Module 4 Game Theory

1. Module 4 Game Theory Prepared by Lee Revere and John Large M4-1

2. Learning Objectives 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. • Solve mixed strategy games when there is no saddle point. M4-2

3. 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 M4-3

4. Introduction • 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 M4-4

5. Introduction(continued) • 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. M4-5

6. 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. M4-6

7. Language of Games (continued) 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. M4-7

8. Language of Games (continued) Note: Although X is considering the advertisements (therefore the results favor X), Y must play the game. M4-8

9. 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. M4-9

10. 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. M4-10

11. The Minimax Criterion (continued) Lower Value of the Game: Maximum of the minimums Upper Value of the Game: Minimum of the maximums M4-11

12. The Minimax Criterion (continued) 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. Saddle point: Both upper and lower values are 3. M4-12

13. The Minimax Criterion (continued) Let’s look at a second example of a pure strategy game. Lower value Saddle point Upper value M4-13

14. 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. M4-14

15. Mixed Strategy Games (continued) 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 P Set these two equations equal to each other and solve for Q M4-15

16. Mixed Strategy Games (continued) • 4P + 2(1-P) = 1P + 10(1-P)4P – 2P – 1P + 10P = 10 – 2P = 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 – 1Q = 9/11 and 1-Q = 2/11 Expected payoff: 2Q + 10(1-Q) = 2(9/11) + 10(2/11) = 3.46 M4-16

17. 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. M4-17

18. Dominance (continued) Initial game X3 is a dominated strategy Game after removal of dominated strategy M4-18

19. Dominance (continued) Initial game Game after removal of dominated strategies M4-19