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Supplement 1 Game Theory

Supplement 1 Game Theory. 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.

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Supplement 1 Game Theory

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  1. Supplement 1 Game Theory S-1

  2. 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. S-2

  3. 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 S-3

  4. 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 S-4

  5. Payoff Table Game player Y’s strategies Game Player X’s Strategies + entry, s X wins and Y loses - entry, Y wins and X loses S-5

  6. Outcomes S-6

  7. Minimax Criterion • In a zero-sum game, each person can choose the strategy that minimizes the maximum loss S-7

  8. Minimax Criterion Maximums of minimums Saddle point Minimum of maximums 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 S-8

  9. 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. S-9

  10. Pure Strategy X’s pure strategy Y’s pure strategy S-10

  11. Pure StrategyMinimax Criterion S-11

  12. Mixed Strategy Game S-12

  13. Solving for P & Q 4P+2(1-P) = 1P+10(1-P) or: P = 8/11 and 1-p = 3/11 Expected payoff: EPX=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: EPY=3.46 S-13

  14. Dominance A strategy can be eliminated if all its game’s outcomes are the same or worse than the corresponding outcomes of another strategy S-14

  15. Domination Initial Game X3 is a dominated strategy Game after removal of dominated strategy S-15

  16. Domination Initial Game Game after dominated strategies are removed S-16

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