Elected politicians are extraordinarily successful

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1 Electoral Selection, Strategic Challenger Entry, and the Incumbency Advantage Scott Ahworth Ethan Bueno de Mequita Princeton Univerity Univerity of Chicago We tudy the comparative tatic of the incumbency advantage in a model of electoral election and trategic challenger entry. The incumbency advantage arie in the model becaue, on average, incumbent have greater ability than challenger. Thi i true for two reaon: high-ability candidate are more likely to win election (electoral election) and high-quality incumbent deter challenger (trategic challenger entry). We how that thi quality-baed incumbency advantage i expected to be greater for high viibility office, in politie with relatively mall partian tide, in unpolarized electoral environment, and in electorate that are relatively balanced in their partian preference. Elected politician are extraordinarily ucceful when they eek reelection. One prominent explanation of thi incumbency advantage i that incumbent have, on average, higher quality than challenger. Thi i true for at leat two reaon. Firt, there i electoral election (Ahworth 2005; Bank and Sundaram 1998; Gowriankaran, Mitchell, and Moro 2008; Londregan and Romer 1993; Samuelon 1984, 1987; Zaller 1998). Voter elect candidate whom they like on ome dimenion (e.g., ability or trutworthine), o, compared to the average challenger, candidate who have won in the pat are relatively more attractive on that dimenion. 1 Second, there i trategic challenger entry (Cox and Katz 1996; Gordon, Huber, and Landa 2007; Jacobon 1980; Jacobon and Kernell 1983; Stone, Maiel, and Maeta 2004). 2 Challenger may be deterred from running againt incumbent who are perceivedtobeofhighability. 3 We refer to the combination of electoral election and trategic challenger entry a the quality difference model of the incumbency advantage. To evaluate the validity of the quality difference model, it i important to know what implication it ha beyond predicting the exitence of an incumbency advantage all theorie of the incumbency advantage make that prediction. The literature to date ha focued on prediction about the dynamic of the incumbency advantage over a politician career: Samuelon (1984) prove that an incumbent expected margin of victory will increae over time in a pure election model. Samuelon (1987) provide a upportive tet of thi prediction, and Zaller (1998) argue that actual electoral career reemble thoe from imulation of a pure election model. Thi claim i confirmed by the tructural etimate of Gowriankaran, Mitchell, and Moro (2008). Thee tudie provide upport for the claim that electoral election i an important part of the incumbency advantage. However, they leave open the quetion of whether the quality difference model can account for the broad range of variation in incumbent ucce that i oberved over time and acro different office and electorate. For intance, Anolabehere, Snyder, and Stewart (2000) demontrate that congreman from more partian ditrict have maller incumbency advantage, while Anolabehere and 1 Diermeier, Keane, and Merlo (2005) tudy a related model in which candidate quality affect reelection probabilitie and trategic deciion to run for higher office or to enter the private ector. 2 Recently, Cox and Katz (2002) uggeted that change in reditricting ince 1964 have contributed to an increaed mimatch between challenger and incumbent, enhancing the incumbency advantage. 3 A cloely related idea i that challenger are le effective than open-eat candidate becaue they have le acce to campaign fund (Gerber 1998). The Journal of Politic, Vol. 70, No. 4, October 2008, Pp doi: /s Ó 2008 Southern Political Science Aociation ISSN

2 electoral election, trategic challenger entry, and the incumbency advantage 1007 Snyder (2002) find that the incumbency advantage i larger for more viible office and ha been increaing over time for all tate-wide office. In light of thee empirical finding, we tudy the comparative tatic of the quality difference model in order to undertand how variation in the office or in the electoral environment affect the magnitude of the advantage. We find that the quality difference model i conitent with the empirical obervation dicued above. Moreover, our comparative tatic yield a variety of additional prediction that contitute novel tetable implication of the quality difference model. In particular, we how that the quality-difference-baed incumbency advantage hould be greater for highviibility office, when partian tide are mall, and in electorate that are either relatively unpolarized, nonpartian, or both. 4 In the concluion, we conider the relationhip between the quality difference model and account of the incumbency advantage baed on contituency ervice, pork barreling, and other factor that are pecific to legilative politic. The Model At each of two date, t 5 1 and t 5 2, a voter mut elect a politician. Each candidate c ha ability u c 2 R, and thee abilitie are independent draw from a normal ditribution with mean 0 and variance 2 u (denoted u c ;N 0; 2 u ). No one ha any private information about thee abilitie the only information about them are the ignal dicued below. In addition to her ability, each candidate ha a policy poition. In particular, there are left (L) and right (R) partie, each with a fixed policy platform in the onedimenional policy pace. We denote thee platform by m p 2 R, where p 2 {L, R}, with m L 5 m R. In the firt election (the open-eat election), each party run one candidate at it location. In the econd election, the incumbent i the candidate for her party, while recruiting a challenger cot the out party k 2 [0, 1/2). A party make it recruitment deciion to maximize the probability of victory minu the recruitment cot, without knowing the realized ability of it candidate. Before the firt election, the voter get a ignal about each candidate ability. Thi ignal for candidate c i 1c 5 u c + e 1c, where e 1c i ditributed Nð0; 2 e 1 Þ and i independent of u c. The winner of thi firt 4 The reult on partian tide formalize a finding from Zaller (1998)) imulation. The other reult have not previouly been derived in any manner. election (the incumbent) erve for the firt period and then might face another candidate (the challenger) in a reelection contet. The voter alo oberve ignal of the candidate abilitie before thi econd election. The econd ignal i 2c 5 u c + e 2c, where e 2c ;Nð0; 2 e 2 Þ. The voter ha preference repreented by the expectation of u (x* x) 2, where x i a policy and x* i the voter ideal point. That i, the voter care both about candidate ability and about the policie enacted. 5 The voter ideal point in each election i an independent draw from a Nðg; 2 x?þ ditribution. A negative value of g indicate an electorate that lean left on average, while a poitive value indicate an electorate that lean right on average. Thu g repreent peritent partian leaning, while the reidual x* g repreent hort-term partian tide the voter ideal point can change from election to election due to idioyncratic factor. Thi will lead to the partian tide oberved in empirical tudie of election. Table 1 ummarize the notation introduced in thi ection. In everal place in the text and concluion we alo explore the implication of variou extenion of the model. In particular, we dicu implication of the voter receiving an extra ignal about the incumbent quality by oberving the incumbent while in office, party primarie a an additional ource of electoral election, and endogenou, reputation-building effort by the incumbent while in office. Preliminarie Our main goal i to derive the comparative tatic of a quality-difference-baed incumbency advantage. In thi ection, we attend to two preliminarie. Firt, we derive the voter optimal voting rule. Second, we define the incumbency advantage in term of the model. The Voter Deciion Rule In each election, a candidate i characterized by the voter belief about her ability. The updating rule follow from tandard reult on Baye rule with a 5 There are everal poible interpretation of the ability term, u, including the ability of a candidate to ecure pork-barrel pending for her ditrict, charima, or demagogic ability. It i important to note that one poible interpretation of ability, namely ability to advance the party platform, i not conitent with the model et-up, ince ability and policy concern are eparable in the voter utility function.

