Assignment4_MATH4321_12S

# Assignment4_MATH4321_12S - MATH 310 Game Theory Spring 2012...

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Game Theory Spring 2012 Assignment 4 1. Problems II.1.5.1, II.1.5.2, II.1.5.3 2. Problems II.2.6.2, II.2.6.4, II.2.6.5, II.2.6.6, II.2.6.7, II.2.6.8, II.2.6.10 3. Prove that for an mxn matrix game, any two saddle points have the same value. Mission : Proof of the Minimax Theorem Let A be the payoff matrix of a 0-sum 2-person game. Let V ( A ) denote the upper value and V ( A ) denote the lower value of game with payoff matrix A . We have proved that V ( A ) ³ V ( A ) . Minimax Theorem (1928, John von Neumann): V ( A ) = V ( A ) Follow the following steps to prove the Minimax Theorem. Definition: gap( A )= V ( A ) - V ( A ) . It is clear that gap( A )is nonnegative. Theorem: (Minimax Theorem) gap( A )=0. Lemma: If A has more than one element, then there exists a row or a column such that gap( A Ù ) ³ gap( A ), where A Ù is obtained from A by deleting the given row or column. We break down the proof of the Lemma into steps.

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## This note was uploaded on 03/13/2012 for the course MATH 4321 taught by Professor Cheng during the Spring '12 term at HKUST.

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Assignment4_MATH4321_12S - MATH 310 Game Theory Spring 2012...

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