American Political Science Review (2018)112.4.826-843 doi:10.1017/S0003055418000515 American Political Science Association 2018 Primaries and Candidate Polarization:Behavioral Theory and Experimental Evidence JONATHAN WOON University of Pittsburgh o primary elections cause candidates to take extreme,polarized positions?Standard equilibrium analysis predicts full convergence to the median voter's position with complete information, but behavioral game theory predicts divergence when players are policy-motivated and have out-of-equilibrium beliefs.Theoretically,I show that primary elections can cause greater extremism or moderation,depending on the beliefs candidates and voters have about their opponents.In a controlled incentivized experiment,I find that candidates diverge substantially and that primaries have little effect on average positions.Voters employ a strategy that weeds out candidates who are either too moderate or too extreme,which enhances ideological purity without increasing divergence.The analysis highlights the importance of behavioral assumptions in understanding the effects of electoral institutions. The partisan primary system,which favors more ideolog- Political scientists have tested this argument,find- ically pure candidates,has contributed to the election of ing that while there is some evidence to suggest more extreme officeholders and increased political polar- that primary elections promote extremism,the em- ization.It has become a menace to governing. pirical record is mixed.Extremists are more likely to Sen.Charles Schumer(D-NY)! win congressional primaries than moderates (Brady. Han,and Pope 2007),and legislators elected under he divergence between candidates and legisla- closed primaries take more extreme positions than tors from the two major parties is an enduring feature of the American political landscape(An- legislators elected under open primaries(Gerber and Morton 1998).But other analyses find that polariza- solabehere,Snyder,and Stewart 2001;Bonica 2013; tion is largely unrelated to the introduction of direct Poole and Rosenthal 1984,1997).and the fact that polarization is at historically high levels is a signifi- primaries (Hirano et al.2010)and to the variation in the openness of primaries across states(McGhee et al. cant concern for scholars and observers of democratic governance,representation,and public policy(Hacker 2014).At best,primaries may cause polarization un- and Pierson 2006;Mann and Ornstein 2013;McCarty, der limited circumstances(Bullock and Clinton 2011), and despite the divergence of candidate positions,gen- Poole,and Rosenthal 2006,2013).Indeed,politicians eral elections nevertheless exert nontrivial pressure on and the popular press often lay much of the blame for candidates to moderate (Hall 2015:Hirano et al.2010) this phenomenon on partisan primary elections,typ- These findings seem puzzling in light of the basic theory ically employing a simple,intuitively appealing argu- of representation at the heart of this literature that the ment:Candidates take extreme positions because they extremity of primary electorates should directly affect must appeal to partisan primary voters,whose prefer- the extremity of party candidates. ences are more extreme than those of voters in the gen- How,then,should we understand the causal relation- eral election ship between primary elections and candidate position- ing?I examine the connection,both theoretically and Jonathan Woon is a Professor,Department of Political Science,De- partment of Economics(secondary),and Pittsburgh Experimenta experimentally,by comparing elections with and with- Economics Laboratory.4437 Wesley W.Posvar Hall,Pittsburgh,PA. out primaries while holding other features of the elec- 15260(woon@pitt.edu). toral environment constant,including preferences and Thanks to Keith Dougherty,Sandy Gordon,Greg Huber,Scott information.The analysis focuses on a particular as- Moser,Charlie Plott,Danielle Thomsen,Alan Wiseman,the editor and anonymous reviewers,participants at the Yale CSAP American pect of primary elections-how the introduction of vot- Politics Conference.seminar participants at Washington University ers in the candidate selection process affects strategic in St.Louis,University of Oxford (Nuffield College CESS).IC3JM competition between parties-while abstracting away (Juan March Institute),and the Pittsburgh Experimental Economics from many other considerations that might also affect Lab for helpful comments and feedback.I am also indebted to Kris polarization.2 Kanthak for vigorous discussions during the early stages of this project.Previous versions of the paper were presented at the 2014 The contribution of this paper is to develop a more Annual Meeting of the American Political Science Association,the nuanced theoretical understanding of the relationship 2016 Public Choice Society Meeting,and the 2016 Midwest Polit- between primaries and polarization than is portrayed ical Science Association Conference.This research was supported in the existing literature.The key theoretical innova- by the National Science Foundation(SES-1154739)and was ap tion is to move beyond preference-based explanations proved by the University of Pittsburgh Institutional Review Board under protocol PRO14060001.Replication material is available on by treating beliefs as a primitive of the model in a the American Political Science Review Dataverse:https://doi.org/10. way that is ruled out by standard equilibrium analysis 7910/DVN/YUNDMQ. Received:June 27 2016;revised:September 11,2017;accepted:July 12,2018.First published online:September 10,2018. 2 Such considerations include candidate valence,turnout,activists. or campaign contributions (Adams and Merrill 2008:Callander Charles E.Schumer,"Adopt the Open Primary,"New York Times, and Wilson 2007:Hirano,Snyder,and Ting 2009;Hummel 2013: July21,2014. Meirowitz 2005:Snyder and Ting 2011). 826
American Political Science Review (2018) 112, 4, 826–843 doi:10.1017/S0003055418000515 © American Political Science Association 2018 Primaries and Candidate Polarization: Behavioral Theory and Experimental Evidence JONATHAN WOON University of Pittsburgh Do primary elections cause candidates to take extreme, polarized positions? Standard equilibrium analysis predicts full convergence to the median voter’s position with complete information, but behavioral game theory predicts divergence when players are policy-motivated and have out-of-equilibrium beliefs. Theoretically, I show that primary elections can cause greater extremism or moderation, depending on the beliefs candidates and voters have about their opponents. In a controlled incentivized experiment, I find that candidates diverge substantially and that primaries have little effect on average positions. Voters employ a strategy that weeds out candidates who are either too moderate or too extreme, which enhances ideological purity without increasing divergence. The analysis highlights the importance of behavioral assumptions in understanding the effects of electoral institutions. The partisan primary system, which favors more ideologically pure candidates, has contributed to the election of more extreme officeholders and increased political polarization. It has become a menace to governing. — Sen. Charles Schumer (D-NY)1 The divergence between candidates and legislators from the two major parties is an enduring feature of the American political landscape (Ansolabehere, Snyder, and Stewart 2001; Bonica 2013; Poole and Rosenthal 1984, 1997), and the fact that polarization is at historically high levels is a significant concern for scholars and observers of democratic governance, representation, and public policy (Hacker and Pierson 2006; Mann and Ornstein 2013; McCarty, Poole, and Rosenthal 2006, 2013). Indeed, politicians and the popular press often lay much of the blame for this phenomenon on partisan primary elections, typically employing a simple, intuitively appealing argument: Candidates take extreme