3 1008 cott ahworth and ethan bueno de mequita TABLE 1 Symbol introduced in Section 1 Symbol Interpretation Symbol Interpretation u Ability of a k Recruitment cot candidate 2 u Prior variance of ability m p Policy platform of party p e i Obervation noie in election i 2 {1, 2} x* Voter ideal point 2 e i Variance of e i 2 Variance of x* x? g Voter partian leaning normal prior and normal ignal (DeGroot 1970) and are formally decribed in the appendix (ection A.1). Intuitively, the voter poterior belief about a candidate ability are normally ditributed with a mean that i a weighted average of the voter prior mean belief and the ignal the voter oberve. Thu, the better the ignal of candidate ability, the higher quality the voter believe the candidate to be. Moreover, the noiier the ignal, the le weight the voter put on it. The mean of the voter belief about a candidate c ability i denoted m c. The voter chooe a candidate by comparing the expected utility of each choice. Hi deciion rule i characterized by the following reult, the proof of which follow from a direct comparion of the voter expected utility for each candidate. Lemma 1 The voter vote for candidate L if and only if m L m R $ 2(m R m L )x*. The voter voting rule take into conideration two factor: hi belief about the candidate abilitie (m L, m R ) and hi policy preference. The term 2(m R m L )x* repreent the policy component of the voter choice. Since m R m L i alway poitive (i.e., the right-wing party i to the right of the left-wing party), when the voter i right-wing (x*. 0), thi term i poitive right-leaning voter are more likely to elect right-wing candidate. Similarly, when the voter i left-wing (x*, 0), thi term i negative left-leaning voter are more likely to elect left-wing candidate. Intuitively, if the voter were exactly indifferent between the two partie on policy (x* 5 0), then the voter would vote for the left-wing candidate if and only if hi expectation of the left-wing candidate ability (m L ) wa higher than that of the right-wing candidate ability (m R ). Similarly, if the voter believed the two candidate were of identical ability (m L 5 m R ) he would vote for the right-wing candidate if he wa a right-leaning voter (x*. 0) and vote for the left-leaning candidate if he wa a left-leaning voter (x*, 0). 6 Since it will play a major role in the analyi, we introduce the following notational implification: h(x*) [ 2(m R m L )x*. When no confuion can reult, we imply write h, which we refer to a the policy factor in the voter deciion. The Incumbency Advantage The incumbency advantage i a meaure of the improvement in a party expected electoral performance when it candidate i an incumbent rather than a candidate in an open-eat election. Definition 1 The incumbency advantage for party p i the expected difference between the probability party p win the election with an incumbent and the probability party p win an open-eat election. Definition 2 The incumbency advantage i a weighted average of the incumbency advantage for left- and right-wing incumbent, where the weight on a party p incumbent i given by the probability that party p win the open-eat election. To ee the intuition for thee definition, conider a imple example. Suppoe that the probability of the left-wing party winning an open-eat election i 48% and the probability of the right-wing party winning an open-eat election i 52%. Further, uppoe that the probability of a left-wing incumbent winning i 54% and the probability of a right-wing incumbent winning i 56%. The incumbency advantage for the left-wing party i 54% 48% 5 6%, while the incumbency advantage for the right-wing party i 56% 52% 5 4%. However, the right-wing party will hold office more frequently than the left-wing party, in expectation. Thu, the overall incumbency advantage i.48 ( ) +.52 ( ) %. The extra probability of winning aociated with being an incumbent i not the only way to define the incumbency advantage. Indeed, empirical work often etimate the advantage a the difference in the incumbent hare of the two-party vote and the incumbent party hare of the two-party vote in open-eat election (the normal vote). Our reult alo apply to thi definition, at leat in a imple onedimenional model of voter heterogeneity. Aume 6 Notice that the voting rule implie that, for any given difference in the candidate expected abilitie, if the hock to the voter ideal point i large enough (and in the correct direction), he will vote for the candidate with lower expected ability. Thi i a pecial feature of quadratic utility. Mot of our reult do not rely on it. Intead, our reult rely on the exitence of a tradeoff between policy and ability, which will exit quite generally.

