# Consider the Hotelling model of spatial competition under plurality rule

Consider the Hotelling model of spatial competition under plurality rule. Voters’
ideal platforms are distributed across uniformly on the interval [0,1]. Voters always vote
for the closest candidate. Suppose there are three candidates, A, B and C. We saw how,
under the assumption that candidates are “share maximizers”, no Nash equilibrium exists.
Let’s make an alternative assumption: candidates believe that winning is the only thing that
matters and any win is a good win. That is, they prefer to win over lose, but are indifferent
between any two winning outcomes (regardless of the margin of victory). Similarly, they are
indifferent between any two outcomes where they lose.
(i) Suppose candidates A and B are located at the position 1/4 and candidate C is at
position 3/4, as depicted below. fraction of the voters vote for candidates A, B
and C? Who currently wins?
B
1/4
1/2
3/4
(ii) Could candidate C do anything to be better off, holding fixed the platforms of the
other candidates?
3
(iii) Holding fixed the other candidates, what fraction of voters would vote for candidate
A if candidate A moved to the following platforms: slightly to the left of 1/4; slightly
to the right of 1/4; the platform 3/4 (exactly).
(iv) Assuming no other potential moves need to be considered, is this configuration a Nash
equilibrium?
(v) Do you think the assumption we’ve made about candidate objectives is a realistic
assumption? with Parts 1- 3