Think Like a Game Theorist

In my last column, I provided an introduction to game theory as it applies to poker. Now I'd like to look at a particular no-limit hold'em situation and discuss how to think about it the way a game theorist would.

Most poker strategies try to answer the question, "How should I play this hand?" The game theorist asks himself, "How should I play poker?" The somewhat subtle distinction between these two questions may be difficult to grasp, at first; but it represents a crucial philosophic difference between a game theoretic approach to poker and most other strategic approaches to poker. Let me illustrate through an example.

You're playing $5-$5 blinds no-limit hold'em and you called a $20 raise preflop. The flop comes 9 7 7 and the raiser leads out for a $50 bet. You have $250 in your stack, and the raiser has you covered. What's your play with the J 10? What's your play with pocket nines?

A typical hand-by-hand strategy would say that with the J 10, you absolutely have to raise. Unless your opponent has flopped a full house or quads, you will never be a big underdog with your gutshot-straight-flush draw. And, since any reasonable raise will commit you, you should move all in. By making that play, you force your opponent to make a tough decision even with some strong hands. At the very least, you can expect a hand like A-K offsuit to fold.

This argument for raising all in with the J 10 is pretty sound, and I think even game theorists would agree with it. They would agree, however, only with the caveat that they also would raise all in with other, stronger hands in the same situation. By way of explanation, let's look at what you should do with the two nines.

A typical hand-by-hand strategy would say that with nines full in this spot, you absolutely have to flat-call. There are so many hands that will just fold if you raise, that you have to call and let them catch up. If you raise, you miss the chance to get paid off by an opponent who flops nothing, but makes a pair or perhaps a small flush on the turn. Furthermore, anyone with a big hand will get all of the chips in later anyway, so you are essentially risking nothing by flat-calling the flop.

Again, this isn't a bad argument, and again it's possible that game theorists would agree with it. But they would agree only with the added stipulation that they also would have to flat-call in this situation with other, weaker hands. For the game theorist, the question of "How should I play this hand in this situation?" is secondary to "How should I play my range of hands in this situation?" And here's how a game theorist might answer that question.

First, he'd list the kinds of hands with which he'd want to continue: maybe flush draws (for example, the Q J) and other semibluffing hands (for example, J-10 offsuit), medium-strength hands (for example, 8-8), and strong hands (for example, A-7 suited, 9-9). Next, he'd try to come up with an overall strategy that enables him to play all of these hands in the best way. Maybe he'd say, "The conventional wisdom is to move in with my draws and flat-call with my big hands and my medium-strength hands. I'll start with that strategy." Then he would put himself in the mind of his opponent. "My opponent," he'd say, "will eventually realize that every time I move in here, I have a draw. So, I'm never going to get him to fold when I have a draw. Furthermore, he'll realize that every time I have a big hand, I slow-play, and that might cost me some bets, as well. It's clear, therefore, that I sometimes have to move in with big hands along with my draws. That way, my opponent can't profitably call me every time I move in, but he won't want to fold every time, either. In fact, he'll have a very difficult decision, and that's what I want."

Then the game theorist would take the final step and ask himself with what hands he should flat-call his opponent's bet. He'll have to keep in mind that there are more betting rounds to come, and that he won't want to give away too much information about his hand with his flop call. Maybe he'll decide he wants to slow-play some of the time with his big hands, because it might make his play on the turn easier. If his opponent bets, he can continue with his big hands and muck most of his medium-strength hands. Remember that game theory allows for playing the same hand in a different manner in order to maintain a balanced strategy. In this example, the game theorist might decide to flat-call with his big hands half the time, and move in with them the other half of the time.

I hope by now that the distinction between the questions "How do I play this hand?" and "How do I play poker?" is clear. From the game theorist's standpoint, every hand we play affects every other hand. No hand is played in a vacuum. "Fine," you might say, "but most of my opponents don't pay attention to what I'm doing, and I usually don't play with the same people for more than an hour or two, anyway. So, what good is game theory, then?"

My response is that you shouldn't worry about balancing your strategy against truly clueless opponents. Just make the best exploitive play – the play that will earn the most money on that individual hand. But lots of players do pay attention. Even if they don't pay attention to you, they've probably at least thought about poker before, which means they're going to make certain assumptions about how you play. Let's put it this way, against a player you've never seen before, what's the first hand you're going to put him on when he moves in on the 9 7 7 board? Without other information, the first hand I'd give my opponent is a flush draw. That's because most people don't worry about balancing their play – and I can take advantage of that even against a total stranger.

So, now everyone has seen some of the ways I use game theory to think about poker. Notice that I didn't come up with a "right" answer to the example situation, one in which the hero moves in with big hands X percent of the time and calls with them (100 – X) percent of the time. That would require some serious math, and I wanted to present the ideas first before getting into the numbers. But if you want to take on the problem, please do so. I think the results might be interesting, as would the thought process required to get the results. If you try it and would like to share the results, please send an e-mail to jacksup@mattmatros.com. If I get some good answers, maybe I'll put them into a column.

Matt Matros is the author of The Making of a Poker Player, which is available at www.CardPlayer.com.