Campaign Rhetoric and the Hide-&-Seek Game

Transcription

1 Campaign Rhetori and the Hide-&-Seek Game Sourav Bhattaharya Department of Eonomis, University of Pittsburgh July, 04 Abstrat We present a model of politial ampaigning where a andidate hooses between promoting oneself (positive ampaign) or attaking the rival (negative ampaign). Campaign hoies determine the subjet of publi deliberation: If a andidate runs a positive ampaign and his rival a negative ampaign, the voters learn the quality of the foal andidate. Thus, negative ampaigns may be used either to expose the rival andidate (informative role) or to turn attention away from oneself (non-informative role). The e et of negative ampaigns depends on whether it is faed with another negative ampaign (ross talk) or a positive ampaign (fruitful debate). We suggest that in order to asertain the e et of negative advertising, studies should take into aount the ampaigns employed by both andidates. Voter expetation about andidate quality plays a major role in ampaign seletion: while the inidene of negative ampaigning goes down as the expeted prior improves, the probability of seletion of the orret andidate is non-monotoni in the said prior. Introdution Eletoral ampaigns are possibly the most important element of voters information about the andidates. However, ampaigns do not ontain disinterested, impartial information. Candidates hoose their rhetori strategially in order to in uene voter pereption. One aspet of suh strategi hoie that has reeived onsiderable attention both in the media and in the aademy is the hoie between positive and negative advertising: whether to highlight one s own quality or to fous on why the rival is unsuitable for o e. While every eletoral ampaign involves a I thank David Austen-Smith, Allen Brierly, Steve Callander, Joyee Deb, Alexandre Debs, Tim Feddersen, Yuk- Fai Fong, Sean Gailmard, Bard Harstad, Jaehoon Kim, Tapas Kundu, Mariano Sinisalhi, seminar partiipants at MPSA onferene (006), Mihigan State University and CIDE, Mexio for omments and suggestions. I also aknowledge the hospitality and nanial support from Wallis Institute of Politial Eonomy, University of Rohester that I enjoyed while preparing part of the manusript. Any error that remains in the paper is my responsibility.

2 large number of positive, negative and (espeially) omparative messages, we shall haraterize the overall theme of a ampaign as broadly positive or negative, and it is the hoie of this theme that we shall examine in this paper. In this paper, we think of ampaigns as publi debates in whih the voting publi forms its judgement about the andidates by omparing and ontrasting their ampaign statements. Thus, the nature and extent of information revealed about andidates depends jointly on the ampaign themes hosen by both andidates. In partiular, our main premise is that If one andidate deides to highlight why he is suitable (positive ampaign) and the other argues why his rival is unsuitable (negative ampaign), then the former andidate is the foal one in the eletoral rae: the publi learns more about his quality than about the other one. This reates the main tension that we study in the paper: while good andidates try to reveal information through foality and the bad ones trying to avoid foality. The analysis borrows its basi idea from the literature on issue hoie in ampaign rhetori (Simon, 00; Kaplan et al, 006; and Egorov, 0): more information is revealed about an issue if the two andidates disuss both sides of the issue rather than if only one andidate presents his side of the issue. In this paper, we study the andidate s deision to fous on one of two deidedly aggregate issues: matters pertaining to one s own suitability for o e vs those pertaining to the rival s quality. Suh a model allows us to analyze onditions under whih negative advertising is used to inform the eletorate or to muddle the debate. Opinions are deeply divided regarding whether the pratie of negative ampaigning improves or degrades the eletoral proess. While some view negative ampaigns as essentially viious mudslinging with the e et of reduing turnout (Ansolabehere and Iyengar 995, Ansolabehere et al 994, 999), destroying voters trust in government and undermining the e ay of the politial system (Brader 005), others (Geer 006) have pointed out its role in providing genuine information and disiplining the andidates, and even in stimulating turnout (Freedman and Goldstein 999, 00, Finkel and Geer 998, Kahn and Kenney 999, Wattenberg and Brians 999). Over the last two deades, there has been a profusion of empirial and experimental researh examining these e ets. However, in a meta-study overing 0 artiles, dissertations and books published on the topi between 987 and 007, Lau et al (007) nds that there is no onsensus in the literature on the role of negative advertising (see also Lau et al 999). All empirial or experimental work aimed at unovering the role of negativity in advertising has so far looked at the e et of individual messages/ampaigns in isolation - our model postulates that we may get a learer and more nuaned piture if future work takes into aount both the ampaign themes (or messages) in ontention. There is a wide variation in the de nition of negative advertising employed in the literature. While some of the literature has looked at the role of individual messages, a large hunk of the empirial literature has oded ampaigns as either primarily positive or primarily negative or the extent to whih a whole ampaign is negative. Ansolabehere et al, in their seminal study of turnout in 34 senate eletions in 99, use the binary haraterization and others (e.g., Brooks (006) and Wattenberg and Brians (999)) have used the same data.

