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Problem 6. Let the 0-sum 2-player normal form game be given by the (m × n)-bimatrix (A, B) and denote by SA {1,2,...,m} and SB = {1,2,...,n} the pure strategies of Alice and Bob, respectively. (a) State the definition of a maximin strategy i* E SA of Alice (in terms of the matrix A) and the definition of a minimax strategy j € SÅ of Bob (in terms of the matrix A). Furthermore state the definition of a saddlepoint (i*,j*) E SAX SB. (b) Show that the following two statements are equivalent: (A) (i*, j*) ¤ SAX SB is a Nash equilibrium. (B) (i*, j*) ¤ SAX SB is a saddle point. (c) State an example of a 0-sum 2-player normal form game which does not have a saddle point in the pure strategies. State a criteria which ensures that such a game has a saddle point in the pairs of pure strategies.

Problem 6. Let the 0-sum 2-player normal form game be given by the (m × n)-bimatrix (A, B) and denote by SA {1,2,...,m} and SB = {1,2,...,n} the pure strategies of Alice and Bob, respectively. (a) State the definition of a maximin strategy i* E SA of Alice (in terms of the matrix A) and the definition of a minimax strategy j € SÅ of Bob (in terms of the matrix A). Furthermore state the definition of a saddlepoint (i*,j*) E SAX SB. (b) Show that the following two statements are equivalent: (A) (i*, j*) ¤ SAX SB is a Nash equilibrium. (B) (i*, j*) ¤ SAX SB is a saddle point. (c) State an example of a 0-sum 2-player normal form game which does not have a saddle point in the pure strategies. State a criteria which ensures that such a game has a saddle point in the pairs of pure strategies.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 42EQ
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Problem 6. Let the 0-sum 2-player normal form game be given by the (m x n)-bimatrix (A, B) {1,2,...,m} and SB = {1,2,...,n} the pure strategies of Alice and Bob, and denote by SA = respectively. (a) State the definition of a maximin strategy i* E SA of Alice (in terms of the matrix A) and the definition of a minimax strategy j E SB of Bob (in terms of the matrix A). Furthermore state the definition of a saddlepoint (i*,j*) € SAX SB. (b) Show that the following two statements are equivalent: (A) (i*, j*) ¤ SA X SB is a Nash equilibrium. (B) (i*, j*) ¤ SA X SB is a saddle point. (c) State an example of a 0-sum 2-player normal form game which does not have a saddle point in the pure strategies. State a criteria which ensures that such a game has a saddle point in the pairs of pure strategies.
Transcribed Image Text:Problem 6. Let the 0-sum 2-player normal form game be given by the (m x n)-bimatrix (A, B) {1,2,...,m} and SB = {1,2,...,n} the pure strategies of Alice and Bob, and denote by SA = respectively. (a) State the definition of a maximin strategy i* E SA of Alice (in terms of the matrix A) and the definition of a minimax strategy j E SB of Bob (in terms of the matrix A). Furthermore state the definition of a saddlepoint (i*,j*) € SAX SB. (b) Show that the following two statements are equivalent: (A) (i*, j*) ¤ SA X SB is a Nash equilibrium. (B) (i*, j*) ¤ SA X SB is a saddle point. (c) State an example of a 0-sum 2-player normal form game which does not have a saddle point in the pure strategies. State a criteria which ensures that such a game has a saddle point in the pairs of pure strategies.
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