Bayesian Poker for Humans

So, how do you use this idea of Bayesian poker to play in real games?  Well, you can think of it in terms of "putting them on a hand".  Rarely can you put someone exactly on a hand, unless they are a very predictable player.  Often you can put them on a range of hands, which lets you know the probability of your winning or losing or making a big pot.  Note that generally the worse the opponent, the less precision you will have in your estimation, because they play is very erratic.  One of the biggest gotchas is just that very bad players can make huge mistakes, so when you're thinking about their play, you have to think that they just may be making a huge mistake.  For example, if the board has two-paired, something like TT977, you bet the pot size with J8, holding a straight, and they raise you all-in.  For a good player to make that raise he likely has a full house, but a bad player may not understand that with two-pair on the board, you need a house to have a good hand.  Generally the most predictable players are mediocre players who have read some Sklansky books and are playing by the formula.  These players are often very easy to beat, because you can tell exact when they're "semi-bluffing" or "buying a free card", etc.  Top players are again very hard to read exactly because they mix up their play, but the principles of Bayesian poker do apply well to them since they are smart and logical.  You simply have to use a opponent model that includes the concept of mixing it up.  They will make plays based on implied odds and bluffing and deception, but they generally won't make purely bad plays.  With a top player, it's hard to put them on a hand when they bet or raise, since it could be the nuts or it could be a pure bluff, but they often play predictably when just calling.

So, the concept of putting someone on a hand in Bayesian poker is just to divide all the hole cards into groups and estimate a probability that they would have made the plays you've already seen with those cards.  We humans generally find it easier to work on the 169 suited/offsuit hole cards rather than the 1326 full list of holes.  At the start of every hand, the full list of holes are all equally likely, but the 169 holes already have some different a-priori probabilities.  There are 6 of each pocket pair, 4 of each suited hand, and 12 of each offsuit hand, so those are the initial probabilities.  As each player folds, you don't really care what hand they have.  To put someone on cards you have to think about what they would do in that position with various holes, would they have made the play that they did?  You need to use a model that's specific to each inidividual - are they loose, aggressive, do they mix it up, are they just a bad player that plays every hand, etc.

Note that when you do the estimation for him, you need to think about the situation he was in when he made the decision.  Obviously he doesn't know your hole cards, he's only seen the betting action up to that point.  So imagine what he would think in that situation, and what kind of hole cards would he play the way that he did.  Also, you can estimate what he will have in the future if you make a certain action and he makes some response.  For example, you can think about "if I bet, what hands will he call with?".  You simply imagine being in his shoes after you have bet - now what hole cards would he call with? what percentage of the time would he likely call with those cards, rather than raising or folding?  This gives you an estimate of what hands you will be facing in the future if you make certain moves now.

The way we simplify Bayesian Poker to do in our heads is just by ruling out the very unlikely holes, or grouping them all together; you can always revise your estimate if you later see actions that make you think they're now likely.  For example, there's a huge number of holes that are just considered "junk", like 92o, 85o, T4o, etc. etc. we usually just lump those together and think of them as a whole and say "there's a 10% chance he's on pure junk" or something like that.

Let's say a good player open for a raise from early position.  That's a dangerous position, so he probably has pretty good cards.  He wouldn't play so fast with his very best hands, like AA and KK, so we rule those out for now.  He would play AK, AQ, AJs, QQ, JJ, TT.  We guess those are his most likely hands, and that there's a 10% chance of other hands.  Now, you're in middle position with AQo.  What should you do?  Well, you look at all the hands he may have, the chance of each, and how your hand stacks up against them.  His probability of each starts from the number of combinations of each, multiplied by the chance he would have made that play with them.  We've got :

AK : 16 ways : dominates AQo, you must hit a Q, and you pay him off if an A comes
AQ : 16 ways : even, though he may get a flush draw
AJs : 4 ways : you dominate him
QQ : 6 ways : dominates you, you must hit an A
JJ & TT : 12 ways : a race, 50/50 shot, though he has an advantage in the way it will play out
Other : you may be dominated (AA,KK) or he may be on a bluff

So, should you play AQo ?  Well, we can group his hands :

Dominate you : AK, AQ : 32 ways
You dominate : AJs : 4 ways
Race (50/50) : AQ, JJ, TT : 28 ways

Obviously the answer is no, you should not play, because you are only favored against a few of the hands he would play, against a bunch of them you are even, which is no reason to play a pot you haven't entered yet.  Note that if he also played AJo, and all AT's (not realistic for a good player, but bear with me), then it starts to swing the other way - now there are 32 hands that you dominate, so it's more of an even choice, and you have the positional advantage over him, which may swing the EV in your direction.

