your stack in the pot that is the worst spot to be in.

Additionally, if you are playing exploitative play and trying to reach post flop equilibrium, your hand range may be entirely different from opponent such that you will HAVE to be tighter even if it means giving up chips because you took on that decision when you started that you expected to be profitable before the flop and now continuing half the time may not be a strong enough hand range to do anything but get in a larger pot coming from behind with your range. Converely, in exploitative poker you may actually stay in more because in order to make up for sub-equilibrium decisions preflop, you may need to deviate from it more the other way. OR in some cases you will be even tighter to exploit opponent who overcommits or overbets at some stage.

The key is to understand how the relationship changes. As you tighten up preflop from equilibrium you will need to do MORE with it somehow after the flop in order to justify deviating from equilibrium. That could mean either looser than equilibrium or tighter depending upon opponents. As you loosen up beyond equilibrium preflop, you also will have to find a way to make more which could mean looser or tighter as well. But why would you tigthen up preflop? If your opponent plays aggressive or will call you down every street but not fold, you probably want either a better hand range to start with OR a hand that can draw depending on HOW much you can get out of it. If you are going to get a few large bets, but not a monster payoff, then you need to be tighter preflop. If you are going to be able to control the pot size at any point and get action when you hit, AND stacks are deep enough then you absolutely want to play lots of drawing hands.

If opponent has a clear mistake where he folds too much after the flop you absolutely want to play a lot of hands against him and make mathematically correct decisions with positve expectations. (i.e. you win with a 1/3rd pot sized bet 52% of the time.

Then there are the specifics of what hand. If opponent overplays his draws, you want to come in with ATs+ and A2s-A5s and any two broadway and T9 and 98 looking to make the big call with a higher flush draw and/or better straight draw or made straight. You want to call him with pairs and maybe even ace high and king high depending upon if he plays like that with his big hands too or not.

There are so many subtle adjustments that can be made, it's a matter of how to deviate from equilibrium, how to recognize whether someone is overbetting, and how to recognize their weaknesses.

EXPLOITING opponents for fun and profit

Exploitative strategy matters. there are many ways to do it.


First a bit on postflop game theory. If you are going to "exploit" you are going to do it by deviating in a direction away from "equilibrium" The more aggressively you "exploit" the more vulnerable you become to counter exploitation.

A very basic and not entirely correct, but close enough rule of thumb on post flop game theory to establish close enough to equilibrium to be effective (if you intend on an exploitative strategy) is:
1)half the time you check, half the time you bet.
2)half the time you check, do it with a hand you can check raise with.
3)half the time you make a decision to bet, do it with a hand ahead, the other half do with a mixture of semibluffs and bluffs.
4)Make sure that the group of hands you check with are on average equally strong.
5)adjust decision according to the flop on over 60% but less than 80%. (i.e. check more often with nothing and monsters on flops condusive to slow playing since you may get opponent to check behind).

ALSO
1)When facing a bet of the pot size, assuming equal hand ranges, you can call half the time to keep opponent honest. (inverted amount of odds he's laying you or 2:1 =1/2 in this case).
2)When facing a bet, you want to call OR raise half the time, including with not much of a hand and the intention of potentially bluffing on some flops (and giving up on others).
3)On the turn you will also make this decision about half the time.
4)The introduction of the option to raise, call, or fold makes it a bit different than half the time, and I will stay in a bit more often than half, and of the percentage I stay in I will raise a bit more often than half the time I stay in and bet or raise a bit more than half the time on the turn. As a result I want to start with a better hand range than my opponent.
5)Given that you raise, the bet size should be mixed up often enough (min raises with both big hands and junk, larger raises).

AND
1)Your BLUFFS in equilibrium should be breaking EVEN if your opponents do everything right as well.
2)You should BLUFF according to the percentage of chips you risk relative to the pot relative to stack sizes and percentage you have a hand. If for example you have a hand on the turn that will hit on the river 10% of the time, and you will want to bet the full pot or 1:1, you will want to have it so your BETS are composed of a bluff that is equal to 1 divided by the ratio plus 1 of your raising range. In other words 1/1+1=1/2. In the case that you will hit a GOOD hand on the river 10% of the time, that means you should BLUFF 5% of the time for a total BETTING of 15% of the time. If you plan to bet 9 times the pot, you should bluff 1/9+1=1/10=10% times the amount you bluff If you have 10% chance of having a hand you can bluff like this, then you should bluff only 1% of the time.


ALL in!

