《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

American Political Science Review (2018)112,4,809-825 doi:10.1017/S0003055418000503 American Political Science Association 2018 Political Competition in Legislative Elections STEFAN KRASA University of Illinois MATTIAS K.POLBORN Vanderbilt University re develop a theory of electoral competition in multidistrict legislative elections when nomina- tion decisions are made by local policy-motivated party members,and voters care about both local and national positions.We show that the asymmetry generated by different national party positions reduces or even entirely removes the competitive pressure to nominate moderate candidates The model has important implications for our understanding of policy divergence and,in particular,of the effects of gerrymandering. INTRODUCTION a position to determine the unilaterally decisive policy maker in the nation,but rather just one of many repre- n the basic model of representative democracy,vot- sentatives who interacts with other representatives in ers elect legislative representatives whose positions the determination of policy. reflect the preferences of their respective districts Our model is based on two realistic ingredients: median voters.These representatives convene in an First,the majority party in a legislature is an impor- amorphous assembly-one in which parties do not play tant power center influencing the crafting of policy,and an important role-and national policy is set to cor- so voters naturally care not only about the positions respond to the preferences of the median representa- espoused by their local candidates,but also about the tive in this assembly.Thus,the legislature is composed national positions of the parties with whom these lo- of representatives who are more moderate than the 4号元 cal candidates are affiliated.A potential microfounda- voters who elect them,and actual policy and legisla- tion for why voters care about national party positions tion reflects the most moderate position in this assem- is that no legislator is a specialist in all policy areas, bly of moderates(a prediction that appears somewhat and therefore they all have to rely sometimes on the incorrect). expertise of their fellow party members (Shepsle and There is,of course,a large body of literature model- Weingast 1987;Gilligan and Krehbiel 1989).The impor- ing interactions of representatives in a legislature and tance of the majority party for law-making creates im- the effects of legislative institutions such as the power portant spillover effects between the candidates of the of specialized committees,but this literature takes the same party who run in different districts.Second,leg- set of legislators as given.There is also a large litera islative candidates are nominated by policy-motivated ture on political competition,but that literature gener- 是 primary voters who,like the general election median ally assumes that candidates either unilaterally choose voter,care about both national party and local candi- or are exogenously endowed with policy positions that date positions,but have more extreme ideal positions. enter voters'utility functions directly and exclusively. A central question in our analysis is how national party That is,voters care only about the positions of those positions affect the competitive pressure to nominate candidates that they personally can decide between.In moderate candidates,which plays a central role both this paper,we build a model of electoral competition in the classical Downsian model,and in the policy- that combines these two strands of literature:When motivated candidates model. voting for their local representative,the voters in our model explicitly take into account that they are not in While parties play a crucial role in the legislature for shaping politics and policy,there is surprisingly lit- tle analysis of how the fact that each candidate is con- Stefan Krasa is a Professor,Department of Economics,University of nected to a party and thus,implicitly,to the positions of Illinois,1407 W.Gregory Dr.,Urbana,IL 61801(skrasa@uiuc.edu). Mattias K.Polborn is a Professor.Department of Economics and candidates of that party from other districts influences Department of Political Science,Vanderbilt University,Nashville nomination decisions,as well as election outcomes in TN,37235(mattias.polborn@vanderbilt.edu). different legislative districts. We are very grateful for the comments of three referees and the Applying the simplest Downsian model naively to editor that helped us to improve the paper.Both authors gratefully Congressional elections-which much of the empirical acknowledge financial support from National Science Foundation Grant SES-1261016.Any opinions,findings,and conclusions or rec- literature implicitly does-generates empirically incor- ommendations expressed in this paper are those of the authors and rect predictions:Since,in the Downsian model,all can- do not necessarily reflect the views of the National Science Founda- didates adopt the preferred position of their district's tion or any other organization.We are thankful for helpful comments median voter,all voters should be policywise indiffer- by seminar participants at USC,Caltech,NYU.MPI Munich.ITAM ent between the Democratic candidate and his Repub- Toulouse,Illinois.Princeton.Western Illinois,lowa.Michigan,Notre Dame,UC Irvine,Illinois State,Mannheim,ETH Zurich,Brigham lican opponent.Thus,Republicans in New England or Young,New Economic School,Higher School of Economics,Yale, Democrats in rural Western districts should have a sub- Southern Methodist and Vanderbilt,as well as Odilon Camara. stantial chance to be elected to Congress if only they Tiberiu Dragu,Bernie Grofman,Matthias Messner,and Arturas match their opponent's policy platform.Furthermore, Rozenas.Replication files are available on the American Political Science Review Dataverse:https://doi.org/10.7910/DVN/SNW8PV. Received:August 1,2017;revised:April 3,2018;accepted:July 11. See Table 1 in Winer,Kenny,and Grofman(2014)for evidence 2018.First published online:August 22,2018. that a significant share of U.S.Senate elections are noncompetitive. 809

American Political Science Review (2018) 112, 4, 809–825 doi:10.1017/S0003055418000503 © American Political Science Association 2018 Political Competition in Legislative Elections STEFAN KRASA University of Illinois MATTIAS K. POLBORN Vanderbilt University We develop a theory of electoral competition in multidistrict legislative elections when nomina￾tion decisions are made by local policy-motivated party members, and voters care about both local and national positions. We show that the asymmetry generated by different national party positions reduces or even entirely removes the competitive pressure to nominate moderate candidates. The model has important implications for our understanding of policy divergence and, in particular, of the effects of gerrymandering. INTRODUCTION I n the basic model of representative democracy, vot￾ers elect legislative representatives whose positions reflect the preferences of their respective districts’ median voters. These representatives convene in an amorphous assembly—one in which parties do not play an important role—and national policy is set to cor￾respond to the preferences of the median representa￾tive in this assembly. Thus, the legislature is composed of representatives who are more moderate than the voters who elect them, and actual policy and legisla￾tion reflects the most moderate position in this assem￾bly of moderates (a prediction that appears somewhat incorrect). There is, of course, a large body of literature model￾ing interactions of representatives in a legislature and the effects of legislative institutions such as the power of specialized committees, but this literature takes the set of legislators as given. There is also a large litera￾ture on political competition, but that literature gener￾ally assumes that candidates either unilaterally choose or are exogenously endowed with policy positions that enter voters’ utility functions directly and exclusively. That is, voters care only about the positions of those candidates that they personally can decide between. In this paper, we build a model of electoral competition that combines these two strands of literature: When voting for their local representative, the voters in our model explicitly take into account that they are not in Stefan Krasa is a Professor, Department of Economics, University of Illinois, 1407 W. Gregory Dr., Urbana, IL 61801 (skrasa@uiuc.edu). Mattias K. Polborn is a Professor, Department of Economics and Department of Political Science, Vanderbilt University, Nashville, TN, 37235 (mattias.polborn@vanderbilt.edu). We are very grateful for the comments of three referees and the editor that helped us to improve the paper. Both authors gratefully acknowledge financial support from National Science Foundation Grant SES-1261016. Any opinions, findings, and conclusions or rec￾ommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Founda￾tion or any other organization.We are thankful for helpful comments by seminar participants at USC, Caltech, NYU, MPI Munich, ITAM, Toulouse, Illinois, Princeton, Western Illinois, Iowa, Michigan, Notre Dame, UC Irvine, Illinois State, Mannheim, ETH Zurich, Brigham Young, New Economic School, Higher School of Economics, Yale, Southern Methodist and Vanderbilt, as well as Odilon Camara, Tiberiu Dragu, Bernie Grofman, Matthias Messner, and Arturas Rozenas. Replication files are available on the American Political Science Review Dataverse: https://doi.org/10.7910/DVN/SNW8PV. Received: August 1, 2017; revised: April 3, 2018; accepted: July 11, 2018. First published online: August 22, 2018. a position to determine the unilaterally decisive policy maker in the nation, but rather just one of many repre￾sentatives who interacts with other representatives in the determination of policy. Our model is based on two realistic ingredients: First, the majority party in a legislature is an impor￾tant power center influencing the crafting of policy, and so voters naturally care not only about the positions espoused by their local candidates, but also about the national positions of the parties with whom these lo￾cal candidates are affiliated. A potential microfounda￾tion for why voters care about national party positions is that no legislator is a specialist in all policy areas, and therefore they all have to rely sometimes on the expertise of their