4 Game Theory To accompany Quantitative Analysis for Management, Twelfth Edition, by Render, Stair, Hanna and Hale Power Point slides created by Jeff Heyl.

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4 Game Theory To accompany Quantitative Analysis for Management, Twelfth Edition, by Render, Stair, Hanna and Hale Power Point slides created by Jeff Heyl Copyright ©2015 Pearson Education, Inc.

LEARNING OBJECTIVES After completing this module, students will be able to: Understand the principles of zero-sum, two-person games. Analyze pure strategy games and use dominance to reduce the size of a game. Solve mixed strategy games. Copyright ©2015 Pearson Education, Inc.

MODULE OUTLINE M4.1 Introduction M4.2 Language of Games M4.4 Pure Strategy Games M4.5 Mixed Strategy Games M4.6 Dominance Copyright ©2015 Pearson Education, Inc.

Introduction Game theory is one way to consider the impact of the strategies of others on our strategies and outcomes A game is a contest involving two or more decision makers, each of whom wants to win Game theory is the study of how optimal strategies are formulated in conflict Game models classified by number of players, sum of all payoffs, and number of strategies employed Two-person and zero-sum games Copyright ©2015 Pearson Education, Inc.

GAME PLAYER Y’s STRATEGIES Language of Games Two lighting fixture stores A duopoly Advertising strategies change Table M4.1 – Store X’s Payoff Matrix GAME PLAYER Y’s STRATEGIES Y1 (Use radio) Y2 (Use newspaper) GAME PLAYER X’s STRATEGIES X1 3 5 X2 1 –2 Copyright ©2015 Pearson Education, Inc.

Language of Games Game outcomes STORE X’s STRATEGY STORE Y’s OUTCOME (% CHANGE IN MARKET SHARE) X1 (use radio) Y1 (use radio) X wins 3 and Y loses 3 Y2 (use newspaper) X wins 5 and Y loses 5 X2 (use newspaper) X wins 1 and Y loses 1 X loses 2 and Y wins 2 Copyright ©2015 Pearson Education, Inc.

Dominance The principle of dominance can be used to reduce the size of games by eliminating strategies that would never be played Copyright ©2015 Pearson Education, Inc.

Dominance Reduce the size of this game Can be reduced to Y1 Y2 X1 4 3 20 X3 1 Can be reduced to Y1 Y2 X1 4 3 X2 2 20 Copyright ©2015 Pearson Education, Inc.

Dominance Reduce the size of this game Can be reduced to Y1 Y2 Y3 Y4 X1 –5 4 6 –3 X2 –2 2 –20 Can be reduced to Y1 Y4 X1 –5 –3 X2 –2 –20 Copyright ©2015 Pearson Education, Inc.

Mixed Strategy Games Players will play each strategy for a certain percentage of the time Called a mixed strategy game Commonly solved using the expected gain or loss approach Each player plays a strategy a particular percentage of the time so that the expected value of the game does not depend upon what the opponent does Copyright ©2015 Pearson Education, Inc.

Mixed Strategy Games For Player Y solve 4P + 2(1 – P) = 1P + 10(1 – P) Table M4.4 – Game Table for Mixed Strategy Game PLAYER Y’s STRATEGIES Y1 Y2 PLAYER X’s STRATEGIES X1 4 2 X2 1 10 For Player Y solve 4P + 2(1 – P) = 1P + 10(1 – P) P = 8/11 1 – P = 1 – 8/11 = 3/11 Copyright ©2015 Pearson Education, Inc.

Mixed Strategy Games For Player X solve 4Q + 1(1 – Q) = 2Q + 10(1 – Q) Table M4.4 – Game Table for Mixed Strategy Game PLAYER Y’s STRATEGIES Y1 Y2 PLAYER X’s STRATEGIES X1 4 2 X2 1 10 For Player X solve 4Q + 1(1 – Q) = 2Q + 10(1 – Q) Q = 9/11 1 – Q = 2/11 Copyright ©2015 Pearson Education, Inc.

Mixed Strategy Games Y1 Y2 P 1 – P Expected gain X1 Q 4 2 Table M4.5 – Game Table for Mixed Strategy Game with Percentages (P, Q) Shown Y1 Y2 P 1 – P Expected gain X1 Q 4 2 4P + 2(1 – P) X2 1 – Q 1 10 1P + 10(1 – P) 4Q + 1(1 – Q) 2Q + 10(1 – Q) Copyright ©2015 Pearson Education, Inc.