Information Revelation and Pandering in Elections

Transcription

1 Information Revelation and Pandering in Elections Navin Kartik Francesco Squintani Katrin Tinn March 18, 2013 Abstract Does electoral competition promote efficient aggregation of politicians policy-relevant information? Or do office-motivated politicians pander to the electorate s beliefs in lieu of using their own information? This paper studies a two-candidate Downsian model in which each politician privately holds imperfect but socially valuable information about policy consequences. Under general conditions, we establish that voter welfare is uniquely maximized by equilibria in which one candidate is always elected and chooses policy based on his information alone. Thus, office motivation precludes efficient information aggregation. However, the inefficiency is not necessarily due to politicians incentives to pander. For familiar information structures, we show that office motivation impels politicians to overreact to their information relative to the electorate s prior belief, i.e. to anti-pander. Such behavior is compatible with equilibrium. Moreover, we find that an appropriate degree of pandering would improve voters welfare instead of harming it. For helpful comments and discussions, we thank Nageeb Ali, Kfir Eliaz, Andrea Galeotti, Matias Iaryczower, Alessandro Lizzeri, Massimo Morelli, David Rahman, several seminar and conference audiences, and, especially, Alfonso Goncalves da Silva, Johannes Hörner, and Balazs Szentes. Kartik gratefully acknowledges financial support from the Sloan Foundation, the National Science Foundation (Grant SES ), and the hospitality of and funding from the University of Chicago Booth School of Business. Columbia University, Department of Economics. nkartik@columbia.edu. Warwick University, Department of Economics. F.Squintani@warwick.ac.uk. Imperial College, Business School. k.tinn@imperial.ac.uk.

2 1 Introduction The efficiency of a representative democracy turns on whether the policies of elected representatives reflect the citizens best interests. Dating back to at least Downs (1957), there is a view that elections function well because office-seeking politicians are impelled to choose policies that best promote a median or representative voter s interests. Furthermore, while citizens may themselves be poorly informed on policy issues as posited by Downs in his rational ignorance hypothesis and since documented by numerous studies starting with Campbell et al. (1960) political candidates devote substantial resources and have broad access to policy experts and think tanks. Information garnered by politicians may be conveyed to the electorate through their policy positions and electoral campaigns. 1 Indeed, in an influential article titled Why Democracies Produce Efficient Results, Wittman (1989, p. 1400) argued that political competition benefits the electorate because there are returns to an informed political entrepreneur from providing the information to the voters, winning office, and gaining the [...] rewards of holding office. Concurrently, there are also charges both in popular circles and in some recent academic work which we discuss subsequently that competitive pressures drive office-motivated politicians to pander to voters opinions. After all, the argument goes, it is hard to win an election by campaigning on policies with recondite merits; a politician is better off just promising to do whatever voters believe to be best. Such pandering is viewed as inefficient because it would lead to policies that do not efficiently aggregate the politicians socially-valuable information because of their distortion toward less-informed opinions. This paper examines the efficiency of elections when office-seeking politicians possess policyrelevant information. We study the classic model of representative democracy, a two-candidate Hotelling-Downs election, with one twist: each of the politicians has imperfect (and private) infor- 1 There is evidence that voters learn, update, or refine their beliefs/preferences during elections. See, for example, experiments on deliberative polling by Fishkin (1997); empirical studies on the effects of information on voters opinions by Zaller (1992), Althaus (1998) and Gilens (2001); studies on framing in polls such as the ones by Schuman and Presser (1981); and experiments on priming by Iyengar and Kinder (1987). 1

3 mation about policy consequences, and hence socially valuable information about which policy would be best for the representative or median voter. In a nutshell, our general findings are three-fold: (i) elections can not efficiently aggregate both politicians private information; (ii) the voter s welfare in any electoral equilibrium is equivalent to policy being chosen based on just one politician s information (while not necessarily using this information efficiently); and (iii) maximum voter welfare is attained only by non-competitive electoral outcomes in which one politician always wins and uses his information efficiently. We show that the reason why only one source of information can be aggregated into policy despite the presence of two information sources is the politicians office motivation. Furthermore, for a familiar class of information structures and voter preferences, we find that: (iv) competitive pressures drive politicians to champion policies that are excessively distant from the voter s prior belief about optimal policy, i.e. to anti-pander rather than pander; and (v) rather than being harmful, an appropriate degree of pandering by each politician can be socially desirable. Altogether, these findings imply that both viewpoints mentioned at the outset are, at best, incomplete, and, at worst, simply incorrect. The baseline model we develop in Section 2 features two purely office-motivated candidates, each of whom receives a noisy private signal about some policy-relevant but unobservable state of the world. A (median) voter desires policies that are closest to the expected state. For concreteness, we suppose the state is drawn from a normal distribution and each candidate s private signal is the true state plus a mean-zero random shock that is also normally distributed. 2 Both candidates simultaneously commit to policy platforms, whereafter the voter updates her beliefs about the true state and elects the candidate whose policy is closest to the expected state. The elected candidate then implements his platform. We study the (Perfect Bayesian) equilibria of this Downsian game with expert politicians. 2 We will explain how our main welfare results (points (i) (iii) above) generalize to virtually any statistical structure and voter preferences; and how the pandering results (points (iv) and (v) above) generalize to a broad class of statistical structures. 2