3 At one extreme, Ansolabehere and Iyengar (995) equates negative advertising with mudslinging and ad hominem attaks; and at the other extreme, Polborn and Yi (006), heneforth referred to as PY, has de ned negative advertising exlusively as provision of valuable eletoral information about the rival andidate. We use a broad de nition of negative and positive ampaigns that is adopted from Finkel and Geer (998): [P]ositive appeals are ones that andidates o er to promote themselves on some issue or trait. ; and Negative appeals are attaks [or ritiism] leveled at the opposition. While this de nition literally applies to spei messages, we adjust this de nition for aggregate ampaign themes In our model, a positive ampaign may ontain messages about one s own strength of harater, values, reords and arguments about why one s position on di erent issues is lose to that of the median voter. On the other hand, a negative ampaign may ontain a series of messages regarding the opponent s features (harater aws, vies, inonsistenies, past failures) and arguments why his politial positions are not lose to that of the median voter. In sum, the ruial di erene between a positive and negative ampaign is whether the informational spotlight is foused on oneself or on the rival. Aording to this de nition, there is nothing inherently negative about talking about the rival (or positive about foussing on oneself). However, it is never in the interest of a andidate to highlight his own failings or his rival s virtues: therefore, self-foussed ampaigns typially ontain positive information and rival-foussed ampaigns ontain negative information. Given this ategorization of ampaign themes, whether negative advertising is useful or harmful to the voter arises endogenously from andidate hoies. When one andidate goes positive and the other negative, the former andidate s quality is the foal issue: the andidates engage in an ative debate about this issue. The voters learn about the foal andidate by omparing laims and ounterlaims. Most of the pre-eletoral disussions in the publi sphere are entered on the foal andidate. Two reent examples of foal andidates in high pro le eletions are Barak Obama in the Amerian Presidential Eletion of 008 and Narendra Modi in the Indian Lok Sabha eletion of 04. Compared to the foal andidate, a lot less is learnt about the rival in suh eletions. Not every eletion has a foal andidate, however. If both andidates run positive ampaigns or both employ negative ampaigns, they are essentially talking past eah other. There is a lot less issue engagement in this ase. Sine there are only one-sided laims whih are less redible, the voters learn a lot less about either andidate in this ase. As the ampaign wathdog, the media has an important role to play in our story. Following PY, we think of the andidates as setting the agenda for the media through their hoie of ampaign themes. The media heks the veraity of statements made by the andidates, ompares laims and ounterlaims and produes a judgement on the issue being debated. Given the onstraints on the time, spae and resoures of the media, fat-heking more e ient when there is onentrated engagement about one andidate rather than a di used disussion about both andidates. Sholars studying Amerian politial ampaigns have long stressed the need for issue engagement (or dialogue ) for enabling the voting publi to make an informed deision. Berelson et al 3

4 (954), Kelley (960), and more reently Petroik (996) and Sellers (998) doument and bemoan the tendeny of andidates to talk past eah other. Simon (00) de nes dialogue ours only when two andidates address the same subjet...the opposite of dialogue is ignoring, responding by disussing a di erent subjet. Emphasizing that the level of dialogue approximates the quality of the ampaign, he goes on to observe that the deision to dialogue an be onstrued as a strategi hoie. This strategi hoie is formally modelled by Simon (00), Kaplan et al (006) and Egorov (0) where they assume that voters learn more about issues over whih andidates hoose to engage in a dialogue. Simon (00) and Kaplan et al (006) also estimate the extent of dialogue over di erent issues in several empirial and laboratory settings. In the urrent paper, we extend the logi of issue hoie to the spei ase of positive vs. negative ampaign themes. Geer (006), in his detailed historial study of Amerian Presidential ampaigns, informally makes an argument similar in spirit to what has been modeled in this paper. While positive ampaigns allow andidates to "talk past eah other" and voters would "assume innoene unless proved guilty", negative ampaigns arry the risk of being aught "making unsubstantiated harges" (Geer, 006; page 5). We allow, in addition, the possibility that even negative ampaigns, when arried on by both sides, may not reveal muh information either.. Model: Brief Disussion In our formal model, eah of the two andidates has a privately known quality - either good or bad. At the beginning of the ampaign, eah andidate has the option to learn about the quality of the rival by investing a ost. Suh ostly learning aptures a range of things from researhing the rival andidate s voting reords and personal history to privately polling setions of the eletorate in order to nd out about how the publi views the rival s positions on salient issues. Armed with information about their own quality and possibly the rival s quality, andidates ommit to a ampaign theme: positive or negative. Suh preommitment is a simplifying assumption made in order to ignore the dynami aspet of ampaigns where a andidate make ounter-laims in response to the rival s laims, and is entirely standard in the literature on issue hoie. In the model, we blakbox the atual proess of voter opinion formation in what we all the information revelation protool. If one andidate hooses a positive ampaign and the other a negative one, we say that there is a fruitful debate, otherwise there is ross talk. In ase of a fruitful debate, the andidate whose quality is disussed is said to be foal while the other andidate is non-foal. In ase of ross-talk, both andidates are foal. In our benhmark ase, we assume a deterministi information revelation protool: a andidate s true quality is revealed if only he is foal, and the ampaign reveals no information about non-foal andidates. In a latter extension, we assume a probabilisti information revelation protool: the qualuty of foal andidate is revealed with a higher probability than that of non-foal andidates. Voters vote based on information revealed through the protool under onsideration. All voters have the same preferene and prefer to vote 4

