HW XXXXXXXXXXSpatial Competition 1
XXXXXXXXXXDowns’ XXXXXXXXXXmodel of electoral competition assumes that candidates
care only about winning the election. One reason for this is the conjecture that
such a candidate has a competitive advantage over opponents with other
motivations. To evaluate this possibility, consider an electoral contest over a
one-dimensional policy space [−1,1] between candidate A, who desires only to
win an election, and candidate B, who cares only about the policy outcome.
Specifically, B’s policy preferences are single-peaked, with an ideal point at 0.9.
Either candidate may adopt any policy platform in the policy space, but must
implement that policy if she wins office. Suppose that there are 21 voters,
whose preferences are single-peaked, with ideal points located at
−1.0,−0.9,−0.8, … ,0.8,0.9, and 1.0, and that each votes for the candidate
whose policy platform is closer to his/her ideal point (flipping a coin if they are
equidistant).
The candidates must announce their platform positions simultaneously.
An equili
ium in this context consists of a pair (????, ????) of policy platforms such
that, given the position of the other candidate, neither candidate regrets her
own platform choice. Answer the following, and explain your reasoning.
a. (+2) Does (0,0) constitute an equili
ium?
. (+2) Are there any equili
ia in which candidate ?? receives a higher
vote share than candidate ???
c. (+2) Are there any equili
ia in which 0.9 is the policy outcome?
d. (+2) Identify one pair of equili
ium platforms other than any that
have been noted above, or state that none exists and explain your
easoning.
HW XXXXXXXXXXSpatial Competition 2
XXXXXXXXXXConsider the following variation on the original Downsian model. There
are infinitely many voters, with ideal points distributed uniformly on [−1,1]. If
policy ?? is implemented, then a voter with ideal point ??�?? derives utility ????(??) =
−(?? − ??�??)2. Suppose further that voters abstain if they feel indifferent or
alienated. Specifically, a voter votes ?? if both of the following conditions are
satisfied,
(1) |????(????) − ????(????)| > .1
(2) ????(????) > −.3
and votes ?? if symmetric conditions are satisfied but abstains otherwise. Now
suppose that candidates ?? is an incumbent and has committed to implement
policy ???? = −0.5. If candidate ?? adopts the median position, what will be each
candidate’s vote share?
HW XXXXXXXXXXMultidimensional Spatial Competition
XXXXXXXXXXConsider spatial voting over a two-dimensional issue by five citizens, A, B,
C, D, and E, who have single-peaked preferences (with circular indifference
curves), with ideal policy bundles (-2,0), (-1,0), (0,1), (1,0), and (2,0), respectively.
The utility ???? = −??2 from policy pair (??1, ??2) to a voter with ideal point (??1,??2)
decreases quadratically in the distance ?? = �(??1 − ??1)2 + (??2 − ??2)2 between
(??1, ??2) and (??1,??2). These citizens must vote for candidate ?? or candidate ??,
who are office motivated and who each commit to platforms consisting of one
policy in each dimension.
a. (+2) Do ???? = (0,0) and ???? = (0,0) together constitute an equili
ium?
Why or why not?
. (+4) Now assume that candidate R has a valence advantage of . 9 over
candidate S. If candidate R adopted the policy bundle ???? = (0,0), is R
guaranteed to win the election, or is there some policy bundle ?? can
adopt in response, and win? What if R adopts the policy bundle ???? =
(0,1)? Explain your answer.
XXXXXXXXXXConsider a probabilistic voting model with three citizens, who possess
wealth ??1 = 0, ??2 = 2, and ??3 = 10. The preferences of agent ?? over a public
good ?? and a private good ???? are
???? = ???? + �??
where the public good ?? = (0?? + 2?? + 10??) = 12?? must be financed by a tax ?? ∈
[0,1] on wealth, leaving each individual with private consumption ???? = ????(1 −
??).
a. (+4) What tax rates ????∗ do each of the three individuals prefer?
. (+4) If a (utilitarian) social planner were to choose a tax rate ??∗ to
maximize welfare ??(??) = ∑ ????(??)3??=1 , what tax rate would the planner
choose?
Now suppose that candidates ?? and ?? have proposed to implement tax rates ????
and ????, respectively. A citizen who is unbiased would vote for candidate ?? if
????(????) > ????(????), and vote for candidate ?? otherwise. However, each citizen has
an additive bias ???? in favor of candidate ?? (where ???? may be negative, implying
that ?? actually has a bias in favor of candidate ??), for reasons unrelated to tax
policy. Candidates observe voters’ tax preferences, but cannot observe voters’
iases.
