Debreu's social equilibrium existence theorem.

Game theory is the study of strategic interaction: how a person should and will behave if her actions affect others and their actions affect her. (We use the term “person” here, but “agent” might be more appropriate, because, in the literature, game theory applies to firms, nations, or even genes, as well as to individual people.) Among fields of application, the theory has had its biggest influence on economics (but has also had considerable sway over political science, evolutionary biology, and computer science, among other disciplines). Economists are interested in predicting (or at least explaining) behavior in economic environments, for example, what products firms will offer for sale, which goods and services households will purchase, or what securities (e.g., stocks or bonds) investors will trade. To see how game theory has contributed to this effort—and, in fact, has revolutionized economics—think about trying to analyze the behavior of firms in the American automobile industry. To simplify, let’s suppose that the industry is dominated by three companies, General Motors, Ford, and Chrysler (these three did indeed dominate for many years). Now, for General Motors to figure out what models to launch, how many units of each model to produce, and what prices to set for them, it has to anticipate what Ford and Chrysler will be doing. However, the same sort of anticipation is required of Ford and Chrysler. That is, Ford’s best course of action will depend on what it thinks GM and Chrysler are going to do. Actually, it’s even more complex than that, because GM will understand that Ford is trying to forecast GM’s behavior. Therefore, to predict Ford’s actions, GM also has to anticipate Ford’s anticipation, and so on, to higher and higher levels of prediction. Such mutual forecasting is clearly crucial to an analysis of the automobile market but is potentially extremely complex.