5 for the good rather than the bad andidate. Given the information revelation protool, if a good andidate is faing a bad andidate, the former would prefer that either andidate be foal while the latter would prefer that neither is foal. This means that the two andidates have opposite preferenes over message pro les: the good andidate prefers to go positive (negative) when the bad andidate goes negative (positive), and the bad andidate prefers to go positive (negative) when the good andidate goes positive (negative), leading to a mathing pennies game between the good and bad types. Sine the bad andidate wants to hide (oneal information) and the good andidate wants to hase (reveal information), we title the ompetitive proess of information provision as a Hide and Seek game. The unique symmetri equilibrium outome of this game desribes andidate behavior and determines the extent of information revealed in eletorate. The introdution of unertainty between andidates and the option of ostly searh is a ruial innovation in this paper. Between-andidate unertainty aptures a fundamental asymmetry between positive and negative ampaigns that annot be obtained in models whih start out assuming that andidates know about eah others type (e.g. PY). While information used in a positive ampaign is free for a andidate, truth in a negative ampaign is ostly to him. As a onsequene, positive ampaigns ontain more information than negative ampaigns. Moreover, this feature allows us to expliitly distinguish between truthful negative ampaigns and false attaks. However, it must be remembered that false attaks also play a vital role in information revelation: if a good andidate an reveal his type to the voter by defending himself suessfully against false attaks.. Main Results Beause of inomplete information between andidates, the prior belief of a andidate being good plays a major role in determining the nature of debate. Using this prior as a proxy for average (expeted) andidate quality, we fous on understanding the strategi behavior of andidates as a funtion of this average (expeted) andidate quality and the welfare impliations of suh behavior. The model on rms the onventional wisdom that the inidene of negative advertising inreases as the average quality worsens. However, the hide-and-seek nature of the rae provides an advantage to the rarer type. In partiular, a good andidate an most e etively reveal his type through positive advertising when the average andidate quality is low and there is a lot of negative ampaigning. At the other extreme, when the average andidate quality is very high, all andidates engage in more positive ampaigning: but this allows the bad andidates to e etively hide by avoiding debate with the good types. Thus, while an eletorate with low expeted andidate quality is ompetitive but e ient for the purpose of andidate seletion, one with a high average quality is ine ient. Consequently, voter welfare in terms of the ex-post expeted quality of the winning andidate may atually go down with an inrease in the ex-ante expeted 5

6 andidate quality. The model also allows us to talk about a novel role of ampaign resoures. Snyder 989, for example, examines the role of ampaign resoures in nding information about the opponent. The searh ost an be seen as an index of the extent of asymmetry in information between the andidates. While a lowering of this ost improves the information ontent of negative advertising, we show that the bad type has a stronger inentive to engage in searh than the good type. Therefore, the expeted welfare of the voter is not monotoni in the ost of searh. In partiular, if a soial planner ould "selet" the ost of searh in order to maximize the probability of orret seletion, then she would set the ost to zero when the good type is more ommon and set the ost very high when the bad type is more ommon. Similarly, the voter s welfare is not monotoni in the payo from winning o e..3 Related formal work As mentioned before, the urrent paper brings together two strands of theoretial work on politial ampaigns: one on issue engagement and the other on positive vs. negative advertising. Simon (00) presents a model where andidates alloate an advertising budget over many issues, and shows that eah andidate has an inentive to emphasize a di erent set of issues rather than engaging in a dialogue with the rival on any given issue. Aording to Simon, a dialogue an be expeted on an issue only if (i) it has overriding saliene, (ii) mass media fores dialogue, or (iii) andidates behave irrationally for whatever reason. Kaplan et al (006) introdues unertainty about voter preferenes over andidates positions and obtains dialogue in Simon s model in a mixed strategy equilibrium. In our model, the between-andidate unertainty performs a similar role as andidate unertainty over voter preferenes in Kaplan at el (006) and leads to fruitful debate in equilibrium. Earlier formal work on hoie between positive and negative advertising (Skaperdas and Grofman 995, Harrington and Hess 996) takes as primitive an in uene funtion that assumes that negative advertising redues support both for its sponsor and target. Skaperdas and Grofman (995) nds that the trailing andidate engages in more negative advertising than the frontrunner, and Harrington and Hess (996) nds that the andidate with less attrative personal attribute runs a more negative ampaign. In the equilibrium of our model too, the bad andidate is weakly more likely to engage in a negative ampaign than the good andidate (and stritly so unless the prior expeted andidate quality is very high). PY onsiders a signaling model where andidates rationally hoose the ampaign theme and voters infer andidate quality by observing the hosen theme. 3 The authors nd that a andidate Desposato (008) studies the inidene of negative advertising in Latin Amerian ountires in a omparative perspetive by using similar in uene funtions, but the outome on voting probabilities is random. 3 Mattes (008) is a very similar model with hoie between informative advertising on two dimensions: negative vs. positive and issue vs. harater. 6