4 electoral election, trategic challenger entry, and the incumbency advantage 1009 that there i a continuum of voter, with policy preference y i ditributed Nðx? ; 2 yþ. We refer to thi variant a a model with a heterogeneou electorate. A before, the median voter ideal point, x*, i itelf a random variable, with ditribution Nð0; 2 x?þ. Each voter payoff i (m p y i ) 2 + m p. In thi model, a voter will vote for the left-wing candidate if m L m R. 2ðm R m L Þy5y, m L m R 2ðm R m L Þ [ y: The left-wing vote hare i the percent of the population to the left of y. Note that thi i the ame rule a in Lemma 1, with x* replaced by y o that we can conider the percent of the vote rather than the probability of winning. Thi definition will not lead to numerically identical reult, ince typically varðyþ 5 2 y þ 2 x?. 2 x?. Nonethele, the common tructure of the two model mean that the comparative tatic will be the ame whether we meaure the incumbency advantage with repect to vote hare or probability of winning. A Baeline Model Thi ection tudie the pecial cae of the model which et both the partian bia (g) and the recruitment cot (k) to zero. We call thi pecification the baeline model. We tart with thi cae for two reaon. Firt, the ymmetry of the baeline model allow for a particularly clear development of the intuition behind our characterization of the incumbency advantage. Second, in thi cae we are able to derive analytically a rich collection of comparative tatic reult. The next ection relaxe the aumption of the baeline model and ue a mix of analytical reult and imulation to tudy the incumbency advantage and it comparative tatic, along the way howing that the reult of the current ection are robut. The Incumbency Advantage Exit Since the candidate have identical ex ante expected abilitie and the baeline model electorate i evenly balanced between the partie on average, each candidate in the open-eat election win with probability one half. Moreover, becaue the recruitment cot i zero, there will alway be a challenger in the econd election. The incumbency advantage, then, i the extent to which an incumbent probability of defeating a challenger i greater than one half. Conider an election with a left-wing incumbent whoe poterior mean ability i m. By Lemma 1, he win reelection exactly when m L m R h $ 0: The left-hand ide of thi inequality i a mean m normal random variable; let be it variance (defined formally in Appendix A.1). Thu the L incumbent win with probability 1 F m 5F m, where F i the cumulative ditribution function of the tandard normal. To calculate the incumbency advantage, we average thi probability conditional on m over the ditribution of incumbent abilitie. Since the innovation to the voter belief about the incumbent a a reult of the econd ignal i independent of m, 7 we can ue the tandard convolution formula for the um of independent random variable to write the probability that an incumbent win a ð F m f ðmþ dm; ð1þ where f i the ditribution of the poterior mean. Thi formula let u formalize the idea that incumbent do well in their reelection contet becaue, on average, they are of higher ability than challenger. Specifically, f i better than the prior ditribution in the ene of firt-order tochatic dominance. Thu it more likely that incumbent win than that a candidate of the ame party win an open-eat election. Propoition 1 In the baeline model, the incumbency advantage exit an incumbent win with probability greater than one half. Thi i a pecial cae of Propoition 6, o we do not give an independent proof. Comparative Static Having etablihed the exitence of the incumbency advantage within the baeline model, we can now turn to our main tak: exploring how change in the electoral environment change it magnitude. All of the comparative tatic dicued in thi ection reult from the interaction of two factor, reflecting the two factor in the integral (1). The firt i the degree of electoral election. Change in the environment that enhance the degree of election for ability increae the incumbency advantage, other thing equal. The econd factor involve change to 7 Thi i a tandard reult on the normal ditribution.

5 1010 cott ahworth and ethan bueno de mequita the probability that an incumbent win, conditional on her reputation following election. We call thi the incumbent inulation, ince it i the probability that an incumbent with fixed reputation retain office how well he i inulated from high-quality challenger and partian tide. With low cot of entry (and conequently incumbent running, on average, a favorite), the more inulated the incumbent, the larger the incumbency advantage. Our firt two comparative tatic are driven by the following intuition. Firt, voter who are cloe to indifferent between the two partie are more likely to vote baed on the quality dimenion, rather than the policy dimenion. When thi i the cae, voting in the open-eat election repond more to quality, leading to a larger electoral election effect and, thu, a greater expected difference in quality between incumbent and challenger in the econd election. Second, when voter are expected to be cloe to indifferent between the partie, it i le likely that an incumbent with aboveaverage quality will be replaced a a reult of partian tide. That i, the incumbent are more inulated. Hence, any factor that increae the likelihood that the voter i cloe to indifferent between the two partie on the policy dimenion tend to increae the incumbency advantage through both the election and inulation effect. (Omitted proof for thi ection are in Appendix B) Lemma 2 In the baeline model, the greater i the variance of the policy term ( 2 h ), the maller i the incumbency advantage. Partian Tide Recall that policy entered into the voter deciion through the term h 5 2(m R m L )x*. Thi policy factor ha two component: the polarization of the partie (m R m L ) and the voter ideal point (x*). Thi ideal point i ditributed normally with mean 0 and variance 2 x?. We can think of the variance of thi ditribution a a meaure of the average ize of hock to partianhip. When the partian hock i large, voter are trong partian and, thu, are particularly likely to bae their vote on policy. Thi implie that, on average, voter deciion are le likely to be influenced by quality conideration when partian tide are expected to be large than when they are expected to be mall, diminihing the quality difference baed incumbency advantage. Thu, we have the following reult. Propoition 2 In the baeline model, the larger i the average partian hock ( 2 ), the maller i the x? incumbency advantage. Proof The reult follow from Lemma 2 and the fact that 2 h 5 4ðm R m L Þ 2 2 x?: Thi reult implie that larger partian tide decreae the incumbency advantage. 8 A far a we know, thi relationhip ha not been tudied empirically. But the mechanim behind the reult eem to work in actual election. 