*Ken Binmore*

- Published in print:
- 2007
- Published Online:
- May 2007
- ISBN:
- 9780195300574
- eISBN:
- 9780199783748
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195300574.003.0007
- Subject:
- Economics and Finance, Microeconomics

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 ...
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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.Less

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.

*João P. Hespanha*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175218
- eISBN:
- 9781400885442
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175218.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

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 ...
More

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.Less

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.