(25 points) A legislature consists of two political parties,LandS, who play an infinitely repeated game. In each round, a policy proposal,x[0,1] is put forward. Both parties simultaneously vote whether to approve the proposal. If both parties approve, then the proposal passes. If at least one party rejects, then the proposal fails. If the proposal passes, the parties payoffs areuL=xanduS= 1x. Thus, partyLbenefits more from "larger" policies, and partySbenefits more from "smaller" policies. If the proposal fails, then both parties obtain a payoff of 1/4, associated with the current "default" policy. Thus, the stage game between the parties is represented by the following matrix

Party S

Approve Reject

Approvex,1x14,14Reject1,1 1,1

Consider one round of the stage game (no repetition). The following parts are worth 2.5 points each.

1. (a)Supposex[0,1]. Is it a Nash equilibrium for both parties to reject? Explain in a sentence or two.
2. (b)Supposex(1/4,3/4). Is it a Nash equilibrium for both parties to approve? Explain in a sentence or two.
3. (c)Supposex(0,1/4). Is it a Nash equilibrium for both parties to approve? Explain in a sentence or two.
4. (d)Supposex(3/4,1). Is it a Nash equilibrium for both parties to approve? Explain in a sentence or two.

Consider an infinitely repeated version of the previous game, with discount factor(0,1). Suppose that at the beginning of periodk, the policy to be considered in that period, denotedxk, is drawn (independently) from a uniform distribution on [0,1] (i.e. Pr(xky) =yfory[0,1]). Policyxkis observed by both parties in periodk, but the parties do not know which policies will be considered in the future. Thus, the parties expect that on average, the expected value of future policies is 1/2.

We will analyze whether the following strategies are a Subgame Perfect Nash equilib- rium of the infinitely repeated game.

• Approve in period 1 for allx1[0,1].
• Approve in periodkfor allxk[0,1], as long as in all past periods the outcome
• was (Approve, Approve).
• Reject in periodkif in any past period a policy was Rejected.
• We will focus on the decision of partyL(partySis symmetric). The following are worth 2.5 points each.

1. (e)Suppose that at some point in the past, a policy was rejected. In this case, the strategy specifies that both parties reject in every subsequent period. Is this credible? (Hint: see part (a))
2. (f)Suppose that in all past periods the outcome was (Approve, Approve). Given that today's policy proposal isx, what is partyL's lifetime payoff of approving? (Hint:Lgetsxtoday, and in all future periods, it expectsE[x] = 1/2).
3. (g)Suppose that in all past periods the outcome was (Approve, Approve). Given that today's policy proposal isx, what is partyL's lifetime payoff of rejecting? (Hint:Lgets 1/4 today and in all future periods).
4. (h)For which values ofis the proposed strategy a Subgame Perfect Nash equilib- rium? (Hint: it is most-tempting for partyLto reject in periodkifxk= 0)