Jonathan Woon candidate's best position is the one that maximizes the point and the median voter,G(c)will be the symmetric same expected utility function EU(cR)mentioned pre- triangular distribution in the middle-right of Figure 2. viously in Figure 1. Note that while the mean of this distribution is the In a two-stage election,it is not the original distribu- same as the original distribution F(ci),it has lower vari- tion of candidates F(ci)that matters,but beliefs about ance.13 Primary elections may therefore also have the which candidate will emerge from the primary election. effect of reinforcing ideological purity (i.e.,increasing Let G(ci)denote this latter set of beliefs about the can- homogeneity)within parties even when there is no dis- didate selected by the opposing party's primary-the cernible effect on average candidate positions (i.e.,ab- candidate that i expects to face in the general election. sent changes in polarization). We can think of the primary election as a selection In contrast to standard equilibrium analysis.which mechanism or filtering process that affects whether a predicts full convergence,a simple model with out- party's candidate is systematically more or less extreme of-equilibrium beliefs generates divergence in candi- than the party's initial set of candidates. date positions,even in the absence of primaries and More precisely,suppose that both of the opposing with complete information about preferences.More- party's candidates are independently drawn from F(c). over,the effect of primaries varies with candidates'ex- Now consider how primary voting behavior affects pectations about the opposing party's voting behavior G(c)and,in turn,candidates'positions.If i's primary Primaries can indeed cause greater polarization (but voters unconditionally select the more extreme candi- only if primary voters select sufficiently extreme can- date(as they would if they voted sincerely),then party didates),cause greater moderation (if primary voters i's candidate in the general election will be the more ex- select moderates),or cause increased intraparty ho- treme of two independent draws from F(ci).This results mogeneity (if voters select on the basis of intermedi in a distribution G(ci)that is skewed more toward i's ate belief-induced ideal points).Thus,the behavioral own ideal point than F(ci),as shown by the triangular theory identifies how the effect of primaries depends distribution in the upper-right of Figure 2 when F(ci) on the connection between beliefs and behavior rather 4号元 is uniform.When voters choose extremists,primaries than on preferences alone. generate incentives for greater extremism than in one- stage elections,as illustrated by the upper-dashed ex- Prediction 3.If candidates have out-of-equilibrium be- pected utility function EU(cR)in Figure 1. liefs about the distribution of opposing candidates,then The flip-side of this is that if i's primary voters select (a)candidate positions will diverge from the median the more moderate candidate (as they would in equi- voter's ideal point in both one-stage and two-stage elec- librium),they generate incentives for greater modera- tions,(b)the direction of the effect of primary elec- tion than in one-stage elections.This is because party tions on candidate polarization depends on expectations i's general election candidate will be the more mod- about voting behavior,and (c)polarization in two-stage erate of two independent draws from F(ci),resulting elections is increasing in the expected extremity of can- in a distribution of beliefs G(ci)that is skewed more didates selected by the opposing party's primary voters. toward the median voter than F(ci).This is shown in Figure 2 by the triangular distribution in the bottom right.2 When the probability of facing an extremist op- EXPERIMENTAL ANALYSIS ponent is lower,a candidate must moderate their posi- The theoretical analysis generated a set of competing tion in response,which is shown by the lower-dashed predictions about the effect of primaries as a function expected utility function EU(cR)in Figure 1. of alternative behavioral assumptions.If all players are These are not the only possibilities,as primary voters fully strategic,then we should observe full convergence might also behave in other ways.For example,a fairly to the median voter's position and primaries should sophisticated voter might reason in the same way as a have no effect (Prediction 1).If candidates are strate- candidate and form the same beliefs G(c;)based on ex- gic but voters are not,then we should observe can- pectations about opposing primary voters.To general- didate divergence only in two-stage elections but not ize this idea,suppose that a voter has a belief-induced in one-stage elections(Prediction 2).If the behavioral ideal point c and always votes for the candidate in theory has merit and candidates have subjective be- the primary whose position is closest to c(sometimes liefs about their opponents'positions,then polariza- this will be the moderate and sometimes the extremist). tion,moderation,or increased homogeneity are possi- The result is a distribution of primary candidates G(ci) ble depending on voter behavior(Prediction 3).Which that has greater mass closer to c than F(c)does.If c set of assumptions is a better reflection of how humans happens to be the midpoint between the voter's ideal behave is ultimately an empirical question,and thus,I 1 If Lparty voters select the extremist,then cL-min (cLi.cL2 and Ge(cL)is first order stochastically dominated by F(cL).If cL and cL2 13 The best response,however.is not exactly the same in the 2S elec- are independently drawn from 0-1,0,then G(cL)has density tion for the symmetric triangular distribution as it is for the uniform selecting the moderate=maand s)first distribution in the 1S election.Nevertheless,there does exist a belief. induced ideal point(via an intermediate value theorem argument order stochastically dominates F(cL).In the example where the uni- which generates an asymmetric distribution)such that candidates form distribution is Ul-1,0],G"(cL)has density g"(cL)=2cL+2 optimal positions diverge from the median voter by the same amount for CL∈[-1,0 (i.e.,the best responses are identical)in both 1S and 2S elections. 