《社会保障概论》课程教学资源（课后阅读资料）APSR_Volume112. Issue 4.Nov 2018_Political_Competition_in_Legislative_Elections

Stefan Krasa and Mattias K.Polborn Republican candidate could increase his winning prob- since it shows how political polarization on the national ability by a large amount by moderating slightly un level can spill over to local races.Polarization between til the Democrat's winning probability is zero.Also,in the two national parties renders the candidates'party equilibrium,it cannot be possible for the Democrat to affiliations more relevant for voters and thus leads,al- select a different candidate that would move A strictly most everywhere,to an intensification of the respective above u;-c,else,the Democrat's winning probabil district median voter's party preference. ity would become strictly positive,making the Demo- Our results are relevant for the large empirical lit- crat strictly better off.In other words,for given xi.R,the erature that analyzes how the ideological composition function (xi.R)assumes its maximum at xi.D,and so of districts and,especially,the partisan gerrymandering the first-order condition a(xi.D,xi.R)/axi.D =0 must of districts affects the ideological positions of represen- be satisfied.Using the conditions that the cutoff voter tatives in Congress.In particular,McCarty,Poole,and is at ui -c and that a0(xi.D,xi.R)/axi.D =0 implies Rosenthal (2009a.2009b)claim that,while Congress Equation(7) has become more polarized in a time during which We can interpret the square root term on the right- electoral districts became more heterogeneous due to hand side of Equation(7)as the Republican leader's gerrymandering,this is merely a temporal coincidence leeway in district i.in the sense that it measures 'Political scientists have demonstrated that whenever the extent to which the Republican candidate can be a congressional seat switches parties,the voting record more conservative than district i's median voter with- of the new member is very different from that of the out being too extreme and losing to the Democratic departing member,increasing polarization.In other 元 candidate. words,it is becoming more common to observe a very The leeway in Equation (7)is increasing in those liberal Democrat replaced by a very conservative Re- factors that amplify the preference of district i's me- publican (and vice versa)."They argue that,since these dian voter for the Republican national position.First switches happen in relatively competitive districts,this if the Republican party position is closer to the me- effect cannot be explained by gerrymandering.Further, dian voter,and the median voter starts to care more they argue that a similar increase in polarization has about national party positions rather than local ones been observed in the Senate that is not subject to ger- (i.e.,y),then the Republican candidate in the district rymandering,and hence gerrymandering cannot be the can exploit this preference increase.For example,sup- prime reason for increased political polarization pose that the public expects that one of the major issues An important insight from the spillover effect in in the next Congress will be filling several Supreme our model is that this argument is somewhat flawed Court vacancies.This issue conceivably increases the because the“treatment'”also affects the“untreated.” importance of national party positions for voters(y), Thus,if gerrymandering affects national party positions relative to local candidate positions.In this environ because more extreme representatives are elected ment,we would therefore expect that the ideologically from gerrymandered districts,there is also an indirect advantaged party in moderately competitive districts effect on the equilibrium positions of candidates in is able to win with more extreme candidates than in moderately competitive districts,even if these districts the absence of such a high-stakes issue,leading to in- themselves were not directly gerrymandered.We will creased polarization. return to analyze this subject more closely in Proposi- S5.501g Second,the more voters in a district perceive that tion 4 below. their district might be decisive for party control in the legislature (pit),the more they will take their prefer- ence for the national party positions into account when Equilibrium when National Concerns Are choosing between local candidates.Further below,we Dominant(y≈o) will see that pi increases for noncentrist districts if there We now turn,in this subsection and the next one,to are fewer centrist districts,which results in more parti- limit cases in which we can characterize the equilib- san candidates in the noncentrist districts rium for an arbitrary degree of uncertainty about the Third,the leeway is increasing in the distance be- median voter's ideal position.We start with the case tween the median voter's preferences from the mid- that all actors care primarily about national positions point of the two party platforms.In a very moderate (y≈0) district,that is,one in which the median voter is close Observe first that,when y=0,that is,neither vot- to indifferent between the parties,there is not much ers nor party leaders care at all about their local can- 四 of an asymmetry that the local leader can exploit-if didates'positions,then payoffs are independent of ac- he tried too much,his party's candidate would lose.In tions and therefore any behavior is an equilibrium.For contrast,the favored party's local leader in districts that a meaningful analysis,we therefore need to look at the are more conservative (or liberal,on the other side of case that y is small,but positive.The following Proposi- the political spectrum)can force a more extreme can- tion 2 shows that an equilibrium exists,and that,in dis- didate down the district median voter's throat because tricts where the median voter is,in expectation,more the median voter is not comfortable to vote for the other party's candidate because of his association with ideologically extreme,the winning probability of the ideologically favored party's candidate increases. (locally)unpopular national positions. Finally,the leeway is increasing in the degree of na- Proposition 2 Let xiD(y),xi.R(y)denote the equi- tional polarization (XR-Xp).This is a crucial effect, librium strategies in district i when the preference 814