fellow party members (Shepsle and Weingast 1987;Gilligan and Krehbiel 1989).The impor￾tance of the majority party for law-making creates im￾portant spillover effects between the candidates of the same party who run in different districts. Second, leg￾islative candidates are nominated by policy-motivated primary voters who, like the general election median voter, care about both national party and local candi￾date positions, but have more extreme ideal positions. A central question in our analysis is how national party positions affect the competitive pressure to nominate moderate candidates, which plays a central role both in the classical Downsian model, and in the policy￾motivated candidates model. While parties play a crucial role in the legislature for shaping politics and policy, there is surprisingly lit￾tle analysis of how the fact that each candidate is con￾nected to a party and thus, implicitly, to the positions of candidates of that party from other districts influences nomination decisions, as well as election outcomes in different legislative districts. Applying the simplest Downsian model naively to Congressional elections—which much of the empirical literature implicitly does—generates empirically incor￾rect predictions: Since, in the Downsian model, all can￾didates adopt the preferred position of their district’s median voter, all voters should be policywise indiffer￾ent between the Democratic candidate and his Repub￾lican opponent. Thus, Republicans in New England or Democrats in rural Western districts should have a sub￾stantial chance to be elected to Congress if only they match their opponent’s policy platform.1 Furthermore, 1 See Table 1 in Winer, Kenny, and Grofman (2014) for evidence that a significant share of U.S. Senate elections are noncompetitive. 809 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

Stefan Krasa and Mattias K.Polborn in this framework,gerrymandering districts affects only lican candidate as a 'supporter'of the 'Bush agenda'or candidate positions in the gerrymandered districts but as a 'rubberstamp,'Democrats believe the President's does not help a party to increase their expected rep- low approval ratings are a stone they can use to sink resentation in Congress.These predictions are cer- their opponents [...Democratic Sen.Hillary Clinton tainly empirically incorrect,but understanding why is of New York got the most mentions in Republican ads challenging. holding forth the supposed horrors of a Democratic- In our model,voters'utility depends on both their controlled Senate [...The runner-up is'San Francisco local representative's position,and the position of the Liberal Nancy Pelosi,'who is mentioned in at least 6 majority party.In the general election,voters take into GOP ads as a reason not to vote for a Democrat who account the two local candidates'positions,as well as would in turn vote to make her Speaker of the House."5 the chance that the election outcome in their district In contrast to the classical one-district spatial model, may change which party is the majority party in the the ideological composition of districts in our model legislature. does not only influence the ideological position of The latter effect implies that,in most districts,the elected candidates,but also the chances of parties to median voter cares not just about the local candidates' win,thus increasing partisan incentives for gerryman- positions when deciding whom to vote for,but also dering.Gerrymandering or,more generally,the intensi- about their party labels,as they are associated with fication of the median ideological preferences in some different national positions.The favored party's pri- districts,affects the political equilibrium even in those mary voter can exploit this situation by nominating a districts where the median voter preferences remain 元 more extreme candidate than the general election me- the same as before.Our results imply that testing for dian voter would prefer.In particular,if voters care the causal effect of gerrymandering on polarization in sufficiently strongly about national positions relative Congress is more complicated than the existing litera- to local candidate positions,then the favored party's ture has recognized. primary voter can simply nominate his own preferred 4 candidate and still win,generally even with a strict su- permajority of votes. RELATED LITERATURE The local general election loses some of its disci- Ever since Downs's(1957)seminal work,candidates' plining force because the voters'national preference position choice is a central topic in political economy factors in their vote choice.The electoral prospects While the classical median voter framework identi- of candidates in a given district are influenced by the fies reasons for platform convergence,many subse- expected ideological position of their parties'winning quent electoral competition models develop different candidates elsewhere.The association with a party that reasons for policy divergence,including policy moti- is not attuned with a district's ideological leanings may vation (Wittman 1983:Calvert 1985:Londregan and be poisonous for a candidate,even if his own policy po- Romer 1993:Osborne and Slivinski 1996:Besley and 是 sitions are tailor-made for his district. Coate 1997:Martinelli 2001:Gul and Pesendorfer Consider,for example,Lincoln Chafee,the former 2009);entry deterrence (Palfrey 1984;Callander 2005); Republican U.S.senator from Rhode Island,who had agency problems(Van Weelden 2013);incomplete in- taken a number of moderate and liberal positions that formation among voters or candidates (Castanheira 5.501g brought him in line with voters in his state.2 As the New 2003;Bernhardt,Duggan,and Squintani 2007;Callan- York Times reported,in the 2006 election,"exit polls der 2008):and differential candidate valence (Bern- gave Senator Lincoln Chafee a 62 percent approval hardt and Ingberman 1985;Groseclose 2001;Krasa and rating.But before they exited the polls,most voters re- Polborn 2010b,2012:Bierbrauer and Bover 2013). jected him,many feeling it was more important to give Most of the literature looks at isolated one-district the Democrats a chance at controlling the Senate."3 elections.Exceptions are Austen-Smith(1984);Snyder His Democratic challenger Whitehouse "succeeded (1994);Ansolabehere,Leblanc,and Snyder(2012)and by attacking the instances in which Chafee supported subsequent work by Polborn and Snyder (2017).In his party's conservative congressional leadership Austen-Smith(1984),the party that wins the majority (whose personalities and policies were very unpopular, of n districts implements an aggregate of its candidates' state-wide).'4 positions.Each district candidate chooses his position In a review of 2006 campaign ads,factcheck.org sum- to maximize his chance of winning.If an equilibrium marized:"President Bush was far and away the most exists,then both party positions fully converge to the frequent supporting actor in Democratic ads [...The median voter in the median district,even though indi- strategy is clear:whether they're referring to a Repub- vidual candidates'positions differ.In contrast,in our model,positions are chosen by policy-motivated pri- In 29.4%of U.S.Senate elections between 1922 and 2004 without an mary voters,and voters care about both national party incumbent running,the winner received a vote share that was at least positions and local candidates.In our equilibrium,na- 20 percentage points larger than the loser's vote share. tional party positions diverge,and we can analyze the 2 For example,Chafee was pro-choice,anti-death-penalty,supported effects of gerrymandering and of more or less radical gay marriage,and voted against the Irag war (see http://en.wikipedia primary voters. org/wiki/Lincoln_Chafee). 3"A GOP Breed loses its place in New England,",New York Times, November 27 2006. See http://en.wikipedia.org/wiki/Lincoln_Chafee See https://www.factcheck.org/2006/11/our-2006-awards. 810

Stefan Krasa and Mattias K. Polborn in this framework, gerrymandering districts affects only candidate positions in the gerrymandered districts but does not help a party to increase their expected rep￾resentation in Congress. These predictions are cer￾tainly empirically incorrect, but understanding why is challenging. In our model, voters’ utility depends on both their local representative’s position, and the position of the majority party. In the general election, voters take into account the two local candidates’ positions, as well as the chance that the election outcome in their district may change which party is the majority party in the legislature. The latter effect implies that, in most districts, the median voter cares not just about the local candidates’ positions when deciding whom to vote for, but also about their party labels, as they are associated with different national positions. The favored party’s pri￾mary voter can exploit this situation by nominating a more extreme candidate than the general election me￾dian voter would prefer. In particular, if voters care sufficiently strongly about national positions relative to local candidate positions, then the favored party’s primary voter can simply nominate his own preferred candidate and still win, generally even with a strict su￾permajority of votes. The local general election loses some of its disci￾plining force because the voters’ national preference factors in their vote choice. The electoral prospects of candidates in a given district are influenced by the expected ideological position of their parties’ winning candidates elsewhere. The association with a party that is not attuned with a district’s ideological leanings may be poisonous for a candidate, even if his own policy po￾sitions are tailor-made for his district. Consider, for example, Lincoln Chafee, the former Republican U.S. senator from Rhode Island, who had taken a number of moderate and liberal positions that brought him in line with voters in his state.2 As the New York Times reported, in the 2006 election, “exit polls gave Senator Lincoln Chafee a 62 percent approval rating. But before they exited the polls, most voters re￾jected him, many feeling it was more important to give the Democrats a chance at controlling the Senate.”3 His Democratic challenger Whitehouse “succeeded by attacking the instances in which Chafee supported his party’s conservative congressional leadership (whose personalities and policies were very unpopular, state-wide).”4 In a review of 2006 campaign ads, factcheck.org sum￾marized: “President Bush was far and away the most frequent supporting actor in Democratic ads [...] The strategy is clear: whether they’re referring to a Repub￾In 29.4% of U.S. Senate elections between 1922 and 2004 without an incumbent running, the winner received a vote share that was at least 20 percentage points larger than the loser’s vote share. 