positions because they must appeal to partisan primary voters, whose preferences are more extreme than those of voters in the general election. Jonathan Woon is a Professor, Department of Political Science, Department of Economics (secondary), and Pittsburgh Experimental Economics Laboratory, 4437 Wesley W. Posvar Hall, Pittsburgh, PA, 15260 (woon@pitt.edu). Thanks to Keith Dougherty, Sandy Gordon, Greg Huber, Scott Moser, Charlie Plott, Danielle Thomsen, Alan Wiseman, the editor and anonymous reviewers, participants at the Yale CSAP American Politics Conference, seminar participants at Washington University in St. Louis, University of Oxford (Nuffield College CESS), IC3JM (Juan March Institute), and the Pittsburgh Experimental Economics Lab for helpful comments and feedback. I am also indebted to Kris Kanthak for vigorous discussions during the early stages of this project. Previous versions of the paper were presented at the 2014 Annual Meeting of the American Political Science Association, the 2016 Public Choice Society Meeting, and the 2016 Midwest Political Science Association Conference. This research was supported by the National Science Foundation (SES-1154739) and was approved by the University of Pittsburgh Institutional Review Board under protocol PRO14060001. Replication material is available on the American Political Science Review Dataverse: https://doi.org/10. 7910/DVN/YUNDMQ. Received: June 27, 2016; revised: September 11, 2017; accepted: July 12, 2018. First published online: September 10, 2018. 1 Charles E. Schumer, “Adopt the Open Primary,” New York Times, July 21, 2014. Political scientists have tested this argument, finding that while there is some evidence to suggest that primary elections promote extremism, the empirical record is mixed. Extremists are more likely to win congressional primaries than moderates (Brady, Han, and Pope 2007), and legislators elected under closed primaries take more extreme positions than legislators elected under open primaries (Gerber and Morton 1998). But other analyses find that polarization is largely unrelated to the introduction of direct primaries (Hirano et al. 2010) and to the variation in the openness of primaries across states (McGhee et al. 2014). At best, primaries may cause polarization under limited circumstances (Bullock and Clinton 2011), and despite the divergence of candidate positions, general elections nevertheless exert nontrivial pressure on candidates to moderate (Hall 2015; Hirano et al. 2010). These findings seem puzzling in light of the basic theory of representation at the heart of this literature that the extremity of primary electorates should directly affect the extremity of party candidates. How, then, should we understand the causal relationship between primary elections and candidate positioning? I examine the connection, both theoretically and experimentally, by comparing elections with and without primaries while holding other features of the electoral environment constant, including preferences and information. The analysis focuses on a particular aspect of primary elections—how the introduction of voters in the candidate selection process affects strategic competition between parties—while abstracting away from many other considerations that might also affect polarization.2 The contribution of this paper is to develop a more nuanced theoretical understanding of the relationship between primaries and polarization than is portrayed in the existing literature. The key theoretical innovation is to move beyond preference-based explanations by treating beliefs as a primitive of the model in a way that is ruled out by standard equilibrium analysis 2 Such considerations include candidate valence, turnout, activists, or campaign contributions (Adams and Merrill 2008; Callander and Wilson 2007; Hirano, Snyder, and Ting 2009; Hummel 2013; Meirowitz 2005; Snyder and Ting 2011). 826 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000515
Primaries and Candidate Polarization in complete information environments.I show that The key finding from the experiment is that pri- primaries can cause polarization or moderation,de- maries appear to cause a kind of ideological purity pending on candidates'beliefs about opposing voters rather than greater extremism.Regardless of whether strategic behavior-even when preferences are held there is a primary election or not,I find that subjects constant.To generate this insight,I rely on ideas from take positions that diverge significantly from the me- behavioral game theory,which retains much of the the dian voter's position.This finding lends support for the oretical apparatus from standard game theory while al- behavioral theory.However,the extent to which pri- lowing for key departures(Camerer 2003).Specifically, maries cause polarization is limited.Greater polariza I allow players to have"incorrect"or"nonequilibrium tion only occurs without feedback such that candidates beliefs about others'actions (Crawford.Costa-Gomes. cannot learn about the behavior of others,and this po- and Iriberri 2013).but assume they are nevertheless larization happens because voters tend to select ex- strategic in the sense that they best respond to what tremists over moderates,even though candidate posi- they think other players do (Camerer.Ho.and Chong tions do not vary with the election format.More pre- 2004;Nagel 1995;Stahl and Wilson 1995).The analysis cisely,the analysis reveals that voters do not support demonstrates that changes in preferences alone are party extremists or party moderates unconditionally. not the only cause of polarization.Instead,beliefs and Instead,they select candidates with intermediate posi- expectations about the strategic behavior of others tions,consistent with their own subjective beliefs about play important roles in conditioning the effect of optimal candidate positions,approximately halfway institutions. between the median voter and their own party's ideal I then turn to the laboratory and conduct a series of point.This behavior generates a greater concentra- experiments to test the effects of primaries on candi- tion of candidate positions around an average that di- date positions and to distinguish between behavioral verges from the median voter.Hence,greater ideolog- assumptions.The chief advantage of the laboratory for ical homogeneity reinforces,rather than exacerbates, theory testing is control(Aldrich and Lupia 2011;Falk polarization 4号元 and Heckman 2009;Morton and Williams 2010),so we can be confident that the observed behavior oc- curs under the conditions specified by the theoretical RELATED LITERATURE model.Importantly,subjects face the same key trade- My analysis follows a long tradition of using spatial off in the experiment as the actors do in the theoretical voting models to understand elections.Although ex- model between increasing the favorability of winning isting spatial models predict candidate divergence in outcomes versus increasing the probability of winning. elections with primaries,they do so in isolation and In the lab,theoretically relevant quantities of interest do not compare them explicitly to elections without that are difficult to measure using observational data primaries (Aronson and Ordeshook 1972;Coleman with any accuracy or without strong assumptions (in 1972:Owen and Grofman 2006).These models also as- particular,preferences and positions)are also known sume that general election outcomes are probabilistic, exactly.Furthermore,experimental manipulations per- which is theoretically consequential because the mech- mit tests of mechanisms not possible using observa- tional data.Thus,laboratory experiments are ideal for anism they rely on to produce divergence is the com- theory testing given their high internal validity.3 bination of policy motivations and uncertainty about which candidate will win the general election-the same forces that generate incentives for candidate di- 3 The main question of interest for theory testing.as