Question

Suppose 2 candidates are vying for election by trying to position themselves along a discrete political spectrum 0 1 2 3 4 5 6 7 8 9 . Ten percent of the votes are at each location on the spectrum. Each candidate wants to maximize her share of the votes by choosing her position on the spectrum; voters vote for the candidate closest to their position on the spectrum, and if there is a tie in distance they split their vote 50-50 between the two candidates.

A. Put the game in normal form.

B. Show that for player 1, position 1 dominates position 0 and that 8 dominates 9.

C. Find the rationalizable strategies for both players using iterative elimination of dominated strategies.

Answer 1

A.

B.

If player 1 is at position 1, for all positions of player 2, the vote share of player 1 is higher compared to if he is at position 0. Similarly, for position 8. Hence, position 1 dominates position 0 and position 8 dominated position 9 for player 1.

C.

For player 1, position 1 dominates position 0. Hence we eliminate position 0 (shown in yellow).

Position 8 dominated position 9. Hence we eliminate position 9 (green).

Similarly, for player 2, position 0 is dominated by position 1. Hence we eliminate position 0 (orange). Position 9 is dominated by 8 (blue).

The uncolored boxes represent rationalizable strategies.