Strategic interaction with simultaneous choices

The invention of game theory may come to be seen as the most important single advance of the social sciences in the twentieth century. The value of the theory is partly explanatory, but mainly conceptual. In some cases it allows us to explain behavior that previously appeared as puzzling. More important, it illuminates the structure of social interaction. Once you see the world through the lenses of game theory – or “the theory of interdependent decisions,” as it might better be called – nothing looks quite the same again.

I first consider games in which agents make simultaneous decisions. The goal is to understand whether and how n agents or players may achieve an unenforced coordination of their strategies. Often, we shall look at the special case of n = 2. A strategy can be the simple choice of an action, as when a poker player decides to bluff after looking at his hand. It can also be the choice of a rule, as when he decides to bluff only when he is dealt the seven of hearts. Finally, a player can use a mixed strategy, that is, assign probabilities to the possible actions and choose one of them with the assigned probability. In shooting a penalty, a soccer player might, for instance, mentally flip a coin between aiming at the right or the left side of the goal. The goal keeper might use the same procedure to decide whether to go to the right or the left.

The players may be able to communicate with each other, but not to enter into binding agreements. (They can make promises, but the decision whether to keep the promise will only recreate the game.) To any n-tuple of strategies, one chosen by each agent, there corresponds an outcome. Each agent ranks the possible outcomes according to his or her preference order. When needed, we shall assume that the conditions for representing preferences as cardinal utilities are satisfied (Chapter 13). The reward structure is the function that to any n-tuple of strategies assigns an n-tuple of utilities. Although the word “reward” may suggest a monetary outcome, the word will be used to refer to psychological outcomes (utilities and ultimately preferences).