Question

Hotelling’s location game. Recall the voting game discussed in class. There are two candidates, each of whom chooses a position from the set S = (1,2,...10). The voters are equally distributed across these ten positions. Voters vote for the candidate whose position is closest to theirs. If the two candidates are equidistant from a given position, the voters at that position split their votes equally. First, unlike in the game analyzed in class, assume that both candidates only care about winning or losing, not about the winning margin. Let the payoff for winning be 100 and the payoff for losing be 0. A tied election yields a payoff of 50.

(a) Find all strategies that strictly dominate strategy 10, and all strategies that weakly dominate 10. Explain your answer.

Next, switch back to the payoff functions we considered in class (candidates care about vote shares) but assume that there are three candidates, instead of two.

(b) Is strategy 10 dominated, strictly or weakly, by strategy 9? How about by strategy 8? Explain.

Answer 1

(a) Since voters vote for the candidate whose position is closest to theirs, so strategies that strictly dominate strategy 10 are strategies 7,8 and 9. The logic behind this as following:

Assuming that the two candidates are A and B. Let A chooses strategy 10. Then, if B chooses strategy 9, he is strictly better off because A will then get 10 percent of votes while B will secure 90 percent of the total votes. Next, given A chooses strategy 10, if B chooses strategy 8, B will get 85 percent of voting while A will earn only 15 percent of the voting. If B chooses strategy 7, given A chooses strategy 10, A will get 20 percent votes while B will get as much as 80 percent votes. And the strategy that weakly dominates strategy 10 is strategy 10 itself because if both the candidates use strategy 10, then the voters at that position will split their votes equally (50 percent) to each candidate.

(b) Let the third candidate be C. With three candidates, if both A and B choose strategy 10 and C chooses startegy 9, then it is obvious that C will win with majority shares. In other words, Strategy 10 will be strictly dominated by startegy 9 and 8.