Aldrich and Lu- vergence in the absence of primaries (Calvert 1985: pia (2011.90)put it,is"Will people who are in the situations you de- Wittman 1983).Thus,it is unclear from the literature scribe in your model act as you predict?"For related discussions,see whether polarization can be traced to any distinctive Dickson(2011).Palfrey(2006).and especially.Morton and Williams features of primaries per se,as electoral institutions. (2010).While the question of external validity ("To what extent can we generalize from a particular sample?")is an enduring source of By explicitly comparing institutions,my analysis speaks controversy in political science,Falk and Heckman(2009)argue in directly to the connection between primaries and their insightful defense of the value of lab experiments in social sci- polarization. ence that "Behavior in the laboratory is reliable and real:Participants in the lab are real human beings who perceive their behavior as rel- evant,experience real emotions,and take decisions with real eco. nomic consequences"(536).Indeed,there are many precedents for testing theories of elite behavior using laboratory experiments(e.g., strategic than ordinary people (because they have more experi eys Aragones and Palfrey 2007:Frechette,Kagel,and Lehrer 2003:Mor- ence,access to advice,information,etc.),there is surprisingly little ton 1993).Moreover,Druckman and Kam(2011)note that there is evidence to support such claims.To the contrary,recent compar- nothing inherently problematic with using student samples,and there isons by Sheffer et al.(2018)demonstrate that politicians are just as is little evidence to suggest that using undergraduates as stand-ins (and sometimes more)susceptible to choice anomalies than ordinary for elites biases the results in any particular direction (see Morton citizens and Williams 2010,343-7).For example,Potters and van Winden An exception is Jackson,Mathevet,and Mattes (2007),who com- (2000)find significant,but small,differences between students and pare alternative nomination systems in a citizen-candidate frame- lobbyists:Fatas,Neugebauer,and Tamborero(2007)find elites do work.In their model,primary elections affect whose preference is not fit prospect theory as well as students,while studies by Belot decisive in nominating candidates and have no effect if party leaders Duch,and Miller (2015),Cooper et al.(1999),and Mintz,Redd,and and the median party voter have the same preferences.Other formal Vedlitz(2006)suggest that student samples provide a lower bound to models of primary elections largely focus on considerations of voter departures from rational decision-making.Despite the common as. uncertainty,incomplete information,and signaling along with issues sertion that politicians must be better decision makers and more of candidate valence and distributional concerns. 827
Primaries and Candidate Polarization in complete information environments. I show that primaries can cause polarization or moderation, depending on candidates’ beliefs about opposing voters’ strategic behavior—even when preferences are held constant. To generate this insight, I rely on ideas from behavioral game theory, which retains much of the theoretical apparatus from standard game theory while allowing for key departures (Camerer 2003). Specifically, I allow players to have “incorrect” or “nonequilibrium” beliefs about others’ actions (Crawford, Costa-Gomes, and Iriberri 2013), but assume they are nevertheless strategic in the sense that they best respond to what they think other players do (Camerer, Ho, and Chong 2004; Nagel 1995; Stahl and Wilson 1995). The analysis demonstrates that changes in preferences alone are not the only cause of polarization. Instead, beliefs and expectations about the strategic behavior of others play important roles in conditioning the effect of institutions. I then turn to the laboratory and conduct a series of experiments to test the effects of primaries on candidate positions and to distinguish between behavioral assumptions. The chief advantage of the laboratory for theory testing is control (Aldrich and Lupia 2011; Falk and Heckman 2009; Morton and Williams 2010), so we can be confident that the observed behavior occurs under the conditions specified by the theoretical model. Importantly, subjects face the same key tradeoff in the experiment as the actors do in the theoretical model between increasing the favorability of winning outcomes versus increasing the probability of winning. In the lab, theoretically relevant quantities of interest that are difficult to measure using observational data with any accuracy or without strong assumptions (in particular, preferences and positions) are also known exactly. Furthermore, experimental manipulations permit tests of mechanisms not possible using observational data. Thus, laboratory experiments are ideal for theory testing given their high internal validity.3 3 The main question of interest for theory testing, as Aldrich and Lupia (2011, 90) put it, is “Will people who are in the situations you describe in your model act as you predict?” For related discussions, see Dickson (2011), Palfrey (2006), and especially, Morton and Williams (2010). While the question of external validity (“To what extent can we generalize from a particular sample?”) is an enduring source of controversy in political science, Falk and Heckman (2009) argue in their insightful defense of the value of lab experiments in social science that “Behavior in the laboratory is reliable and real:Participants in the lab are real human beings who perceive their behavior as relevant, experience real emotions, and take decisions with real economic consequences” (536). Indeed, there are many precedents for testing theories of elite behavior using laboratory experiments (e.g., Aragones and Palfrey 2007; Frechette, Kagel, and Lehrer 2003; Morton 1993). Moreover, Druckman and Kam (2011) note that there is nothing inherently problematic with using student samples, and there is little evidence to suggest that using undergraduates as stand-ins for elites biases the results in any particular direction (see Morton and Williams 2010, 343–7). For example, Potters and van Winden (2000) find significant, but small, differences between students and lobbyists; Fatas, Neugebauer, and Tamborero (2007) find elites do not fit prospect theory as well as students, while studies by Belot, Duch, and Miller (2015), Cooper et al. (1999), and Mintz, Redd, and Vedlitz (2006) suggest that student samples provide a lower bound to departures from rational decision-making. Despite the common assertion that politicians must be better decision makers and more The key finding from the experiment is that primaries appear to cause a kind of ideological purity rather than greater extremism. Regardless of whether there is a primary election or not, I find that subjects take positions that diverge significantly from the median voter’s position. This finding lends support for the behavioral theory. However, the extent to which primaries cause polarization is limited. Greater polarization only occurs without feedback such that candidates cannot learn about the behavior of others, and this polarization happens because voters tend to select extremists over moderates, even though candidate positions do not vary with the election format. More precisely, the analysis reveals that voters do not support party extremists or party moderates unconditionally. Instead, they select candidates with intermediate positions, consistent with their own subjective beliefs about optimal candidate positions, approximately halfway between the median voter and their own party’s ideal point. This behavior generates a greater concentration of candidate positions around an average that diverges from the median voter. Hence, greater ideological homogeneity reinforces, rather than exacerbates, polarization. RELATED LITERATURE My analysis follows a long tradition of using spatial voting models to understand