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.

Candidate 2 0 2 4 5 6 7 8 9 00.5,0.5 0.1,0.9 0.15,0.85 0.20,0.8 0.25,0.75 0.3,0.7 0.35,0.65 0.40,0.6 0.45,0.55 0.5,0.5 10.9,0.1 0.5,0.5 0.2,0.8 0.25,0.75 0.3,0.7 0.35,0.65 0.4,0.6 0.45,0.55 0.5,0.5 0.55,0.45 20.85,0.15 0.8,0.2 0.5,0.5 0.30,0.70.35,0.65 0.4,0.6 0.45,0.55 0.5,0.5 0.55,0.45 0.6,0.4 30.8,0.2 0.75,0.25 0.7,0.30.5,0.5 0.4,0.6 0.45,0.55 0.5,0.5 0.55,0.45 0.6,04 0.65,0.35 40.75,0.25 0.7,0.3 0.65,0.35 0.6,0.4 0.5,0.5 0.5,050.55,0.45 0.6,0.4 0.65,0.35 0.7,0.3 5 0.7,0.3 0.65,0.35 0.6,0.4 0.55,0.45 0.5,0.5 0.5,0.5 0.6,0.4 0.65,0.35 0.7,0.3 0.75,0.25 60.65,0.35 0.6,0.4 0.55,0.45 0.5,0.5 0.45,0.55 0.4,0.6 0.5,0.50.7,0.3 0.75,0.25 0.8,0.2 7|0.6,0.4 0.55,0.45|0.5,0.5 |0.45,0.55|0.4,0.6 |0.35,0.65 0.3,0.7 |0.5,0.5 |0.8,0.2 |0.85,0.15 80.55,0.45 0.5,0.5 0.45,0.55 0.4,0.60.35,0.65 0.3,0.7 0.25,0.75 0.2,0.8 0.5,0.50.9,0.1 90.5,0.5 0.45,0.55 0.4,0.6 0.35,0.55 0.3,0.7 0.25,0.75 0.2,0.80.15,0.85 0.1,0.9 0.5,0.5 Candidate 1