Game Theory concepts are discussed including players, strategies, payoff tables, assumptions of rationality. Various game scenarios are presented involving dominance, saddle points, and mixed strategies. Graphical and linear programming solutions are provided for determining optimal strategies in games with mixed strategies where players' expected payoffs differ.

1. Game Theory

3. Pure strategy Row min Column max 1 2 3 4 1 2 3 Player B Player A Maximin = 4 Minimax = 4 Optimal strategy = (2,2)  Saddle point. Value of the game = 4 -3 6 -3 2 6 7 7 4 4 1 8 4 7 6 4 5 8 1 7

5. No dominant strategy Row min Column max 1 2 3 1 2 3 Player A Maximin = 0 Minimax = 0 Optimal strategy = (2,2)  Saddle point. Value of the game = 0  Fair game Player B -4 -4 -2 5 0 -3 6 0 5 2 0 2 6 -2 -3

6. Mixed strategy Row min Column max 1 2 3 1 2 Player A Maximin = 2 Minimax = 5 Player B Maximin ≠ Minimax  No saddle point, unstable solution Maximin ≤ Value of the game ≤ Minimax 11 5 8 2 1 2 5 8 11 3 1

8. Graphical solution -2x 1 +5 2 9x 1 +2 3 -7x 1 +8 1 A’s expected payoff B’ s pure strategy

9. Graphical solution 1 3 2 1/4 3/4 1 1/2 1 2 3 4 5 6 7 8 x 1 A’s expected payoff Maximin x 1 = 3/11, x 2 = 8/11, y 1 =0 Value of the game = 49/11

10. Graphical solution y 2 = 9/11, y 3 = 2/11 3y 2 +2 2 -8y 2 +11 1 A’s expected payoff A’ s pure strategy

11. Graphical solution 1 2 3 1 2 Player A Player B Solve: 2 6 5 3 6 1

12. LP solution 1 (y 1 ) 1 (x 1 ) 2 (x 2 ) Player A Player B m (x m )  2 (y 2 ) n (y n )      a mn a 2n a 1n a m2 a m1 a 22 a 21 a 12 a 11

13. LP solution

14. LP solution

15. 1 2 3 1 2 Player A Player B LP solution Example 2 5 8 11 3 1

16. LP solution