2 For example, Chafee was pro-choice, anti-death-penalty, supported gay marriage, and voted against the Iraq war (see http://en.wikipedia. org/wiki/Lincoln_Chafee). 3 “A GOP Breed loses its place in New England,”, New York Times, November 27, 2006. 4 See http://en.wikipedia.org/wiki/Lincoln_Chafee. lican candidate as a ‘supporter’ of the ‘Bush agenda’ or as a ‘rubberstamp,’ Democrats believe the President’s low approval ratings are a stone they can use to sink their opponents [...] Democratic Sen. Hillary Clinton of New York got the most mentions in Republican ads holding forth the supposed horrors of a Democratic￾controlled Senate [...] The runner-up is ‘San Francisco Liberal Nancy Pelosi,’ who is mentioned in at least 6 GOP ads as a reason not to vote for a Democrat who would in turn vote to make her Speaker of the House.”5 In contrast to the classical one-district spatial model, the ideological composition of districts in our model does not only influence the ideological position of elected candidates, but also the chances of parties to win, thus increasing partisan incentives for gerryman￾dering.Gerrymandering or,more generally, the intensi￾fication of the median ideological preferences in some districts, affects the political equilibrium even in those districts where the median voter preferences remain the same as before. Our results imply that testing for the causal effect of gerrymandering on polarization in Congress is more complicated than the existing litera￾ture has recognized. RELATED LITERATURE Ever since Downs’s (1957) seminal work, candidates’ position choice is a central topic in political economy. While the classical median voter framework identi￾fies reasons for platform convergence, many subse￾quent electoral competition models develop different reasons for policy divergence, including policy moti￾vation (Wittman 1983; Calvert 1985; Londregan and Romer 1993; Osborne and Slivinski 1996; Besley and Coate 1997; Martinelli 2001; Gul and Pesendorfer 2009); entry deterrence (Palfrey 1984; Callander 2005); agency problems (Van Weelden 2013); incomplete in￾formation among voters or candidates (Castanheira 2003; Bernhardt, Duggan, and Squintani 2007; Callan￾der 2008); and differential candidate valence (Bern￾hardt and Ingberman 1985;Groseclose 2001;Krasa and Polborn 2010b, 2012; Bierbrauer and Boyer 2013). Most of the literature looks at isolated one-district elections. Exceptions are Austen-Smith (1984); Snyder (1994); Ansolabehere, Leblanc, and Snyder (2012) and subsequent work by Polborn and Snyder (2017). In Austen-Smith (1984), the party that wins the majority of n districts implements an aggregate of its candidates’ positions. Each district candidate chooses his position to maximize his chance of winning. If an equilibrium exists, then both party positions fully converge to the median voter in the median district, even though indi￾vidual candidates’ positions differ. In contrast, in our model, positions are chosen by policy-motivated pri￾mary voters, and voters care about both national party positions and local candidates. In our equilibrium, na￾tional party positions diverge, and we can analyze the effects of gerrymandering and of more or less radical primary voters. 5 See https://www.factcheck.org/2006/11/our-2006-awards. 810 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 Snyder (1994)considers a dynamic setting in which heira and Crutzen (2010),Eguia (2011a,2011b)and voters care only about national party positions tha Diermeier and Vlaicu(2011)provide theories of en- are chosen by the party's representatives in the pre- dogenous institution choice leading to powerful par- election legislature to maximize their individual reelec- ties.All these models of the importance of parties in tion chances.In Ansolabehere,Leblanc,and Snyder Congress take the distribution of legislator preferences (2012),a special version of this model,the left and the as exogenously given,while our model provides for right party locate at the 25th and 75th percentile of an electoral model and thus endogenizes the types of the district median distribution.Polborn and Snyder elected legislators. (2017)analyze a model of legislative competition in which only the two parties'national positions matter MODEL for voters and are determined by the median caucus member.Their main focus is on deriving comparative A polity is divided into a set of districts /where #is static predictions about the effects of idiosyncratic (i.e. odd.Each district i contains three strategic agents:a lo- candidate-specific)valence uncertainty and systematic cal Democratic leader.a local Republican leader,and 4 electoral shifts (as in wave elections,where one party a general election median voter. does better in most districts than in previous elections) In the first stage,in each district i,the local leader of on polarization,and testing them empirically.To focus each party P=D,R chooses the position xiP ER of 4 on these comparative static predictions,their model of party P's candidate in district i.The local party lead- candidate competition is much simpler than ours,es- ers can be thought of as a shorthand for the decisive sentially assuming that the local candidates perfectly voter in the respective party's primary election that de- represent the median voter in their respective districts. cides which candidate to nominate.We therefore as- In contrast.our focus is on the effect of the nomina- sume that these local party leaders are not interested in tion process in which candidates are chosen by policy- winning per se,but rather,like any other voter,derive motivated primary voters,who,as we show,generally utility from policy(with details explained below). 4号元 select nominees who are more extreme than the gen- In the second stage,there is a general election in all eral election median voter of the district would prefer districts.In addition to their local candidates'positions. In the influential models of Erikson and Romero each party has a national policy Xp that it can imple- (1990)and Adams and Merrill (2003),voters receive, ment if it receives a majority in the legislature.The na- in addition to the payoff from the elected candidate's tional policy position is not a strategic choice by any position,a "partisan"payoff from his party affiliation, particular player,but rather some aggregate of the po- which,however,is exogenous and orthogonal to his sitions of a party's legislators,discussed further below. policy position.Our model provides a microfoundation The utility of a voter with ideal position 0 from for these partisan payoffs,and shows how they depend district i is on the equilibrium polarization between the parties' candidates in other districts.and how they,in turn,af- g(Xp,x.Q)=-(1-y)Xp-9)2-y(c.o-8)2, fect the candidates'equilibrium positions Probabilistic voting models (e.g.,Lindbeck and (1) Weibull 1987;Dixit and Londregan 1995),as well as differentiated candidates models (DCMs;Aragones where y e(0,1)is the voter's weight on the local rep- and Palfrey 2002:Soubeyran 2009:Krasa and Polborn resentative's position,the policy of the district's elected 2010a,2010b,2012,2014;Camara2012),often con- representative is xi.o,where O[D,R]is the represen- 8 sider an exogenous valence dimension.In the spirit tative's party,and the policy of the majority party P in of the DCM,one can interpret party affiliation in our the legislature is Xp. model as a fixed characteristic,but in contrast to exist- If voters only care about the policy implemented by ing DCMs,voters'preferences over characteristics(i.e., the legislature,then the value of y is zero.There are the candidates'party affiliations)depend on national at least two conceptually distinct reasons why y might party positions and therefore,ultimately,on positions be positive.First,voters may attach an expressive value of candidates in other districts. to their actual vote;that is,a voter may derive utility Our model assumes that national party positions from voting for a local candidate whose position he matter for voters,and a significant number of mod- likes (or opposing one that he dislikes)even if he rec- els explains why this is so.Conditional party govern- ognizes that national policy is determined by national ment theory (Rohde 2010;Aldrich 1995)and endoge- party positions.Second,representatives may have spe- nous party government theory (Volden and Bergman cial influence on policy that is particularly relevant for 2006;Patty 2008)argue that party leaders can use in- their district,for example,through funding projects in centives and resources to ensure cohesiveness of their their district whose payoffs depend on ideology.Note party.Procedural cartel theory (Cox and McCubbins that the case where each election is completely inde- 2005)argues that party leadership can at least en- pendent of what happens in the rest of the country(i.e., force voting discipline over procedural issues.Castan- where literally "all politics is local"and nobody cares about national legislation)corresponds to y =1. Halberstam and Montagnes(2015)provide empirical evidence of spillovers from national presidential campaigns on Senate elections 7 In principle,we could also allow for y to vary between districts, and the positions of candidates in those elections. and all of our results would generalize(with y being replaced by yi 811