From a candidate’s perspective, each citizen’s bias is drawn
independently from a uniform distribution on the interval [−1,1]. The cdf of a
uniform distribution can be written as ??(??) = Pr (???? < ??) =
??+1
2
, so the
probability with which a citizen votes ?? can be written as ????(??) = Pr[????(????)
????(????) + ????] = Pr[???? < ????(????) − ????(????)] = ??[????(????) − ????(????)] =
????(????)−????(????)+1
2
.
Let ?????? denote a binary random variable that equals one if ?? votes for
candidate ?? and zero otherwise, and let ???? = ∑ ??????3??=1 denote the total number
of votes for candidate ??. The expected number of votes for ?? is then given by
??(????) = ∑ ??(??????)3??=1 = ∑ ????(??)3??=1 . The expected number ??(????) of votes for ??
can be defined analogously.
c XXXXXXXXXXImplicitly, ??(????) and ??(????) depend on the tax rates ???? and ????
proposed by the two candidates. Suppose that candidate ?? treats ???? as
given, and chooses ???? to maximize ??(????). What tax rate ????∗ should ??
adopt?
d. (+2) Suppose that ????∗ is determined analogously, and compare ????∗ and
????∗ with the three voters’ prefe
ed tax rates ??1∗, ??2∗, and ??3∗, and the
planner’s prefe
ed tax rate ??∗.
HW XXXXXXXXXXEntry
XXXXXXXXXXThere are a continuum of voters, with single-peaked utility ????(??) =
−|?? − ??�??| and ideal points ??�?? distributed uniformly from −1 to 1. In a first round
of play, each voter simultaneously decides whether to enter a political campaign
as a candidate, at cost ?? = .1, or exit (i.e. stay out of the race). In a second stage,
every voter votes sincerely for the candidate whose ideal point is closest to his
own. The candidate ?? with the most votes (
eaking ties, if necessary, with equal
probability) then takes office, implements her ideal policy ??�??, and receives bonus
utility ?? = .4. For this game, there are (perhaps multiple) equili
ia in which
exactly two candidates run for office. Let ???? and ???? denote the platforms of
these two candidates, where (without loss of generality) ???? ≤ ????. What is the
furthest left that ???? might be? What is the furthest right that ???? might be?
Explain your answer.
XXXXXXXXXXConsider the following spatial model of candidate entry. First, parties A
and B commit to policy positions in the interval [-1,1], where voter ideal points
are distributed uniformly over this interval (i.e., and the median voter’s ideal
point is therefore at 0). After these “frontrunner” parties have committed to
policy positions, party C has the option of either staying out of the race or paying
a cost ?? > 0 to enter at any position. Citizens then each vote sincerely for the
candidate (of those in the race) whose platform they prefer. None of the
candidates have policy preferences; each merely wants the benefit ?? of winning
office (where you may assume that 1
3
?? > ??).
a. (+8) Consider first the behavior of candidate C, in the subgame after
candidates A and B have already taken positions ???? ≤ ????. For what
platform pairs should C enter the race, and which policy should C adopt
in these cases (if any)?
. (+4) If the front-runner candidates A and B expect candidate C to
ehave as you have predicted above, tell what types of policy pairs
(????, ????) these candidates might adopt in a (subgame-perfect, pure-
strategy) equili
ium, or explain why no such equili
ium exists.
HW XXXXXXXXXXIdeology as Opinion
XXXXXXXXXXAn electorate can implement any policy ?? ∈ [−1,1], but share a common
interest in implementing a policy as close as possible to the policy ?? that is best
for society.
a. (+4) Suppose that voter utility ??(??, ??) = −(?? − ??)2 is simply given by
the quadratic distance between ?? and ??. In that case, show that the
policy that maximizes expected utility ??[??(??, ??)] is simply the
expectation ??∗ = ??(??) of the optimum. Identical reasoning implies that,
for a citizen with private information (????, ????), expected utility
??[??(??, ??)|???? , ????] is maximized at the conditional expectation ??∗ =
??(??|????, ????).
. (+2) Now consider the case of binary truth, meaning that the optimal
policy ?? ∈ {−1,1} is known to lie at one of the two extremes of the policy
interval, and suppose that each citizen observes a binary private signal
???? ∈ {−1,1} that is co
elated with ??. Specifically, let ???? ∈ [0,1] denote
the co
elation coefficient between ???? and ??. It can then be shown that
??(????|????, ??) =
1
2
(1 + ??????????). By Bayes’ rule, the updated distribution of ??,
conditional on the private signal, is then given by the same function:
??(??|????, ????) =
1
2
(1 + ??????????). Given this information, derive the policy
????∗ = ??(??|????, ????) that is optimal in expectation, as