Stefan Krasa and Mattias K. Polborn Republican candidate could increase his winning prob￾ability by a large amount by moderating slightly un￾til the Democrat’s winning probability is zero. Also, in equilibrium, it cannot be possible for the Democrat to select a different candidate that would move θ strictly above μi − c, else, the Democrat’s winning probabil￾ity would become strictly positive, making the Demo￾crat strictly better off. In other words, for given xi, R, the function θ( ·, xi, R) assumes its maximum at xi, D, and so the first-order condition ∂θ(xi, D, xi, R)/∂xi, D = 0 must be satisfied. Using the conditions that the cutoff voter is at μi − c and that ∂θ(xi, D, xi, R)/∂xi, D = 0 implies Equation (7). We can interpret the square root term on the right￾hand side of Equation (7) as the Republican leader’s leeway in district i, in the sense that it measures the extent to which the Republican candidate can be more conservative than district i’s median voter with￾out being too extreme and losing to the Democratic candidate. The leeway in Equation (7) is increasing in those factors that amplify the preference of district i’s me￾dian voter for the Republican national position. First, if the Republican party position is closer to the me￾dian voter, and the median voter starts to care more about national party positions rather than local ones (i.e., γ ↓), then the Republican candidate in the district can exploit this preference increase. For example, sup￾pose that the public expects that one of the major issues in the next Congress will be filling several Supreme Court vacancies. This issue conceivably increases the importance of national party positions for voters (γ ↓), relative to local candidate positions. In this environ￾ment, we would therefore expect that the ideologically advantaged party in moderately competitive districts is able to win with more extreme candidates than in the absence of such a high-stakes issue, leading to in￾creased polarization. Second, the more voters in a district perceive that their district might be decisive for party control in the legislature (pi↑), the more they will take their prefer￾ence for the national party positions into account when choosing between local candidates. Further below, we will see that pi increases for noncentrist districts if there are fewer centrist districts, which results in more parti￾san candidates in the noncentrist districts. Third, the leeway is increasing in the distance be￾tween the median voter’s preferences from the mid￾point of the two party platforms. In a very moderate district, that is, one in which the median voter is close to indifferent between the parties, there is not much of an asymmetry that the local leader can exploit—if he tried too much, his party’s candidate would lose. In contrast, the favored party’s local leader in districts that are more conservative (or liberal, on the other side of the political spectrum) can force a more extreme can￾didate down the district median voter’s throat because the median voter is not comfortable to vote for the other party’s candidate because of his association with (locally) unpopular national positions. Finally, the leeway is increasing in the degree of na￾tional polarization (XR − XD). This is a crucial effect, since it shows how political polarization on the national level can spill over to local races. Polarization between the two national parties renders the candidates’ party affiliations more relevant for voters and thus leads, al￾most everywhere, to an intensification of the respective district median voter’s party preference. Our results are relevant for the large empirical lit￾erature that analyzes how the ideological composition of districts and, especially, the partisan gerrymandering of districts affects the ideological positions of represen￾tatives in Congress. In particular, McCarty, Poole, and Rosenthal (2009a, 2009b) claim that, while Congress has become more polarized in a time during which electoral districts became more heterogeneous due to gerrymandering, this is merely a temporal coincidence. “Political scientists have demonstrated that whenever a congressional seat switches parties, the voting record of the new member is very different from that of the departing member, increasing polarization. In other words, it is becoming more common to observe a very liberal Democrat replaced by a very conservative Re￾publican (and vice versa).” They argue that, since these switches happen in relatively competitive districts, this effect cannot be explained by gerrymandering. Further, they argue that a similar increase in polarization has been observed in the Senate that is not subject to ger￾rymandering, and hence gerrymandering cannot be the prime reason for increased political polarization. An important insight from the spillover effect in our model is that this argument is somewhat flawed because the “treatment” also affects the “untreated.” Thus,if gerrymandering affects national party positions because more extreme representatives are elected from gerrymandered districts, there is also an indirect effect on the equilibrium positions of candidates in moderately competitive districts, even if these districts themselves were not directly gerrymandered. We will return to analyze this subject more closely in Proposi￾tion 4 below. Equilibrium when National Concerns Are