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8. (15 points) Consider the following model of electoral accountability. There are two periods. In period 1, an incumbent politician I exerts effort .91 that is benecial to a voter V. V observes el. Then there is an election in which either V reelects I, or V replaces I with a challenger C. In period 2, whichever politician wins the election chooses a level of effort e2 and the game ends. In each period the effort level e, is either 0 (\"low\") or 1 (\"high\"). Each politician can be either \"diligent\" d or \"lazy\" I. Each politician is diligent with probability p > 1/2 and lazy with probability 1 p. Each politician's type is her private information. It is drawn at the start of the game and xed for both periods. For a diligent politician, both effort levels cost 0 jollies. For a lazy politician, high effort costs 2 jollies and low effort costs 0 jollies Vot- ers obtain utility each period equal to the politician's effort level that period. The politician in ofce in a period obtains a payoff equal to their effort level from that period, plus a benet R for holding ofce in each period, less the cost of effort for that period. The politician out of ofce in a period obtains 0 payoff in that period. Total payoffs in the game for each player are the sum of payoffs across the periods. Players do not discount future payoffs. (a) What action does the lazy type of either politician take in period 2 in any PBE? What action does the diligent type of either politician take in period 2 in any PBE? (b) Derive conditions on R such that there is a PBE in which both types of the incumbent take high effort in period 1, and V reelects I if and only if I exerts high effort