elections. Although existing spatial models predict candidate divergence in elections with primaries, they do so in isolation and do not compare them explicitly to elections without primaries (Aronson and Ordeshook 1972; Coleman 1972;Owen and Grofman 2006).4 These models also assume that general election outcomes are probabilistic, which is theoretically consequential because the mechanism they rely on to produce divergence is the combination of policy motivations and uncertainty about which candidate will win the general election—the same forces that generate incentives for candidate divergence in the absence of primaries (Calvert 1985; Wittman 1983). Thus, it is unclear from the literature whether polarization can be traced to any distinctive features of primaries per se, as electoral institutions. By explicitly comparing institutions,my analysis speaks directly to the connection between primaries and polarization. strategic than ordinary people (because they have more experience, access to advice, information, etc.), there is surprisingly little evidence to support such claims. To the contrary, recent comparisons by Sheffer et al. (2018) demonstrate that politicians are just as (and sometimes more) susceptible to choice anomalies than ordinary citizens. 4 An exception is Jackson, Mathevet, and Mattes (2007), who compare alternative nomination systems in a citizen-candidate framework. In their model, primary elections affect whose preference is decisive in nominating candidates and have no effect if party leaders and the median party voter have the same preferences. Other formal models of primary elections largely focus on considerations of voter uncertainty, incomplete information, and signaling along with issues of candidate valence and distributional concerns. 827 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000515
Jonathan Woon Existing theoretical models of two-stage elections fice motivations.Parties are completely homogeneous also typically maintain the assumption that all politi- as candidates and voters belonging to the same party cal actors,candidates as well as voters,are strategic and both care about policy and have the same ideal point. forward-looking (e.g..Owen and Grofman 2006).Sev- Thus,there are three ideal points in the model:0,for eral models consider the issue of raiding and crossover members of Party L,Og for members of Party R,and voting in open primaries (Cho and Kang 2014;Chen eM for the electorate's median voter,where <M< and Yang 2002:Oak 2006).which requires a fairly high R.I assume that preferences are symmetric and single- degree of strategic sophistication,but this kind of be- peaked.Specifically,in the experimental implementa- havior is outside the scope of my analysis.My results tion,all actors have linear loss utility functions,ui(w) also differ from Adams and Merrill (2014).who find =K-w -6il,for i[L,M,R)and some constant that strategic versus expressive voting both generate K>0.Preferences are also common knowledge,so the divergence,but in their model candidates are office- election takes place under conditions of complete in- motivated and vary in their campaign skills.In contrast formation. to the preponderance of existing formal models,I take There are two types of elections.In one-stage elec- a behavioral (i.e.,bounded rationality)approach advo- tions(1S),each party has one candidate and their posi- cated by Simon (1955).Ostrom(1998).Bendor (2010). tions are cL and cR.respectively,and there is one round and others.I do so by explicitly allowing for sincere of majority rule voting to select the winning candidate (myopic)voting as well as subjective beliefs that are In two-stage elections (2S),each party has two candi- inconsistent with observed behavior. dates (denoted cLi and cL2 for Party L,cRI and cR2 This paper is also related to two distinct literatures for Party R)who first compete in intraparty elections in experimental political science.The experimental lit- (the primaries).The candidates who win their respec- erature on candidate positioning in two-party elections tive party primaries then compete in a second-round finds a strong tendency for candidates and election out- election (the general election)to select the winning comes to converge to the median voter's position and, policy w.In other words,the parties hold simultaneous more generally,to the Condorcet winner under a va- "closed"primaries in which the voter with ideal point riety of conditions,including incomplete information eL chooses cL E [CLI,CL2)for Party L while the voter (Collier et al.1987:McKelvey and Ordeshook 1982 with ideal point OR chooses cR E[CR1,cR2 for Party R.In 1985).An exception is Morton (1993),in which can- the general election,the median voter with ideal point didates are ideological and voting is probabilistic.The ey chooses the election outcome from the two candi- other related literature,on strategic voting,generally dates selected by the parties'respective median voters finds little (at best,mixed)evidence for voter sophisti- w∈{CL,CR}. cation in the early stages of a multistage voting agenda To generate predictions about candidate positioning or election contest (Cherry and Kroll 2003;Eckel and and to identify the effects of the election format,I con- Holt 1989;Herzberg and Wilson 1988;McCuen and sider a variety of alternative behavioral assumptions.I Morton 2010:Plott and Levine 1978:Van der Straeten begin with standard game theoretic analysis,applying et al.2010).5 Taken together,these previous studies Nash equilibrium as the solution concept.Since I am raise doubts that voters will be highly strategic (even interested in making behavioral predictions,the inter- if candidates are),calling into question theories predi- pretation of Nash equilibrium is worth a brief discus- cated on voter rationality and strategic sophistication. sion.One way to interpret Nash equilibrium is to think of it as an idealized set of assumptions such that actors THEORETICAL FRAMEWORK AND are not only fully rational but also that their rationality ANALYSIS is common knowledge(Aumann and Brandenburger 1995).In this interpretation,we can think of political I consider an environment with two parties,Party L and actors as forming beliefs about others'current and fu- Party R,competing to win a single office.Candidates ture behavior (as well as beliefs about beliefs and ra- choose positions in a one-dimensional policy space, tionality,and so on)that are fully consistent with play- and the winning candidate's position is implemented ers'actual strategies and behavior.Alternatively,Nash as the policy outcome.In the electorate,there are an equilibrium can be interpreted as merely representing equal number of voters in each party and a set of inde- a stable outcome in which strategies are mutual best pendent,nonpartisan "swing"voters.Candidates and responses,without necessarily invoking an epistemic or voters alike are entirely policy-motivated,caring only belief-based justification of how individuals make deci- about the location of the policy outcome we R.The in- sions in games.The latter approach,however,does not centive to win office is therefore purely instrumental in make clear cut predictions about how games are played this model,which departs from the usual Downsian of- before an equilibrium state is reached.Nevertheless under a wide variety of learning models,experience can lead play to converge to Nash equilibrium(Fudenberg Smirnov (2009),who studies endogenous agendas and finds behav- and Levine 1998).and the role of experience can be ior consistent with sophisticated expected utility maximization,is an investigated experimentally. exception.There is stronger experimental evidence for other kinds Relaxing the Nash assumption of the mutual con- strategic voting.however,such as coordinating on a less-preferred candidate in multicandidate contests (Rietz 2008),and in incom sistency of beliefs and actions generates an interest- plete information pivotal voter settings(e.g.,Battaglini,Morton,and ing variety of behavioral possibilities.In my analysis Palfrey 2010). I first explore the implications of voter sophistication 828
