The Ideal Way to Play Hold'Em

Let's start by describing how you would play Hold'Em if you had all the skills of a top player and also an unlimitted amount of processing power.

First, you need to put your opponents on various cards. You don't actually put them on certain cards, rather you assign them a probability of having each possible hole. There are (52*51)/2 holes before you see any cards, and the number goes down as more cards are revealed. Basically you are doing a Bayesian estimation here - given their actions, which you have seen, what is the probability they would have acted that way with a given hole? Since all holes are a-priori equally likely, the probability of a hole is just the probability that they would have done those actions with that hole. In conditional probability this can be written :

P(H|A) = P(A|H)

Here "H" is some hole (two cards) and "A" is the set of their actions you have seen (eg. check, bet, call, etc.). The "|" means "given", so this reads "The Probability of hole H given actions A is equal to the probability of actions A given hole H".

Let's consider this in practice. At the start of action pre-flop everyone has equal probability of every hole. Now, you can see your own hole, so all holes containing your two cards are impossible, so they get a probability of zero. Now imagine the player under the gun limps in. What is the probability he would do that with various cards? Well, with something like the 24o (two-four-offsuit), it's very unlikely indeed. So, you adjust your probability, like : P(24o) = 0.000001 , or something. With something like AA (aces) it's pretty likely he would limp; maybe he would limp 75% of the time with Aces and raise 25% of the time. So P(AA) = 0.75

When the flop comes out on the board, their probabilites don't change, but they do change as they make more actions. Let's say after the flop, they check. So, they're probably weak *or* trapping. So, you can lower the probability of medium level hands, like top pair with a bad kicker or middle pair, and the probability of checking with a very weak hand is roughly 100%, and the probability of checking a good hand is maybe 25% or something; so you multiply that probability for each hole onto the probability you already had.

For example, if the big blind just checks pre-flop, that doesn't mean that all holes are equally likely. The reason is that he would have raised with some of his better holes, and you've seen that he hasn't raised, so that means he's more likely to have junk, or possibly a monster that he's slow-playing. Simply good holes like AJ or 88 are then less likely than pure junk like 72o or monsters like AA.

Now, assuming you could estimate all these probabilities right, you would have a perfect model of your opponent. That model is simply a probability for each hole. In order to estimate the probabilities, you need a good "opponent model" - that's imagining what your opponent would do in all situations. Of course, what your opponent will do is based on what he thinks you will do, so you have to imagine "what does he think that I think?".

Given this opponent model, playing poker is pretty easy. As each opponent acts, you adjust their probabilities. On your turn, you always have three choices - { fold, call/check , bet/raise }. I'll refer to these as just FCR - {fold,call,raise}. To choose one, you want the one that gives you the highest EV. To compute EV, you try each one, and then simulate what your opponents will do for all their hole cards, weighted by the probability of all their future actions. That is, you form the full EV equation :

add/subtract the cost of my action (calling/raising/etc)
for the next opponent
sum over all possible hole cards by him; weighted by the probability of each hole
he can choose {f,c,r}
for that hole, compute his probabilities of each P(f),P(c),P(r)
branch 3 ways and sum with the weights of his probabilities of each
adjust the pot for his actions in each case

In most cases this will come back to me and I'll have a choice again. Each point in the tree where I have a choice, you take the best (highest EV) of the 3 choices. At each point where an opponent has a choice (FCR), you sum all three weighted by the probability of each action given that hole. For all opponents you always do a weighted sum on all the holes they could have.

One nice way to do this weighted sum is to sum over all possible hole card assignments to the entire table; this brings the sum out of the whole thing and you just wind up with this big trinary-tree of branching choices for you and your opponent.

Note that this EV calculation contains every fancy or smart move you can imagine - the idea of semi-bluffing is in there, the idea of betting when you think he will call and you have a better hand, the idea of checking it off if you think he'll bluff and you can beat him, etc. etc. All you do is pick the move with the biggest EV and you're golden.

