1. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-1 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Module 4 Game Theory Prepared by Lee Revere and John Large

2. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-2 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Learning Objectives Students will be able to: 1.Understand the principles of zero-sum, two-person games. 2.Analyze pure strategy games and use dominance to reduce the size of the game. 3.Solve mixed strategy games when there is no saddle point.

3. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-3 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Module Outline M4.1 Introduction M4.2 Language of Games M4.3 The Minimax Criterion M4.4 Pure Strategy Games M4.5 Mixed Strategy Games M4.6 Dominance

4. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-4 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Game theory is the study of how optimal strategies are formulated in conflict. Game theory has been effectively used for:  War strategies  Union negotiators  Competitive business strategies Introduction

5. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-5 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458  Game models are classified by the number of players, the sum of all payoffs, and the number of strategies employed.  A zero sum game implies that what is gained by one player is lost for the other. Introduction (continued)

6. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-6 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Language of Games Consider a duopoly competitive business market in which one company is considering advertising in hopes of luring customers away from its competitor. The company is considering radio and/or newspaper advertisements. Let’s use game theory to determine the best strategy.

7. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-7 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Language of Games (continued) STORE X’s PAYOFFs Y’s strategy 1 (use radio) Y’s strategy 2 (use newspaper) X’s strategy 1 (use radio) 35 X’s strategy 2 (use newspaper) 1-2 Below is the payoff matrix (as a percent of change in market share) for Store X. A positive number means that X wins and Y loses, while a negative number implies Y wins and X loses. Note: Although X is considering the advertisements (therefore the results favor X), Y must play the game.

8. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-8 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Language of Games (continued) Store X’s Strategy Stores Y’s Strategy Outcome (% change in market share) X1: RadioY1: RadioX wins 3 Y loses 3 X1: RadioY2: Newspaper X wins 5 Y loses 5 X2: Newspaper Y1: RadioX wins 1 Y loses 1 X2: Newspaper Y2: Newspaper X loses 2 Y wins 2 Note: Although X is considering the advertisements (therefore the results favor X), Y must play the game.

9. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-9 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Minimax Criterion The minimax criterion is used in a two-person zero-sum game. Each person should choose the strategy that minimizes the maximum loss. Note: This is identical to maximizing one’s minimum gains.

10. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-10 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Minimax Criterion (continued) The upper value of the game is equal to the minimum of the maximum values in the columns. The lower value of the game is equal to the maximum of the minimum values in the rows.

11. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-11 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum X1 (radio) 353 X2 (newspaper) 1-22 Maximum35 The Minimax Criterion (continued) Lower Value of the Game: Maximum of the minimums Upper Value of the Game: Minimum of the maximums

12. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-12 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum X1 (radio) 353 X2 (newspaper) 1-22 Maximum35 The Minimax Criterion (continued) Saddle point: Both upper and lower values are 3. A saddle point condition exists if the upper and lower values are equal. This is called a pure strategy because both players will follow the same strategy.

13. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-13 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum X1 (radio) 106 6 X2 (newspaper) -122 Maximum10 6 The Minimax Criterion (continued) Saddle point Let’s look at a second example of a pure strategy game. Lower value Upper value

14. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-14 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Mixed Strategy Game A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy. Note: The expected gain is determined using an approach very similar to the expected monetary value approach.

15. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-15 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Mixed Strategy Games (continued) Y1 (P) Y2 (1-P) Expected Gain X1 (Q) 424P + 2(1-P) X2 (1-Q) 1101p + 10(1-P) Expected Gain 4Q + 1(1-Q) 2Q + 10(1-Q) Each player seeks to maximize his/her expected gain by altering the percent of time (P or Q) that he/she use each strategy. Set these two equations equal to each other and solve for Q Set these two equations equal to each other and solve for P

16. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-16 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Mixed Strategy Games (continued)  4P + 2(1-P) = 1P + 10(1-P) 4P – 2P – 1P + 10P = 10 – 2 P = 8/11 and 1-P = 3/11 Expected payoff: 1P + 10(1-P) = 1(8/11) + 10(3/11) = 3.46  4Q + 1(1-Q) = 2Q + 10(1-Q) 4Q – 1Q – 2Q + 10Q = 10 – 1 Q = 9/11 and 1-Q = 2/11 Expected payoff: 2Q + 10(1-Q) = 2(9/11) + 10(2/11) = 3.46

17. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-17 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Dominance Dominance is a principle that can be used to reduce the size of games by eliminating strategies that would never be played. Note: A strategy can be eliminated if all its game’s outcomes are the same or worse than the corresponding outcomes of another strategy.

18. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-18 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Dominance (continued) Y1Y2 X143 X2220 X311 Y1Y2 X143 X2220 Initial game X3 is a dominated strategy Game after removal of dominated strategy

19. To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-19 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Dominance (continued) Y1Y2Y3Y4 X1-546-3 X2-262-20 Initial game Game after removal of dominated strategies Y1Y4 X1-5-3 X2-2-20