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