9 Conider, for example, the congreional election of The ubtantial partian tide in favor of the Republican wept out an unuual number of incumbent Democrat. And the mall meaured incumbency advantage through the ret of the 1990 reflect, on thi view, the fact that the tide wept in many below-average-quality Republican. Propoition 2 i jut the tatement that, when the typical partian hock i large, thee two mechanim keep the incumbency advantage mall. From thi perpective, the fortune of the congreional cla of 1974 eem anomalou the Democrat who rode into office on the Watergate-inpired partian tide fared extremely well in their reelection effort. But the quality difference model ha potential reource to account for thee candidate a well. The partian tide of 1974 wa well anticipated, and, a a conequence, the Democratic challenger and openeat candidate were above average in quality (Jacobon and Kernell 1983). Comparing the election of 1974 and 1994 illutrate the importance of our aumption that potential candidate in the openeat election have no information about the realization of the partian hock. Polarization of the Partie The other component of the policy factor in vote choice i the polarization of the partie. When the partie are highly polarized, voter are more likely to have trong preference for one party or the other. Thi diminihe the quality advantage of incumbent, on average, becaue voter are willing to accept candidate with ubpar quality if they are of the favored party and are willing to replace high quality candidate if they are of the wrong party. Thu, a highly polarized party ytem weaken the incumbency advantage in the quality-difference-baed model. Propoition 3 In the baeline model, the more polarized the two partie are the larger i (m R m L ), the maller the incumbency advantage. Proof The reult follow from Lemma 2 and the fact that 2 h 5 4ðm R m L Þ 2 2 x?. Thi reult i not driven by quadratic utility over policy any trictly concave utility function would 8 Thi reult i imilar to Zaller (1998) comparative tatic on electoral luck. 9 We thank an anonymou referee for prompting thi and the following paragraph.

6 electoral election, trategic challenger entry, and the incumbency advantage 1011 lead to the ame reult. Intead, the key aumption behind thi reult i that the uncertainty driving the probabilitic voting i over the voter ideal point, rather than over an additive popularity hock for one of the partie. A dicued in more detail by Ahworth and Bueno de Mequita (2007), the random ideal point model doe, and the random popularity hock model doe not, make the policy/quality tradeoff a function of platform polarization. Viibility of the Office. Election for ome office, uch a governor or U.S. Senator, attract more media and voter attention than do election for other office, like tate auditor. The natural way to formalize thi in our model i to aume that voter receive more informative ignal about high-viibility office eeker. Formally, thi mean that the ignal of candidate ability have lower variance for more viible office. Conider firt the impact of more informative ignal in the open-eat election. When thee ignal are more informative, the voter i more likely to correctly identify the higher quality candidate. Thu, the election effect i heightened. There i another, reinforcing, effect. When the voter get better information, he become confident about hi belief more quickly, leading him to be le reponive to ubequent ignal. Thu, once he ha choen a candidate, future ignal are le likely to convince him to change hi mind, which help the incumbent (who run a the favorite) by increaing the inulation effect. While the voter i le reponive to information in the econd election, if thi information i, for ome reaon, ignificantly more precie than the information in the firt election, then the voter will take note. Thi can only hurt the incumbent, a it decreae inulation. Thu, the incumbency advantage i increaing in the informativene of the ignal in the firt election and decreaing in the informativene of the ignal in the econd election. It may make intuitive ene to think that, on average, the firt and econd election are equally informative (holding the office in quetion contant). That i, race for, ay, governor in a given tate attract about the ame level of media attention from election to election. 10 (Thi doe not mean that the voter ha the ame amount of information about both candidate, ince he ha oberved two ignal on the incumbent veru only one for the challenger.) If the two election 10 On the other hand, many office, e.g., legilative race, probably attract more attention in open-eat race. We can capture thi by letting the open-eat ignal have variance 2 e and the econd election ignal have variance b 2 e, with b, 1. The ame logic that applie to the cae of equal variance would then imply that the incumbency advantage i increaing in 2 e. are equally informative, then the original argument above implie that the incumbency advantage i increaing in the overall informativene of the ignal. Thee reult are tated in the following propoition. 11 Propoition 4 In the baeline model, aume that there i no uncertainty about the voter ideal point ( 2 h 5 0). The incumbency advantage i increaing in the informativene of the firt election ignal ( 1 ) and 2 e 1 i decreaing in the informativene of the econd election ignal ( 1 ). If both election are equally 2 e 2 informative, then the incumbency advantage i increaing in the informativene of the election ignal. Propoition 4 predict that high-viibility office will have larger incumbency advantage than low-viibility office. 12 Anolabehere and Snyder (2002) oberve exactly thi pattern in their empirical tudy of the incumbency advantage for all tatewide elected office. It i alo worth noting that thi reult allow u to conider what would happen to the incumbency advantage and our comparative tatic if the voter oberved an extra ignal about the incumbent that he did not oberve about the challenger (ince the voter oberve the incumbent during her term in office). Formally, thi would be equivalent to decreaing the variance of the econd ignal for the incumbent. Such a change would decreae the ize of the incumbency advantage. Thi i for the ame reaon that the incumbency advantage i decreaing in the informativene of the econd period ignal on average, the incumbent run a a favorite in the econd election, o it i not in her interet for the voter to learn more information, which might change the tatu quo. However, thi extra ignal would have no affect on the tructure of the underlying caue of the quality difference baed incumbency advantage. Thu, none of our comparative tatic would change. Polarization of the Electorate. Earlier, we dicued an interpretation of our model that meaure the incumbency advantage a the increae in hare of the vote won by an incumbent, rather than a the increae in the probability of winning. In thi verion, 11 The proof of thee reult relie on the pecial aumption that 2 h 5 0. However, we demontrate the robutne of the reult to relaxing thi aumption in our computational imulation in the next ection. 12 The reult about variation in the firt-period ignal variance would be enhanced if the voter wa rik-avere over ability then he would prefer the candidate with lower poterior variance, all ele equal. In fact, we could get thoe reult in a model driven entirely by rik averion, without electoral election. But rik averion alone i inufficient to derive the other comparative tatic in the paper, ince the other change we conider do not change the poterior variance.

