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...

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Only do HW 17 on page 2- thoroughly please. Thanks!


HW 16 (+8) Spatial Competition 1 1. (+8) Downs’ (1957) model 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 equilibrium 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 equilibrium? b. (+2) Are there any equilibria in which candidate ?? receives a higher vote share than candidate ??? c. (+2) Are there any equilibria in which 0.9 is the policy outcome? d. (+2) Identify one pair of equilibrium platforms other than any that have been noted above, or state that none exists and explain your reasoning. HW 17 (+8) Spatial Competition 2 1. (+8) Consider 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 18 (+20) Multidimensional Spatial Competition 1. (+6) Consider 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 equilibrium? Why or why not? b. (+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. 2. (+14) Consider 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? b. (+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’ biases. 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.="" (+4)="" implicitly,="" (????)="" 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’="" preferred="" tax="" rates="" 1∗,="" 2∗,="" and="" 3∗,="" and="" the="" planner’s="" preferred="" tax="" rate="" ∗.="" hw="" 19="" (+20)="" entry="" 1.="" (+8)="" there="" 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="" (breaking="" 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)="" equilibria="" 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.="" 2.="" (+12)="" consider="" 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)? b. (+4) If the front-runner candidates A and B expect candidate C to behave as you have predicted above, tell what types of policy pairs (????, ????) these candidates might adopt in a (subgame-perfect, pure- strategy) equilibrium, or explain why no such equilibrium exists. HW 20 (+10) Ideology as Opinion 1. (+10) An 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 ??∗ = ??(??|????, ????). b. (+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 correlated with ??. Specifically, let ???? ∈ [0,1] denote the correlation 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

Answer To: HW XXXXXXXXXXSpatial Competition 1 XXXXXXXXXXDowns’ XXXXXXXXXXmodel of electoral competition assumes...

Komalavalli answered on Mar 14 2022
102 Votes
HW17
XA = -0.5
If candidate B adopts the median position, then he has committed to implement polic
y 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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