Minus the all in strategies for equalibrium we have covered all we need to know about equilibrium. As equilibrium you can really gain an edge from opponents that do not acheive equilibrium. But the strategies after the flop are pretty complicated since there are so many different types of flop structure, different hand ranges playing each other based upon different positions and such that at the moment I don't have a real good rule of thumb. A few different times I constructed BASED upon certain ranges how I could come up with a strategy that basically broke even that involved pushing with draws as well as monsters. I made one where I push the flop and another the turn but it dependend upon stack sizes. But the strategy changed so much based upon hand ranges, and as I learned more, I realized it was too tight of a strategy in that it would have bennefitted more from pushing more of my decent to good hands in which I had made a top pair or middle pair, and as a result loosen up just a bit more on the draws. It also would have been best to add in some sort of randomness in which sometimes I would not shove all in. Even so, I learned it better to be on the turn.

I don't still have that work saved, or if I do I havent found it. so I will postpone the "all in" portion of equilibrium. There are some great tables on the book "kill everyone" and the book "The Raiser's Edge" that can help you get more specifics of various "close to equilibrium" strategies based upon different stack sizes.

NOW... finally with that out of the way we can talk about exploiting opponents.

Maximal exploitation is as far away from equilibrium as you get and aim at a specific thing opponent is not doing according to the above. The hand range preflop depends on how big the players weaknesses on and after the flop are. If your opponent folds more than 50% to a flop sized bet or less you can exploit him by being the first one in to bluff. If he folds more than 50% on the turn, you may even actually want to build up a pot in order to make a more profitable bet on the turn. Same thing on the river.

If he bets on every flop, you should almost ALWAYS check to him UNLESS you feel there is a greater hole in his game such as how he responds to a bet. You can control the pot on a draw if he is the type that won't raise when faced with a bet, or will fold too much. You can overbet the pot if he calls or raises too much once you have a hand. But normally you check to him. This way YOU have positional advantage and can check raise him, or check call with the intention of check raising him. or can call 80% of all flops and then start the game on the turn. If opponent folds at any stage of the game too often, you exploit. If opponent fears a raise too much, you exploit. If opponent is loose and bluffs a lot but avoids all in, you exploit!

Maximally exploiting may in some cases mean ALWAYS betting to obtain the profitable expectation if opponent folds too much.

BUT I don't go for "maximally exploitative strategies" Instead, I seek to find a somewhat aggressive balance between equilibrium and exploitative play so that I have a balance. More occasionally than not (say 20-40% of the time I am playing for an extended period of time) I will swing back the other way and be too tight but still aggressive to protect against the possibility of my opponent counter exploiting and also decieving my opponent as to just how aggressive I am being, vs the potential of just picking up a good run of cards. Too aggressive too often and it may be too obvious. But switching to play tight to much tighter every so now and then can really help.

Certainly there are times when I will go for maximal exploitation, but usually only after an extended run of building up an image of playing tight on top of thinking my opponent has shown no capability of adjusting... or perhaps if he is playing a volatile strategy and I think he may leave soon I want to try to exploit him to the max until he leaves. Other times when I will not. But the key is to be seeking out a strategy that will hold up in the long run. If that strategy is too aggressively exploiting opponents, there is a good chance your opponent will notice and play much differently against you than he does against everyone else.

The good thing is an equilibrium strategy leaves people thinking you are a bit of a maniac because of the nature and ability to push all in with a wider range as discussed in the last post. But when you shift to exploitative, you are even more loose aggressive, and can usually tighten up or loosen up on the all in front where needed. Some opponents you build up a huge pot and rob blind, and others you play small pots with and only go with the major all in bets when you have a monster hand and the occasional monster draw where you are better than 50% to win. The KEY to playing this strategy is often being very aggressive in the right spots in terms of bet frequency, but bet size should be such that it much more often than not corresponds to the strength of the hand. In other words, the decision is much more often than not consistent with how you can get the most chips possible in the middle when you have the nuts, and how you can win on a bluff or semibluff with the minimum amount of risk when you do not. Equilibrium would suggest that the strategy has an ideal amount of percentage of times you should be bluffing with that same bet, but when exploiting it should probably be less than HALF of the suggested amount.

Nutball Examples

It folds to you or the action is weak behind you. Such as full of limpers. You might raise because you have position. I would raise with just about any two if it folds to me. Not every time so it looks like I am more willing to fold than I am, but very frequently. Sometimes I mix in a limp if I am at the point I know my opponent is going to call if I want to keep the pot small or get my opponent to keep betting into me every hand. But VERY frequently I will limp behind limpers and raise behind folders.

