1 Probabilistic voting Probabilistic voting is a way of modelling the voting process that introduces continuity into individuals’ voting decisions. In this way, calculus-type derivations become...

1 Probabilistic voting Probabilistic voting is a way of modelling 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 (labelled
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 θ
A i
and θ
B i
, where
i
= 1, … ,
n. The probability that a given voter will vote for candidate
A
is given by π
A i
=
f
[Ui(θ
A i
) −
Ui(θ
B i
)], where
f
ʹ > 0 >
f
ʹʹ. The probability that the voter will vote for candidate
B
is π
B i
= 1 − π
A i
.

a. How should each candidate choose his or her platform so as to maximise the probability of winning the election subject to the constraint Θi B
= 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 maximising a utilitarian social welfare? [Social welfare is given by
SW
= ai Ui(θi).]