1. A public facility needs to be located on a street , which I denote by the interval [0, 1].

   In the city there are n voters. Each voter i has an ideal location pi, 0 pi 1, where 1

she wants the facility to be located. If the facility is located at l, 0 l 1, then the

utility of voter i is

(pi l)2

The following voting game is played to decide on the location. Every citizen/ voter i votes for a location xi, where 0 xi 1. Given the voted location profile, x = (x1, , xn) the facility is chosen at location W (x) where W (x) is a rule that picks a location for every voted location profile x. We call W(x) a voting rule. The specific W (x) that the city uses, is the following:

W(x) = W(x1, ,xn) = min{x1, ,xn}

1. (a) Define the game in normal form. (5 points)

2. (b) (8 points) Is it the case that, it is a weakly dominant strategy for each agent to vote for her ideal location, given the voting rule described above. Explain why, or why not.

3. (c) (7 points) If instead the city used the voting rule, V (x) = V (x1, , xn) = max{x1, , xn}

   Would you answer to part (b) change ? Why, or why not ?