Political Competition in Legislative Elections Snyder (1994) considers a dynamic setting in which voters care only about national party positions that are chosen by the party’s representatives in the pre￾election legislature to maximize their individual reelec￾tion chances. In Ansolabehere, Leblanc, and Snyder (2012), a special version of this model, the left and the right party locate at the 25th and 75th percentile of the district median distribution. Polborn and Snyder (2017) analyze a model of legislative competition in which only the two parties’ national positions matter for voters and are determined by the median caucus member. Their main focus is on deriving comparative static predictions about the effects of idiosyncratic (i.e., candidate-specific) valence uncertainty and systematic electoral shifts (as in wave elections, where one party does better in most districts than in previous elections) on polarization, and testing them empirically. To focus on these comparative static predictions, their model of candidate competition is much simpler than ours, es￾sentially assuming that the local candidates perfectly represent the median voter in their respective districts. In contrast, our focus is on the effect of the nomina￾tion process in which candidates are chosen by policy￾motivated primary voters, who, as we show, generally select nominees who are more extreme than the gen￾eral election median voter of the district would prefer. In the influential models of Erikson and Romero (1990) and Adams and Merrill (2003), voters receive, in addition to the payoff from the elected candidate’s position, a “partisan” payoff from his party affiliation, which, however, is exogenous and orthogonal to his policy position.Our model provides a microfoundation for these partisan payoffs, and shows how they depend on the equilibrium polarization between the parties’ candidates in other districts, and how they, in turn, af￾fect the candidates’ equilibrium positions. Probabilistic voting models (e.g., Lindbeck and Weibull 1987; Dixit and Londregan 1995), as well as differentiated candidates models (DCMs; Aragones and Palfrey 2002; Soubeyran 2009; Krasa and Polborn 2010a, 2010b, 2012, 2014; Camara 2012), often con￾sider an exogenous valence dimension. In the spirit of the DCM, one can interpret party affiliation in our model as a fixed characteristic, but in contrast to exist￾ing DCMs, voters’ preferences over characteristics (i.e., the candidates’ party affiliations) depend on national party positions and therefore, ultimately, on positions of candidates in other districts. Our model assumes that national party positions matter for voters,6 and a significant number of mod￾els explains why this is so. Conditional party govern￾ment theory (Rohde 2010; Aldrich 1995) and endoge￾nous party government theory (Volden and Bergman 2006; Patty 2008) argue that party leaders can use in￾centives and resources to ensure cohesiveness of their party. Procedural cartel theory (Cox and McCubbins 2005) argues that party leadership can at least en￾force voting discipline over procedural issues. Castan- 6 Halberstam and Montagnes (2015) provide empirical evidence of spillovers from national presidential campaigns on Senate elections and the positions of candidates in those elections. heira and Crutzen (2010), Eguia (2011a, 2011b) and Diermeier and Vlaicu (2011) provide theories of en￾dogenous institution choice leading to powerful par￾ties. All these models of the importance of parties in Congress take the distribution of legislator preferences as exogenously given, while our model provides for an electoral model and thus endogenizes the types of elected legislators. MODEL A polity is divided into a set of districts I, where #I is odd. Each district i contains three strategic agents: a lo￾cal Democratic leader, a local Republican leader, and a general election median voter. In the first stage, in each district i, the local leader of each party P = D, R chooses the position xi,P ∈ R of party P’s candidate in district i. The local party lead￾ers can be thought of as a shorthand for the decisive voter in the respective party’s primary election that de￾cides which candidate to nominate. We therefore as￾sume that these local party leaders are not interested in winning per se, but rather, like any other voter, derive utility from policy (with details explained below). In the second stage, there is a general election in all districts. In addition to their local candidates’ positions, each party has a national policy XP that it can imple￾ment if it receives a majority in the legislature. The na￾tional policy position is not a strategic choice by any particular player, but rather some aggregate of the po￾sitions of a party’s legislators, discussed further below. The utility of a voter with ideal position θ from district i is uθ (XP, xi,Q) = −(1 − γ )(XP − θ ) 2 − γ (xi,Q − θ ) 2 , (1) where γ ∈ (0, 1) is the voter’s weight on the local rep￾resentative’s position, the policy of the district’s elected representative is xi, Q, where Q ∈ {D, R} is the represen￾tative’s party, and the policy of the majority party P in the legislature is XP. If voters only care about the policy implemented by the legislature, then the value of γ is zero. There are at least two conceptually distinct reasons why γ might be positive. First, voters may attach an expressive value to their actual vote; that is, a voter may derive utility from voting for a local candidate whose position he likes (or opposing one that he dislikes) even if he rec￾ognizes that national policy is determined by national party positions. Second, representatives may have spe￾cial influence on policy that is particularly relevant for their district, for example, through funding projects in their district whose payoffs depend on ideology. Note that the case where each election is completely inde￾pendent of what happens in the rest of the country (i.e., where literally “all politics is local” and nobody cares about national legislation) corresponds to γ = 1.7 7 In principle, we could also allow for γ to vary between districts, and all of our results would generalize (with γ being replaced by γ i 811 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

Stefan Krasa and Mattias K.Polborn Ex ante,there is uncertainty about the ideal position Let the indifferent voter type for whom Equation(2) 马 of district i's median voter,described by a cdf ()that holds as equality be denoted by is symmetric about ui.Let pi denote the probability that district i is decisive in determining which party has a majority in the legislature(i.e.,pi is the probability 1(1-y)p:(X3-X3)+y(xR-D) that both parties win the same number of representa- 0(xL.D.*R)=(1-Y)p:(XR-Xp)+y(iR-x tives in all other districts ji).Note that pi can either (3) be derived by the equilibrium played in other districts Then Mi prefers to vote for the Democrat if and only or can simply be thought of as reflecting the(not neces- if Mis(xi.D,xi.R).The probability that the Democrat sarily rational)perception of the voters in district i that wins in district i is therefore (xiD,xi.R)). their district is pivotal. For example,if the district cannot be pivotal(pi=0) In summary,the game proceeds as follows: then Equation (3)implies that 0()=(xi.R +x.D)/2 that is,the median voter chooses the candidate whose 1.In each district i.the local Democratic leader with platform is closest to Mi.The probability that the ideal point di selects the Democratic candidate's Democrat wins is thereforeΦ，(ktD). position to maximize his expected utility,taking as Now consider the first stage of the game.The Demo- given the probability that the district is decisive,pi, cratic leader with ideal point di chooses xi.p to maxi- and the party policies,Xp and XR.Similarly,the lo- mize his expected payoff,that is, cal Republican leader with ideal point ri selects xi.R, with an analogous objective. 2.In each district i,the median voter Mi is realized,and max-Φ，(e(cD,.R)(1-y) votes for his preferred candidate,if any.If M;is indif- ferent between Democrat and Republican,he votes ×p(Xn-d2+yxD-d)2） 4号 for the candidate of the party whose national posi- tion he prefers(if any),or otherwise randomizes.10 -(（1-,(6(cD,xR(1-y) EQUILIBRIUM ×p(XR-d)2+ycR-d)P） Our analysis starts with stage 2 of the game.The local -(1-p)1-y)Erl(Xp-d)2] (4) median voter Mi has expected utility piuM,(XR,xi.R)+ (1-pi)Ep[uM,(Xp,xi.R)]if the Republican candidate Similarly,the Republican primary voter solves wins,where Ep[.is the expectation over which party wins a majority,given that district i is not pivotal.Sim- ilarly,if the Democrat wins,Mi's expected payoff is max-,(8.D,R)1-y） PiuM(XD.xi.D)+(1-Pi)Ep[uM (Xp,xi.D)].Using the utility function in Equation(1),it follows that median voter Mi prefers the Democrat to the Republican if x pi(Xp-n)2+y(xi.D-n)2) -(1-Φ，(e(x.D,xR)(1-y) -pi(1-y)(Xp Mi)2-y(xD.i-Mi)2 -P:(1-Y)(XR-M)2-y(xR.i-M)}2.(2) ×p:(XR-n)2+Y(x.R-r)） -(1-p)1-y)E[(Xp-r)2] (5) Definition 1 Policies xi.p and xi.R are an equilibrium in in most formulas).We refrain from doing so explicitly to keep the notation simpler. district i if and only if xi.D solves Equation (4)and xi R If citizens vote only because of the probability that their vote makes solves Equation (5). a difference for the election outcome,actual participation levels in large elections can,with any positive cost of voting.only be rational- If pi=0,then all terms that reflect the parties'na- ized if voters mistakenly believe that the pivot probability is much tional positions drop out and the optimization problem higher than it actually is.For example,in the entire history of U.S. reduces to that of a standard Calvert-Wittman model. elections,no single voter has ever been pivotal for the outcome in a Congressional race.In a similar vein,voters may also overestimate As mentioned above,in this case (xi.D,xi.R)=(xi.D the pivot probability of their own district.(Of course,participation +xi.R)/2.Thus,after substituting pi=0,dropping the in elections can also be rationalized through a sufficiently large civic third(constant)term in Equation(5),and dividing by benefit from the act of voting,or a rule utilitarian paradigm (Coate y,the Republican primary voter's problem is equiva- and Conlin 2004;Feddersen and Sandroni 2006).) lent to After the main analysis,we will endogenize the party policies XD and XR as aggregations of a party's successful candidates'policies. 10 We will point out below where we use this tie-breaking assumption that,in case of indifference,the median voter votes for the candidate 2D-)2 /:sony of the party whose national position he prefers.Other tie-breaking assumptions would be slightly more cumbersome to work with,but not lead to qualitatively different results. (-( .R-n)2; 812

