Question

Suppose voters are uniformly distributed along a continuum between 0 and 1. There are two candidates. Voters will vote for the candidate who locates closes to them. Candidates only care about receiving more votes than the other candidate (and prefer a tie to losing).What is the rationalizable set of locations for each candidate?

Answer 1

Both the candidates will choose to locate at the median of the continuum of 0 and 1 i.e at 0.5 and end up in a tie.This is a strictly dominating strategy for both the politicians. Thus the set of rationalizable locations is (0.5, 0.5)

For example, politician 1 considers shifting to 0.4 when politician 2 is at 0.5. Not the politician 2 will get all the voters that are located above 0.5. Further, he will also get voters located > 0.45 as he is closer to those voters than politician 1. Thus politician 1 gets lesser votes and he will stay at 0.5 where he ties the election rather than losing. This is the case for all points below 0.5. Similarly, consider the politician 1 shifting to a point 0.6. Now the politician 2 will get all votes less than 0.5 and further will get votes < 0.55. Thus, Thus politician 1 gets lesser votes and he will stay at 0.5. This is the case fo all points above 0.5. Thus, staying at 0.5 is a strictly dominating strategy for player 1

The exact same reasoning makes locating at 0.5 the strictly dominating strategy fro player 2 as well. No other point can be a strictly dominating strategy, as one of the politicians can always move to gain more votes.