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 815Political 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