Game Theory and Bluffing

Game theory sounds like a theory about games, but it is actually a branch of mathematics dealing with the decision-making process. While it applies to games, as we shall see, it also applies to such disciplines as economics, international relations, social science, and military science. Essentially game theory attempts to discover mathematically the best strategies against someone also using the best strategies. Against an opponent you think is weaker than you are — and it can be in any game whatsoever — you would usually rely on your judgment rather than on game theory. However, against an opponent you think is better than you or against an opponent you don't know, game theory can sometimes enable you to overcome the other's judgmental edge.

To show how game theory can work in this regard, we'll employ the children's game of odds and evens. Each of two players puts out one or two fingers. If the total is even, one player wins; if the total is odd, his opponent wins. Now mathematically this is an absolutely even game. However, over a long series it is possible for one person to gain an edge by outwitting the other, by deciding whether to put out one or two fingers on the basis of what the other person put out in the previous round or rounds, by picking up patterns — in a word, by figuring out what his opponent is thinking and then putting out one or two fingers in order to foil him.8

Figuring out what the other person is thinking is, of course, a crucial aspect of poker. See Chapter Twenty-three, "The Psychology of Poker."

Suppose someone challenges you to this game. Feeling confident about his judgment and ability to outguess you, he is willing to lay you $101 to $100 per play. We'll assume you too feel your challenger has the best of it in terms of judgment. Nevertheless, by employing game theory, you can gladly accept the proposition with the assurance that you have the best of it. All you have to do is flip a coin to decide whether to put out one or two fingers.

If the coin comes up say, heads, you put out one finger; if it comes up tails, you put out two fingers. What has this procedure done? It has completely destroyed your opponent's ability to outguess you. The chances of your putting out one or two fingers are 50-50. The chances of a coin coming up heads or tails are 50-50. However, instead of your thinking about whether to put out one or two fingers, the coin is making the decisions for you, and most importantly it is randomizing the decisions. Your opponent might be able to outguess you, but you are forcing him to outguess an inanimate object, which is impossible. One might as well try to guess whether a roulette ball is going to land on the red or the black.

Since your opponent is laying you $101 to $100, by using game theory you have assured yourself of an 0.5 percent mathematical advantage (or a 50-cent positive expectation per bet). You have removed whatever advantage your opponent might have had in out-thinking you and given yourself an insuperable edge over the long run. Only if you thought you could out think your opponent would you be better off using your judgment instead of a coin flip.

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