Question 1. (30 points)
Consider a voting body with five equal-sized blocs of voters, labeled 1 through 5.
They have ideal points in a two-dimension issue space, given by $(x_1, y_1) = (0,5)$,
$(x_2, y_2) = (6,6)$, $(x_3, y_3) = (2, 4)$, $(x_4, y_4) = (0,0)$, and $(x_5, y_5) = (2,0)$, where
the first coordinate gives a voter's ideal policy on the $x$ dimension and the sec-
ond coordinate gives the ideal policy on the $y$ dimension.
(1) Assume that each voter's indifference curves are all circular with center
at his/her ideal point (that is, voter $i$'s utility from policy $(x,y)$ is $u_i(x,y) =$
$-[(x-x_i)^2 + (y - y_i)^2]$). Which of the following policies is (are) in the majority
winset of (2,4)? (i) (1, 3.2); (ii) (3, 2); (iii) (3,3)
(2) Two candidates, A and B, compete for office. A's policy platform is fixed at
(0,0), and B's policy is fixed at (6,6). If the voters vote only on the $x$ dimension,
what is A's vote share and what is B's vote share? Answer the same question
if the voters vote only on the $y$ dimension.
(3) The two candidates are the same as in (2), but now assume that all voters
have weighted quadratic loss utility, that is, voter $i$'s utility from policy $(x,y)$
is $u_i(x, y) = -[\alpha(x - x_i)^2 + (1 - \alpha)(y - y_i)^2]$, where $0 \leq \alpha \leq 1$ represents the
relative salience of the $x$ dimension. A voter will vote for the candidate whose
policy gives her a higher utility (and toss a coin if indifferent). Calculate A's
vote share as a function of $\alpha$, and explain the implication on a candidate's cam-
paign strategy in an election.
