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ch11_solutions_solved edit

ch11_solutions_solved edit - Solutions to Chapter 11...

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Solutions to Chapter 11 Exercises SOLVED EXERCISES S1. False. The players are not assured that they will reach the cooperative outcome. Rollback reasoning shows that the subgame-perfect equilibrium of a finitely played repeated prisoners’ dilemma will entail constant cheating. S2. (a) The payoffs are ranked as follows: high payoff from cheating (72) > cooperative payoff (64) > defect payoff (57) > low payoff from cooperating (20). This conforms to the pattern in the text so the game is a prisoners’ dilemma, as can also be seen in the payoff table: KID’S KORNER High price Low price CHILD’S PLAY High price 64, 64 20, 72 Low price 72, 20 57, 57 If the game is played once, the Nash equilibrium strategies are (Low, Low) and payoffs are (57, 57). (b) Total profits at the end of four years = 4 57 = 228. Firms know that the game ends in four years so they can look forward to the end of the game and use rollback to find that it’s best to cheat in year 4. Similarly, it is best to cheat in each preceding year as well. It follows that it is not possible to sustain cooperation in the finite game. (c) The one-time gain from defecting = 72 – 64 = 8. Loss in every future period = 64 – 57 = 7. Cheating is beneficial here if the gain exceeds the present discounted value of future losses or if 8 > 7/ r. Thus, r > 7/8 (or d > 8/15) makes cheating worthwhile, and r < 7/8 lets the grim strategy sustain cooperation between the firms in the infinite version of the game. If r = 0.25, cooperation can be sustained. (d) Total profits after four years = 4 64 = 256. With no known end of the world, the firms can sustain cooperation if r < 7/8 as in part c. This answer is different from that in part (b) because the firms see no fixed end point of the game and can’t use backward induction. Instead, they assume the game is infinite and use the grim strategy to sustain cooperative outcome. (e) A 10% probability of bankruptcy translates into a 90% probability that the game continues, so p = 0.9. Then, for r = 0.25 ( d = 0.8), R = 39%. This rate would need to exceed 7/8 before Solutions to Chapter 11 Solved Exercises 1 of 5
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cheating was worthwhile, so the firms will still cooperate in this case. For a 35% probability of
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