Game Theory and Poker: An Introduction

In previous columns, I introduced some basic terms used by the math weenies of the poker world. These terms included equity, pot odds, probability, range of hands, combinations, and outs. I hope by now that you know what these things mean.



In this column, I'd like to take an introductory look at how game theory applies to poker.



For those who don't know, game theory is a branch of mathematics that deals with decision-making in situations in which two or more players have competing interests. In a poker game, everyone at the table is competing for everyone else's chips, so I'd say the players have competing interests.



Word has gotten out that I'm a game theory expert. It might have started when I mentioned game theory in my World Poker Tour interview, and the television producers decided to run with it. They went so far as to create a computer-generated graphic consisting of numbers racing across the screen, and text that read, "Game Theory," all of which floated over my head when the interview aired. The secret is that I'm not a game theory expert. I never even took a class in game theory. In fact, all that I know of game theory are the bits and pieces that apply to poker. And I have to credit other people – namely, poker players Bill Chen and Jerrod Ankenman – for introducing me to game theory at all.Bill says that at this point, I've learned enough to call myself a game theorist. Since Bill is one of the foremost of all poker thinkers, I'll take his word for it, but I'm still no expert. Bill and Jerrod have written a soon-to-be-released book called The Mathematics of Poker, which I've read in drafts and highly recommend, as it will enable anyone who reads it to suddenly consider himself a game theorist.



The pioneer in applying game theory to poker was former World Champion Chris "Jesus" Ferguson. Chris would tell you that he learned poker not from getting years and years of experience playing back-alley games, but by working out the optimal strategies with pencil and paper. Opponents and chips were not necessary.



This idea of learning poker without playing poker may sound strange. Perhaps it will help if I explain a little more of what game theory is all about. From the game theorist's standpoint, the question of how to play a hand of poker is a mathematical problem, and there is a correct solution to such a problem. Anti-math types complain that you have to play the same hand in different ways at the poker table so that you don't become too predictable, and therefore anyone locked into a mathematical approach to the game is doomed. What these naysayers don't realize is that game theory allows for players to play the same hand differently in its solutions. In fact, there is even a name for this: It's called a mixed strategy. Let's say, hypothetically, that if everyone folds to you on the button and you have A-J, you're going to move all in half the time and fold the other half of the time. That's a mixed strategy. The opposite of a mixed strategy is a pure strategy, in which you do the same thing with a specific hand every time. For example, if you move in with aces every time you get them, you're playing a pure strategy with that particular hand.



The solution to a given poker problem might be a pure strategy, or it might be a mixed strategy. The point is, game theory allows for the idea that you have to mix up your play in order to be successful. In fact, mixing up your play is a lot of what game theory is about, as we'll soon see.



There are two types of "correct" strategies in poker game theory. There is the optimal strategy, and the exploitive strategy. The optimal strategy is when you could tell your opponent your plan and there wouldn't be anything he could do to change your expectation in the game. Of course, you couldn't tell him what cards you were holding as you played against him, but you could say, for example, "I'm going to be bluffing 40 percent of the time and value betting 60 percent of the time. What are you going to do about it?" A person playing optimal strategy, and executing it without giving off any tells, cannot be beaten in the long run. If you were to play such an opponent heads up for a few thousand hours, your best case would be that you both lose money to the house.



You might be thinking, "Great, let's learn optimal strategy!" I wish we could, but even if we did, optimal strategy isn't always best. Let's say you're playing no-limit hold'em against a calling station, who never folds preflop no matter what the bet is, but will sometimes fold after the flop if he misses completely. He just insists on seeing the flop. Now let's say you're dealt two aces and you each have a few thousand blinds in front of you. The optimal strategy is probably to make a small raise, both building a pot and disguising your hand. But with this player in the game, a much better play is to move all in, knowing that he'll call you. To take maximum advantage of this terrible opponent, you need to employ an exploitive strategy. Sure, the optimal strategy would still win you money, but against bad players, other strategies might win you more money, as in the example I just gave. An optimal strategy is designed to protect you against opponents who play well. But when we can find ways to do better than optimal strategy against certain players, we do it.



OK, I've given you a framework for game theory in poker. How would it actually work at the table? Well, for starters, ask yourself if plays you routinely make might be exploitable by savvy opponents. For example, in no-limit hold'em, do you ever open-raise to six or seven big blinds? When you do that, do you always have a very specific hand, or maybe a very small range of hands? In my experience, people who open for an overbet in no-limit hold'em almost always have a medium pair (nines through queens) or A-K. This is information I can use to exploit opponents who make this play.



Let's say, for example, that I'm dealt two eights and I'm on a short stack. If someone opens with a standard raise with two tens, I'll probably move in on them, and they'll get to call me and (usually) bust me. But if someone opens with an overbet with two tens, I'll just fold my eights and the tens will miss out on a nice moneymaking opportunity.

OK, now you know a little something about game theory. In future columns, I'll look at some of the practical insights into poker that game theory has given us.

Matt Matros is the author of The Making of a Poker Player, which is available at www.CardPlayer.com. He can be reached via e-mail at [email protected].