4 A candidate who maximizes the voter s interest based on his information alone would propose the policy that is his Bayesian estimate of the state given his private signal; call this an unbiased strategy. Our first result in Section 3, Proposition 1, is that both candidates cannot play unbiased strategies in equilibrium. Perhaps surprisingly, this is not because candidates have incentives to distort their platforms toward the prior expected state (i.e., to pander to the voter s prior beliefs); rather, a profitable deviation is to anti-pander or overreact to information by choosing a platform that puts more weight on the private signal than what is prescribed by an unbiased strategy. This deviation incentive arises because if both candidates were to use unbiased strategies, it would be optimal for the voter to elect the candidate whose platform is more extreme (i.e., further from the prior expected state), and hence a candidate could increase his probability of winning by choosing a more extreme platform than he is supposed to. 3 Building on this intuition, Proposition 2 demonstrates existence of equilibria in which both candidates overreact to their private information. Information is still revealed in these equilibria, but each candidate champions a policy that is further away from the prior expected state than his unbiased estimate given his signal, i.e. there is anti-pandering rather than pandering. 4 Given these strategic forces stemming from political competition, we then turn to the fundamental question of welfare: how well does an election aggregate the politicians private information? Theorem 1 establishes that in any equilibrium of the Downsian election, the voter s welfare (i.e. ex-ante expected utility) is equivalent to the welfare obtained by always electing one of the two politicians no matter which policies they propose. In other words, the voter s welfare is effectively 3 That the voter would choose the more extreme candidate when they play unbiased strategies is a consequence of Bayesian updating. Intuitively, the voter is able to infer two signals about the state from the pair of candidates platforms, and hence her posterior on the state puts less weight on the prior than does either candidate s individual unbiased estimate; consequently, the expected state following any two platforms is more extreme than the average of the two platforms. Glaeser and Sunstein (2009) and Roux and Sobel (2012) also identify this implication of Bayesian updating, but they are interested in non-strategic group decision-making. 4 There is almost sure policy divergence in these equilibria, but explaining policy divergence is not a focus of this paper. Explanations for policy divergence include policy-motivated candidates (Wittman, 1983; Calvert, 1985; Bernhardt et al., 2009b), candidates trying to signal character or competence (Kartik and McAfee, 2007; Honryo, 2011), heterogeneity in candidates valence (Aragones and Palfrey, 2002; Groseclose, 2001), the threat of entry by a third candidate (Palfrey, 1984), private information about the median voter s location (Bernhardt et al., 2009a), or lack of commitment in citizen-candidate models (Osborne and Slivinski, 1996; Besley and Coate, 1997). 3

5 determined by a single politician s policy function. 5 Since each politician s policy can depend only on his own information, this result uncovers a sharp upper bound on how well elections can aggregate the two politicians information. Theorem 2 verifies that the upper bound is tight by demonstrating (asymmetric) equilibria in which one politician uses an unbiased strategy and is elected with probability one. More generally, the Theorem establishes that only such a non-competitive electoral outcome achieves the maximum voter welfare: any competitive equilibrium one in which both candidates have a positive ex-ante probability of winning delivers a strictly lower level of voter welfare. A novel implication is that elections that are contested by two candidates may provide strictly lower welfare than uncontested elections where only one candidate can feasibly win. The driving force behind these welfare conclusions is that an election between office-seeking politicians is a constant-sum game in terms of their payoffs, and these payoffs are independent of their private information because the voter can only choose whom to elect based on the proposed policies. 6 We prove what appears to be a new result about two-player constant-sum Bayesian games: when payoffs depend only the action profile and not (directly) on the players types, every equilibrium has the property that the distribution of actions played by a type t i of either player i would also be best response for any other type t i. The subtlety here is that types t i and t i will generally hold different beliefs about the opponent s distribution of actions when types are correlated. We leverage this result to show that in our electoral model, any informative equilibrium when at least one politician s policy position reveals something about his information must be an ex-post equilibrium in the sense that the probability of a candidate being elected cannot depend on either s policy position. The conclusions of Theorem 1 and Theorem 2 largely follow from this fact. Finally, from a normative perspective, we explore what pair of candidates strategies would 5 To be clear: which politician this is can depend on the equilibrium. 6 The politicians information is socially valuable because it affects which policies are optimal for the voter. Hence, the voter s decision on whom to elect will depend on (her beliefs about) the politicians strategies; what is relevant to the politicians, however, is the induced game they play holding fixed the voter s strategy. 4