7 uses a negative ampaign when he either does not have too muh to say about himself or he knows that his opponent is very bad. While these authors assume that negative advertising ontains only hard information and is therefore always informative, our framework aounts for both good and bad motivations for attak by onsidering softer information. Consequently, these two models generate opposite empirial preditions about the e etiveness of negative advertising. PY nds that in an environment where most andidates are expeted to run negative ampaigns, when a andidate runs a negative ampaign against one running a positive ampaign, the former is more likely to win. We obtain the exatly opposite onlusion: in suh an environment, the positive ampaigner has an advantage. In our model, when the expeted andidate quality is very low, the bad types use negative ampaigns to turn the fous on the rival - but this allows the rare good type the opportunity to reate foality by running a positive ampaign. We also show that the full knowledge of rival type assumed in PY does not hold, even for very small searh osts. The remainder of the paper is organized as follows. Setion sets up the benhmark model in whih the andidates are either good or bad and Setion 3 haraterizes the equilibrium. Setion 4 disusses how andidate behavior in the benhmark ase hanges as the prior belief and the searh ost hange, and analyses welfare impliations. Setion 5 presents a few extensions and Setion 6 onludes the main body of the paper. Appendix A disusses an extended version of the model with voter inferene and a more general type spae. Most proofs are in Appendix B. Basi Model: Binary Types There are two players: andidates and with a private quality i ; (i = ; ) whih an be either Good (G) and Bad (B): 4 For both andidates, quality follows a ommonly known Bernoulli distribution with the prior probability of a good type (0; ): The andidates run ampaigns (positive or negative) whih reveal information to a voter about quality based on whih she votes for one of the two. While the voter is not modeled expliitly as a player, her ations based on the revealed information are taken into aount in the payo s arising from andidate ations. In appendix A we show exatly the same onlusions hold when we formally model the voter as a player.. Ations Player i has two di erent ation hoies: the ampaign ation M i fp; Ng and the searh ation X i fs; NSg: P denotes a positive ampaign and N denotes a negative one. If player i 4 The quality of a andidate an be thought of in two ways. In a ommon values framework where all voters have the same preferene, quality aptures all harateristis that voters are about. In a private values framework where the two andidates an be assigned two loations on the left-right ideologial ontinuum, quality an simply be thought of as the distane of a andidate from the median voter s ideal point. 7

8 undertakes ation S (searh), he gets to know the type of his rival i with ertainty. If ation NS (no searh) is taken, the rival s type is not known. One an think of the searh ation as being taken before the ampaign hoie, so that the message an be onditioned on the information obtained through searh. But we assume that the searh ation itself is private, i.e. a andidate annot detet whether the rival has searhed or not. Thus, the searh stage and the debate stage an be onsidered simultaneous from the strategi point of view. Campaign hoies by the andidates are also, in the same sense, strategially simultaneous: we assume that a andidate annot ondition his hoie of ampain on the ampaign hosen by the other andidate. 5 The ampaign pro le is denoted by M = fm ; M g; the searh pro le by X = fx ; X g and the type pro le by = f ; g: Campaigns are thought of as publi debates. The payo to player i from an ation and type pro le fm; X; g is assumed to be the payo u i (M; ) from debate less the ost of searh, whih is > 0 if X i = S and zero otherwise. The payo from debate u i (M; ) depends on the information revelation protool. The winner of the debate gets a payo w > 0; while the one who loses the debate gets a payo of 0: We normalize w = : The searh ost is an index of how easy it is for a andidate to nd out detailed information about the other andidate: in this sense it measures the extent of asymmetry of information between andidates. Sine the payo from winning the debate has been normalized, in e et, aptures the ratio of the atual ost of searh to the payo from winning o e. Thus, an inrease in the importane of the ontested o e (with the di ulty of nding information about the rival remaining the same) would imply a drop in searh ost. In the paper, we restrit to lie in the range 0; : In this range, we show that searh ours in equilibrium for moderate values of the prior (; ): While the ase with > an easily be overed by the model, we do not formally study this ase sine there is no information aquisition for any prior belief over andidate quality in equilibrium (please see footnote 8).. Information Revelation Protool The ampaign an be thought of as a publi debate that reveals information to the voter. The atual proess of debate is not modeled: we assume a protool whih determines how information about andidate types is revealed to the voter depending on the ampaign pro le. If one andidate goes positive and the other goes negative (i.e. M 6= M ), then the former andidate is foal in the sense that his quality is subjet to a fruitful debate. If on the other hand, there is ross talk, i.e. if M = M ; there is no foal andidate. We assume that a foal andidate s quality is revealed to the voter and a non-foal andidate s quality is not. A foal andidate wins the eletion if he is revealed to be good and loses the eletion if he is revealed to be bad. If there is ross talk, the winner is hosen randomly with equal probability. 5 Sine the ampaign theme is part of the broader ampaign strategy whih is determined in advane (rather than tatis whih an hange as the ampaign progresses), the simultaneity of message hoie is not too bad an assumption to start with. 8

9 Although simplisti, suh a passive voter response to debate works for a two-type ase beause if one andidate is revealed to be good (bad), the other andidate annot be better (worse). In Appendix A, we show that if we inlude the voter as a rational player in the game, suh passive voting strategies onsidered here arise as equilibrium behavior, and the equilibrium remains unhanged. Assume that the winner of the debate gets a payo normalized to : In this onstant sum game, the utility from debate u (M; ) for player is given by the following table, and that for player is u (M; ) = u (M; ): u (P; P; ) = u (N; N; ) = ; u (P; N; G; ) = u (N; P; ; B) = ; u (P; N; B; ) = u (N; P; ; G) = 0; The rather extreme assumption that the media reveals the true type of the foal andidate and reveals nothing otherwise is not neessary for the qualitative results of the model to go through. It has been made for tehnial onveniene so as to be able to drive the basi point home without using unneessary parameters. In setion 5 we disuss the results in a generalized model where this extreme assumption is relaxed retaining the assumption that a laim reveals information with a higher probability if it is faed with a ounter-laim. In that setion, we use a probabilisti (rather than deterministi) information revelation protool whih assumes that () in a fruitful debate, the quality of the foal andidate is revealed with a higher probability than the non-foal one, and () the probability that the voter learns neither andidate s type is higher under ross talk than under a fruitful debate..3 Inentives To understand the inentives that suh a payo struture reates, it is useful to desribe them in a normal form. There are three possible situations, depending on whether a good type faes a good type, a bad type faes another bad type, or whether the two andidates are of di erent types. Eah ase is desribed in a separate matrix, with the row player s payo shown in the matrix. G type B type P N P N G type G type P N P N 0 B type B type P N P 0 N Suppose for now the andidates knew eah other s types. Then they would know whih one of 9

