Question

1. Is the Grim Strategy played by each an equilibrium in the following PD game, with discount rate for both .7?

       Column   C       D

       Row C   5,5    1,6

                 D   6,1    2,2

2. In the last question, what is the minimum discount rate for which (Grim, Grim) is an equilibrium?

Answer 1

There are two outcome possible for this subgame: (C, C) in for all periods including the current one or (D, D) in all periods as the punishment is given forever. For the first case, a player’s payoff is 5/1-δ for infinite period.  

If he deviates in first period he will be able to secure 6 in that period but will receive only 2 for each period forever. Hence the payoff is 6 + 2δ + 2δ² + ... = 6 + 2δ/(1 - δ) . The player has no incentive to deviate if the payoff from not deviating exceed the payoff from deviating:

5/1-δ ≥ 6 + 2δ/(1 - δ)

5 - 2δ/1-δ ≥ 6

5 - 2δ ≥ 6 – 6δ

4δ ≥ 1

δ ≥ 0.25

Hence 0.7 can be a Grim strategy for player 1 Game is symmetric so it is also true for player

2) The minimum discount rate is 0.25.