6 maximize voter welfare in the electoral game form. Proposition 3 shows that voter welfare would be strictly improved if candidates were to choose policies suitably closer to the prior expected state than individual unbiased estimates. In other words, candidates should pander to an appropriate extent! The intuition stems from a winner s curse that under unbiased strategies, a candidate would win precisely when his signal is more extreme than his opponent s, and hence voter welfare would be improved by moderating one s policy position. Proposition 4 establishes that, within a class of strategy profiles, such pandering by both candidates is the socially-efficient way to aggregate information in the election game. Subsection 4.1 explains why our main results regarding efficiency and voter welfare hold broadly across both voter preferences and information structures: roughly speaking, all that is needed is that the politicians signals should not be independent; this is naturally satisfied whenever each politician s signal is informative about the true state that affects the voter s policy preference. We also show through a formal continuity result (Theorem 4) that our welfare conclusions also hold when there are small departures from the assumption that candidates are purely office-motivated. Further, our (anti-)pandering results hold so long as the voter prefers policies that are closer to the expected state and the information structure is conjugate and exponential, i.e. the posterior distribution of the state given a politician s signal is in an exponential family and the prior is conjugate with the posterior. As is well-known, this class subsumes a variety of commonly-used information structures. 7 For most of the paper, we take the Hotelling-Downs game form in which candidates commit to platforms as given. This is because we believe it is important to understand how candidates policyrelevant private information is aggregated in the canonical model of representative democracy. A quite different perspective would be that of mechanism design: which electoral regimes would foster aggregation of office-motivated politicians private information (cf. Li et al., 2001)? In particular, suppose one drops the assumption of commitment, so that policy platforms become cheap talk. The 7 Exponential families include a variety of familiar discrete and continuous distributions with bounded and unbounded supports, such as normal, exponential, gamma, beta, chi-squared, Binomial, Dirichlet, and Poisson. 5

7 cheap-talk game has an equilibrium that efficiently aggregates information: candidates make truthful announcements, the voter flips a coin between candidates no matter their announcements, and the electoral winner implements a policy that is optimal for the voter given both candidates information. A significant result in Subsection 4.2 is that is these cheap-talk equilibria are implausible. Intuitively, if platforms are non-binding and one candidate reveals his information, then the other candidate has an incentive to not reveal his information. By withholding his information, he becomes the only politician who can implement the efficient policy, and hence withholding his information would ensure that he wins the election. We formalize this intuition by using a version of a cheap-talk refinement due to Farrell (1993), neologism proofness. Theorem 3 establishes that the only cheap-talk equilibria that satisfy the refinement are uninformative equilibria. Thus, the maximal voter welfare is identical in the Downsian model and in neologism-proof equilibria of the cheap-talk model. The same conclusion applies to many other game forms, e.g. if candidates were to first make cheap-talk statements and then commit to policies before the voter elects one of them. Related Literature. Our work ties most closely into the literature on electoral competition when candidates have policy-relevant private information. 8 Heidhues and Lagerlof (2003) illustrate why candidates may have an incentive to pander to the electorate s prior belief; their setting is one with binary policies, binary states, and binary signals. We find that in our richer setting, precisely the opposite is true for a broad class of information structures. Plainly, with binary policies, one cannot see the logic of why and how candidates may wish to overreact to private information. Loertscher (2012) maintains the binary signal and state structure, but introduces a continuum policy space. His results are more nuanced, but at least when signals are sufficiently precise, the conclusions are similar to Heidhues and Lagerlof (2003). 9 8 There are, of course, contexts where candidates have private information that is not policy relevant for voters. For instance, the strategic effects of private information about the location of the median voter are studied by Chan (2001), Ottaviani and Sorensen (2006), and Bernhardt et al. (2007, 2009a). 9 In Appendix D, we provide an example with a binary signal but where the policies and the state lie in the unit interval (the prior on the state has a Beta distribution) and explicitly show that there is also an incentive to overreact or anti-pander here. This is a special case of the aforementioned Exponential family, but allows a closer comparison 6