Jonathan Woon Existing theoretical models of two-stage elections also typically maintain the assumption that all political actors, candidates as well as voters, are strategic and forward-looking (e.g., Owen and Grofman 2006). Several models consider the issue of raiding and crossover voting in open primaries (Cho and Kang 2014; Chen and Yang 2002; Oak 2006), which requires a fairly high degree of strategic sophistication, but this kind of behavior is outside the scope of my analysis. My results also differ from Adams and Merrill (2014), who find that strategic versus expressive voting both generate divergence, but in their model candidates are officemotivated and vary in their campaign skills. In contrast to the preponderance of existing formal models, I take a behavioral (i.e., bounded rationality) approach advocated by Simon (1955), Ostrom (1998), Bendor (2010), and others. I do so by explicitly allowing for sincere (myopic) voting as well as subjective beliefs that are inconsistent with observed behavior. This paper is also related to two distinct literatures in experimental political science. The experimental literature on candidate positioning in two-party elections finds a strong tendency for candidates and election outcomes to converge to the median voter’s position and, more generally, to the Condorcet winner under a variety of conditions, including incomplete information (Collier et al. 1987; McKelvey and Ordeshook 1982, 1985). An exception is Morton (1993), in which candidates are ideological and voting is probabilistic. The other related literature, on strategic voting, generally finds little (at best, mixed) evidence for voter sophistication in the early stages of a multistage voting agenda or election contest (Cherry and Kroll 2003; Eckel and Holt 1989; Herzberg and Wilson 1988; McCuen and Morton 2010; Plott and Levine 1978; Van der Straeten et al. 2010).5 Taken together, these previous studies raise doubts that voters will be highly strategic (even if candidates are), calling into question theories predicated on voter rationality and strategic sophistication. THEORETICAL FRAMEWORK AND ANALYSIS I consider an environment with two parties,Party L and Party R, competing to win a single office. Candidates choose positions in a one-dimensional policy space, and the winning candidate’s position is implemented as the policy outcome. In the electorate, there are an equal number of voters in each party and a set of independent, nonpartisan “swing” voters. Candidates and voters alike are entirely policy-motivated, caring only about the location of the policy outcome w ∈ R. The incentive to win office is therefore purely instrumental in this model, which departs from the usual Downsian of- 5 Smirnov (2009), who studies endogenous agendas and finds behavior consistent with sophisticated expected utility maximization, is an exception. There is stronger experimental evidence for other kinds strategic voting, however, such as coordinating on a less-preferred candidate in multicandidate contests (Rietz 2008), and in incomplete information pivotal voter settings (e.g., Battaglini, Morton, and Palfrey 2010). fice motivations. Parties are completely homogeneous, as candidates and voters belonging to the same party both care about policy and have the same ideal point. Thus, there are three ideal points in the model: θL for members of Party L, θR for members of Party R, and θM for the electorate’s median voter, where θL < θM < θR. I assume that preferences are symmetric and singlepeaked. Specifically, in the experimental implementation, all actors have linear loss utility functions, ui(w) = K − |w − θi|, for i ∈ {L, M, R} and some constant K > 0. Preferences are also common knowledge, so the election takes place under conditions of complete information. There are two types of elections. In one-stage elections (1S), each party has one candidate and their positions are cL and cR, respectively, and there is one round of majority rule voting to select the winning candidate. In two-stage elections (2S), each party has two candidates (denoted cL1 and cL2 for Party L, cR1 and cR2 for Party R) who first compete in intraparty elections (the primaries). The candidates who win their respective party primaries then compete in a second-round election (the general election) to select the winning policy w. In other words, the parties hold simultaneous “closed” primaries in which the voter with ideal point θL chooses cL ∈ {cL1, cL2} for Party L while the voter with ideal point θR chooses cR ∈ {cR1, cR2} for Party R. In the general election, the median voter with ideal point θM chooses the election outcome from the two candidates selected by the parties’ respective median voters, w ∈ {cL, cR}. To generate predictions about candidate positioning and to identify the effects of the election format, I consider a variety of alternative behavioral assumptions. I begin with standard game theoretic analysis, applying Nash equilibrium as the solution concept. Since I am interested in making behavioral predictions, the interpretation of Nash equilibrium is worth a brief discussion. One way to interpret Nash equilibrium is to think of it as an idealized set of assumptions such that actors are not only fully rational but also that their rationality is common knowledge (Aumann and Brandenburger 1995). In this interpretation, we can think of political actors as forming beliefs about others’ current and future behavior (as well as beliefs about beliefs and rationality, and so on) that are fully consistent with players’ actual strategies and behavior. Alternatively, Nash equilibrium can be interpreted as merely representing a stable outcome in which strategies are mutual best responses, without necessarily invoking an epistemic or belief-based justification of how individuals make decisions in games. The latter approach, however, does not make clear cut predictions about how games are played before an equilibrium state is reached. Nevertheless, under a wide variety of learning models, experience can lead play to converge to Nash equilibrium (Fudenberg and Levine 1998), and the role of experience can be investigated experimentally. Relaxing the Nash assumption of the mutual consistency of beliefs and actions generates an interesting variety of behavioral possibilities. In my analysis, I first explore the implications of voter sophistication 828 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000515
Primaries and Candidate Polarization for candidate positioning while holding candidate ra- be specified.Given a set of candidate positions and vot- tionality constant.If voting is "sincere,"then primary ers'expectations that the general election median voter elections produce more polarized candidates than vot- will choose the more moderate of the parties'candi ing that follows an equilibrium strategy.I then consider dates,a primary voter's strategy is to choose the can- another departure from standard assumptions:beliefs didate closest to her ideal point as long as she believes that some players make mistakes in choosing their po- the candidate will also win the general election (and sitions.They might do so for any number of reasons, in equilibrium,the voter's beliefs about which candi- such as miscalculating the optimal position.misjudg date will win are correct).Because candidates and vot- ing or underestimating the rationality of others,or hav- ers have the same preferences,the incentives guiding ing preferences over outcomes of the game that are optimal candidate strategies in the one-stage election not fully captured by their material payoffs.Strategi- are similar to those that guide rational voting behav- cally sophisticated players,recognizing that there are ior in two-stage elections:if offered the same choices. other players who make mistakes,will then choose po- candidates