7 1012 cott ahworth and ethan bueno de mequita we can alo ak what happen to the incumbency advantage a the electorate become more polarized. A more polarized electorate correpond, in the model, to a greater diperion of individual voter policy preference ( 2 y ). Thi increae in polarization doe not affect the degree of quality difference becaue the winner of the open-eat election i determined by the median voter. However, more polarization doe affect the hare of the vote received by an incumbent of any given ability. When the electorate become more polarized, policy conideration loom large for a greater number of voter, diminihing quality-baed voting. Thu, a polarization increae, the incumbency advantage meaured a additional vote hare decreae. Propoition 5 In the baeline model with a heterogeneou electorate, the more polarized the electorate ( 2 y ), the maller i the incumbency advantage in term of vote hare. Two Meaure of the Incumbency Advantage. A we noted earlier, there are at leat two reaonable meaure of the incumbency advantage: increaed probability of winning and increaed hare of the vote. Further, a we dicued above, while the two meaure hare comparative tatic, they may not generate identical point prediction. Becaue of thi, our model ha implication for how etimate of the incumbency advantage will differ depending on which meaure i employed. The intuition i imple the probability of winning definition correpond to the vote hare definition for an electorate that i identical in every repect except that polarization of the electorate i et to zero. Thu, Propoition 5 implie the following reult. Corollary 1 In the baeline model, the incumbency advantage meaured by probability of winning will be greater than the incumbency advantage meaured by hare of vote. And, indeed, thi difference i oberved acro empirical analye of the incumbency advantage. For example, Gowriankaran, Mitchell, and Moro (2008) find that incumbency increae the probability of winning a Senate election by approximately 15 percentage point, while Anolabehere and Snyder (2002) report that incumbency increae vote hare in Senate election by approximately 9 percentage point. The Full Model Now we relax the baeline model aumption that the ditrict i perfectly balanced between the two partie and that there i no recruitment cot. Becaue the model i ymmetric, we can focu attention on L incumbent. The Incumbency Advantage Exit A before, an L incumbent facing a challenge win reelection exactly when m L m R h $ 0: There are two difference between reelection probabilitie in the general cae and the baeline cae. Firt, the left-hand ide of the inequality now ha mean m 2ðx R x L Þg [ m h; yielding a reelection probability conditional on a challenge of F m h. Second, not all incumbent will face challenger. Intead, the R party will chooe to field a candidate if and only if the probability he will win exceed the recruitment cot. Thu the R party will mount a challenge if and only if m h 1 F $ k: The left-hand ide i trictly decreaing in the incumbent expected ability (m), o there i a cutoff m* uch that there i a challenge if and only if the incumbent reputation i le than or equal to m*. Thi cutoff can be written a a function of the cot of mounting a challenge (k), the partian leaning (h), and the variance (): m? ðk; h; Þ 5 F 1 ð1 kþþh: Since k, 1/2, we have m?. h. Thu the marginal challenger run a an underdog in the ene of having lower expected ability (net of expected partianhip) than the incumbent. In addition, thi function i differentiable in k and, 0: Thee comparative tatic are intuitive: higher cot directly make running le attractive, while a higher variance increae the likelihood of an upet, making running more attractive for the marginal (underdog) challenger. Let Vðm; m? ; hþ be the probability an incumbent with expected ability m i reelected, given a cutoff for mounting a challenge of m*. We have

8 electoral election, trategic challenger entry, and the incumbency advantage 1013 ( Vðm; m ; hþ 5 F m h if m # m? : 1 if m. m?? Becaue F(m/), 1 for all m, the function V i decreaing in m* in the following ene:??? 0 m. m Vðm; m ; hþ # Vðm; m ; hþ with trict inequality if m 2ðm????? ; m That i, the higher m* (othemorelikelythereito be a challenger), the le likely the incumbent i to be elected. Thi i becaue, if no challenge i mounted, the incumbent win for ure, wherea, if there i a challenge, thereiomechancethattheincumbentloe. A before, we define the incumbency advantage a the weighted average of the incumbency advantage for each party, with weight given by the probability that each party win an open-eat election. Recall that the incumbency advantage for party p i the difference between the probability that an incumbent from party p win reelection and the probability that a candidate from party p win the open-eat election. In the baeline model, the probability of a party winning the open-eat election wa 1/2, the ame for each party. In the current framework, the probability that a party win the open-eat election depend on the voter partian leaning. Of coure, thee partian leaning alo affect the probability that an incumbent i reelected. A in the previou ection, we can ue the tandard convolution formula for the um of independent random variable to write the probability that an incumbent win a ð Vðm; m ; hþ f g ðmþ dm; ð2þ? where f g i the ditribution of the poterior mean (derived formally in Appendix A.1). Uing thi formula, we have the following (ee Appendix A.2 for the proof). Propoition 6 1. The incumbency advantage exit the probability an incumbent win i greater than the probability the incumbent party win the open-eat election. 2. The incumbency advantage i continuou in the entry cot (k) and the partian bia (g). 