Voting Suppose the spectrum of political positions is described by the closed interval (line segment) Œ0; 5. Voters are uniformly distributed over Œ0; 5. There are two candidates, who may occupy any...

Voting

Suppose the spectrum of political positions is described by the closed interval (line segment) Œ0; 5. Voters are uniformly distributed over Œ0; 5. There are two candidates, who may occupy any of the positions in the set {0; 1; 2; 3; 4; 5}. Voters will always vote for the nearest candidate. If the candidates occupy the same position they each get half of the votes. The candidates simultaneously and independently choose positions. Each candidate wants to maximize the number of votes for him/herself. Only pure strategies are considered.

(a) Model this situation as a bimatrix game between the two candidates.

(b) Determine the best replies of both players.

(c) Determine all Nash equilibria (in pure strategies), if any.

We now change the situation as follows. Candidate 1 can only occupy the positions 1; 3; 5, and candidate 2 can only occupy the positions 0; 2; 4.

(d) Answer questions (a), (b), and (c) for this new situation.

(e) The two games above are constant-sum games. How would you turn them into zero-sum games without changing the strategic possibilities (best replies, equilibria)? What would be the value of these games and the (pure) optimal strategies?