and voters would choose the same position sitions that differ from the Nash predictions-in the di- (the only difference is that candidates can choose any rection of their parties'ideal points-but that are op- position while primary voters'choices are constrained). timal given their own beliefs about the distribution of In any equilibrium of the two-stage election game, opponents'positions.Introducing noise or the possibil- there must be at least one candidate from each party ity of mistakes generates divergence in both one-stage located at eM,so primary voters will always be ob- and two-stage elections,despite complete information served choosing the moderate candidate along the path about preferences. of play.If so,both parties'primary voters will select a With noise,the effect of introducing a primary elec- candidate at the median voter's ideal point and the pol- tion is more nuanced.Similar to the case in which can- icy outcome is therefore w=0M.Ruling out other pos- didates do not make mistakes,the optimal positions sible outcomes then follows from the same logic as in depend critically on the behavior of primary voters.If the nonprimary election.Assuming fully strategic be- 4r元 voters choose moderate primary candidates,then two- havior from voters therefore predicts full convergence stage elections will generate greater convergence of to the median voter's position in both one-stage and candidate positions than in one-stage elections.How- two-stage elections. ever,if voters choose extreme primary candidates,then candidates in two-stage elections will be more polar- Prediction 1.If voters and candidates are rational, ized than candidates in one-stage elections.There is forward-looking agents and form correct beliefs about also a third possibility:If voters form their own be- others'behavior,then (a)the moderate candidates from liefs about the position most likely to maximize their each party will adopt the median voter's position and expected utility and vote for candidates closest to this (b)primaries will have no effect on the polarization of position.then the degree of candidate divergence in candidates in the general election. two-stage elections is increasing in what we might call voters'belief-induced ideal points.Behavioral game Candidate Equilibrium with Sincere Voters theory thus establishes a critical link between candi- dates'beliefs about opponents'primary voting behav- I next consider the possibility that primary voters are ior and the effect of primaries. myopic and vote"sincerely."6 I assume that sincere vot- ers simply vote for the candidate closest to their ideal points,so they are myopic in the sense that they fail to recognize that the candidate's chances of winning the Candidate Equilibrium with Fully Strategic general election affect the policy outcome(and hence Voters their payoffs).With myopic voters,the two-stage elec- Standard equilibrium analysis leads to identical predic- tion game has multiple equilibria in which candidates tions for both one-stage and two-stage elections.This take divergent positions while the equilibrium of the is because,in any equilibrium,the winning candidate's one-stage election game remains the same(full conver- position is the median voter's ideal point.In one-stage gence,since there are no primary voters) elections,the logic is straightforward.The median voter In any equilibrium of the two-stage election game chooses the party candidate closest to his or her ideal with sincere voters,candidates within each party must point as the winning candidate,so if one candidate adopt the same position,and opposing party can- didates must be equidistant from the median voter. 四 adopts 0w as a campaign position,no other position can defeat it.In the unique equilibrium of the one-stage Specifically,an equilibrium is characterized by the election game,both parties'candidates must choose condition that cLI CL2 =0M-8 and CRI CR2 cL=CR =0M.If not,either the winning party's candi- e+8,where 8>0 denotes some amount of diver- date could do better by finding a position closer to her gence between candidates.The median voter's strategy ideal point while still winning the election or the los ing candidate can find a position that wins the election While the overall level of voter "rationality"remains an ongoing thereby obtaining a better policy outcome for herself. subject of debate,the assumption that voters are myopic is consistent Thus,w=6M is the unique equilibrium policy outcome. with recent observational and experimental research on accountabil- ity (e.g..Healy and Malhotra 2009;Huber,Hill,and Lenz 2012;Woon In two-stage elections,the outcome is the same,but 2012a).A theory of elections with boundedly rational,behavioral the equilibrium strategies of the primary voters must voters is also worked out by Bendor et al.(2011). 829
Primaries and Candidate Polarization for candidate positioning while holding candidate rationality constant. If voting is “sincere,” then primary elections produce more polarized candidates than voting that follows an equilibrium strategy. I then consider another departure from standard assumptions: beliefs that some players make mistakes in choosing their positions. They might do so for any number of reasons, such as miscalculating the optimal position, misjudging or underestimating the rationality of others, or having preferences over outcomes of the game that are not fully captured by their material payoffs. Strategically sophisticated players, recognizing that there are other players who make mistakes, will then choose positions that differ from the Nash predictions—in the direction of their parties’ ideal points—but that are optimal given their own beliefs about the distribution of opponents’ positions. Introducing noise or the possibility of mistakes generates divergence in both one-stage and two-stage elections, despite complete information about preferences. With noise, the effect of introducing a primary election is more nuanced. Similar to the case in which candidates do not make mistakes, the optimal positions depend critically on the behavior of primary voters. If voters choose moderate primary candidates, then twostage elections will generate greater convergence of candidate positions than in one-stage elections. However, if voters choose extreme primary candidates, then candidates in two-stage elections will be more polarized than candidates in one-stage elections. There is also a third possibility: If voters form their own beliefs about the position most likely to maximize their expected utility and vote for candidates closest to this position, then the degree of candidate divergence in two-stage elections is increasing in what we might call voters’ belief-induced ideal points. Behavioral game theory thus establishes a critical link between candidates’ beliefs about opponents’ primary voting behavior and the effect of primaries. Candidate Equilibrium with Fully Strategic Voters Standard equilibrium analysis leads to identical predictions for both one-stage and two-stage elections. This is because, in any equilibrium, the winning candidate’s position is the median voter’s ideal point. In one-stage elections, the logic is straightforward.The median voter chooses the party candidate closest to his or her ideal point as the winning candidate, so if one candidate adopts θM as a campaign position, no other position can defeat it. In the unique equilibrium of the one-stage election game, both parties’ candidates must choose cL = cR = θM. If not, either the winning party’s candidate could do better by finding a position closer to her ideal point while still winning the election or the losing candidate can find a position that wins the election, thereby obtaining a better policy outcome for herself. Thus,w = θM is the unique equilibrium policy outcome. In two-stage elections, the outcome is the same, but the equilibrium strategies of the primary voters must be specified.Given a set of candidate positions and voters’ expectations that the general election median voter will choose the more moderate of the parties’ candidates, a primary voter’s strategy is to choose the candidate closest to her ideal point as long as she believes the candidate will also win the general election (and in equilibrium, the voter’s beliefs about which candidate will win are correct). Because candidates and voters have the same preferences, the incentives guiding optimal candidate strategies in the one-stage election are similar to those that guide rational voting behavior in two-stage elections: if offered the same choices, candidates and voters would choose the same position (the only difference is that candidates can choose any position while primary voters’ choices are constrained). In any equilibrium of the two-stage election game, there must be at least one candidate from each party located at θM, so primary voters will always be observed choosing the moderate candidate along the path of play. If so, both parties’ primary voters will select a candidate at the median voter’s ideal point and the policy outcome is therefore w = θM. Ruling out other possible outcomes then follows from the same logic as in the nonprimary election. Assuming fully strategic behavior from voters therefore predicts full convergence to the median voter’s position in both one-stage and two-stage elections. Prediction 1. If voters and candidates are rational, forward-looking agents and form correct beliefs about others’ behavior, then (a) the moderate candidates from each party will adopt the median voter’s position and (b) primaries will have no effect on the polarization of candidates in the general election. Candidate Equilibrium with Sincere Voters I next consider the possibility that primary voters are myopic and vote “sincerely.”6 I assume that sincere voters simply vote for the candidate closest to their ideal points, so they are myopic in the sense that they fail to recognize that the candidate’s chances of winning the general election affect the policy outcome (and hence their payoffs). With myopic voters, the two-stage election game has multiple equilibria in which candidates take divergent positions while the equilibrium of the one-stage election game remains the same (full convergence, since there are no primary voters). In any equilibrium of the two-stage election game with sincere voters, candidates within each party must adopt the same position, and opposing party candidates must be equidistant from the median voter. Specifically, an equilibrium is characterized by the condition that cL1 = cL2 = θM − δ and cR1 = cR2 = θM + δ, where δ ≥ 0 denotes some amount of divergence between candidates. The median voter’s strategy 6 While the overall level of voter “rationality” remains an ongoing subject of debate, the assumption that voters are myopic is consistent with recent observational and experimental research on accountability (e.g.,Healy and Malhotra 2009;Huber,Hill, and Lenz 2012;Woon 2012a). A theory of elections with boundedly rational, behavioral voters is also worked out by Bendor et al. (2011). 829 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000515
Jonathan Woon is to select the candidate closest to her own ideal point. To model this,let candidate i's beliefs about the po- breaking ties in favor of each party with equal proba- sitions of candidates from the opposing party ji be bility.The result of the general election is therefore a given by the cumulative distribution F(ci).Importantly, lottery over w [eM-8,eM+8),and the expected these beliefs need not be accurate.For instance,if i's value of the outcome is the median voter's position true position is ci=0,candidate i might believe that E[w]=0M.Any candidate who adopts a more extreme c;is uniformly distributed between-1 and 1.We can position would,at best,be able to win their own pri- think of the distribution F(ci)as representing subjec- mary but then would lose the general election with cer- tive beliefs that will typically not satisfy the equilibrium tainty.Moving to a more moderate position would not consistency requirement. change the result of the primary and thus would not By relaxing the standard equilibrium assumption of change the general election result either.Since no can- belief consistency,an otherwise expected utility max- didate can obtain a better policy outcome by unilater- imizing candidate will choose a position that diverges ally adopting a different position,campaign promises from the median voter's ideal point.The reasoning is as characterized by intraparty convergence and interparty follows.If a candidate believes there is some possibil- symmetric divergence constitute an equilibrium of the ity that the opposing candidate's position diverges from primary election game with sincere voters.The basic the median voter,then it cannot be optimal for a policy- intuition underlying this result is that due to the myopic motivated candidate to choose a platform exactly at behavior of sincere primary voters,intraparty compe- the median voter's ideal point.Instead,the candidate tition limits any one candidate's ability to moderate will choose a position that trades off some probabil- 元 their party's position in the general election.Thus,in ity of winning against potential policy gains obtained contrast to full convergence in one-stage elections,any from choosing a position closer to his or her own ideal amount of divergence can be supported in two-stage point.To illustrate this concretely,suppose that oR= elections. 1,the left party's ideal point is L=-1,the median is 令 eM =0,and F(cL)is a uniform random variable,cL~ 4号元 Prediction 2.If candidates are rational and forward- UI-1,0].With linear loss utility,the optimal position looking but primary voters "sincerely"select candidates closest to their own ideal points,then (a)candidates from that balances this tradeoff is c=.This is illustrated each party will take positions that diverge from the me- by the solid line showing the expected utility function dian voter by the same amount in two-stage elections, EU(cR)in Figure 1.10 While this logic is similar to the and(b)winning candidates will be weakly more polar- tradeoff found in Calvert (1985)and Wittman(1983), ized in two-stage elections than in one-stage elections, the important distinction is that the source of uncer- while candidates in the latter will converge to the median tainty in this model is entirely about opponents'behav- voter. ior rather than about voters or preferences.Moreover, candidate positions are responsive to beliefs such that when a candidate is more likely to expect her oppo- Candidate Best Responses to nent to be extreme (i.e.,when F(c)puts more weight Out-of-Equilibrium Beliefs on extreme positions),then she herself will take a posi- The previous sections assumed that candidates cor- tion with greater divergence from the median voter in rectly anticipate whether voters use either Nash or sin- response cere voting strategies and that their beliefs about other Next,I consider how these beliefs about opposing candidates are consistent with those candidates'actual candidates'positions interact with the type of elec- behavior.That is,if candidate j chooses the platform tion.The main result is that the effect of primaries will cj,then candidate i must believe with certainty that ci depend on the candidates'beliefs about the opposing party's primary voters.The baseline for comparison is a must really be i's position.However,this mutual con- sistency of candidates'beliefs and actions might break one-stage election with opponents drawn from the be- down in a number of ways.Candidates are likely to lief distribution F(ci).For the purposes of exposition, face cognitive constraints,they may engage in incom- suppose that F(cj)is uniform as in the example just plete strategic reasoning,or they may doubt the ratio- given and as shown in the left side of Figure 2,so the nality of other candidates.In this section,I apply the notion of limited strategic sophistication motivated by level-k models in behavioral game theory (Crawford levels of sophistication or reasoning as modeled explicitly in the level-k framework. 