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). 829Primaries and Candidate Polarization for candidate positioning while holding candidate ra￾tionality constant. If voting is “sincere,” then primary elections produce more polarized candidates than vot￾ing that follows an equilibrium strategy. I then consider another departure from standard assumptions: beliefs that some players make mistakes in choosing their po￾sitions. They might do so for any number of reasons, such as miscalculating the optimal position, misjudg￾ing or underestimating the rationality of others, or hav￾ing preferences over outcomes of the game that are not fully captured by their material payoffs. Strategi￾cally sophisticated players, recognizing that there are other players who make mistakes, will then choose po￾sitions that differ from the Nash predictions—in the di￾rection of their parties’ ideal points—but that are op￾timal given their own beliefs about the distribution of opponents’ positions. Introducing noise or the possibil￾ity 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 elec￾tion is more nuanced. Similar to the case in which can￾didates do not make mistakes, the optimal positions depend critically on the behavior of primary voters. If voters choose moderate primary candidates, then two￾stage elections will generate greater convergence of candidate positions than in one-stage elections. How￾ever, if voters choose extreme primary candidates, then candidates in two-stage elections will be more polar￾ized than candidates in one-stage elections. There is also a third possibility: If voters form their own be￾liefs 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 candi￾dates’ beliefs about opponents’ primary voting behav￾ior and the effect of primaries. Candidate Equilibrium with Fully Strategic Voters Standard equilibrium analysis leads to identical predic￾tions 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 candi￾date could do better by finding a position closer to her ideal point while still winning the election or the los￾ing 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 vot￾ers’ expectations that the general election median voter will choose the more moderate of the parties’ candi￾dates, a primary voter’s strategy is to choose the can￾didate 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 candi￾date will win are correct). Because candidates and vot￾ers have the same preferences, the incentives guiding optimal candidate strategies in the one-stage election are similar to those that guide rational voting behav￾ior 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 ob￾served 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 pol￾icy outcome is therefore w = θM. Ruling out other pos￾sible outcomes then follows from the same logic as in the nonprimary election. Assuming fully strategic be￾havior 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 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 general election affect the policy outcome (and hence their payoffs). With myopic voters, the two-stage elec￾tion game has multiple equilibria in which candidates take divergent positions while the equilibrium of the one-stage election game remains the same (full conver￾gence, 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 can￾didates 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 diver￾gence 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 accountabil￾ity (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