Stefan Krasa and Mattias K. Polborn Ex ante, there is uncertainty about the ideal position of district i’s median voter, described by a cdf i(·) that is symmetric about μi. Let pi denote the probability that district i is decisive in determining which party has a majority in the legislature (i.e., pi is the probability that both parties win the same number of representa￾tives in all other districts j = i). Note that pi can either be derived by the equilibrium played in other districts or can simply be thought of as reflecting the (not neces￾sarily rational) perception of the voters in district i that their district is pivotal.8 In summary, the game proceeds as follows: 1. In each district i, the local Democratic leader with ideal point di selects the Democratic candidate’s position to maximize his expected utility, taking as given the probability that the district is decisive, pi, and the party policies, XD and XR. 9 Similarly, the lo￾cal Republican leader with ideal point ri selects xi, R, with an analogous objective. 2. In each district i, the median voter Mi is realized, and votes for his preferred candidate,if any. If Mi is indif￾ferent between Democrat and Republican, he votes for the candidate of the party whose national posi￾tion he prefers (if any), or otherwise randomizes.10 EQUILIBRIUM Our analysis starts with stage 2 of the game. The local median voter Mi has expected utility piuMi(XR, xi,R) + (1 − pi)EP[uMi(XP, xi,R)] if the Republican candidate wins, where EP[ · ] is the expectation over which party wins a majority, given that district i is not pivotal. Sim￾ilarly, if the Democrat wins, Mi’s expected payoff is piuMi(XD, xi,D) + (1 − pi)EP[uMi(XP, xi,D)]. Using the utility function in Equation (1), it follows that median voter Mi prefers the Democrat to the Republican if −pi(1 − γ )(XD − Mi) 2 − γ (xD,i − Mi) 2 ≥ − pi(1 − γ )(XR − Mi) 2 − γ (xR,i − Mi) 2 . (2) in most formulas). We refrain from doing so explicitly to keep the notation simpler. 8 If citizens vote only because of the probability that their vote makes a difference for the election outcome, actual participation levels in large elections can, with any positive cost of voting, only be rational￾ized if voters mistakenly believe that the pivot probability is much higher than it actually is. For example, in the entire history of U.S. elections, no single voter has ever been pivotal for the outcome in a Congressional race. In a similar vein, voters may also overestimate the pivot probability of their own district. (Of course, participation in elections can also be rationalized through a sufficiently large civic benefit from the act of voting, or a rule utilitarian paradigm (Coate and Conlin 2004; Feddersen and Sandroni 2006).) 9 After the main analysis, we will endogenize the party policies XD and XR as aggregations of a party’s successful candidates’ policies. 10 We will point out below where we use this tie-breaking assumption that, in case of indifference, the median voter votes for the candidate of the party whose national position he prefers. Other tie-breaking assumptions would be slightly more cumbersome to work with, but not lead to qualitatively different results. Let the indifferent voter type for whom Equation (2) holds as equality be denoted by θ (xi,D, xi,R) = 1 2 (1 − γ )pi(X2 R − X2 D) + γ (x2 i,R − x2 i,D) (1 − γ )pi(XR − XD) + γ (xi,R − xi,D) . (3) Then Mi prefers to vote for the Democrat if and only if Mi θ(xi, D, xi, R). The probability that the Democrat wins in district i is therefore i(θ(xi, D, xi, R)). For example, if the district cannot be pivotal (pi = 0), then Equation (3) implies that θ(·) = (xi, R + xi, D)/2, that is, the median voter chooses the candidate whose platform is closest to Mi. The probability that the Democrat wins is therefore i xi,R+xi,D 2 . Now consider the first stage of the game. The Demo￾cratic leader with ideal point di chooses xi, D to maxi￾mize his expected payoff, that is, max xi,D − i(θ (xi,D, xi,R)) (1 − γ ) × pi(XD − di) 2 + γ (xi,D − di) 2  −  1 − i(θ (xi,D, xi,R))(1 − γ ) × pi(XR − di) 2 + γ (xi,R − di) 2  − (1 − pi)(1 − γ )EP  (XP − di) 2 . (4) Similarly, the Republican primary voter solves max xi,R − i(θ (xi,D, xi,R)) (1 − γ ) × pi(XD − ri) 2 + γ (xi,D − ri) 2  −  1 − i(θ (xi,D, xi,R))(1 − γ ) × pi(XR − ri) 2 + γ (xi,R − ri) 2  − (1 − pi)(1 − γ )EP  (XP − ri) 2 . (5) Definition 1 Policies xi, D and xi, R are an equilibrium in district i if and only if xi, D solves Equation (4) and xi, R solves Equation (5). If pi = 0, then all terms that reflect the parties’ na￾tional positions drop out and the optimization problem reduces to that of a standard Calvert-Wittman model. As mentioned above, in this case θ(xi, D, xi, R) = (xi, D + xi, R)/2. Thus, after substituting pi = 0, dropping the third (constant) term in Equation (5), and dividing by γ , the Republican primary voter’s problem is equiva￾lent to max xi,R − xi,D + xi,R 2 (xi,D − ri) 2 −  1 −  xi,D + xi,R 2 (xi,R − ri) 2 ; (6) 812 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 and similar for the Democrat.Formally,this is equiva- didate positions xi.D,then a Republican candidate at lent to the standard problem of policy motivated candi- Xi R =i wins. dates with ideal points ri and di,respectively,who trade It is useful to define the average party policy,that off the probability of winning and selecting a policy is,the voter type who is indifferent between the two that is closer to their ideal point.1 national party positions,as=.If x is suffi- If,instead,pi>0,there is no closed form solution ciently far from the possible location of district i's me- for the equilibrium in this case.2 To learn something dian voter,then the candidate choice of the disadvan- about the structure of equilibrium,we proceed with our taged party does not constrain the advantaged party,so analysis by varying the importance of national and lo- that electoral competition has no moderating effect.In cal issues for voters,y.Note that this importance can the remainder of this section,we now turn to the other plausibly vary over time and between different cham- case.We say that candidates face effective competition bers.3 In the first of the following subsections,we deal if a Democrat located at di or a Republican located at with the case that the uncertainty about the median r;would lose with positive probability voter's position is described by a uniform distribution with sufficiently small range.In the second and third Assumption 1 There exist xi.D,xi.R ER such that ui- subsection,we analyze the limit cases of y0(i.e. c0 such that,if c ui+c for all possible positions xi.R of the Republican candidate,4 then the Democrats can win 2.If X>ui then the Democrat wins with prob- for sure in district i by nominating a candidate with po- ability one in any pure strategy equilibrium, and the equilibrium candidate positions in sition xi D =di(which,clearly,is optimal in this case). Similarly,if (xi.D,r)ui-c and c is small,then the 813

Political Competition in Legislative Elections and similar for the Democrat. Formally, this is equiva￾lent to the standard problem of policy motivated candi￾dates with ideal points ri and di, respectively, who trade off the probability of winning and selecting a policy that is closer to their ideal point.11 If, instead, pi > 0, there is no closed form solution for the equilibrium in this case.12 To learn something about the structure of equilibrium,we proceed with our analysis by varying the importance of national and lo￾cal issues for voters, γ . Note that this importance can plausibly vary over time and between different cham￾bers.13 In the first of the following subsections, we deal with the case that the uncertainty about the median voter’s position is described by a uniform distribution with sufficiently small range. In the second and third subsection, we analyze the limit cases of γ ≈ 0 (i.e., voters care mostly about national party positions) and γ ≈ 1 (the Calvert-Wittman benchmark); we can do this without imposing any assumptions on the uncer￾tainty about the median voter positions. Intermediate Weights on Local vs. National Positions We start with the general case that voters care about both local candidate positions and national party po￾sitions in a way that both components have a non￾negligible weight in their utility function (i.e., γ takes an intermediate value).Tractability requires us to focus on the case where the distribution of the median voter is uniform on some interval [μi − c,μi + c], with c small. Given that the uniform distribution has bounded support, in sufficiently extreme districts, the favored party’s advantage is so large that they can win in dis￾trict i even if they choose a candidate who is located at the respective leader’s ideal point. For example, if θ(di, xi, R) > μi + c for all possible positions xi, R of the Republican candidate,14 then the Democrats can win for sure in district i by nominating a candidate with po￾sition xi, D = di (which, clearly, is optimal in this case). Similarly, if θ(xi, D, ri) 0 such that, if c μi then the Democrat wins with prob￾ability one in any pure strategy equilibrium, and the equilibrium candidate positions in district i are xi,D = μi + c − 1 − γ γ 2pi(XR − XD)(X − μi − c), xi,R = μi + c. (8) 3. If X = μi and ri − μi = μi − di then xi, R = −xi, D = μi + ric/(ri + c), and both can￾didates win with equal probability. Note first that,if national party positions differ and if no two districts have the same expected median voter positions, then either all or almost all districts are in cases 1 or 2. Consider the case that X μi − c and c is small, then the 813 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

