19.12 Probabilistic voting Probabilistic voting is a way of modeling the voting process that introduces continuity into individuals’ voting decisions. In this way, calculus-type derivations become possible. To take an especially simple form of this approach, suppose there are n voters and two candidates (labeled A and B) for elective office. Each candidate proposes a platform that promises a net gain or loss to each voter. These platforms are denoted by uA i and uB i , where i ¼ 1, ... , n. The probability that a given voter will vote for candidate A is given by pA i ¼ f ½UiðuA i Þ ' UiðuB i Þ(, where f 0 > 0 > f 00. The probability that the voter will vote for candidate B is pB i ¼ 1 ' pA i .

a. How should each candidate chose his or her platform so as to maximize the probability of winning the election subject to the constraint P i uA i ¼P i uB i ¼ 0? (Do these constraints seem to apply to actual political candidates?)

b. Will there exist a Nash equilibrium in platform strategies for the two candidates?

c. Will the platform adopted by the candidates be socially optimal in the sense of maximizing a utilitarian social welfare?

[Social welfare is given by SW ¼ P i UiðuiÞ.]