8 Laslier and Van de Straeten (2004) show that if voters in the Heidhues and Lagerlof (2003) model are endowed with sufficiently precise private information about the policy-relevant state, then there are equilibria in which candidates fully reveal their private information; see also Klumpp (2011) and Gratton (2013). By contrast, we are interested in settings in which any information that voters have that candidates do not have is relatively imprecise. While we make the extreme assumption that voters have no private information, our main themes are robust to small variations on this dimension. Schultz (1996) studies a model in which two candidates are perfectly informed about the policyrelevant state but are ideologically motivated. He finds that when the candidates ideological preferences are sufficiently extreme, platforms cannot reveal the true state; however, because of the perfect information assumption, full revelation can be sustained when ideological preferences are not too extreme. Martinelli (2001) and Martinelli and Matsui (2002) derive further results with ideologically motivated candidates who are perfectly informed about a policy-relevant variable. There are other models of politics in which policy distortions arise because office-motivated politicians wish to influence voters beliefs; the mechanisms, however, are less related to ours because these models are generally of a single incumbent with career concerns to build reputation for either competence or aligned preference. 10 While most of these papers focus on pandering toward the electorate s prior, anti-pandering arises in Prendergast and Stole (1996) and Levy (2004). 2 A Model of Expert Politicians An electorate is represented in reduced-form by a single median voter, whose preferences depend upon the implemented policy, y R, and an unknown state of the world, θ R. We assume that the voter s preferences can be represented by a von-neumann utility function, U(y, θ) = (y θ) The with the setting of Heidhues and Lagerlof (2003) and Loertscher (2012). 10 Harrington (1993) and Cukierman and Tommasi (1998) are early contributions in this vein; see also Canes-Wrone, Herron and Shotts (2001), Majumdar and Mukand (2004), Maskin and Tirole (2004), Prat (2005), and more recently, Morelli and van Weelden (2011a,b), Acemoglu et al. (forthcoming). 11 A simple way in which this represents a median voter s preferences is if the electorate consists of a set of voters, J, with the preferences of each j J represented by U j(y, θ) = (y θ b j) 2 for some constant b j R. Our analysis 7

9 state θ is drawn from a Normal distribution with mean 0 and a finite precision α > 0 (i.e. variance 1/α). There are two candidates: A and B, each of whom gets a utility of 1 if elected and 0 otherwise; hence they are purely office motivated and maximize the probability of winning the election. Each candidate i privately observes a signal θ i = θ + ε i, where each ε i is drawn independently of any other random variable from a Normal distribution with mean 0 and finite precision β > After privately observing their signals, both candidates simultaneously choose platforms, y A R and y B R respectively. Upon observing the pair of platforms, the voter updates her belief about the state and then elects one of the two candidates. The elected candidate implements his platform as final policy, i.e., platforms are policy commitments in the Downsian tradition. All aspects of the model except the candidates privately observed signals are common knowledge, and players are expected-utility maximizers. With some abuse of notation, a pure strategy for a candidate i will be denoted as a (measurable) function y i ( ) : R R, so that y i (θ i ) is the platform chosen by i when his signal is θ i. 13 A mixed strategy for the voter is a (measurable) function p( ) : R 2 [0, 1], where p(y A, y B ) represents the probability with which candidate A is elected when the platforms are y A and y B. We are interested in perfect Bayesian equilibria of the electoral game (including those in which candidates play mixed strategies), which implies that the voter elects candidate i if y i is strictly preferred to y i, where the subscript i refers to candidate i s opponent. As is common, we require that the voter randomize with equal probability between the two candidates if she is indifferent between y A and y B. 14 Furthermore, for technical reasons, we restrict attention to equilibria in which for any applies so long as there is some voter, m, such that b m is the median of the distribution of b j s; it is then without loss of generality to normalize b m = The Normal-Normal structure is a well-known family of conjugate distributions. We discuss in Subsection 4.1 how our results hold for broader classes of distributions. The assumption that both candidates receive equally precise signals is for expositional simplicity only; the results extend to the case in which one candidate is known to receive a more precise signal than the other. Such an asymmetric competence can serve to capture incumbency advantage. 13 A mixed strategy for candidate i is a mapping σ i : R (R), where (R) denotes the set of probability measures on R. All our formal results and proofs cover mixed strategies for candidates; however, since mixing by candidates does not play an essential role in the analysis, we focus the exposition in the main text on candidates pure strategies for simplicity. 14 This does not play a significant role but simplifies matters as it pins down voter behavior on any equilibrium path. 8