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| 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. 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Stefan Krasa and Mattias K.Polborn ui)to nominate a more extreme candidate. voter is to the left of the expected median voter,hence As Xg increases further beyond ui,the Republican ad- with probability 1/2. vantage in district i decreases which generally (for s not What happens when y is now slightly decreased? too small)makes a somewhat more moderate candi- For concreteness,suppose that the expected median date optimal for Republicans.When Xg becomes large, voter in district i prefers the Democratic national po- district i starts to favor the Democratic national posi- sition.If y<1 and the realized median voter is equal tion (which will move xi.D toward di),and this forces to the expected one,then the Democrat wins,so that the Republican candidate toward(Xp +XR)/2. the indifferent voter type is now more conservative than in the case of y=1.The effect on the strate- Equilibrium when Local Concerns are gic location incentives of both local leaders is as fol- Dominant(y≈1) lows:For the same extent of moderation,the indiffer- ent voter type is more responsive to movements in the We finally turn to the other polar case,namely that vot- Democratic candidate's position than to that of the Re- ers care primarily about the two local candidates'posi- publican because the indifferent voter is farther away tions(y-1).The limit case ofy =1 is the well-known from the Democrat's position than from the Republi- Calvert-Wittman model,so the main point of interest can's position.15 Thus,the Democrat's marginal benefit in this section is how a minimal voter concern about from moderation increases,and the Republican's bene- national party positions affects the parties'strategic lo- fit from moderation decreases.Since the marginal cost cation incentives,relative to the Calvert-Wittman case. of moderation is unchanged for both,this means that Superficially,and from the results derived so far,it the Democratic equilibrium position will become more 4 may seem as if voters'national concerns should al- moderate,and the Republican one less so. ways lead to more polarization,relative to a standard This intuition is similar to the one in Groseclose Calvert-Wittman model in which each party trades off (2001).who analyzes the effect of a candidate's va more distance from their bliss point against some gain lence advantage on the positions chosen by the (policy- in the probability of winning.As y decreases from 1,the motivated)candidates and finds that a small valence median voter's responsiveness to local positions is low- advantage induces the favored candidate to move to- ered,and consequently nominating a candidate closer ward the expected median voter,and the disadvan- to the party's ideal point becomes less costly taged candidate to move away,with the latter move- However,nominating a more extremist candidate ment being bigger so that polarization,as measured by also becomes less attractive for the local leader because the distance between the candidates,increases.How- his utility also increasingly derives from national posi- ever,there is an interesting difference:While valence tions,and winning the local race may determine which in Groseclose's model is uniformly appreciated by all party has the majority in the legislature.From this per- voter types,the extent and even the sign of the Demo- spective,nominating a more moderate candidate ap- crat's net advantage varies with the preference type pears more attractive,and the net effect of y on polar- of the realized median voter in our model.This is the ization close to y =1 is therefore unclear.Proposition 3 reason why the polarization result is different in our analyzes this case. model.that is,for small changes of y away from 1,the distance between the equilibrium Democratic and Re- Proposition 3 Consider a district where the positions of publican position is unchanged(to the first order). the local party leaders are symmetric around the median voter's expected position (i.e.,ri-ui=ui-di).Suppose that,starting from a situation in which all voters only A CLOSER LOOK AT THE EFFECTS OF care about local candidates'positions (i.e.,y =1),y de- GERRYMANDERING creases slightly.Then,the position of the candidate of the advantaged party (i.e,the one whose national position In this section.we analyze how a change in the distri- the expected median voter prefers)moves toward ui,and bution of district median voters affects the equilibrium the opponent's position moves away from ui.Moreover, degree of polarization,in particular in those districts the first-order change in local polarization is zero,i.e., that are not directly affected by the preference change 品(cR-.D川y=1=0. For our model,it does not matter whether the change in the district median distribution was brought about Proof.See Appendix. ■ intentionally,through gerrymandering,or unintention- ally through voter sorting (say,conservatives moving With the symmetry assumption with respect to to conservative states,and liberals to liberal states).In the two local leaders,it is easy to characterize the the next subsection,we focus on the effect that works L equilibrium of the Calvert-Wittman model.Leaders choose positions to trade off the benefit of choos- ing a more moderate position-an increased winning probability-with the cost of a more moderate posi- 1s Remember that,with astrictly concave utility function,the median voter appreciates the same amount of policy moderation the more, tion,namely that the policy,if the candidate wins,is the farther away a candidate is. 816

Stefan Krasa and Mattias K. Polborn μi) to nominate a more extreme candidate. As XR increases further beyond μi, the Republican ad￾vantage in district i decreases which generally (fors not too small) makes a somewhat more moderate candi￾date optimal for Republicans.When XR becomes large, district i starts to favor the Democratic national posi￾tion (which will move xi, D toward di), and this forces the Republican candidate toward (XD + XR)/2. Equilibrium when Local Concerns are Dominant (γ ≈ 1) We finally turn to the other polar case, namely that vot￾ers care primarily about the two local candidates’ posi￾tions (γ → 1). The limit case of γ = 1 is the well-known Calvert-Wittman model, so the main point of interest in this section is how a minimal voter concern about national party positions affects the parties’ strategic lo￾cation incentives, relative to the Calvert-Wittman case. Superficially, and from the results derived so far, it may seem as if voters’ national concerns should al￾ways lead to more polarization, relative to a standard Calvert-Wittman model in which each party trades off more distance from their bliss point against some gain in the probability of winning.As γ decreases from 1, the median voter’s responsiveness to local positions is low￾ered, and consequently nominating a candidate closer to the party’s ideal point becomes less costly. However, nominating a more extremist candidate also becomes less attractive for the local leader because his utility also increasingly derives from national posi￾tions, and winning the local race may determine which party has the majority in the legislature. From this per￾spective, nominating a more moderate candidate ap￾pears more attractive, and the net effect of γ on polar￾ization close to γ = 1 is therefore unclear. Proposition 3 analyzes this case. Proposition 3 Consider a district where the positions of the local party leaders are symmetric around the median voter’s expected position (i.e., ri − μi = μi − di). Suppose that, starting from a situation in which all voters only care about local candidates’ positions (i.e., γ = 1), γ de￾creases slightly.Then,the position of the candidate of the advantaged party (i.e., the one whose national position the expected median voter prefers) moves toward μi, and the opponent’s position moves away from μi. Moreover, the first-order change in local polarization is zero, i.e., ∂ ∂γ (xi,R − xi,D)|γ =1 = 0. Proof. See Appendix. With the symmetry assumption with respect to the two local leaders, it is easy to characterize the equilibrium of the Calvert-Wittman model. Leaders choose positions to trade off the benefit of choos￾ing a more moderate position—an increased winning probability—with the cost of a more moderate posi￾tion, namely that the policy, if the candidate wins, is farther away from the leader’s preferred position. The two candidates’ equilibrium positions are symmetric about the position of the expected median voter, and the Democrat wins if and only if the realized median voter is to the left of the expected median voter, hence with probability 1/2. What happens when γ is now slightly decreased? For concreteness, suppose that the expected median voter in district i prefers the Democratic national po￾sition. If γ < 1 and the realized median voter is equal to the expected one, then the Democrat wins, so that the indifferent voter type is now more conservative than in the case of γ = 1. The effect on the strate￾gic location incentives of both local leaders is as fol￾lows: For the same extent of moderation, the indiffer￾ent voter type is more responsive to movements in the Democratic candidate’s position than to that of the Re￾publican because the indifferent voter is farther away from the Democrat’s position than from the Republi￾can’s position.15 Thus, the Democrat’s marginal benefit from moderation increases, and the Republican’s bene￾fit from moderation decreases. Since the marginal cost of moderation is unchanged for both, this means that the Democratic equilibrium position will become more moderate, and the Republican one less so. This intuition is similar to the one in Groseclose (2001), who analyzes the effect of a candidate’s va￾lence advantage on the positions chosen by the (policy￾motivated) candidates and finds that a small valence advantage induces the favored candidate to move to￾ward the expected median voter, and the disadvan￾taged candidate to move away, with the latter move￾ment being bigger so that polarization, as measured by the distance between the candidates, increases. How￾ever, there is an interesting difference: While valence in Groseclose’s model is uniformly appreciated by all voter types, the extent and even the sign of the Demo￾crat’s net advantage varies with the preference type of the realized median voter in our model. This is the reason why the polarization result is different in our model, that is, for small changes of γ away from 1, the distance between the equilibrium Democratic and Re￾publican position is unchanged (to the first order). A CLOSER LOOK AT THE EFFECTS OF GERRYMANDERING In this section, we analyze how a change in the distri￾bution of district median voters affects the equilibrium degree of polarization, in particular in those districts that are not directly affected by the preference change. For our model, it does not matter whether the change in the district median distribution was brought about intentionally, through gerrymandering, or unintention￾ally through voter sorting (say, conservatives moving to conservative states, and liberals to liberal states). In the next subsection, we focus on the effect that works 15 Remember that, with a strictly concave utility function, the median voter appreciates the same amount of policy moderation the more, the farther away a candidate is. 816 Downloaded from https://www.cambridge.org/core. 