Question

In: Economics

Consider the infinitely repeated game with discount factor E[0,1] of the following variant of the Prisoner's...

Consider the infinitely repeated game with discount factor E[0,1] of the following variant of the Prisoner's Dilemma game:

Player 2
L C R
T (6, 6) (-1, 7) (-2, 8)
Player 1 M (7, -1) (4, 4) (-1, 5)
B (8, -2) (5, -1) (1,1)

A) For which values of the discount factor E[0,1] can the players support the pair of actions (M, C) played in every period?

B) For which values of the discount factor E[0,1] can the players support the pair of actions (T,L) played in every period? Why is your answer different in part A) above?

Solutions

Expert Solution

Assume that both the players follow the grim trigger strategy. This implies punsihment is given forever once any of the player deviates. Hence there are two outcome possible for this subgame: (M, C) in for all periods including the current one or (B, R) in all periods as the punishment is given forever.

a) For the first case, a player’s payoff is 4 + 4? + 4?2+ ... = 4/1?? . If he deviates in first period he will be able to secure 8 in that period by choosing B but will receive only 0 for each period forever when player 2 starts choosing R as a punishment. Hence the payoff is 8 + 0? + 0?2 + ... = 8 . The player has no incentive to deviate if the payoff from not deviating exceed the payoff from deviating:

4/1?? ? 8

? ? 1/2

Hence for ? ? 1/2, players always choose M, C

b) Now we have two outcome possible for this subgame: (T, L) in for all periods including the current one or (B, R) in all periods as the punishment is given forever.

For the first case, a player’s payoff is 6 + 6? + 6?2 + ... = 6/1?? . If he deviates in first period he will be able to secure 8 in that period by choosing B but will receive only 0 for each period forever when player 2 starts choosing R as a punishment. Hence the payoff is 8 + 0? + 0?2 + ... = 8 . The player has no incentive to deviate if the payoff from not deviating exceed the payoff from deviating:

6/1?? ? 8

? ? 1/4

Hence for ? ? 1/4, players always choose (T, C)

The discount factor is reduced here because the total payoff from current strategy is more.

Rahul Sunny answered 2 years ago
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