3. The incumbency advantage i increaing in the entry cot, k.? Þ: A before, the exitence of the incumbency advantage i driven by two effect: election and inulation. The inulation effect i now augmented (relative to the baeline model) becaue ufficiently trong incumbent (m. m*) deter a challenger from entering and o are fully inulated. The incumbency advantage i increaing in the cot of challenger entry becaue thi entry deterrence component of the inulation effect grow moothly with thi cot. Do Cot of Challenger Entry Change the Comparative Static? In the baeline model we derived our comparative tatic under the retrictive aumption that there were no cot of challenger entry. Propoition 6 demontrate that the incumbency advantage i continuou in the cot of entry. Hence, the comparative tatic clearly hold for ufficiently mall entry cot. The quetion remain a to what happen when the cot of entry are large. To ee what i at iue, recall that all of the comparative tatic in the baeline model followed the ame baic logic ome exogenou change in a parameter increae the incumbency advantage becaue it enhance both electoral election and inulation. But when recruitment cot are great enough, the change in thee effect tart to work in oppoing direction. The reaon i that, for ufficiently high cot, mot incumbent who face a challenge will run a underdog. To ee thi mot clearly, conider the cae of maximal cot of entry (k 5 1/2). In thi cae, there will be a challenge if and only if the incumbent expected ability net of partian bia i le than the prior mean of zero (i.e., m h, 0). Thu, for maximal cot, all challenged incumbent will run a underdog becaue high cot deter potential candidate from challenging thoe incumbent who would run a favorite. For lightly lower cot, mot challenged incumbent will run a underdog. Unlike an incumbent running a a favorite, an incumbent who run a an underdog benefit from increaed variability in the outcome of the election. If the tatu quo favor the challenger, anything that decreae the likelihood that the outcome will reflect the tatu quo will help the incumbent. 13 Thu, when recruitment cot are ufficiently high, change in 13 It i important to note that we are not arguing that the incumbency advantage can decreae a the cot of entry increae. (Indeed, the incumbency advantage increae becaue more incumbent run unconteted.) Rather, we are arguing that, a the cot of entry increae, the other comparative tatic might change ign.

9 1014 cott ahworth and ethan bueno de mequita FIGURE 1 Simulated comparative tatic on k and 2 h, with g 5 0. Each frame repreent a vector of parameter value ( 2 u, 2 e ). The x-axi i the recruitment cot (k) and the y-axi i the incumbency advantage. Thu, moving along the x-axi i equivalent to increaing the cot of mounting a challenge. Each eparate curve in a given cell repreent a value for 2 h (the variability of the policy component of the voter deciion). The value of 2 h i decreaing a the curve increae on the vertical axi. A reveral of the comparative tatic on 2 h would involve thee curve croing, which they never do for the parameter value explored. σ 2 θ = 1, σ2 ε = 1 σ 2 θ = 1.5, σ2 ε = 1 σ 2 θ = 2, σ2 ε = 1 Incumbency Advantage Recruitment Cot Recruitment Cot Recruitment Cot Incumbency Advantage Incumbency Advantage Incumbency Advantage Incumbency Advantage Incumbency Advantage σ 2 θ = 2.5, σ2 ε = 1 σ 2 θ = 3, σ2 ε = 1 σ2 θ = 3.5, σ2 ε = Recruitment Cot Recruitment Cot Recruitment Cot parameter value that ued to increae both election and inulation, now increae election but decreae inulation. Increaing electoral election tend to increae the incumbency advantage. But decreaing inulation tend to decreae the incumbency advantage. Thu, thee effect are in tenion with one another. If the magnitude of the decreae in inulation i larger than the magnitude of the increae in electoral election, our overall comparative tatic will be revered. Unfortunately, we cannot give a imple analytic characterization of when the comparative tatic might revere. However, we explore thi quetion computationally, through a erie of imulation. A Figure 1 make clear, our main comparative tatic (on 2 h ) i robut to the incluion of a large recruitment cot. Each cell in the figure repreent a different vector of parameter. A one move up the y-axi within a cell, each curve repreent a lower value of 2 h that i, the incumbency advantage i decreaing in 2 h, jut a in Lemma 2. Moreover, for the parameter value explored here, the comparative tatic never revere, even for very high value of k (i.e., the curve within a cell never cro). Thu, at leat for thee parameter value, the increae in inulation aociated with an increae in 2 h (for high value of k) i never large enough to dominate the decreae in electoral election aociated with an increae in 2 h. It i, of coure, poible to find cae where the decreae in inulation doe become trong enough to revere a comparative tatic. The eaiet way to do thi i to allow the variance of the two election ignal to differ. Recall that the variance of the econd ignal ( 2 e 2 ) only affect inulation (ince election happen in the firt election). A a reult, for low value of k a move from a mall to a large 2 e 2 increae the incumbency advantage by increaing inulation. However, for high value of k, thi comparative tatic revere making the econd election le informative actually decreae the incumbency advantage by