Dominant (γ ≈ 0) We now turn, in this subsection and the next one, to limit cases in which we can characterize the equilib￾rium for an arbitrary degree of uncertainty about the median voter’s ideal position. We start with the case that all actors care primarily about national positions (γ ≈ 0). Observe first that, when γ = 0, that is, neither vot￾ers nor party leaders care at all about their local can￾didates’ positions, then payoffs are independent of ac￾tions and therefore any behavior is an equilibrium. For a meaningful analysis, we therefore need to look at the case that γ is small, but positive.The following Proposi￾tion 2 shows that an equilibrium exists, and that, in dis￾tricts where the median voter is, in expectation, more ideologically extreme, the winning probability of the ideologically favored party’s candidate increases. Proposition 2 Let xi, D(γ ), xi, R(γ ) denote the equi￾librium strategies in district i when the preference 814 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000503

Political Competition in Legislative Elections parameter is y,and let X=(XR Xp)2 be the voter Proof of Corollary 1.See Appendix. ■ type who is indifferent between the two national party positions.Let hi(x)=bi(x)/(1-i(x))be the hazard The first claim is very intuitive and simply means rate in district i,and suppose that is symmetric around that candidates nominated in more conservative dis- its mean wi.Ify is close to zero,then the following results tricts espouse more conservative positions. hold: The second claim states that polarization between the local candidates.measured as the distance between 1.There exists an equilibrium in which the posi- their positions,increases as the advantage of one of the tions of the candidates in district i are parties increases.Conversely,in the most moderate dis- trict,electoral competition works best in terms of forc- limx.D(y）=d ing both candidates toward a moderate position. y40 The third claim states that x,R→ri and x,D→X h(-X+2)(d-X)2 in a district where Republicans almost certainly have 1-h(-X+2u)(d-X) (9) an ideological advantage.To get some intuition,nor- malize X to 0,and consider a conservative district ui> 0.Let s->0.which means that actors become more limxi..R(y）=rn and more confident that district i's realized median y40 voter in fact prefers the Republican national party po- h(X)(-X)2 sition to the Democratic one,though not necessarily by (10) 1+h,(X)(r:-X) much.In this case,the hazard rates in Equations(9)and (10)go to infinity and zero,respectively,which implies 2.The Democratic winning probability in dis- that x.r→ri andx.D→0：Thus,both parties'candi- trict i converges to i(X),and the Republican dates choose positions that are different from the posi- one to1-ΦX). tion preferred by district i's median voter.Most signif- icantly,the Republican candidate-who is most likely 'asn The candidate positions are more moderate than the to be elected because of the median voter's preference respective local party leaders'ideal positions.The ex- for the national Republican position-is at his (local) & act positions depend on the hazard rates in Equations party's ideal position. (9)and (10),which capture the degree of uncertainty Interestingly,the Democrats also do not choose to about district i's median voter's position,as h(t)dt is the position their candidate at the expected median voter probability that the median voter's position is in the position ui.The reason is that,if the realized median (small)interval [t,t+dt],conditional on being greater voter is at or close to ui,the Democrats still would not win in district i,even with a local candidate close to than t. that position competing against a relatively misaligned 是 To illustrate Proposition 2,consider the following ex- ample where the median voter's position follows a lo- Republican,because the median voter's preference for gistic distribution with parameter s.(The parameter s the Republican party's national position outweighs his of a logistic distribution is proportional to the standard local candidate preference.For the Democrats to have a chance of winning in district i,the realized median 55.501g deviation of the distribution,sa/v3.) voter must be close to 0(i.e..indifferent between the Corollary 1 Suppose that the median voter's position national party positions),and a position near 0 is the is distributed logistically,with cumulative distribution most competitive in this contingency.In contrast to the given byΦ，(r)）=te--nm,so that the hazard rate is Democratic leader,the Republican leader expects to win with a high probability in district i,so compromis- hi(x)Then the following holds for y ing by nominating a more moderate candidate is more close to zero: costly in terms of expected utility,and therefore,the Republican candidate's equilibrium position is close 1.Both xi.D and xi.R increase in ui. to ri. 2.Suppose that di and ri are symmetric around Suppose that in this conservative district the me- X.Then increasing wi from i =X strictly in- dian voter's position,ui,is closer to the median pri- creases local polarization,that is,xiR-xi.D mary voter's position,ri,than to zero.Then,in addi- increases. tion to being disadvantaged by its national position,the 3.Suppose that the expected median voter in dis- Democrats may also seem "ideologically stubborn"by trict i has a strict preference for the Republican nominating an "inappropriate"candidate for district i. national party position over the Democratic That is,if the variance is small,then the realized median one (that is,ui -XRl ui -XpD).Further, voter will almost always prefer the national Republi- suppose that the variance,s2n2,converges to can position (over the national Democratic one)and zero.Then xi.R→riand xiD→X,and the Re- the local Republican candidate over his Democratic publican's winning probability converges to 1. challenger. 4.For s >0,changes in the national party posi- Finally,the last point in Corollary 1 considers the tions Xp and XR have a nonmonotone effect effect of a change in the national party positions.In on the equilibrium local candidate positions general,this effect is nonmonotone,as the following xi.D and xi.R. thought experiment shows.Suppose that,initially,XD 815