10 the above three situations they were in. If a good andidate were faing another good andidate, then it is stritly dominant for both to run a positive ampaign: both players would want to disuss their own qualities. If a bad andidate were faing another bad one, then it is stritly dominant to use a negative ampaign: neither player wants the fous on himself. When a good andidate faes a bad andidate, there is a mathing pennies game: the good andidate wants to reveal information about either andidate: thus he wants either to be foal himself (P; N) or the rival to be foal (N; P ). The bad andidate, on the other hand, wants to oneal information by induing ross talk (P; P ) or (N; N).What makes this game interesting is that there is inomplete information about the rival s type, and eah andidate has an inentive to invest in researh about the rival s quality before deiding the ampaign theme. At this point, two omments are in order about some of the assumptions of the model. In a formal sense, the ampaign hoie M i by andidate i f; g is a message about his private information i : There are two traditions in eonomis about the relationship between messages and private information, and both have been applied to the analysis of politial advertising. At one extreme, messages are thought to be heap talk, i.e., ompletely unveri able (Prat 004). At the other extreme, messages are onsidered to be ompletely veri able signals about private information, and therefore by de nition truthful (PY). We reognize that while the ontent of andidate statements often arry valuable information about andidate types, suh statements also inlude half-truths, unveri able laims and outright lies. Even a fat-heking media may not always be able to asertain the truth in all ases. We therefore assume that messages are only onditionally veri able, i.e., i is revealed only when M i = P and M i = N: One interpretation ould be that there is always two kinds of information about a andidate: positive information and negative information. Very often, both kinds of information have to be weighed against eah other to have full piture of the quality of a andidate. Suh a omplete assessment annot be obtained if both sides of the same issue are not debated by the andidates. We assume that searh by one andidate about the other s quality is private. There are two restritions embedded in this assumption. First, while deiding on the ampaign theme, a andidate does not know whether his rival has found the relevant information to asertain his quality. Seond, when there is a fruitful debate, the voting publi annot distinguish whether the negative fats about the foal andidate were unearthed by the media or the rival andidate or were simply revealed through the proess of debate. See footnote 9 for how the resiults would hange when searh is still a ostly option, but a andidate is aware of whether the rival searhed or not. 0

11 3 Equilibrium and its properties Eah andidate has to deide whether to aquire information and to hoose a ampaign theme onditional on the information aquired (if any). Sine a strategy for a player is a map from types to probabilities of eah ation, the strategy set for a andidate of type should inlude the following three elements:. p() : probability of searh.. q() : probability of using a positive ampaign (P ) onditional on not searhing 3. r(; 0 ) : probability of using a positive ampaign (P ) onditional on searhing, and disovering the rival andidate to be of type 0 The equilibrium onept onsidered here is Bayesian Nash equilibrium where andidates engage in type-symmetri strategies. Sine the equilibrium strategies depend only on the types and not on the identities of players, the strategies are not indexed by the identity i of the player. There are, of ourse, multiple equilibria of this model where andidate strategies depend on their identities. However, suh equilibria require us to take an ad-ho stand on whih one among those is going to be foal. Sine we are interested in a theory of foality of andidates anyway, we do not want to assume exogenously that one or the other asymmetri equilibria will be foal. We want foality of andidates to be generated endogenously from quality di erenes in an otherwise symmetri setting. From the previous disussion, we must have r(g; G) = and r(b; B) = 0 (stritly dominant strategies). The remaining elements in the strategy set of a player is the set of probabilities p(g); p(b); q(g); q(b); r(g) and r(b) where, with a slight abuse of notation, r(g) denotes r(g; B) and r(b) denotes r(b; G). It is useful to de ne P 0 as the probability of the event that when the two andidates faing eah other are of types and 0 respetively, the andidate with type uses a positive ampaign. Therefore: P 0 = p()r(; 0 ) + ( p())q(); where ; 0 = G; B () Expanding on equation (); we de ne: BP G = p B r B + ( p B )q B () GP B = p G r G + ( p G )q G Lemma, whih deals with ampaign hoie for eah type of andidate onditional on available information, brings out the hide and seek nature of the game.

