Infinitely Repeated Games Practice

Consider the following normal-form game repeated infinitely many times in an infinitely repeated game.

Assume that x > 12. Assume that Alice and Bob share a discount factor [0, 1).

1. Identify the pure strategy stage-game Nash equilibriums.

2. Answer the following three questions

a. Is the strategy profile Play UL in each round no matter what, a subgame perfect equilibrium in the repeated game? Why?

b. Is the strategy profile Play MC in each round no matter what, a subgame perfect equilibrium in the repeated game? Why?

c. Is the strategy profile Play DR in each round no matter what, a subgame perfect equilibrium in the repeated game? Why?

3. Assume that x = 16. For each strategy profile below, determine the range of values of for which the strategy profile is a subgame perfect equilibrium.

a.

Play MC if nobody has ever deviated from playing MC

Play UL if anybody has ever deviated from playing MC

b.

Play MC if nobody has ever deviated from playing MC

Play DR if anybody has ever deviated from playing MC

c.

Play MC if nobody has ever deviated from playing MC

Alternate between M L and UC in each round forever if anybody has ever deviated from playing MC, beginning with M L

Play UL forever if anybody has ever deviated from alternating M L and UC following a deviation from playing MC

4. Compare your results in parts (3a.), (3b.) and (3c.) above, and explain why the values of that you found for one part differ from the values of that you found for the other parts. Under which strategy profile is it easier to enforce playing MC?

5. Assume that = 0.9. For which values of x > 12 are the strategy profiles described in parts (3a.) and (3b.) subgame perfect equilibriums? Explain why the values of x that you find for one strategy profile differ from the values of x that you find for the other strategy profile.

Bob R U 1,1 0,8| 12,12 | -5, D 0,2| , -5 8,0 2,0 Alice 5,5 Bob R U 1,1 0,8| 12,12 | -5, D 0,2| , -5 8,0 2,0 Alice 5,5