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4 Problem 3 Consider that two political parties compete each other by proposing policies respectively, in order to win the national election. There are two political parties L and R, where L is the Left-wing party and R is the Right-wing party, each of which would propose a policy selected from a policy space T in order to maximize the probability of winning the election, given the opponent party's policy proposal. To simplify the argument, let us assume that T = [0, 1). That is, the range of tax rates also represents the policy space of this society over which the two parties can select their own policy proposals. Let N be the set of voters. Each voter i E N has his/her own preference on alternative policy proposals, which is represented by a utility function u; : T - R which is defined as: for any te T, wi (t ) = - (t -1)2, where t' is the ideal policy for voter i. The ideal policy is to give this voter the maximal level of utility if it is implemented. Moreover, the longer the distance between the policy t and his ideal policy ' is, the lower the utility of this voter characterized this ideal policy t' is. Let the distribution of voters' types in this society be characterized by a cumulative distribution function F : T - [0, 1]: for each t E T, F (t) = Vi. Note that F (t) represents the percentage of the voters whose ideal tax rates { } are less than the tax rate t. Given the environment (N;T; F; (u;)ien), let us define the probability of wining the election for party L. Let to be a policy proposed by L in the national election, and to be a policy proposed by R in the national election. Then, let I(tz ,tR ) = Late. Let the percentage of the voters who prefer to to to in this society be denoted by P(tz , tR ). Then, given the feature of the environment, P(tz , tR ) is specified as follows: F (I(tz ,tR )) if tL 0.5 II (tL , tR ) if P(tL , tR ) = 0.5 if PP(tL , tR )