Let's do some more examples.

Let's say you flop bottom two pair, and a guy moves all in ahead of you.  You estimate that he will make that play with AA,KK, or any two pair or set.  Let's assume first that he would have played any two cards to this point, like maybe he was in the big blind and just called.  You beat AA,KK, tie another bottom-two-pair, and lose to his other hands.  Should you call?  There are 6 each of AA and KK.  The other bottom two pair can be made in 4 ways (2*2).  The top two-pairs can be made in 6 ways each, (3*2), 12 total.  The top set can be made in 3 ways (3*2/2); the bottom sets each can be made in 1 way.  So, the totals are :

you beat AA & KK : 12 ways
you tie bottom two pair : 4 ways
you lose to better two pairs and any set : 12+5 = 17 ways

So, you should probably fold (depending on the size of the pot vs. the size of the bet).  Similarly, if you have top two pair you should clearly call.  Also, if for some reason you think the two pairs are unlikely, like if the board is rags and he raised it pre-flop, you can also call.  Let's do that case in more detail.

He is one before the button and raises preflop to 3x the big blind.  You call in the big blind with the 56s.  The flop comes 56T.  Now you check, slow-playing your two pair, and he moves all in.  Should you call?  Well, he would have played 55 or 66 or TT or KK or AA this way.  Let's assume he wouldn't go all-in with JJ or QQ.  He wouldn't have played 56 or 5T or 6T for a raise preflop, so you can pretty much rule those out.  That makes it a clear call.  You beat 12 holes and lose to 5, greatly in your favor.

How about when the flop comes AAK.  Let's say you have K6 and you were in the big blind.  Only the button is in, and you put him on having any decent hand, since he likes to limp from the button with any K or any A.  You check and he bets.  Are you beat?  Let's assume he would bet with any Ace, and he would be 50% of the time with any king or pocket pair.  There are 2*45 holes with an ace, and 2*43 holes with a king-high.  There are also roughly 11*6 pocket pairs.  So, 90 holes have an ace, 86 have a king, 66 have a pocket pair.  Almost any king beats you, so you are clearly beat most of the time.  If we add in some chance that he's bluffing, it might bias you towards a call, but basically you can't beat many legitimate hands here, so you're basically just looking at beating a bluff.

Let's look at some more simple cases concerning all-ins on the flop.

Say you're playing heads up, he checks the flop and you go all-in.  How will he play it?  Well, that of course depends on how you play and how he thinks you play.  But assuming you're both average players, he can assume you probably don't have a real monster, like a flopped straight or something like that.  You could have top pair with an Ace, or an over-pair, or perhaps just a flush draw and you're semi-bluffing.  He know you're a good player so you're probably not on a pure bluff.  So, if he has hands in that class, he could call you.  If the all-in is not much bigger than pot size, this could be a bluff.

Now, what if instead he leads out with a pot-size bet on the flop, and then you go all-in.  Now, he has to assume that you have much better hands.  You knew that he was strong because he led out with a big bet.  Your all-in after his bet says that you have something better than top pair.  Again you probably don't have a huge monster hand, you would have slow-played that, but you may have two pair or a set or something like that.  Your move is the same (all-in), but because you did it in a different situation, he has a different estimation of what your hand is.  The all-in player could be on a bluff here, because he knows that the initial pot-size bet is probably just a pair, since the original bettor would have slow-played a monster.

Consider one more.  The flop is seen three way.  Player 1 checks, Player 2 bets pot-size, Player 3 calls, Player 1 goes all in.  What kind of hand does the all-in player have?  In this case he probably has a very good hand indeed.  Player 2 has shown strength by betting the pot, probably something like top pair.  Player 3 has just called, so he may have a monster that he's slow-playing, or he may be on something like a flush draw.  Player 1 is almost surely not bluffing, because there are two players behind him, one of them is likely to call, he can't be counting on them both folding.  And Player 1 surely has something better than top pair.  An over pair is possible, but a set or two pair is more likely.  He's also check-raised here which shows a lot more strength than leading out betting, it means he has a hand big enough to possibly let his opponents see a free card.