If it folds to me one off or two off the button, I will often raise to get opponents in position to fold so I gain position. This especially works well if there are limpers and I raise a near minimum amount. But I will be a bit more selective about the hands I play at this point.

But lets say I pick up 45s and it folds to me on the button...
PERFECT:

probability of flopping an 8+out draw	21.390%
probability of flopping 2 pair or better	5.582%
probability of flopping a pair	26.940%
probability of gutshot draw	14.370%
straight flush draw	1.020%

Now basically I have a REAL good chance of being able to stay in the hand and play a style where I do not have to give up equity in the hand. In other words, about 70% of the time I have a good enough hand where I feel I can impliment a strategy that OVERALL is better than break even.

I may have a particular move (out of position) such as a check raise and a shove on the turn, or a bet and then a check raise with gutshots that is incredibly unprofitable mixed in there. But it is infrequent enough (as I also can take free cards and play a smaller pot), combined with the fact that I could just as easily be pushing with 2 pair or better, a pair WITH outs or a more premium draw. PLUS my opponent may not be able to call. Additionally, I make sure I am VERY aware of stack sizes and amount in the pot to ensure that my overall strategy is profitable. It HAS to be profitable enough that 70% of the time to offset the 30% of the time that I miss the flop completely. Yet even then I feel I can gain a small amount of equity back by playing small pot poker and making some good small bets, and gain a TON from playing a nutball strategy when the situation calls for it

Additionally the odds against my opponent hitting anything is good. PLUS I am real good at recognizing what kind of flops are likely to hit my opponent. Add on to that the supreme power of position. In other words, I can take a free card if I miss and hit a monster because they checked. Or I can minraise or flat call in position on a back door draw or with complete air to potentially make a small bluff or big semibluff on the turn.... I say potentially because there are many times when I will just take the free card and represent one draw while I have another and make a play on the river, I may also just try to check the turn and fold the river or shove all in if I hit. I also in some cases will call with the intention of only bluffing if I improve. I have already represented strength. I will also play monster hands like this over half the time I have them on a similar flop by just calling with a set then when a potential draw hits I will shove. on a board with 2 suits I also would just call and then shove a flush. I will sometimes even play top pair like this to extract more chips. My game has "balance" even though it is still pretty far away from "equilibrium".

Why? People to combat my style would have to start making crazy calls or else avoiding pots all together. If they start avoiding pots, I change my style to be much more cautious since I usually can recognize what THEY are doing, yet THEY can't recognize what I am since I play every opponent differently.

BUT IF people start growing a pair, they would have to start to call with hands like ace high to prevent me from robbing them blind in some cases. And if they do I have pretty good outs at worst, and they are drawing dead in some cases. If they are not, they are drawing to 1 out as I will be shoving with a pair AND a draw there a decent percentage of the time.


Let me give you the example provided in the book "kill everyone" of why this strategy is so difficult to play against. Say you've raised it to 4 big blinds or 800 in 100/200 blinds.

Flop comes down a face card and two blanks... in this case an ace. As 7h 2d

SB is short stacked now with only 5400 left and he pushes all in to a 1800 sized pot. You have him covered but would be a large part of your stack.

What if opponent puts your raise range (correctly) on 22+,A2s,A5+,K9+, QT+,J9s, T8s+, 97s+, 86s+, 75s+, 64s, 54s....

If your call standards are top pair with ten kicker or better, then you are allowing it to be profitable for SB to push with ANY TWO CARDS! He could push with Q3!

If you lowered your standards to A8 he could "tighten up" and only push with All pairs, all wheel draws, and any pocket pair.

Think that's something? Just wait!
If the flop is 7,5,2 rainbow what should you call with facing a 3x pot size bet?
How weak can you go?
If you called with 44, your opponent is still profitable by calling preflop and then pushing with ANY TWO (assuming he doesn't actually slow play his big hands)!
You would actually even have to call with 33,A3 and even AK to prevent opponent from pushing with any two. Beyond that, if opponent had even a SLIGHT ability to "tighten up" preflop, although there would be gains from the blind steals, once you got to the flop you would most likely have to call even with a hand like AJ. If he went even tighter preflop plus a few occasional check folds and the occasional check raise all in mixed in (which could give him more equity), then you might have to call him with ANY ace high. And what if your raising range preflop is even looser? There are a handful of players that would raise from the button with about any two if it's folded to them! Now you may have to call with king high!

HA! So how many poker players at your game is there that can play that loose to play against your style?

BUT! What if the flop is something like QQ4? This is fun! You would have to call with 77 or higher to prevent the "any two" push. If opponent tightened up just a bit preflop and then pushed with this flop every time, you may have to call with an even looser hand.