12 Lemma For fg; Bg, the ampaign hoie q() onditional on not searhing, and the ampaign hoie r() onditional on nding the rival type to be di erent from own type, is determined as follows: (a) On nding the rival to be a bad type, the good type uses a positive ampaign (r(g) = ) if BP G < and a negative ampaign (r(g) = 0) if BP G > : Conditional on not searhing, the good type employs a positive ampaign (q(g) = ) if Pr(B) BP G < and a negative ampaign (q(g) = 0) if Pr(B) BP G > ; where Pr(B) = : (b) On nding the rival to be a good type, the bad type uses a positive ampaign (r(b) = ) if GP B > and a negative ampaign (r(b) = 0) if GP B < : Conditional on not searhing, the bad type employs a positive ampaign (q(b) = ) if Pr(G) GP B > and a negative ampaign (q(b) = 0) if Pr(G) GP B < ; where Pr(G) = : Proof. See Appendix B. Aording to Lemma, the ampaign hoie of a good type depends only on BP G; the probability with whih she expets a bad type of the rival to employ the positive ampaign against her. If the bad type goes positive with a high probability, the good type prefers to go negative and expose the bad type, and if the bad type goes negative with a high probability, the good type prefers to go positive and reveal her true type. In the same way, the message of the bad type depends only on GP B; the probability that the good type of the rival uses a positive ampaign against him. The bad type always prefers to avert a fruitful debate with the good type, and therefore tries to mimi her ampaign hoie. As an impliation of Lemma part (a), onditional on not searhing, the good type has a stritly dominant ampaign hoie (P ) if it is the more ommon type. Similarly, Lemma part (b) implies that onditional on not searhing, the bad type nds it stritly dominant to hoose a negative ampaign (N) if the bad type is more ommon. In other words, > ) q(g) = < ) q(b) = 0 Proposition demonstrates the equilibrium behavior of andidates. 6 Proposition The unique equilibrium for di erent values of the prior is as follows: (i) If < ; there is a fully separating equilibrium where no type searhes, the good type uses the positive ampaign and the bad type attaks i.e. p(g) = p(b) = 0; q(g) = ; q(b) = 0; and (o equilibrium), r(g) = r(b) = : 6 We do not onsider the ase = sine it is non-generi. In this ase, we an have a ontinuum of equilibria. However, the equilibria disussed in the proposition extended to! still exist in the limit = : The ase of = is not onsidered due to the same reasons. ) (3)

13 (ii) If (; ); there is a partially separating equilibrium where both types searh with positive probability, i.e. p(g) = ( ); p(b) = ( + ):The good type employs a positive ampaign when she does not searh and a negative ampaign when she searhes and nds the rival to be a bad type, i.e. q(g) = ; and r(g) = 0: The bad type employs a negative ampaign when he does not searh and a positive ampaign when he searhes and nds the rival to be a good type, i.e. q(b) = 0 and r(b) = : (iii) If > there is a fully pooling equilibrium where both types send the positive ampaign and neither type searhes, i.e. p(g) = p(b) = 0; q(g) = q(b) = ; and (o equilibrium), r(g) = 0; r(b) = : Proof. See appendix B. If either type is too rare (when < or > ), then there is no inentive for searh, and eah andidate ats as if the rival is of the ommon type. When < ; the bad type nds it stritly dominant to employ a negative ampaign (by equation (3)).Sine is no searh, the ation of the bad type is preditable, i.e. BP G = 0; and by Lemma part (a), the good type employs a positive ampaign: Thus, when expeted andidate quality is very low, we have a ompetitive eletorate, there is a lot of negative advertising, but the good type an always separate itself from the bad type by ensuring a fruitful debate. At the other extreme, when > ; the good type always goes positive (equation (3)), and Lemma part (b) ditates that then the bad type will use a positive ampaign too, and suessfully ensure ross talk. Therefore, when the expeted andidate quality is very high, we have a onservative eletorate, where there is only positive advertising, and it is impossible to distinguish the good type from the bad. Notie that the no-searh ase demonstrates that the rarer type has the advantage in the hide-and-seek game. 7 Searh is undertaken only when neither type is very rare, i.e., the expeted andidate quality is moderate ( < < ). The good andidate always provides arguments supporting himself (positive ampaign) unless he is sure that the rival is a bad type, in whih ase he tries to expose the rival by going negative. The bad andidate on the other hand has a default ampaign whih is negative, but when he is sure that the rival is a good type, he tries to ensure ross talk by defending himself (positive ampaign), hoping that the rival has not searhed and is going to employ a positive ampaign too. This equilibrium is supported by the fat that the good andidate searhes less frequently than the bad type. To see the tehnial intuition for the equilibrium with searh, notie that if a andidate were to invest in searh with any positive probability, he must play di erent ations with di erent types of the rival. Sine the good type plays P if the rival is also good, he must play N when searh reveals the rival to be bad, i.e. r(g) = 0: In the hide-and-seek game with the bad type, the good type 7 If > ; then we have either < or < : Whenever > ; there is no searh. For < ; the equilibrium is fully separating with the good type going positive and the bad type going negative; and for > ; the equilibrium is fully pooling with both types going positive. 3