2003;Nagel 1995;Stahl and Wilson 1995),positing that 9 I assume that the density f(ci)has full support over the interval candidates have some(possibly arbitrary)beliefs and between median voter M and the opposing party 0j.The distribution analyze the best response to such beliefs.s F(cj) can also be interpreted as an objective probability distribution if candidates'choices are noisy and F(c)reflects the true distribution of candidate positions. 7 Note that it is also possible to construct equilibria in which the me- 10 Formally,given beliefs with density f(cL),the expected utility func- dian voter has a bias for one of the parties (i.e.,breaks ties in fa L vor of one party rather than randomizing),but this would not affect N-CR the equilibrium positions of the candidates.Thus,even though the EU(CR) u(CR)f(CL)dcL+ u(cL)f(CL)dcL, random tie-breaking rule matches the experimental setup,it is not 28M-ER necessary for the results. where the integral on the left is the expected utility if cR is closer s While level-k models are a subset of the class of models that as- to the median voter and wins while the integral on the right is the sume out-of-equilibrium beliefs,my theory does not rely on different expected utility if the opposing candidate cL is closer to the median. 830
Jonathan Woon is to select the candidate closest to her own ideal point, breaking ties in favor of each party with equal probability.7 The result of the general election is therefore a lottery over w ∈ {θM − δ, θM + δ}, and the expected value of the outcome is the median voter’s position, E[w] = θM. Any candidate who adopts a more extreme position would, at best, be able to win their own primary but then would lose the general election with certainty. Moving to a more moderate position would not change the result of the primary and thus would not change the general election result either. Since no candidate can obtain a better policy outcome by unilaterally adopting a different position, campaign promises characterized by intraparty convergence and interparty symmetric divergence constitute an equilibrium of the primary election game with sincere voters. The basic intuition underlying this result is that due to the myopic behavior of sincere primary voters, intraparty competition limits any one candidate’s ability to moderate their party’s position in the general election. Thus, in contrast to full convergence in one-stage elections, any amount of divergence can be supported in two-stage elections. Prediction 2. If candidates are rational and forwardlooking but primary voters “sincerely” select candidates closest to their own ideal points,then (a) candidates from each party will take positions that diverge from the median voter by the same amount in two-stage elections, and (b) winning candidates will be weakly more polarized in two-stage elections than in one-stage elections, while candidates in the latter will converge to the median voter. Candidate Best Responses to Out-of-Equilibrium Beliefs The previous sections assumed that candidates correctly anticipate whether voters use either Nash or sincere voting strategies and that their beliefs about other candidates are consistent with those candidates’ actual behavior. That is, if candidate j chooses the platform cj, then candidate i must believe with certainty that cj must really be j’s position. However, this mutual consistency of candidates’ beliefs and actions might break down in a number of ways. Candidates are likely to face cognitive constraints, they may engage in incomplete strategic reasoning, or they may doubt the rationality of other candidates. In this section, I apply the notion of limited strategic sophistication motivated by level-k models in behavioral game theory (Crawford 2003; Nagel 1995; Stahl and Wilson 1995), positing that candidates have some (possibly arbitrary) beliefs and analyze the best response to such beliefs.8 7 Note that it is also possible to construct equilibria in which the median voter has a bias for one of the parties (i.e., breaks ties in favor of one party rather than randomizing), but this would not affect the equilibrium positions of the candidates. Thus, even though the random tie-breaking rule matches the experimental setup, it is not necessary for the results. 8 While level-k models are a subset of the class of models that assume out-of-equilibrium beliefs, my theory does not rely on different To model this, let candidate i’s beliefs about the positions of candidates from the opposing party j = i be given by the cumulative distribution F(cj). Importantly, these beliefs need not be accurate. For instance, if j’s true position is cj = 0, candidate i might believe that cj is uniformly distributed between −1 and 1. We can think of the distribution F(cj) as representing subjective beliefs that will typically not satisfy the equilibrium consistency requirement.9 By relaxing the standard equilibrium assumption of belief consistency, an otherwise expected utility maximizing candidate will choose a position that diverges from the median voter’s ideal point. The reasoning is as follows. If a candidate believes there is some possibility that the opposing candidate’s position diverges from the median voter, then it cannot be optimal for a policymotivated candidate to choose a platform exactly at the median voter’s ideal point. Instead, the candidate will choose a position that trades off some probability of winning against potential policy gains obtained from choosing a position closer to his or her own ideal point. To illustrate this concretely, suppose that θR = 1, the left party’s ideal point is θL = −1, the median is θM = 0, and F(cL) is a uniform random variable, cL ∼ U[ − 1, 0]. With linear loss utility, the optimal position that balances this tradeoff is c∗ R = 1 3 . This is illustrated by the solid line showing the expected utility function EU(cR) in Figure 1. 10 While this logic is similar to the tradeoff found in Calvert (1985) and Wittman (1983), the important distinction is that the source of uncertainty in this model is entirely about opponents’ behavior rather than about voters or preferences. Moreover, candidate positions are responsive to beliefs such that when a candidate is more likely to expect her opponent to be extreme (i.e., when F(cj) puts more weight on extreme positions), then she herself will take a position with greater divergence from the median voter in response. Next, I consider how these beliefs about opposing candidates’ positions interact with the type of election. The main result is that the effect of primaries will depend on the candidates’ beliefs about the opposing party’s primary voters.The baseline for comparison is a one-stage election with opponents drawn from the belief distribution F(cj). For the purposes of exposition, suppose that F(cj) is uniform as in the example just given and as shown in the left side of Figure 2, so the levels of sophistication or reasoning as modeled explicitly in the level-k framework. 9 I assume that the density f(cj) has full support over the interval between median voter θM and the opposing party θj.The distribution F(cj) can also be interpreted as an objective probability distribution if candidates’ choices are noisy and F(cj) reflects the true distribution of candidate positions. 10 Formally, given beliefs with density f(cL), the expected utility function is given by EU(cR) = 2θM−cR θL u(cR)f(cL)dcL + θM 2θM−cR u(cL)f(cL)dcL, where the integral on the left is the expected utility if cR is closer to the median voter and wins while the integral on the right is the expected utility if the opposing candidate cL is closer to the median. 830 Downloaded from https://www.cambridge.org/core. Shanghai JiaoTong University, on 26 Oct 2018 at 03:53:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0003055418000515