Now consider me, the nutball poker play as your opponent. I typically avoid these kind of out of position moves. Yet in position I build up a pot, BUT am still a bit selective about my preflop range. I might play connected hands, suited connectors and suited one gap hands (i.e. 79s), and suited aces and suited kings plus occasional combinations of 2 face cards, any pair. I might play over 25% of all hands, and occasionally that may include "any two" if I am in position and feel I can make up for it after the flop. There is always a CHANCE I have any two cards even if I on average might play 25% of hands in middle position. Tighter out of position usually, sometimes much looser out of position (but not always).

Now how do you handle my aggressive move? But hold on... I am not just blindly shoving the flop. I will sometimes shove the flop, sometimes the turn, sometimes the river and the range of hands I may do that with have a good mixture of bluffs, semibluffs, monsters and even thin value bets. AND about 60% of the time I may switch to a "small ball" approach and control the size of the pot.

My shove range absolutely is tighter than "any two" preflop, and I absolutely am not always shoving. Additionally I am focused on YOU, and your particular playing style.

How do you do it?

The perception is PROBABLY that I am like the typical random maniac and that you tighten up and wait for a hand. Meanwhile I rob you blind preflop and on the flop and then you finally get your hand and I may not even be in it. If I am, I may be able to get away from it if I notice the number of hands you play is small.

AND about 20-40% of the time I will just decide to SWITCH styles completely and go back to playing a strategy where I ONLY come in with a superior starting hand range than you. Then I make decisions based upon not only your hand range, but the percentage of the time that you call on the flop (actually the fold percentage is where my aggression comes from), the percentage of the time you call on the turn, and river. If you have a leak in your own game I will aggressively attack it in either case, BUT this time I am doing it with a superior hand. Even if you wait for a hand that beats me it will be too late. If you play too loose I will have a better hand anyways.


You see... of this "nutball" strategy, I also have a large handful of variations of it that respect the game theory and mathematics.

Another amazing thing about the "kill everyone" book is that on a draw you can actually call a nearly double the pot size bet on the TURN on a draw where you are only maybe 24% to win IF you in turn have an equilibrium all in strategy on the river. In other words, you will have over 10 times the amount in the pot on the turn, you are faced with a bet twice the pot, and as a result, you have the all in bet on the river that is twice the pot. You might bluff 16% of the time, bet your real hands 24% of the time, for a total betting 40% of all time and a ratio of 60% of your bets you have a real hand and 40% of the time you are bluffing. But put in the ability to size up opponent along with that respect for math, game theory, equilibrium and odds, and suddenly you may even be able to call with a much wider range with only a 20% chance to win. And since people won't bet double the pot, and you can make other decisions, in some cases you can call with maybe a 15% chance to win or around 8 outs on the turn.

And then there's the book "raiser's edge" that actually attempts to estimate some more exact "equilibrium" strategies. Now most work in books is befrore the flop, but I haved a real good understanding AFTER the flop. I have an even better understanding of different exploitative approaches but we will cover that next.

nutball poker

I have a confession to make. I play poker like a "nutball". This is not someone who sits around and waits for the nuts, but someone who plays LIKE a maniac. Yet, I do so in a mathematically sound way. I don't just shove on every hand, I don't always shove preflop, I don't always on the flop, I don't always on the turn, and I don't always on the river. I also mix in small ball poker concepts to be able to control the potsize and win showdowns to the river.

But most people watching me play would say I play like a nutball, a maniac, etc. GOOD... that is exactly what I want...

I assure you, my approach is very mathematical and based upon game theory. Yet I mostly seek exploitative strategies.

The thing about my approach is you could easily take a more patient approach with much less swings and the probability of you being profitable is much greater, but much smaller potential. You could take somewhere between the two and take an equilibrium strategy WITH a lot of pushes all in that by itself can many times be quite "nutty" and you would probably be very profitable as well. But I am seeking to create more future profitable situations as well as exploit opponents more aggressively with the downside being the risk of an opponent completely flipping their style around and setting a "playing style trap". i.e. they bet draws and bluffs one way and big hands the other and once you go to pounce on them, they anticipate it and flip their strategy around. Or perhaps they play a wide hand range preflop and on flop and fold a lot on the turn only to set up a series of check raise bluffs and aggressive play. If your range is "maximally exploitative" you would probably play every hand that person is in and be aggressive on the flop and turn. But if opponent knows it he can start to play super aggressive back to counter exploit you. With an equilibrium strategy he cannot.

As such, I try to walk the fine line and flip back to an equilibrium strategy.

The next post will get into more mathematical examples.