1. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Supplement 1 Game Theory

2. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Learning Objectives After completing this supplement, 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 the game. Solved mixed strategy games when there is no saddle point.

3. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Supplement Outline S1.1 Introduction S1.2 Language of games S1.3 Minimax Criterion S1.4 Pure strategy games S1.5 Mixed strategy games S1.6 Dominance

4. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Introduction Game: a contest involving two or more decision makers, each of whom wants to win. Game theory: the study of how optimal strategies are formed in conflict Games classified by: Number of players Sum of all payoffs Number of strategies employed Zero-sum game: the sum of the losses must equal the sum of the gains

5. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Payoff Table Y 1 Use radio Y2 Use newspaper X1 Use radio 35 X2 Use newspaper 1-2 Game player Y’s strategies Game Player X’s Strategies + entry, s X wins and Y loses - entry, Y wins and X loses

6. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Outcomes Store X’s Strategy Store Y’s Strategy Outcome % change in market share) X 1 RadioY 1 RadioX wins 3 Y loses 3 X 1 RadioY 2 Newspaper X wins 5 Y loses 5 X 2 Newspaper Y 1 RadioX wins 1 Y loses 1 X 2 Newspaper Y 2 Newspaper X loses 2 Y wins 2

7. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Minimax Criterion In a zero-sum game, each person can choose the strategy that minimizes the maximum loss

8. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Minimax Criterion Y1Y1 Y2Y2 Minimum X1X1 353 X2X2 1-2 Maximum35 Minimum of maximums Maximums of minimums Saddle point Note: an equilibrium or saddle point exists if the upper value of the game is equal to the lower value of the game. This is called the value of the game. This is a “pure strategy” game

9. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Whenever a saddle point is present, the strategy a player should follow will always be the same, regardless of the strategy of the other player.

10. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Pure Strategy Second Player’s (Y) Strategies Y1Y1 Y2Y2 First Player’s (X) Strategies X1X1 35 X2X2 1-2 Y’s pure strategy X’s pure strategy

11. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Pure Strategy Minimax Criterion Player Y’s Strategies Minimum Row Number Y1Y2 Player X’s strategies X11066 X2-122 Maximum Column Number 106

12. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Mixed Strategy Game Y1 P Y2 1-P Expected Gain X1 Q 424P+2(1-P) X2 1-Q 1101P+10(1-p) 4Q+1(1-Q)2Q+10(1-q)

13. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Solving for P & Q 4P+2(1-P) = 1P+10(1-P) or: P = 8/11 and 1-p = 3/11 Expected payoff: EP X =1P+10(1-P) =1(8/11)+10(3/11) = 3.46 4Q+1(1-Q)=2Q+10(1-q) or: Q=9/11 and 1-Q = 2/11 Expected payoff: EP Y =3.46

14. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Dominance A strategy can be eliminated if all its game’s outcomes are the same or worse than the corresponding outcomes of another strategy

15. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Domination Y1Y1 Y2Y2 X1X1 43 X2X2 220 X3X3 11 Initial Game Y1Y1 Y2Y2 X1X1 43 X2X2 220 X3 is a dominated strategy Game after removal of dominated strategy

16. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Domination Y1Y1 Y2Y2 Y3Y3 Y4Y4 X1X1 -546-3 X2X2 -262-20 Initial Game Y1Y1 Y4Y4 X1X1 -5-3 X2X2 -2-20 Game after dominated strategies are removed