1. Game Theory Warin Chotekorakul MD 1/2004

2. Introduction A game is a contest involving to or more players, each of whom wants to win. Game theory is the study of how optimal strategies are formulated in conflict. Game theory is one way to consider the impact of the strategies of others on our strategies and outcomes.

3. What type of Games we focus? Two-person, zero-sum games

4. EXAMPLE StrategyY1 (Use Radio) Y2(Use newspaper) X1 (Use Radio) 35 X2 (Use Newspaper) 1-2 Game Player Y’s Strategies Game Player X’s Strategies

5. Minimax The Minimax Criterion A player using the minimax criterion will select the strategy that minimizes the maximum possible loss. A player using the minimax criterion will select the strategy that minimizes the maximum possible loss. 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. saddle point Pure Strategy An equilibrium or saddle point condition 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. Hence, Pure Strategy is present.

6. Pure Pure Strategy Games A pure strategy exists whenever a saddle point is present. a pure strategy will always be the same regardless of the other player’s strategy. When it is a pure strategy, the strategy each player should follow will always be the same regardless of the other player’s strategy.

7. Mixed Mixed Strategy Games When there is no saddle point, a mixed strategy game exists. For mixed strategies, players will play each strategy for a certain percentage of the time. no matter what the other player’s strategy happens to be. Each player desires a strategy that will result n the most winnings no matter what the other player’s strategy happens to be.

8. EXAMPLE StrategyY1 (Q) Y2 (1-Q) X1 (P)42 X2 (1-P)110 Game Player Y’s Strategies Game Player X’s Strategies

9. Mixed Strategies Let  P, 1-P = fraction of the time X plays strategies X1 and X2 respectively  Q, 1-Q = fraction of the time Y plays strategies Y1 and Y2 respectively Game Value = X1,Y1Payoff(P)(Q)+X1,Y2Payoff(P)(1- Q)+X2Y1Payoff(1-P)(Q)+X2Y2Payoff(1-P)(1-Q)

10. Dominance reduce the size Dominance can be used to reduce the size of games by eliminating strategies that would never be played A strategy for a player is said to be dominated if the player can always do as well or better playing another strategy. A strategy for a player is said to be dominated if the player can always do as well or better playing another strategy.

11. Examples StrategyY1Y2 X143 X2520 X32

12. Examples(2) StrategyY1Y2Y3Y4 X1-62-75 X2-34-123

13. Examples (3) StrategyY1Y2Y3 X14-35 X2-47-2 X313 X4052