A Little Game Theory

by Lou Krieger | Published: Jul 11, 2001

Talk to poker theorists long enough and the concept of game theory is bound to pop up, and it's really not a new idea. Nesmith C. Ankeny wrote an entire book about it in 1981, Poker Strategy: Winning With Game Theory. David Sklansky also explained game theory in his seminal work, Winning Poker, two years thereafter.

Game theory, despite its name, isn't about Monopoly, Trivial Pursuit, or other leisure-time diversions. It's a branch of mathematics dealing with decision-making that has proven to be useful in fields as diverse as economics, political science, operations research, military science, and poker – where the idea is to optimize a decision, rather than to maximize or minimize any one of a multitude of potential outcomes in certain situations.

That sounds pretty theoretical, so here's a practical example: After all of the cards have been dealt, let's assume that just you and one other player are contesting the pot. You don't know a thing about your opponent's playing style. You've never played against him before, and haven't picked up even the slightest inkling of a tell. Just for the heck of it, pretend that you're playing against the Invisible Man. You don't have much of a hand. In fact, you have nothing more than a busted flush, and the only way you can win is by bluffing successfully. We're also going to assume that your opponent knows with absolute certainty that you were on a flush draw. Although he cannot beat a flush, his hand is strong enough to beat any busted flush.

Here's where game theory comes into play. Suppose that you bet every flush draw – whether you made it or not. What do you think would happen? If your opponent was very cautious, he'd throw his hand away most of the time and you'd win the pot whenever he did. But if he were a decent player, he'd begin to suspect you of stealing and call with increasing frequency. In fact, if he knew you bluffed every time you failed to make your hand, he'd call each and every time you came out betting. Now the situation has reversed itself. Rather than winning each time you came out betting, you'd lose most of the time. While you'd win with your legitimate hands, each time you tried to bluff, your opponent's call would capture the pot. Since you'd miss those flush draws more often than not, compulsive bluffing would cost you quite a bit of money.

Suppose that you took the opposite tack and never bluffed, betting only when you completed your draw. Just as he did in the case when you bluffed too often, your opponent would soon get wise to your habits. Once he gloms onto the fact that you never bluff, he will adjust his strategy accordingly. Now, he'll fold when you bet, but he'll also show down the best hand and win the pot whenever you check.

Do you see what's happening here? Your opponent has wised up to your pattern of always bluffing each time you have a busted drawing hand, but more importantly, your results are not a function of your actions. Instead, the results you achieve are wholly dependent on the choices your opponent made. You are no longer in charge, and that's a bad thing. Your playing strategy has allowed the locus of control to pass to your opponent, who, by virtue of his decisions about whether to call or fold, is the one who determines how much you win or lose.

It's pretty clear from all of this that you can't be a one-dimensional player – always bluffing or never bluffing. And you didn't have to know about an arcane branch of mathematics called game theory to tell you that. Even if you bluffed once in a blue moon, or refrained from bluffing only every once in a while, you'd create opportunities for your opponent to make errors by forcing him to decide whether or not you held a legitimate hand. When you always bluffed or never bluffed, your opponent was relieved of the responsibility for making a decision. He knew that you bluffed all the time, or perhaps realized that you never bluffed at all, and either way, it made no difference at all. His strategy was easy and obvious, and allowed him to maximize his winnings as a result.

But when you veered away from these polar extremes, your opponent was put to the test by having to answer this question: Do you or don't you have the goods? And you know what, if you give your opponent a chance to make mistakes, he will make some. He will, I will, and every player who's ever lived will err in judgment. There's not a poker player alive who makes the right decision all the time. Poker, after all, is a game of incomplete information, and that means wrong decisions will be made.

Game theory gives one the wherewithal to optimize his play. When you bluff properly – not too often and not too infrequently – it makes no difference how your opponent responds. Game theory allows you to control the outcome of your actions and optimize your results.

Here's how to bluff using game theory: First, make sure the odds against your bluff are equal to the odds your opponent is getting from the pot. Confusing? Not really. Suppose that by betting on the river, you create a situation in which your opponent will be getting 4-to-1 from the pot. That's easy to imagine. The pot contains $300, and by wagering $100, your opponent stands to win $400 for his $100 wager.

Now, let's say that any one of eight available cards would have given you the winning hand. If you bluff whenever two predetermined cards come up in addition to the eight you need, you are bluffing at a frequency that precludes your opponent from taking advantage of your bluffing proclivities – regardless of what he chooses to do.

How easy is that to pull off? You can trigger your bluff vs. no-bluff decision by randomizing it with cards. Suppose that you are looking for either a 7 or a queen to complete your hand. Any one of those eight cards will do; it doesn't matter which one pops out of the deck. Now suppose you tell yourself that you will come out bluffing if your last card is a red deuce instead of the desired 7 or queen. By giving yourself two bluffing cards as well as eight winning cards, in the 4-to-1 ratio of winning cards to your opponent's pot odds, you've optimized your decision-making.

But there is a rub. It's tough to make these kinds of calculations in the heat of battle. Most players don't do this sort of thing, trust me. But you can work out common drawing situations in advance, just as we did here. And you don't even have to be absolutely precise. Oh, sure, it's nice to be right on the money, mathematically speaking. However, as long as you realize that winning poker requires making mistakes at the polar extremes – neither habitually bluffing nor always checking, nor always calling your opponent's wager or folding every time he bets – to avoid making more costly mistakes in the middle, you're on the right track.

When all is said and done, you probably won't use game theory all that often in the heat of battle. And the more you play and the better you become at reading your opponent – by putting him on a hand or picking up tells – the less you'll have to rely on game theory. After all, you usually won't be playing against the Invisible Man. Even if you play online, where your opponents actually are invisible, you can discover tells about opponents and put them on hands based on their proclivities for checking vs. betting, and calling vs. folding.

While game theory is pretty cool stuff, and we all owe a debt of gratitude to Ankeny and Sklansky for presenting it to us so cogently two decades ago, its greatest value probably lies in how well it serves as a parameter that can assist us in learning how, as well as how often, to bluff vs. bet for value, or to fold in the face of a bet vs. trying to pick off a bluff.