832Jonathan Woon candidate’s best position is the one that maximizes the same expected utility function EU(cR) mentioned pre￾viously in Figure 1. In a two-stage election, it is not the original distribu￾tion of candidates F(cj) that matters, but beliefs about which candidate will emerge from the primary election. Let G(cj) denote this latter set of beliefs about the can￾didate selected by the opposing party’s primary—the candidate that i expects to face in the general election. We can think of the primary election as a selection mechanism or filtering process that affects whether a party’s candidate is systematically more or less extreme than the party’s initial set of candidates. More precisely, suppose that both of the opposing party’s candidates are independently drawn from F(cj). Now consider how primary voting behavior affects G(cj) and, in turn, candidates’ positions. If j’s primary voters unconditionally select the more extreme candi￾date (as they would if they voted sincerely), then party j’s candidate in the general election will be the more ex￾treme of two independent draws from F(cj).This results in a distribution G(cj) that is skewed more toward j’s own ideal point than F(cj), as shown by the triangular distribution in the upper-right of Figure 2 when F(cj) is uniform.11 When voters choose extremists, primaries generate incentives for greater extremism than in one￾stage elections, as illustrated by the upper-dashed ex￾pected utility function EUe (cR) in Figure 1. The flip-side of this is that if j’s primary voters select the more moderate candidate (as they would in equi￾librium), they generate incentives for greater modera￾tion than in one-stage elections. This is because party j’s general election candidate will be the more mod￾erate of two independent draws from F(cj), resulting in a distribution of beliefs G(cj) that is skewed more toward the median voter than F(cj). This is shown in Figure 2 by the triangular distribution in the bottom right.12 When the probability of facing an extremist op￾ponent is lower, a candidate must moderate their posi￾tion in response, which is shown by the lower-dashed expected utility function EUm(cR) in Figure 1. These are not the only possibilities, as primary voters might also behave in other ways. For example, a fairly sophisticated voter might reason in the same way as a candidate and form the same beliefs G(ci) based on ex￾pectations about opposing primary voters. To general￾ize this idea, suppose that a voter has a belief-induced ideal point c∗ j and always votes for the candidate in the primary whose position is closest to c∗ j (sometimes this will be the moderate and sometimes the extremist). The result is a distribution of primary candidates G(cj) that has greater mass closer to c∗ j than F(cj) does. If c∗ j happens to be the midpoint between the voter’s ideal 11 If L party voters select the extremist, then cL = min {cL1, cL2} and Ge(cL) is first order stochastically dominated by F(cL). If cL1 and cL2 are independently drawn from U[ − 1, 0], then Ge(cL) has density ge(cL) = −2cL for cL ∈ [ − 1, 0]. 12 In selecting the moderate, cL = max {cL1, cL2} and so Gm(cL) first order stochastically dominates F(cL). In the example where the uni￾form distribution is U[ − 1, 0], Gm(cL) has density gm(cL) = 2cL + 2 for cL ∈ [ − 1, 0]. point and the median voter,G(cj) will be the symmetric triangular distribution in the middle-right of Figure 2. Note that while the mean of this distribution is the same as the original distribution F(cj), it has lower vari￾ance.13 Primary elections may therefore also have the effect of reinforcing ideological purity (i.e., increasing homogeneity) within parties even when there is no dis￾cernible effect on average candidate positions (i.e., ab￾sent changes in polarization). In contrast to standard equilibrium analysis, which predicts full convergence, a simple model with out￾of-equilibrium beliefs generates divergence in candi￾date positions, even in the absence of primaries and with complete information about preferences. More￾over, the effect of primaries varies with candidates’ ex￾pectations about the opposing party’s voting behavior. Primaries can indeed cause greater polarization (but only if primary voters select sufficiently extreme can￾didates), cause greater moderation (if primary voters select moderates), or cause increased intraparty ho￾mogeneity (if voters select on the basis of intermedi￾ate belief-induced ideal points). Thus, the behavioral theory identifies how the effect of primaries depends on the connection between beliefs and behavior rather than on preferences alone. Prediction 3. If candidates have out-of-equilibrium be￾liefs about the distribution of opposing candidates, then (a) candidate positions will diverge from the median voter’s ideal point in both one-stage and two-stage elec￾tions, (b) the direction of the effect of primary elec￾tions on candidate polarization depends on expectations about voting behavior, and (c) polarization in two-stage elections is increasing in the expected extremity of can￾didates selected by the opposing party’s primary voters. EXPERIMENTAL ANALYSIS The theoretical analysis generated a set of competing predictions about the effect of primaries as a function of alternative behavioral assumptions. If all players are fully strategic, then we should observe full convergence to the median voter’s position and primaries should have no effect (Prediction 1). If candidates are strate￾gic but voters are not, then we should observe can￾didate divergence only in two-stage elections but not in one-stage elections (Prediction 2). If the behavioral theory has merit and candidates have subjective be￾liefs about their opponents’ positions, then polariza￾tion, moderation, or increased homogeneity are possi￾ble depending on voter behavior (Prediction 3).Which set of assumptions is a better reflection of how humans behave is ultimately an empirical question, and thus, I 13 The best response, however, is not exactly the same in the 2S elec￾tion for the symmetric triangular distribution as it is for the uniform distribution in the 1S election. Nevertheless, there does exist a belief￾induced ideal point (via an intermediate value theorem argument, which generates an asymmetric distribution) such that candidates’ optimal positions diverge from the median voter by the same amount (i.e., the best responses are identical) in both 1S and 2S elections. 832 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