10 electoral election, trategic challenger entry, and the incumbency advantage 1015 decreaing inulation. 14 Thi reveral of comparative tatic i illutrated in Figure 2. Ditrict with Partian Leaning In thi ection, we explore one more comparative tatic of the quality difference model: change in partian balance. What happen to the magnitude of the incumbency advantage when the electorate become more evenly divided between the two partie? Intuitively, a the voter become more likely to be indifferent between the two partie on the policy dimenion, the probability that he will make hi deciion baed on quality increae, leading to a tronger quality difference baed incumbency advantage. Thi i imilar to the intuition of Propoition 2, but we will ee that peritent partianhip (a oppoed to hort-run partian tide) introduce new complication. To keep the model tractable, we return to the aumption that k 5 0. Although the formula for the incumbency advantage in Appendix A.1 i complicated, we can build ome intuition about how the incumbency advantage change with hift in the partian balance of the electorate. In particular, we will ak what happen to the incumbency advantage for a left-wing incumbent when the electorate become more right-wing (g increae). Since the problem i ymmetric for leftand right-wing candidate, thi will alo tell u about right-wing incumbent. A rightward hift in the voter partian bia ha three effect on the incumbency advantage for a leftwing candidate, formalized in the three following lemmata. The firt effect i that a left-wing candidate ha a harder time winning the open-eat election the more right-wing i the voter. Thi mean that leftwing candidate only win open-eat election in right-wing ditrict if they can convince the voter they are of particularly high ability. Thu, the more right-wing the ditrict, the more electoral election left-wing candidate face, and the higher the voter believe a left-wing incumbent ability to be. Thi increae in election tend to increae the left-wing incumbency advantage. 14 Thi doe not necearily mean that the ubtantive comparative tatic the incumbency advantage hould be greater in more viible office revere. Increaing the viibility of the office mean decreaing both variance, a cae where our imulation do not reveal a reveral. In addition, it i probably eaier for partie to recruit acceptable candidate for le viible office, meaning k hould increae with viibility. Thi ha the direct effect of increaing the incumbency advantage, along with the indirect effect of potentially revering the comparative tatic on the ignal. We thank an anonymou referee for the lat point. FIGURE 2 Incumbency Advantage For low value of k the incumbency advantage i increaing in the econd ignal variance. For high value of k the incumbency advantage i decreaing in the econd ignal variance σ 2 ε 2 = 1 σ 2 ε 2 = 10 σ 2 θ = 10, σ 2 ε 1 = Recruitment Cot Lemma 3 The ditribution of left-wing incumbent abilitie i better (in the ene of firt-order tochatic dominance) the more right leaning i the electorate (higher g). Lemma 3 identifie the effect of partian leaning on electoral election. All ele equal, an increae in electoral election, like that identified in Lemma 3, implie an increae in the incumbency advantage. However, in the current model, increaed partianhip ha an effect not only on election, but on the two probabilitie in the definition of the incumbency advantage. Thee probabilitie are the probability a left-wing candidate win an open-eat election and the probability of a given left-wing incumbent being reelected (i.e., inulation). The next two lemmata decribe the effect on thee probabilitie. A the electorate become more right-wing, the probability the left-wing party win the open-eat election decreae. The left-wing incumbency advantage i the probability a left-wing incumbent win minu the probability the left-wing party win the open-eat election. Hence, thi econd effect alo increae the incumbency advantage. Lemma 4 The more right-wing the voter, the lower the probability the left-wing party win the open-eat election. The third effect i that the left-wing incumbent in a more right-wing ditrict face a more right-wing voter on average. Thu, a the ditrict become more right-wing, the probability a left-wing incumbent

11 1016 cott ahworth and ethan bueno de mequita FIGURE 3 The left-hand panel how how change in partian balance affect the imulated probability a left-wing candidate win a an incumbent and in an open eat election. The right-hand panel how how partian balance affect the left-wing incumbency advantage the difference between the two line from the left-hand panel. The horizontal axi meaure partian balance a the probability that the right-wing candidate win the open eat election. By a light abue of the tandard terminology, we call thi the normal vote ince, in our model, the probability of winning and the expected vote hare have the ame comparative tatic, thi hould not caue any confuion. σ 2 θ = 1, σ 2 ε = 1, σ 2 η = 1 σ 2 θ = 1, σ 2 ε = 1, σ 2 η = 1 probability left wing candidate win open eat left wing incumbent left wing incumbency advantage win reelection decreae, holding her expected ability contant. Thi decreae in inulation tend to reduce the incumbency advantage. Lemma 5 Conider a fixed ditribution of left-wing incumbent expected abilitie. The more right-wing i the voter, the le likely i a left-wing incumbent to win reelection. Thee reult how that changing the partianhip of the voter by making him more right-wing ha competing effect on the left-wing incumbency advantage. On the one hand, it increae the expected quality of left-wing incumbent by increaing electoral election (Lemma 3) and decreae their chance of winning the open eat (Lemma 4) both of which work to increae the left-wing incumbency advantage. On the other hand, it diminihe inulation (Lemma 5) working to decreae the left-wing incumbency advantage. Unfortunately, we cannot give a complete analytic account of how thee competing effect balance out to affect the incumbency advantage. However, we can ue imulation to build on the analyi above. The three effect are illutrated in imulation reported in Figure Conitent with Lemma 4, the imulated probability that the left-wing candidate win the open-eat election decreae a the voter