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This chapter describes the theory of two-person, zero-sum games invented by John Von Neumann in 1928. It begins with an application to the computation of economic shadow prices. It shows that a two-person game is strictly competitive if, and only if, it has a zero-sum representation. Such a game can be represented using only the first player's payoff matrix. The minimax and maximin values of the matrix are defined and linked to the concept of a saddle point. The ideas are then related to a player's security level in a game. An inductive proof of Von Neumann's minimax theorem is offered. The connexion between the minimax theorem and the duality theorem of linear programming is explained. The method of solving certain two-person, zero-sum games geometrically with the help of the theorem of the separating hyperplane is introduced. The Hide-and-Seek Game is used as a non-trivial example.

The movie A Beautiful Mind (2001) portrays the life and work of John F. Nash Jr., who received the Nobel Prize in Economics in 1994. A class of his theories deals with how people should behave in strategic situations that involve what are known as “mixed strategies,” that is, choosing among various possible strategies when no single one is always the best when you face a rational opponent. This chapter uses data from a specific play in soccer (a penalty kick) with professional players to provide the first complete test of a fundamental theorem in game theory: the minimax theorem. The minimax theorem can be regarded as a special case of the more general theory of Nash. It applies only to two-person, zero-sum or constant-sum games, whereas the Nash equilibrium concept can be used with any number of players and any mixture of conflict and common interest in the game.

This chapter deals with the Minimax Theorem and its proof, which is based on elementary results from convex analysis. The theorem states that for every matrix A, the average security levels of both players coincide. In a mixed policy, the min and max always commute. For every constant c, at least one of the players can guarantee a security level of c. The chapter first considers the statement of the Minimax Theorem before discussing the convex hull and the Separating Hyperplane Theorem, one of the key results in convex analysis. It then demonstrates how to prove the Minimax Theorem and presents the proof. It also shows the consequences of the Minimax Theorem and concludes with a practice exercise related to symmetric games and the corresponding solution.

This chapter surveys applications of game theory in literature, and is organized as follows. Section 1.2 sets out the three questions posed to several game theorists who at some time applied game theory to literature. Section 1.3 shows how Arthur Conan Doyle and Edgar Allan Poe side-stepped rather than confronted the consequences of the so-called minimax theorem in their fiction. It then presents an application that illustrates how William Faulkner captured the spirit of the theorem, even invoking a fictitious “Player” to make seemingly random choices, which, according to the minimax theorem, are optimal under certain conditions. Section 1.4 considers problems of coalition formation in zero-sum games. Sections 1.5 and 1.6 review several works of fiction that may be interpreted as nonzero-sum games. Section 1.7 looks at game-theoretic analyses of the devil in Johann Wolfgang von Goethe’s Faust and of God in the Hebrew Bible. Section 1.8 discusses Sir Gawain and the Green Knight, a medieval narrative poem that has been explicitly modeled as a game of incomplete information, while Section 1.9 concludes.

Chapter 9 presents a formal justification of the objective Bayesian approach. The norms of objective Bayesianism are justified on the grounds that they must hold if one is to avoid certain avoidable losses. In particular, they must hold if one is to avoid avoidable sure loss and worst-case expected loss. This line of justification is shown to be robust with respect to various ways in which the underlying assumptions might be relaxed.