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Political Competition in Legislative Elections through affecting the pivot probabilities,and in the fol- political polarization (i.e.,the distance between the lo- lowing subsection on the effect of endogenous party cal candidates).is small. platforms. Now suppose that an even number of centrist dis- tricts are transformed into an equal number of left- Endogenous Pivot Probabilities leaning and right-leaning districts.As the number of centrist districts decreases,p;increases,and Equations So far,we have interpreted pi,the probability with (7)and (8)imply that the distance between the can- which voters in district i believe that their district is didates in the left-and right-leaning districts increases pivotal for the majority in the legislature,as an exoge- and that the advantaged candidates become more ex- nously given parameter.In this section,we endogenize treme.For example,if only one centrist district remains, Pi and show that this gives rise to an externality be then the pivot probability in the left and right-leaning tween districts.Specifically,we show that a decrease in districts increases all the way up to pi=1/2. the number of centrist districts leads to an increase in Note that,in each step of removing centrist districts polarization. and transforming them into left-and right-leaning dis- Consider a symmetric setting with k left-leaning dis- tricts,the candidates become more extreme in both the tricts,k right-leaning districts,and 2m+1 centrist dis- newly-created districts and in those districts that were tricts.In the left-leaning and right-leaning districts,the already left-leaning or right-leaning.Thus,if an empir- median voters are uniformly distributed on [-u-c, ical researcher were to"difference out the time-trend -u+c]and [u-c,u+c],respectively,where u >0. of polarization"(by looking at changes in polarization In the centrist districts,the respective median voter is in newly-gerrymandered districts,versus existing dis- uniformly distributed on [-c,c].Given the symmetry tricts),he would clearly underestimate the effects of of the model,we assume that party positions,Xp and gerrymandering. XR are symmetric around zero,so that X=0.16 Intuitively,in the centrist districts the electorate is Proposition 1 determines the candidates'positions willing to switch between candidates.If there are many 4号元 and winning probabilities (when c is small).In the such swing districts,then the election in the partisan centrist districts,the candidates'equilibrium positions district is less likely to determine the control of the are close to zero,independent of the pivot probability, legislature.As a consequence,voters in a conservative- and each candidate wins with probability 1/2.In right- leaning district would be more willing to give the leaning districts,X=0-u which means erate to be competitive.If,however,there are few cen- that the Democrats win. trist districts,the probability that that particular district To determine the candidates'equilibrium positions is decisive for the majority in the legislature increases in left and right-leaning districts,we now derive the In this case.conservative voters in the same district are pivot probabilities,Pi.Consider a particular right- less likely to support the local Democratic candidate leaning district.In equilibrium,all other right-leaning because they are more concerned that voting for him districts vote for Republicans and all left-leaning dis- will result in Democratic control of the legislature.This, tricts for Democrats.Given that the total number of in turn,means that even a more extreme Republican districts is odd,the pivotal event occurs if exactly m+1 candidate can win,and the Republican base can exploit of the centrist districts vote for the Republican candi- this effect by nominating a more extreme candidate. date-in this case Republicans get m +1+(k-1) legislators,excluding the selected right-leaning district, while Democrats have m+k legislators,that is,there Endogenous Party Positions is a tie.This occurs with probability So far,we have taken the national party positions as exogenous and independent of the election outcome +1) in any specific district.Fixing the party positions may be realistic in the short run if party positions are de- termined by some party elite that the individual repre- Note that p(m)decreases in m and,by Stirling's for- sentative cannot influence,or because,even if the party mula,the pivot probability goes to zero at rate 1/m. position is jointly decided by the party's elected legisla- Suppose we start in a situation with many centrist tors,the party position that voters perceive at the time districts,as well as one left-leaning and one right- of the election is unlikely to be conditional on the out- leaning district.Then,the pivot probabilities for the come of the election in the district in question.In the left-leaning and right-leaning districts are close to zero long run,though,it is useful to consider a situation in and Equations (7)and (8)imply that both candidates which a party's position is an aggregate of the positions are located close to -u-c in the left-leaning district, of its winning candidates. and to u+c in the right-leaning district.Hence,local We interpret XR and Xp as the party policies that voters expect to be implemented if the respective par- 6 In the next subsection.we will analyze a model in which party po ties win a majority.It is plausible that these expected sitions are determined endogenously from the positions of the indi- policies are a function of the positions of the individual vidual representatives. candidates,weighted by their respective probability of 817

Political Competition in Legislative Elections through affecting the pivot probabilities, and in the fol￾lowing subsection on the effect of endogenous party platforms. Endogenous Pivot Probabilities So far, we have interpreted pi, the probability with which voters in district i believe that their district is pivotal for the majority in the legislature, as an exoge￾nously given parameter. In this section, we endogenize pi and show that this gives rise to an externality be￾tween districts. Specifically, we show that a decrease in the number of centrist districts leads to an increase in polarization. Consider a symmetric setting with k left-leaning dis￾tricts, k right-leaning districts, and 2m + 1 centrist dis￾tricts. In the left-leaning and right-leaning districts, the median voters are uniformly distributed on [ − μ − c, −μ + c] and [μ − c, μ + c], respectively, where μ > 0. In the centrist districts, the respective median voter is uniformly distributed on [ − c, c]. Given the symmetry of the model, we assume that party positions, XD and XR are symmetric around zero, so that X = 0.16 Proposition 1 determines the candidates’ positions and winning probabilities (when c is small). In the centrist districts, the candidates’ equilibrium positions are close to zero, independent of the pivot probability, and each candidate wins with probability 1/2. In right￾leaning districts,X = 0 −μ which means that the Democrats win. To determine the candidates’ equilibrium positions in left and right-leaning districts, we now derive the pivot probabilities, pi. Consider a particular right￾leaning district. In equilibrium, all other right-leaning districts vote for Republicans and all left-leaning dis￾tricts for Democrats. Given that the total number of districts is odd, the pivotal event occurs if exactly m + 1 of the centrist districts vote for the Republican candi￾date — in this case Republicans get m + 1 + (k − 1) legislators, excluding the selected right-leaning district, while Democrats have m + k legislators, that is, there is a tie. This occurs with probability p(m) = 2m + 1 m + 1 2−(2m+1). (11) Note that p(m) decreases in m and, by Stirling’s for￾mula, the pivot probability goes to zero at rate 1/ √m. Suppose we start in a situation with many centrist districts, as well as one left-leaning and one right￾leaning district. Then, the pivot probabilities for the left-leaning and right-leaning districts are close to zero, and Equations (7) and (8) imply that both candidates are located close to −μ − c in the left-leaning district, and to μ + c in the right-leaning district. Hence, local 16 In the next subsection, we will analyze a model in which party po￾sitions are determined endogenously from the positions of the indi￾vidual representatives. political polarization (i.e., the distance between the lo￾cal candidates), is small. Now suppose that an even number of centrist dis￾tricts are transformed into an equal number of left￾leaning and right-leaning districts. As the number of centrist districts decreases, pi increases, and Equations (7) and (8) imply that the distance between the can￾didates in the left-and right-leaning districts increases, and that the advantaged candidates become more ex￾treme. For example,if only one centrist district remains, then the pivot probability in the left and right-leaning districts increases all the way up to pi = 1/2. Note that, in each step of removing centrist districts and transforming them into left-and right-leaning dis￾tricts, the candidates become more extreme in both the newly-created districts and in those districts that were already left-leaning or right-leaning. Thus, if an empir￾ical researcher were to “difference out the time-trend of polarization” (by looking at changes in polarization in newly-gerrymandered districts, versus existing dis￾tricts), he would clearly underestimate the effects of gerrymandering. Intuitively, in the centrist districts the electorate is willing to switch between candidates. If there are many such swing districts, then the election in the partisan district is less likely to determine the control of the legislature. As a consequence, voters in a conservative￾leaning district would be more willing to give the Democratic candidate a chance, which in turn means that the Republican candidate must be relatively mod￾erate to be competitive. If, however, there are few cen￾trist districts, the probability that that particular district is decisive for the majority in the legislature increases. In this case, conservative voters in the same district are less likely to support the local Democratic candidate because they are more concerned that voting for him will result in Democratic control of the legislature.This, in turn, means that even a more extreme Republican candidate can win, and the Republican base can exploit this effect by nominating a more extreme candidate. Endogenous Party Positions So far, we have taken the national party positions as exogenous and independent of the election outcome in any specific district. Fixing the party positions may be realistic in the short run if party positions are de￾termined by some party elite that the individual repre￾sentative cannot influence, or because, even if the party position is jointly decided by the party’s elected legisla￾tors, the party position that voters perceive at the time of the election is unlikely to be conditional on the out￾come of the election in the district in question. In the long run, though, it is useful to consider a situation in which a party’s position is an aggregate of the positions of its winning candidates. We interpret XR and XD as the party policies that voters expect to be implemented if the respective par￾ties win a majority. It is plausible that these expected policies are a function of the positions of the individual candidates, weighted by their respective probability of 817 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