14 does not want to play the same ation against the bad type all the time. Hene, the good type uses P onditional on not searhing and N onditional on searhing and disovering the rival to be bad. Similarly, the bad type has r(b) = and q(b) = 0: The searh probabilities are hosen by eah type so as to keep the other type indi erent between searhing and not searhing: searh thus performs the role of mixing between the two di erent ations in the debate. As demonstrated by the above proposition, irrespetive of whether searh ours or not, when two good andidates are in ompetition, there is ross talk with both andidates arguing in support of themselves (positive ampaign). When two bad andidates fae eah other, we again have ross talk, but with negative ampaigns if < and with positive ampaigns otherwise: A fruitful debate an our only between a good and a bad type. Notie that although we have onsidered mehanisti or passive voting, inluding the voter as a rational player does not alter this equilibrium. First, note there is nothing more to learn from ross-talk beause both andidates take the same ation. As long as there is a ommon prior over both andidates, it is rational for the voter to randomly hoose the winner. In equilibrium, fruitful debate ours only between two di erent types of andidates - thus the passive ation is again rational. We make the additional assumption that o the equilibrium, if the voter observes a andidate to be of a type that is not supposed to be observed in equilibrium (e.g. type B in the ase < ), then the voter assumes that eah type of the rival andidate has a small (not neessarily equal) positive probability of having played N; whih implies that there remains an unertainty about the type of the rival. Thus, o the equilibrium path too, the voter stritly prefers to vote for the andidate revealed to be good and against the andidate revealed to be bad. Therefore, the naive voting ation hardwired in the payo s does not hange if we inlude a rational voter in the model. Voter inferene is formally disussed in Appendix A. 3. Properties of Searh A few properties of searh are worth mentioning here:. Given the equilibrium ampaign strategies, searh has the property of strategi substitutability: ertain searh by one type takes away the inentive of the other type to searh. Moreover, the type that does not searh an mix messages in suh a way as to nullify the informational advantage of the type that has searhed. Therefore, no andidate searhes with ertainty, even if searh ost is very low. The result that searh must be probabilisti asts doubt over the assumption of full information between andidates whih is assumed in PY. 8 8 The result that searh is always probabilisti depends on the assumption that searh is private. In a model where eah andidate knows whether the rival has searhed or not and an make their ampaign hoie onditional on suh information, searh is never probabilisti: for all parameter values in a type-symmetri equilibrium, a given type either serhes or does not searh. Mixing happens at the level of ampaign hoie. 4

15 . Searh is reiproal, i.e. if one type searhes with a positive probability, the other type does so too. 3. If searh ours, the bad type searhes with a higher frequeny than the good type. In the equilibrium with searh, we must have p(b) > > p(g): Notie that the ex-ante marginal value of positive advertising is higher to the good type than the bad. It is this advantage that depresses the good type s inentive to searh, and raises the bad type s motivation for the same. The ost of searh drives a wedge between the inentives to searh of the two types: For any given for whih searh is worthwhile in equilibrium; p(b) and p(g) are both inreasing in : This leads to an ine ieny whih we disuss later in the disussion on andidate seletion To drive the point home further, note that as! 0; both p(b) and p(g) onverge to and the ine ieny due to searh vanishes in the limit. 3. Campaigns and andidate quality There is a large empirial/experimental literature disussing the e et of negative advertising. The main nding is that attak advertising hurts both the sponsor and the target of the negative ampaign, although there is onsiderable disagreement on the relative and absolute size of these e ets. Reent formal work by PY and Mattes (008) has provided a theoretial foundation for these e ets by studying voter inferene of andidate qualities from their ampaign strategies. The most important issue highlighted in this paper is that information about andidate quality is revealed by the pro le of ampaign hoies rather than the hoie by an individual andidate: A negative ampaign may have very di erent e ets depending on whether it is met by a positive or a negative ampaign. 3.. Campaign pro les and Information Revelation The pro le of ampaigns reveal information to the voter diretly through the information revelation protool, and additionally through equilibrium inferenes. To analyze voter inferene, one really has to look at the voter as a rational player. We have informally disussed before and formally demonstrated in appendix A that the equilibrium strategies derived earlier in this setion an be used to analyze voter inferene of andidate types from ampaign hoies. A fruitful debate reveals the true quality of the foal andidate. Moreover, the rival andidate s type an also be inferred with ertainty by the rational voters. In equilibrium, a frutiful debate an our only between opposite types: suh a debate has either a good andidate exposing a bad one, or a good andidate suessfully defending herself against attaks by a bad one. In ase of ross talk, there is no new information revealed diretly through debate. In equilibrium, voters annot distinguish between the two andidates as they are observed to take the same ation. However, we an talk about the e et of ampaign hoie in terms of whether the prior, 5

16 i.e., expeted andidate quality is updated upwards or downwards. When there is ross talk with positive ampaigns, either or both of the andidates must be good. On the other hand, when both andidates use negative ampaigns, at least one of the two andidates must be bad, and voters adjust their assessment downwards. This feature of equilibrium leads to voter inferenes under ross talk disussed below. Formally, suppose both andidates hoose the same ampaign M = M = M fp; Ng;and the inferred probability of the andidates being good is b(m). Then,. If (; ), then b(n) < min ; i.e., when both andidates employ negative ampaigns, the assessment of andidate quality goes down. In fat, irrespetive of the prior ; the voters believe that both the andidates are more likely to be bad than good. The opposite onlusions hold when eah andidate runs a positive ampaign, i.e., b(p ) > max ; :. If > there is full pooling: the ampaign hoie is ompletely uninformative about quality, and thus b(p ) = : 3. If < ; we have full separation. Then, anyone using a positive ampaign distinguishes himself as a good type while anyone attaking the rival in debate reveals himself to be a bad type. Formally, in this ase, b(n) = E et of Negative Advertising In order to ompare our results diretly with the existing literature on negative ampaigns, we now turn to the following question: Suppose a andidate is observed to run a negative ampaign. What e et does our model predit about the sponsor and the target? The answer, in our model, depends on what ampaign the target is observed to run. If the rival also goes negative, the expeted quality of both the target and the sponsor goes down. On the other hand, if the rival runs a positive ampaign, the quality assessment of the sponsor of the negative ampaign and that of the target move in opposite diretions. For the ase of a fruitful debate (unmathed messages), there are two ases to onsider. When < ; only the bad type runs a negative ampaign: Thus the sponsor of a negative ampaign is hurt and the target bene ts. When (; ); the e et is ambiguous sine both good and bad types run negative ampaigns. In partiular, the attaker in a fruitful debate is revealed to be good with probability q() and bad with probability q() = p G p B p G p B + ( p G )( p B ) q(), where It an be heked that q() > if and only if > : Remark sums up the e et of a negative ampaign when the rival uses a positive ampaign in equilibrium 6

