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

Answered Same DayMar 13, 2022

HW17

XA = -0.5

If candidate B adopts the median position, then he has committed to implement policy xA =(-1+1/2) = 0

Candidate A’s vote share is ui(xA) = -(-0.5-0)^2

Candidate A’s vote share is ui(xA) = -0.25

Candidate B’s vote share is 0

Reference:

Ocw.mit.edu. 2022. [online] Available at:...

XA = -0.5

If candidate B adopts the median position, then he has committed to implement policy xA =(-1+1/2) = 0

Candidate A’s vote share is ui(xA) = -(-0.5-0)^2

Candidate A’s vote share is ui(xA) = -0.25

Candidate B’s vote share is 0

Reference:

Ocw.mit.edu. 2022. [online] Available at:...

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