Stefan Krasa and Mattias K.Polborn getting elected.For example,a Democratic candidate Proposition 4 Suppose there are 2m 1 centrist dis- in Utah who has an extremely small chance of being tricts,k left-leaning,and k right-leaning district in which elected probably has a much smaller impact on the i are uniform distributions,and Assumption I holds. perception of the Democratic position in the rest of the Further,assume that the position of the local leaders is country than a candidate from the urban Northeast, the same in all districts,and that the national party po- who is almost certain to be elected. sitions are given by a weighted sum of the positions of To formalize this notion,we need a bit of notation. the Democratic and Republican legislators,respectively. Let xL.D and xc.D be the positions of the Democratic Suppose that k is increased and m decreased.Then: candidates in the left-leaning and centrist districts,re- spectively.Similarly,let xc.R,and xr.R be the positions of the Republican candidates in centrist and right- 1.The positions of all candidates in the left- leaning districts.We assume that the party positions are leaning and right-leaning districts become a weighted sum of the positions of representatives from more extreme. these districts,that is. 2.Local polarization,xi.R-xiD in left and right-leaning districts increases; XR a(k,m)xR.R+(1-a(k,m))xC.R. 3.Party polarization,XR-Xp,increases. XD a(k,m)xL.D+(1-a(k,m))xC.D, (12) In summary,the results of this section show two channels through which a change in the distribution of where a(k,m)is decreasing in m (i.e.,the expected district medians can affect local and society-wide po- number of representatives from centrist districts)and larization.First,a reduction in the number of swing increasing in k,the number of representatives from districts increases the pivot probability in each district. more partisan districts.Note that because Republicans are elected with zero probability in left-leaning dis- and thereby makes it easier for more extreme candi- 4号元 tricts,the positions of these candidates do not affect dates to win in competitive districts that lean either XR.Similarly,Xp does not depend on the position of liberal or conservative.Second,a reduction in the num- Democratic candidates who run in right-leaning dis- ber of swing districts implies that each party has fewer tricts and lose with probability one.For example,if members from moderate districts in their caucus,and the weights are solely determined by the probability of this shifts the perceived position of each party to be getting elected then a =k/(k m +1/2).Note that more extreme.This,in turn,again increases the local Equation(12)implies that Xp =-XR.Thus,Proposi- party leaders'leeway to nominate more extreme candi- tion 1 implies dates in ideologically favorable,but(in principle)com- petitive districts =H-C 1-y ak,mR+-ak,m》e (u-c-X). EMPIRICAL IMPLICATIONS We now discuss some evidence regarding two empirical Solving this equation for xi.R yields predictions of our model. xi.R (u-c)(1+2aap Partisanship in Legislative and Executive 平 Elections ap(u-c +apa2)(-c)+(1- Our model presumes that,in legislative elections,vot- ers care not only about the positions of their local can- (13) didates,but also about those of the national parties.In contrast,in elections for executive positions,the win- where a=(1-r)ly,a =a(k,m)and p=p(m).8 ner is much freer to implement his position and,conse- Now suppose again that some centrist districts are quently,his association with a party matters less.Thus, transformed into more partisan ones(m↓，k个).Then voters'ideological preferences should have a much a(k,m)increases.The previous subsection showed that larger effect in legislative elections than in executive 四 p(m)increases.Equation(13)immediately implies that ones. xi.R increases.This is summarized formally in Proposi- To analyze this prediction,we consider Guberna- tion 4. torial and U.S.Senate elections from 1978 to 2012. While both of these types of contests are high-profile, statewide races,gubernatorial elections are for execu- 17 For example,the Republican caucus has k members from conser- tive positions while Senate elections are for legislative vative districts and,in expectation,1/2 of the 2m+1 representatives ones.Consistent with the empirical literature,we from centrist districts. 18 The solution is unique here because XR and Xp are linear func. measure the median state ideology by its Partisan tions of policies.For more general specifications of Equation(12),it Voting Index (PVI).that is,the difference of the state's 士 may be possible that there are multiple equilibrium values for Xk average Republican and Democratic Party's vote share and Xp. in the past U.S.Presidential election,relative to the 818

Stefan Krasa and Mattias K. Polborn getting elected. For example, a Democratic candidate in Utah who has an extremely small chance of being elected probably has a much smaller impact on the perception of the Democratic position in the rest of the country than a candidate from the urban Northeast, who is almost certain to be elected. To formalize this notion, we need a bit of notation. Let xL, D and xC, D be the positions of the Democratic candidates in the left-leaning and centrist districts, re￾spectively. Similarly, let xC, R, and xR, R be the positions of the Republican candidates in centrist and right￾leaning districts.We assume that the party positions are a weighted sum of the positions of representatives from these districts, that is, XR = α(k, m)xR,R + (1 − α(k, m))xC,R, XD = α(k, m)xL,D + (1 − α(k, m))xC,D, (12) where α(k, m) is decreasing in m (i.e., the expected number of representatives from centrist districts) and increasing in k, the number of representatives from more partisan districts. Note that because Republicans are elected with zero probability in left-leaning dis￾tricts, the positions of these candidates do not affect XR. Similarly, XD does not depend on the position of Democratic candidates who run in right-leaning dis￾tricts and lose with probability one. For example, if the weights are solely determined by the probability of getting elected then α = k/(k + m + 1/2).17Note that Equation (12) implies that XD = −XR. Thus, Proposi￾tion 1 implies xi,R = μ − c + 2 1 − γ γ  α(k, m)xi,R + (1 − α(k, m)) rc r + c (μ − c − X ), Solving this equation for xi, R yields xi,R = (μ − c)  1 + 2αap + 2 ap(μ − c)  (α + apα2 )(μ − c) + (1 − α) rc r + c , (13) where a = (1 − γ )/γ , α = α(k, m) and p = p(m).18 Now suppose again that some centrist districts are transformed into more partisan ones (m↓, k↑). Then α(k, m) increases. The previous subsection showed that p(m) increases. Equation (13) immediately implies that xi, R increases. This is summarized formally in Proposi￾tion 4. 17 For example, the Republican caucus has k members from conser￾vative districts and, in expectation, 1/2 of the 2m + 1 representatives from centrist districts. 18 The solution is unique here because XR and XD are linear func￾tions of policies. For more general specifications of Equation (12), it may be possible that there are multiple equilibrium values for XR and XD. Proposition 4 Suppose there are 2m + 1 centrist dis￾tricts, k left-leaning, and k right-leaning district in which i are uniform distributions, and Assumption 1 holds. Further, assume that the position of the local leaders is the same in all districts, and that the national party po￾sitions are given by a weighted sum of the positions of the Democratic and Republican legislators, respectively. Suppose that k is increased and m decreased. Then: 1. The positions of all candidates in the left￾leaning and right-leaning districts become more extreme. 2. Local polarization, xi, R − xi, D in left and right-leaning districts increases; 3. Party polarization, XR − XD, increases. In summary, the results of this section show two channels through which a change in the distribution of district medians can affect local and society-wide po￾larization. First, a reduction in the number of swing districts increases the pivot probability in each district, and thereby makes it easier for more extreme candi￾dates to win in competitive districts that lean either liberal or conservative. Second, a reduction in the num￾ber of swing districts implies that each party has fewer members from moderate districts in their caucus, and this shifts the perceived position of each party to be more extreme. This, in turn, again increases the local party leaders’ leeway to nominate more extreme candi￾dates in ideologically favorable, but (in principle) com￾petitive districts. EMPIRICAL IMPLICATIONS We now discuss some evidence regarding two empirical predictions of our model. Partisanship in Legislative and Executive Elections Our model presumes that, in legislative elections, vot￾ers care not only about the positions of their local can￾didates, but also about those of the national parties. In contrast, in elections for executive positions, the win￾ner is much freer to implement his position and, conse￾quently, his association with a party matters less. Thus, voters’ ideological preferences should have a much larger effect in legislative elections than in executive ones. To analyze this prediction, we consider Guberna￾torial and U.S. Senate elections from 1978 to 2012. While both of these types of contests are high-profile, statewide races, gubernatorial elections are for execu￾tive positions while Senate elections are for legislative ones. Consistent with the empirical literature, we measure the median state ideology by its Partisan Voting Index (PVI), that is, the difference of the state’s average Republican and Democratic Party’s vote share in the past U.S. Presidential election, relative to the 818 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