17 Remark Suppose in equilibrium, one andidate runs a negative ampaign and the rival runs a positive ampaign. The negative ampaigner is more likely to win than lose (and the positive ampaigner is more likely to lose than win) if and only if the bad type is rarer, i.e., > : Later, we observe in remark that when < ; a random andidate is more likely to run a negative ampaign and when > ; he is more likely to run a positive ampaign. Therefore, when a negative ampaign is faed with a positive one, the attaker has the advantage if most andidates are expeted to run positive ampaigns and the attaker has a disadvantage if most andidates are expeted to go negative. In other words, the andidate using the less expeted mode of ampaign has the advantage. PY (Proposition 4) has the opposite nding: when two andidates hoose unmathed messages, the one using the dominant mode of ampaigning is at an advantage. The main reason for the di erene is that while PY onsider only informative ampaigns, we onsider softer information that leads to a hide-and-seek framework. 4 Comparative Statis In this setion, we examine the omparative stati properties of the equilibrium for di erent levels of the searh ost and average andidate quality, and disuss the impliations of suh properties. 4. Welfare Analysis: Candidate Seletion Looking at politial ampaigns as debates between andidates with partial information about eah other helps us understand a few important issues about the e ieny of the ampaign proess espeially in terms of its ability to selet the better andidate. The major nding of the model is that as the prior probability of a andidate being good inreases, the probability of a good andidate being seleted through the eletoral proess may atually go down. Figure plots the total probability of seletion of the good andidate against the prior for some given searh ost 9 : In the graph, there exist downward jumps in otherwise pieewise ontinuous and monotonially inreasing graphs. There are three regimes based on ranges of : Full separation, partial separation and full pooling, and the downwards jumps our when we move from a more e ient regime to a less e ient one as the average andidate quality inreases. Note also that sine an inrease in the searh ost favours the bad type, the downward jumps are larger as the ost of information inreases. 9 Denote the total probability of seletion of the good type given and as f (): It an be dedued from Propositions 8 that: < if < 3 f () = : if < < if > 7

18 Probability of type G winning α α Figure : Total probability of seletion of a good andidate The total probability of seletion may not fully re et the e ieny of debate as a seletion mehanism. Perhaps a better indiator of the e ieny of debates in seleting the right andidate would be a measure of how often the good andidate wins when ompeting against a bad andidate. Hene we look at the equilibrium probability of a fruitful debate onditional on andidates being of di erent types. Denote this probability (; ): Proposition shows how hanges with the parameters of the model. Proposition Suppose one of the two ompeting andidates is a good type and the other is a bad type. Debate is always fruitful if < ; never fruitful if > ; and if (; ) debate is fruitful with a probability (; ) = Proof. Follows from proposition. ( ) : When there is no searh, the rarer type has full advantage in the hide and seek game. For very low priors, debate is fully e ient and for very high priors, debate is fully ine ient. When the prior is moderate, both types searh in equilibrium. In the ase when the searh ost is very low, i.e.! 0; we have p B and p G both onverging to : In this ase, the debate between the good and bad types redues to the mathing pennies game in whih there is no advantage to either type: hene eah message pro le ours with equal probability. Then (; )! for almost all values of the prior. As the searh ost inreases, there is a further ine ieny due to di erential inentives for searh that the two types have. As mentioned before, we have p G < < p B; moreover, both jp G j and jp B 8 j are inreasing in :

19 This ine ieny due to di erential inentives for searh leads to a further welfare loss of ( ) : Figure shows how (; ) hanges for di erent values of. The dashed line shows when! 0: Probability of a fruitful debate ½ ½ ½ α Figure : Probability of orret seletion when two andidates are of di erent types Given an expeted andidate quality, if a soial planner ould hoose the searh ost (or more realistially, the extent of o e perks), what would the best hoie be? The answer depends on whether the good or the bad type is more ommon. If > ; we unambiguously improve welfare by reduing, and the best hoie of would be as lose to zero as possible. On the other hand, if the bad type is more ommon, there are two opposing e ets. When < ; a marginal redution in redues the ine ieny due to searh and inreases. On the other hand, for any given less than ; if we set > ; we get full separation of types. Hene, if the bad type is more ommon, a soial planner an ahieve full separation by su iently reduing the prize from o e. 4. Negative Advertising Next, we look at the type of messages exhanged in the debate as the prior varies. As the average andidate quality improves, the ex-ante probability that a andidate will employ a negative ampaign goes down:in general, the bad type s propensity to attak goes down as the probability of the rival being good goes up. For moderate values of the prior, the propensity of negative ampaigning dereases with beause the good type s inreasing aggressiveness (in searhing and attaking the bad type) is more than ompensated by the bad type s inreasing onservatism (in searhing and sending positive messages). This is illustrated in gure 3 and stated in Remark, 9