become more right-wing (the line marked with olid 15 The R code ued for all imulation i available at home.uchicago.edu/~bdm. circle in the left-hand panel i decreaing). Conitent with Lemma 5, the imulated probability that a left-wing incumbent win reelection (i.e., the level of inulation) alo decreae a the voter become more right-wing (the curve marked with open diamond in the left-hand panel i alo decreaing). However, the curve marked by open diamond in the left-hand panel of the figure, which repreent the probability of an incumbent winning, alo how the election effect from Lemma 3 the more right-wing the electorate, the higher the expected ability of a leftwing incumbent. Hence, in the figure, expected ability i not contant a partian balance change, rather ability i increaing with partianhip. Thu, the curve marked with open diamond ha a hallower lope than the line marked with olid circle. The vertical ditance between the two curve repreent the left-wing incumbency advantage for a given level of partianhip the probability a leftwing incumbent win minu the probability a leftwing candidate win an open-eat election. Thi difference i plotted in the right-hand panel of the figure. Figure 3 how that the left-wing incumbency advantage i maximized with a moderately right-wing voter. Thi maximum i achieved where the decreae in inulation aociated with increaed right-wing partianhip begin to more than compenate for the poitive effect of tronger electoral election and a maller normal vote. The right-wing incumbency advantage, of coure, i the mirror image. Thu the

12 electoral election, trategic challenger entry, and the incumbency advantage 1017 FIGURE 4 Simulated comparative tatic on 2 e and g. Each frame repreent a vector of parameter value ( 2 u, 2 e, 2 h ). The x-axi i the normal vote for a right-wing candidate. Thu, moving along the x-axi i equivalent to changing the level of partian balance (g), where the electorate partian preference are exactly balanced between the two partie at the midpoint. σ 2 θ = 1, σ 2 ε = 1, σ 2 η = 1 σ 2 θ = 1, σ 2 ε = 1.5, σ 2 η = 1 σ 2 θ = 1, σ 2 ε = 2, σ 2 η = 1 σ 2 θ = 1, σ 2 ε = 2.5, σ 2 η = 1 σ 2 θ = 1, σ 2 ε = 3, σ 2 η = 1 σ 2 θ = 1, σ 2 ε = 3.5, σ 2 η = 1 right-wing incumbency advantage i maximized in a moderately left-wing ditrict. The overall incumbency advantage i a weighted average of the left- and right-wing incumbent advantage, weighted by the probability of each party winning an open-eat election. A Lemma 4 how, left-wing candidate will be more likely to win an open-eat election in left-leaning ditrict and rightwing candidate will be more likely to win an open-eat election in right-leaning ditrict. Thu, in calculating the overall incumbency advantage, the majority of weight in any given ditrict will be put on the candidate whoe incumbency advantage i increaing in partian balance. Thi ugget that the overall incumbency advantage will be increaing in partian balance and will be maximized at a normal vote of one half for each party i.e., in a perfectly balanced electorate. Thi intuition i confirmed in the imulation reported in Figure 4 and 5. The incumbency advantage i increaing a the electorate normal vote move cloer to one half, i.e., the partian balance increae. Thi can be een in each frame of both Figure 4 and 5. A expected, for all vector of parameter value that we have invetigated, the incumbency advantage (the y-axi) i increaing a partian balance increae. The imulation alo erve a a check on the robutne of our earlier comparative tatic on the viibility of the office ( 2 e ), expected ize of partian tide ( 2 ), and polarization (m x? R m L ). The comparative tatic for the viibility of office can be een by moving acro the panel of Figure 4, and the comparative tatic for both partian tide and polarization can be een by moving acro the panel of Figure 5 (ince h 5 2(m R m L )x*). In each cae the imulated comparative tatic in the more general model are the ame a the analytically derived reult from the baeline model. The incumbency advantage i decreaing a the viibility of office decreae (Figure 4) and a party polarization and the ize of partian tide increae (Figure 5). Given that we had to ue computational method to explore the interaction of all three effect, one might wonder whether the effect of partian balance on the incumbency advantage i actually caued by the electoral election effect identified in Lemma 3 or whether it i driven by the nonelection baed effect

13 1018 cott ahworth and ethan bueno de mequita FIGURE 5 Simulated comparative tatic on 2 h and g. σ 2 θ = 1, σ2 ε = 1, σ2 η = 1 σ 2 θ = 1, σ2 ε = 1, σ2 η = 1.5 σ 2 θ = 1, σ2 ε = 1, σ2 η = 2 σ 2 θ = 1, σ 2 ε = 1, σ 2 η = 2.5 σ 2 θ = 1, σ 2 ε = 1, σ 2 η = σ 2 θ = 1, σ 2 ε = 1, σ 2 η = 3.5 identified in Lemmata 4 and 5. In order to invetigate thi poibility, we imulate a verion of the model that include the nonelection effect, but exclude electoral election. Thi i done by auming that, after the ignal of candidate ability are revealed in the open-eat election, the winner i choen at random. Thu, there i no election for ability. Thee imulation are hown in Figure 6. FIGURE 6 The left-hand panel how the effect of partian balance on the probability a left-wing incumbent win, if there i no election on quality, and the probability a left-wing candidate win an open eat. The difference between thee (hown in the right-hand panel) how how partian balance affect the left-wing incumbency advantage given the effect identified in Lemmata 4 and 5, but without electoral election. Notice that the right-hand panel ha the ame vertical cale a the right-hand panel of Figure 3. σ 2 θ = 1, σ 2 ε = 1, σ 2 η = 1 σ 2 θ = 1, σ 2 ε = 1, σ 2 η = 1 probability left wing candidate win open eat left wing incumbent left wing incumbency advantage without election