2. This is question 7.9 in the Gehlbach book: Consider the following model, based on Myerson (1993), which explores the impact of electoral rules on political selection. In the model, each voter decides which party to support (the parties are not strategic). In this question, you will be given a possible equilibrium and you must verify that it is (or is not) in fact an equilibrium. Actors: There are N 2 4 voters with N even. Strategies: Each voter can choose from four different parties, each of which has a xed policy platform. Parties 1 and 2 are left parties, whereas parties 3 and 4 are right parties. Beyond these policy differences, parties are heterogeneous in their propensity for corruption, where parties 1 and 3 are honest and parties 2 and 4 are corrupt. Voting is by plurality rule, with ties decided by a fair coin toss. Preferences: Assume that % of these actors (left voters) receive a payoff of one if a left party is elected a payoff of zero otherwise, where the other % actors (right voters) receive a payoff of one if a right party is elected and a payoff of zero otherwise. In addition, assume that any voter bears a cost 7 E [0, 1) if a corrupt party is elected. (a) Show that it is a Nash equilibrium for all left voters to vote for party 1 and all right voters to vote for party 3. [1 point] (b) Show that it is a Nash equilibrium for all left voters to vote for party 2 and all right voters to vote for party 4. [1 point] (c) Now imagine that 'y > 1. Is there still an equilibrium where all left voters vote for party 2 and all right voters vote for party 4? [ungraded] Go back to assuming that 'y E [0,1). Now also assume that voting is by proportional representation, where any party receives seats in parliament in proportion to its share of votes in the election. Voters receive payoffs in proportion to the seat shares of parties elected, given the preferences previously assumed. Thus, for example, if party 1 received one-fourth of the seats party 4 received three-fourths, left voters would receive a payoff of 1 E 'y, whereas right voters would receive a payoff of g 3'7. ((1) Show that it is a Nash equilibrium for all left voters to vote for party 1 and all right voters to vote for party 3. [1 point] (e) Is there any other Nash equilibrium? Why or why not? What do the results say about the effectiveness of electoral rules in preventing corruption? [1 point]