Question

Consider a population of voters distributed in the ideological space [0,1]. If the voters cast their vote for the candidate with the ideology closest to their own and the candidates simultaneously choose their ideologies, obtain the Nash equilibrium of two candidates who are only interested in winning the elections.
Hint: remember the three elements that characterize a game.

Answer 1

This can be explained with the help " The median Voter theorem". To describe in brief, let's start with ' single peaked preferences'.

Preferences are said to be single-peaked if the alternatives can be
represented as points on a line, and each utility function has a maximum thing at
some point on the line and the slopes away from the maximum on either side.

The situation quoted in question having uncertain preferences will eventually lead us to a nash equilibrium in pure strategies.This is because, as one party moves closer to
the other, it becomes worse off in the event that it wins, but at the same
time it increases its probability of winning the contest. Hence, it potentially
faces a trade-off between its ideology and the electoral success, which results
in an equilibrium where parties’ positions are different.

To be breif and precise, there would be 3 assumptions of the model :-

1. Given the proposal profile, each voter votes for the platform
(party) he likes the most. The party that obtains more than half of
the votes wins the election, and its proposed policy is implemented. Ties
are broken by a random draw, so that each party wins with probability one
half in the case of a tie in votes.

2. Parties believe that fraction of the types supporting is in addition to asumption 1.

3.That is, parties have preferences over policy, but also on the office itself.
one can imagine situations where parties care
more about policy if they win the election than if they lose. In addition,
there might be other motivations to consider. For example, a party may
have preferences over its margin of victory, apart from policies and the
office.