Political Competition in Legislative Elections parameter is γ , and let X = (XR + XD)/2 be the voter type who is indifferent between the two national party positions. Let hi(x) = φi(x)/(1 − i(x)) be the hazard rate in district i, and suppose that i is symmetric around its mean μi. If γ is close to zero,then the following results hold: 1. There exists an equilibrium in which the posi￾tions of the candidates in district i are lim γ ↓0 xi,D(γ ) = di + hi(−X + 2μi)(di − X )2 1 − hi(−X + 2μi)(di − X ) , (9) lim γ ↓0 xi,R(γ ) = ri − hi(X )(ri − X )2 1 + hi(X )(ri − X ) . (10) 2. The Democratic winning probability in dis￾trict i converges to i(X), and the Republican one to 1 − i(X). The candidate positions are more moderate than the respective local party leaders’ ideal positions. The ex￾act positions depend on the hazard rates in Equations (9) and (10), which capture the degree of uncertainty about district i’s median voter’s position, as h(t)dt is the probability that the median voter’s position is in the (small) interval [t, t + dt], conditional on being greater than t. To illustrate Proposition 2, consider the following ex￾ample where the median voter’s position follows a lo￾gistic distribution with parameter s. (The parameter s of a logistic distribution is proportional to the standard deviation of the distribution, sπ/√3.) Corollary 1 Suppose that the median voter’s position is distributed logistically, with cumulative distribution given by i(x) = 1 1+e−(x−μi)/s , so that the hazard rate is hi(x) = 1 s(1+e−(x−μi)/s) . Then the following holds for γ close to zero: 1. Both xi, D and xi, R increase in μi. 2. Suppose that di and ri are symmetric around X. Then increasing μi from μi = X strictly in￾creases local polarization, that is, xi, R − xi, D increases. 3. Suppose that the expected median voter in dis￾trict i has a strict preference for the Republican national party position over the Democratic one (that is, |μi − XR| < |μi − XD|). Further, suppose that the variance, s2π2/3, converges to zero.Then xi, R →ri and xi, D →X, and the Re￾publican’s winning probability converges to 1. 4. For s > 0, changes in the national party posi￾tions XD and XR have a nonmonotone effect on the equilibrium local candidate positions xi, D and xi, R. Proof of Corollary 1. See Appendix. The first claim is very intuitive and simply means that candidates nominated in more conservative dis￾tricts espouse more conservative positions. The second claim states that polarization between the local candidates, measured as the distance between their positions, increases as the advantage of one of the parties increases. Conversely, in the most moderate dis￾trict, electoral competition works best in terms of forc￾ing both candidates toward a moderate position. The third claim states that xi, R → ri and xi, D → X in a district where Republicans almost certainly have an ideological advantage. To get some intuition, nor￾malize X to 0, and consider a conservative district μi > 0. Let s → 0, which means that actors become more and more confident that district i’s realized median voter in fact prefers the Republican national party po￾sition to the Democratic one, though not necessarily by much. In this case, the hazard rates in Equations (9) and (10) go to infinity and zero, respectively, which implies that xi, R → ri and xi, D → 0: Thus, both parties’ candi￾dates choose positions that are different from the posi￾tion preferred by district i’s median voter. Most signif￾icantly, the Republican candidate—who is most likely to be elected because of the median voter’s preference for the national Republican position—is at his (local) party’s ideal position. Interestingly, the Democrats also do not choose to position their candidate at the expected median voter position μi. The reason is that, if the realized median voter is at or close to μi, the Democrats still would not win in district i, even with a local candidate close to that position competing against a relatively misaligned Republican, because the median voter’s preference for the Republican party’s national position outweighs his local candidate preference. For the Democrats to have a chance of winning in district i, the realized median voter must be close to 0 (i.e., indifferent between the national party positions), and a position near 0 is the most competitive in this contingency. In contrast to the Democratic leader, the Republican leader expects to win with a high probability in district i, so compromis￾ing by nominating a more moderate candidate is more costly in terms of expected utility, and therefore, the Republican candidate’s equilibrium position is close to ri. Suppose that in this conservative district the me￾dian voter’s position, μi, is closer to the median pri￾mary voter’s position, ri, than to zero. Then, in addi￾tion to being disadvantaged by its national position, the Democrats may also seem “ideologically stubborn” by nominating an “inappropriate” candidate for district i. That is,if the variance is small, then the realized median voter will almost always prefer the national Republi￾can position (over the national Democratic one) and the local Republican candidate over his Democratic challenger. Finally, the last point in Corollary 1 considers the effect of a change in the national party positions. In general, this effect is nonmonotone, as the following thought experiment shows. Suppose that, initially, XD 815 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000503