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. 830Jonathan Woon is to select the candidate closest to her own ideal point, breaking ties in favor of each party with equal proba￾bility.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 pri￾mary but then would lose the general election with cer￾tainty. 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 can￾didate can obtain a better policy outcome by unilater￾ally 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 compe￾tition 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 forward￾looking 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 me￾dian voter by the same amount in two-stage elections, and (b) winning candidates will be weakly more polar￾ized 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 cor￾rectly anticipate whether voters use either Nash or sin￾cere 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 con￾sistency of candidates’ beliefs and actions might break down in a number of ways. Candidates are likely to face cognitive constraints, they may engage in incom￾plete strategic reasoning, or they may doubt the ratio￾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 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 me￾dian voter has a bias for one of the parties (i.e., breaks ties in fa￾vor 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 as￾sume out-of-equilibrium beliefs, my theory does not rely on different To model this, let candidate i’s beliefs about the po￾sitions 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 subjec￾tive beliefs that will typically not satisfy the equilibrium consistency requirement.9 By relaxing the standard equilibrium assumption of belief consistency, an otherwise expected utility max￾imizing 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 possibil￾ity that the opposing candidate’s position diverges from the median voter, then it cannot be optimal for a policy￾motivated candidate to choose a platform exactly at the median voter’s ideal point. Instead, the candidate will choose a position that trades off some probabil￾ity 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 uncer￾tainty in this model is entirely about opponents’ behav￾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￾nent to be extreme (i.e., when F(cj) puts more weight on extreme positions), then she herself will take a posi￾tion with greater divergence from the median voter in response. Next, I consider how these beliefs about opposing candidates’ positions interact with the type of elec￾tion. 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 be￾lief 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 func￾tion 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