The only exception to that is when you may want to maximize your long term EV instead of your EV for just this hand. For example, when you first sit down at the table, you may want to make some sub-optimal moves to establish a certain image. Similarly, you may want to fold slightly less than optimal on the river to prevent people from bluffing you out too often, etc.

Now, the big difficult piece here was the opponent model. If you have a good opponent model, then the EV tree is just a matter of having a huge amount of processing. It is, of course,too much processing in most cases. There are simple cases, like heads-up play, where it can actually be evaluated (or approximated well), but in practice, you wind up doing some heuristics and estimation.

So, this is the "ideal" way to play poker. How do you get close in practice? Well, you usually keep a sort of simplified opponent model going in your head. For example you might have the idea that "this guy probably has Ax, or maybe KQ, and he also has like a 10% chance of any two cards". That's a form of opponent model, and all good poker players are tracking their opponents like that. Then, using that opponent model you can start doing some prediction stuff. Like say if you bet, he'll call with Ax and fold the rest of his hands, and you have A8, so when he calls you beat him about half the time. That makes your bet +EV, though you still need to compare to calling. For most practical purposes you usually don't need to look very far ahead to maximize EV. Top mathematical players do heuristic forms of this analysis.

Stealing from Late Position

In poker you make a lot of money by betting at pots. One of the most common bluffs if the "late position steal". This happens when you are in late position and everyone has shown weakness, such as by checking around. A bet here is almost automatic. The thing is, even when people know you're on a steal raise, frequently they can't call. The two main times this happens are pre-flop when everyone folds around and you're on the button or one off the button, now you're just raising into the blinds who have random hands and will often have to fold. Even if they don't fold pre-flop, you can continue aggression on the flop and probably win then. The steal is also common on the flop when it checks around to you in last position. In this case you need to be a bit worried that someone is slow-playing or trying a check-raise. Obviously you can use their pre-flop action and the texture of the board to decide how likely it is they are trapping. eg. if it was raising up pre-flop and the board comes AQT, then someone probably has something, so the checks are very suspicious.

Now, what kind of hand do you want to steal-raise with? Well, if the table is very weak, you can do it with any hand at all. This is actually pretty rare; at a decent table, people will defend often enough that it's not +EV to steal if that's the only way you have to win. You also need some chance of winning with your hand when you're called. For example, against the blinds you need roughly a 1/3 overall chance of winning for a steal to be profitable. You might succeed in your steal 20% of the time, then you also need to be able to win 15% of the time when you're called. The "when you're called" is very important - you have to judge how your hand does against hands that are likely to call you. Because of that, hands like A2o and K2o are not good steal hands. The problem is that against hands that can call you, you're likely to be dominated, against eg. A6o or K7s. You actually prefer hands that are a little sneaker, like QTo, J9o, 65s, 33, and of course all better hands. These hands are unlikely to be dominated and play well against random hands. Hands like 65s are worth a special note; they work well only because you are going to represent high cards when they flop. That is, if the flop comes A98, you're going to bet as if you have the Ace. Often he won't have the ace and will fold it. Thus you win when the flop hits you (with a straight, flush, or even just a pair), and you win when the flop doesn't hit you (high cards), and when you add them up it can be very profitable. The same thing holds roughly for steals on the flop. You want to have a few outs (at least 4) to make a hand that can beat what he's likely to call with. If it checks around, you make a steal-bet, and someone re-raises you, you want to be able to call. At that point the pot is usually offering around 9:1, so you need roughly 5 outs to call. You get 5 outs with two overcards, any pair (not pocket), gutshot with an overcard, etc. If you just have a gutshot or just an overcard, you can usually call with just those 3 or 4 outs because your implied odds are pretty good - he thinks you're weak and will pay you off if you hit. Of course this is assuming he would check-raise with just top pair. If he check-raises too rarely, eg. only with monsters, then you just fold. Usually if good players see you doing a lot of steal-bets from late position, they will start check-raising more, and even start check-raising as a bluff or semi-bluff.

Charles Blooom [cb][at][cbloom][dot][com]
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