Column Defect Cooperate Defect 1,1 6,0 Row Cooperate 0,6 2,2 Suppose that this stage game is repeated infinite number of times. The payoff of each player from this repeated game is the discounted sum of his payoffs from each period, with discount factor SЄ (0,1). Recall the definitions of three strategies: Always Cooperate, Always Defect, and Grim Trigger. • • Always Cooperate: Cooperate in every period, regardless of what happened before. Always Defect: Defect in every period, regardless of what happened before. • Grim Trigger: In period 1, Cooperate. In period t, Cooperate if no player defected in periods 1 through t - 1; Defect if any player (either the other player or himself or both) defected in at least one of the previous periods. Use the one-shot deviation principle to answer the following questions. (a) Can (Always Cooperate, Always Cooperate) be an SPE? If so, for what values of the discount factor is it an SPE? If not, explain why not. (b) Can (Always Defect, Always Defect) be an SPE? If so, for what values of the discount factor is it an SPE? If not, explain why not. (c) Can (Grim Trigger, Grim Trigger) be an SPE? If so, for what values of the discount factor is it an SPE? If not, explain why not. Suppose now that the same game is played indefinitely, not infinitely. That is, at the end of each period, the game continues to the next period with probability p, while the game ends forever with probability 1-p. If the game ends, then from the next period on both players get a payoff of 0. As before, each player discounts future payoffs with discount factor = (0,1). (d) Suppose that p=0.9. Do you answers to (a), (b), and (c) change? If so, how?
Column Defect Cooperate Defect 1,1 6,0 Row Cooperate 0,6 2,2 Suppose that this stage game is repeated infinite number of times. The payoff of each player from this repeated game is the discounted sum of his payoffs from each period, with discount factor SЄ (0,1). Recall the definitions of three strategies: Always Cooperate, Always Defect, and Grim Trigger. • • Always Cooperate: Cooperate in every period, regardless of what happened before. Always Defect: Defect in every period, regardless of what happened before. • Grim Trigger: In period 1, Cooperate. In period t, Cooperate if no player defected in periods 1 through t - 1; Defect if any player (either the other player or himself or both) defected in at least one of the previous periods. Use the one-shot deviation principle to answer the following questions. (a) Can (Always Cooperate, Always Cooperate) be an SPE? If so, for what values of the discount factor is it an SPE? If not, explain why not. (b) Can (Always Defect, Always Defect) be an SPE? If so, for what values of the discount factor is it an SPE? If not, explain why not. (c) Can (Grim Trigger, Grim Trigger) be an SPE? If so, for what values of the discount factor is it an SPE? If not, explain why not. Suppose now that the same game is played indefinitely, not infinitely. That is, at the end of each period, the game continues to the next period with probability p, while the game ends forever with probability 1-p. If the game ends, then from the next period on both players get a payoff of 0. As before, each player discounts future payoffs with discount factor = (0,1). (d) Suppose that p=0.9. Do you answers to (a), (b), and (c) change? If so, how?
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.9P
Related questions
Question
Transcribed Image Text:Column
Defect Cooperate
Defect
1,1
6,0
Row
Cooperate 0,6
2,2
Suppose that this stage game is repeated infinite number of times. The payoff of each player from
this repeated game is the discounted sum of his payoffs from each period, with discount factor
SЄ (0,1).
Recall the definitions of three strategies: Always Cooperate, Always Defect, and Grim Trigger.
•
•
Always Cooperate: Cooperate in every period, regardless of what happened before.
Always Defect: Defect in every period, regardless of what happened before.
• Grim Trigger: In period 1, Cooperate. In period t, Cooperate if no player defected in
periods 1 through t - 1; Defect if any player (either the other player or himself or both)
defected in at least one of the previous periods.
Use the one-shot deviation principle to answer the following questions.
(a) Can (Always Cooperate, Always Cooperate) be an SPE? If so, for what values of the discount
factor is it an SPE? If not, explain why not.
(b) Can (Always Defect, Always Defect) be an SPE? If so, for what values of the discount factor
is it an SPE? If not, explain why not.
(c) Can (Grim Trigger, Grim Trigger) be an SPE? If so, for what values of the discount factor
is it an SPE? If not, explain why not.
Suppose now that the same game is played indefinitely, not infinitely. That is, at the end of each
period, the game continues to the next period with probability p, while the game ends forever with
probability 1-p. If the game ends, then from the next period on both players get a payoff of 0.
As before, each player discounts future payoffs with discount factor = (0,1).
(d) Suppose that p=0.9. Do you answers to (a), (b), and (c) change? If so, how?
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