Question

Consider the following game: C D C (2,1) (-1,4) D (3,-3) (0,0) Assume that this game is repeated an infinite number of times, and that both the row and column player discount the future with the same discount factor δ. First, suppose that both players follow the following grim-trigger strategy: "play c as long as no one has ever played d; otherwise play d". (a) Find the minimum value of δ such that this is a subgame-perfect equilibrium. Now suppose that the players follow a strategy in which they both start in "phase C" (described below) and then switch "phases" as the instructions dictate: Phase C: Play c provided that, in each period since the (re)start of the phase, either both players chose c or both players chose d. Stay in phase C until one player chooses d while the other chooses c. In this event, if it is row who chose d, go to phase Pr; if it is column who chose d, go to phase Pc. Phase Pr: Play d for Tr periods (regardless of what happens during those periods), then revert to phase C. Phase Pc: Play d for Tc periods (regardless of what happens during those periods), then revert to phase C. (b) Write down expressions in terms of δ for the smallest Tr and Tc such that this is an SPE. [These expressions might be quite ugly, so do not attempt to simplify them much.] For any given δ, which of Tr and Tc is larger? Why? (c) Suppose that row and column are playing some SPE equilibrium of the infinitely repeated game. They may or may not have played according to the equilibrium strategies so far. Let Vr and Vc denote the present discounted values of continuing to play from here on according to the equilibrium strategies. What are the lowest values that Vr and Vc could have? [Not as hard as it looks. Hint: You don't need any calculation to answer this.]

          Consider the following game:

C
D
C
(2,1)
(-1,4)
D
(3,-3)
(0,0)

Assume that this game is repeated an infinite number of times, and that both the row and column player discount the future with the same discount factor δ.

First, suppose that both players follow the following grim-trigger strategy: "play c as long as no one has ever played d; otherwise play d".

(a) Find the minimum value of δ such that this is a subgame-perfect equilibrium.

Now suppose that the players follow a strategy in which they both start in "phase C" (described below) and then switch "phases" as the instructions dictate:

Phase C: Play c provided that, in each period since the (re)start of the phase, either both players chose c or both players chose d. Stay in phase C until one player chooses d while the other chooses c. In this event, if it is row who chose d, go to phase Pr; if it is column who chose d, go to phase Pc.

Phase Pr: Play d for Tr periods (regardless of what happens during those periods), then revert to phase C.

Phase Pc: Play d for Tc periods (regardless of what happens during those periods), then revert to phase C.

(b) Write down expressions in terms of δ for the smallest Tr and Tc such that this is an SPE. [These expressions might be quite ugly, so do not attempt to simplify them much.] For any given δ, which of Tr and Tc is larger? Why?

(c) Suppose that row and column are playing some SPE equilibrium of the infinitely repeated game. They may or may not have played according to the equilibrium strategies so far. Let Vr and Vc denote the present discounted values of continuing to play from here on according to the equilibrium strategies. What are the lowest values that Vr and Vc could have? [Not as hard as it looks. Hint: You don't need any calculation to answer this.]

Added by Brett S.

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