This question explores what happens when political candidates care not only about winning

This question explores what happens when political candidates care not only about winning
but also about the policies they espouse.
The U.S. Congress is up for grabs in the next election. Bipartisanship has again broken
down, and the parties are very polarized. The more liberal faction of the Democratic Party
now dominates that party and a more conservation faction dominates the Republican Party.
As the election approaches, these parties are trying to stake out positions that reflect their
own policy preferences and will attract enough voters to win. To simplify matters, suppose
the parties have to choose a position along a left-right spectrum and can adopt one of the
following positions: Liberal (L), Liberal leaning centrist (LC), Middle of the Road (M),
Conservative Centrist, (CC), Conservative (C). These positions are represented on a line
from 0 to 100 where the L is at the most left position (0), then comes LC (25), then M (50),
then CC (75) and finally C (100).
Since the voters ideal points are evenly distributed along the political spectrum, the party
whose position is closer to the middle of the road (M) wins. If, for example, the Republicans,
R, choose a position of CC and the Democrats, D, announces, L, then R would win because
CC is closer to M. If the parties run on platforms that are equally distant from M, each party
is equally likely to win. Finally, each party chooses its platform secretly.
As noted above, R and D care about policies as well as winning. Ds payoffs are: 5 for winning
with L; 4 for winning with LC; 3 for winning with M; 2 for winning with CC; 1 for winning
with C; -1 for losing with L; -2 for losing with LC; -3 for losing with M; -4 for losing with
CC; and 5 for losing with C.
Rs payoffs are the opposite: 5 for winning with C; 4 for winning with CC; 3 for winning with
M; 2 for winning with LC; 1 for winning with L; -1 for losing with C; -2 for losing with CC;
-3 for losing with M; -4 for losing with LC; and 5 for losing with L.
1. Suppose R adopts position C and D chooses LC. What are their payoffs?
2. Suppose R adopts C and D chooses the more extreme position L. What are the parties
payoffs?
3. Specify the strategic form of this game (write down the table)
4. Solve the game by iterated deletion of dominated strategies. Be sure to indicate the
order of deletion, what dominates what, and whether this is strict or weak dominance.
5. What is/are the pure strategy Nash equilibria of this game?
6. Finally, suppose that there are three parties instead of two. Call the third party S
(for spoiler) and assume that S has the same preferences that D does. Consider the
three-player game in which each player selects his position secretly; each voter votes for
the party whose position is closest to her ideal point; and the party that receives the
most votes wins. If each candidate chooses M, is this a Nash equilibrium of the game?
Explain why or why not.