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Answers to Odd Exercises for Section 9.5 1. A3 dominates A1, B3 dominates B1 3. A1 dominates A2 B3 dominates B4 , B2 dominates B4 5. B2 dominates B3, then A1 dominates A2 and A3, then B2 dominates B1. A will play A1 and B will play B2 with a payoff of -2 to A. 7. A1 dominates A2 and B1 dominates B3. No final solution. 9. A Nash equilibrium exists. A would choose A1 and B would choose B2, with a payoff of -2 to A. 11. A Nash equilibrium exists. A will play A2 and B will play B2 for a payoff of 1 to A. 1 1 2 1 13. P( X 5) 0.333... 33% 6 6 6 3 15. The expected value of X E ( X ) x1 P( X x1 ) x2 P( X x2 ) ...xn P( X xn ) n 1 1 1 1 1 1 1 2 3 4 5 6 21 1 xi P( X xi ) 1 2 3 4 5 6 3 6 6 6 6 6 6 6 6 6 6 6 6 6 2 i 1 17. Max v Subject to: v 4 x1 1x2 1x3 v 3x1 0 x2 2 x3 v 1x1 3x2 1x1 x1 x2 x3 1 x1 , x2 , x3 0 19. Max v v Subject to: 2 x1 1x2 4 x3 v v 0 2 x1 3 x2 2 x3 v v 0 4 x1 5 x2 5 x3 v v 0 x1 x2 x3 1 x1 , x2 , x3 , v , v 0 21. Max v Subject to: v -2 x1 4 x2 2 x3 v 1x1 1x2 3 x3 v 5 x1 3 x2 4 x3 x1 x2 x3 1 x1 , x2 , x3 0 A should play A2 63.6% of the time and A3 36.4% of the time for an expected payoff of 0.455 . 23. Max v v Subject to: 1x1 1x2 3 x3 v v 0 2 x1 1x2 0 x3 v v 0 0 x1 0 x2 2 x3 v v 0 x1 x2 x3 1 x1 , x2 , x3 , v , v 0 A should play A1 and A2 each 50% of the time for an estimated payoff to A of 0. 25. a) Yes; A1 is dominated by A2 . b) No; there is no Nash equilibrium. c) Max v Subject to: v 1x1 6 x2 5 x3 v 2 x1 8x2 15x3 v 6 x1 2 x2 7 x1 x1 x2 x3 1 x1 , x2 , x3 0 You should play man-to-man about 68% (roughly 2/3) of the time, and double-team their star player about 32% (roughly 1/3) of the time, for an expected savings of about 0.84 points (around one point). 27. a) B1 is dominated by B2 , A1 is dominated by A2 . b) Yes; A plays A2 and B plays B2 . c) Yes; A plays A2 and B plays B2 . 29. a) Sbtract 0.50 from all of the entries in the matrix to get the zero-sum payoff matrix, so that they correspond to the amount that the final Pay increase is over $0.50 . b) No; there are no dominated strategies. c) No; there are no pure strategy Nash equilibria. d) Max v Subject to: v 0.50 x1 0.25 x2 v 0.10 x1 0.25 x3 v 0.05 x2 0.50 x1 x1 x2 x3 1 x1 , x2 , x3 0 The Union should propose $0.00 with a probability of 29.4% , $0.50 with a probability of 62.5% , and $1.00 with a probability of 8.1% . The shadow prices indicate that Management should propose $0.00 with a probability of 9.9% , $0.50 with a probability of 58.8% , and $1.00 with a probability of 31.3% . e) Min v v Subject to: 0.50 y1 0.10 y2 v v 0 0.25y1 0.05 y3 v v 0 0.25 y2 0.50 y3 v v 0 y1 y2 y3 1 y1 , y2 , y1 , y4 , v , v 0 The solutions come out the same as in part (d). 31. a) y1 = 0.5 , y2 = 0.5 , ν = -1 b) Min v v Subject to: y1 3 y2 v v 0 -4y1 2 y2 v v 0 y1 y2 1 y1 , y2 , v , v 0 The solution comes out the same as in part (a), with ν+ = 0 and ν- = 1 , so that ν = ν+ - ν- = 0 – 1 = -1 .