Consider the following n-player variant of the game of the Prisoners' Dilemma: Each player i chooses...

 

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Consider the following n-player variant of the game of the Prisoners' Dilemma: Each player i chooses from a set of strate- gies A, = {C, D}. Payoffs are given as follows: . If i plays a, = C and k other players (0 ≤ k ≤ n - 1), j, play a, = D. then i receives 3 - (2). n . If i plays a play aj D and exactly k other players (0 ≤ k ≤n-1), j, D, then i receives frame The idea is that a defecting player gets a good deal if they are the only player providing the evidence, but that deal is not as good as more other players also give evidence to the police. Also, the quality of the evidence the Police have to convict players that choose C (or the chance that a Court will use this evidence) is increasing in the number of people that provide the evidence (i.e., the number that choose D). (You can verify that this game is identical to the two-player Prisoners' Dilemma when 7 = 2.) (a) (2 point) Show that the profile where all players cooperate (C..... C) is not a Nash equilibrium for all n. (b) (3 points) Show that the profile where all players defect (D...., D) is a Nash equilibrium for all n. (Hint: Apply the definition.) (e) (6 points) Using the definition of Nash equilibrium, write two inequalities (expressed in terms of k* and n) that are necessary and sufficient for a profile with exactly k (0 <k* <n) players playing D to be a Nash equilibrium. (Hint: Players that play D must not find it profitable to deviate to C; and the same for those playing D. You do not need to solve these inequalities for k* .) (d) (2 points) Based on your answer in part (e), are there Nash equi- 32 libria with only one (k = 1) player playing D when = When n25? Explain. Consider the following n-player variant of the game of the Prisoners' Dilemma: Each player i chooses from a set of strate- gies A, = {C, D}. Payoffs are given as follows: . If i plays a, = C and k other players (0 ≤ k ≤ n - 1), j, play a, = D. then i receives 3 - (2). n . If i plays a play aj D and exactly k other players (0 ≤ k ≤n-1), j, D, then i receives frame The idea is that a defecting player gets a good deal if they are the only player providing the evidence, but that deal is not as good as more other players also give evidence to the police. Also, the quality of the evidence the Police have to convict players that choose C (or the chance that a Court will use this evidence) is increasing in the number of people that provide the evidence (i.e., the number that choose D). (You can verify that this game is identical to the two-player Prisoners' Dilemma when 7 = 2.) (a) (2 point) Show that the profile where all players cooperate (C..... C) is not a Nash equilibrium for all n. (b) (3 points) Show that the profile where all players defect (D...., D) is a Nash equilibrium for all n. (Hint: Apply the definition.) (e) (6 points) Using the definition of Nash equilibrium, write two inequalities (expressed in terms of k* and n) that are necessary and sufficient for a profile with exactly k (0 <k* <n) players playing D to be a Nash equilibrium. (Hint: Players that play D must not find it profitable to deviate to C; and the same for those playing D. You do not need to solve these inequalities for k* .) (d) (2 points) Based on your answer in part (e), are there Nash equi- 32 libria with only one (k = 1) player playing D when = When n25? Explain.

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aThe profile where all players cooperate is not a Nash equilibrium for all n because there is always ... View the full answer