10 given policy of one candidate, say y A, the voting function p(y A, ) has at most a countable number of discontinuities, and analogously for p(, y B ) for any y B. The notion of welfare we use is the voter s ex-ante expected utility. 2.1 Terminology and Preliminaries A pure strategy y i ( ) is informative if it is not constant and it is fully revealing if it is a one-to-one function, i.e. if the candidate s signal can be inferred from his platform. 15 As is well known (Degroot, 1970), the Normal-Normal information structure implies that the expected value of the state θ given a single signal θ i is E [θ θ i ] = β α + β θ i, (1) whereas conditional on both signals, the expected value is E [θ θ A, θ B ] = 2β α + 2β ( θa + θ B 2 ). (2) Because of quadratic utility, the optimal policy for the voter is the conditional expectation of the state given all available information. Since the only information a candidate has when he selects his platform is his own signal, we refer to the strategy y i (θ i ) = E [θ θ i ] = β α+β θ i as the unbiased strategy. Plainly, this strategy is full revealing. We say that a strategy y i ( ) displays pandering to the voter s beliefs if for all θ i 0, y i (θ i ) is in between 0 and E[θ θ i ]. 16 In other words, a candidate panders if his platform is systematically distorted from his unbiased estimate of the best policy toward the voter s prior beliefs. Similarly, we say that y i ( ) displays overreaction to private information if for all θ i 0, y i (θ i ) is more extreme than E[θ θ i ] relative to 0: for θ i < 0, y i (θ) < E[θ θ i ], while for θ i > 0, y i (θ i ) > E[θ θ i ]. An equilibrium is informative if at least one candidate plays an informative strategy. Observe Moreover, in a full-fledged model with voters of heterogenous ideologies, this property would be necessary whenever the platforms are distinct and yet the median voter is indifferent between them. 15 There are straightforward generalizations of these notions to mixed strategies. 16 More precisely, we require that for θ i < 0, y i(θ i) (E[θ θ i], 0], while for θ i > 0, y i(θ i) [0, E[θ θ i]). 9

11 that for any uninformative pure strategy, there is an equilibrium in which both candidates use that pure strategy due to the latitude in specifying off-path beliefs. Our interest will be in informative equilibria. An equilibrium is fully revealing if both candidates strategies are fully revealing. An equilibrium is symmetric if both candidates use the same strategy, and it is linear if both candidates play linear pure strategies. An equilibrium is competitive if both candidates have an ex-ante positive probability of winning; it is non-competitive if one candidate wins with ex-ante probability one Main Results 3.1 Equilibrium Anti-Pandering Given that the voter desires policies as close as possible to the true state, and that a candidate s only information when choosing his policy is his private signal, one might conjecture that a candidate can do no better than playing an unbiased strategy, particularly if the opponent is also using an unbiased strategy. However: Proposition 1. The profile of unbiased strategies is not an equilibrium: candidates would deviate by overreacting to their information. (The proof of this and all other formal results are in the Appendices.) The incentive to overreact arises because if both candidates were to play unbiased strategies, the voter would optimally select the candidate with a more extreme platform, i.e. the one with larger magnitude. Why? Since unbiased strategies are fully revealing, the voter would infer both candidates signals from their platforms, and accordingly, form a posterior expectation that has the same sign as the average of the two candidates individual posterior expectations but that is more extreme, i.e. has a larger magnitude. This is a direct implication of equations (1) and (2). 18 Since 17 A technical note is that because of the continuum policy space, various statements in the analysis and proofs (e.g. about uniqueness of equilibria) should be understood to hold subject to almost all qualifiers; we supress such caveats unless essential. 18 Indeed, the voter s posterior mean can be larger in magnitude than both candidates platforms (rather than just 10

12 the candidate whose platform is closer to the voter s posterior expectation is elected, it follows that the voter would elect i if and only if y i > y i. Hence, each candidate would like to raise his probability of playing the more extreme platform, which can be achieved by placing more weight on his private signal than what is prescribed by the unbiased strategy. Consequently, a profitable deviation involves overreaction rather than pandering. 19 Despite the incentive to overreact, can information be revealed in equilibrium? Perhaps surprisingly, we find that an appropriate degree of overreaction can support full revelation of information. Proposition 2. There is a symmetric and fully-revealing equilibrium with overreaction where both candidates play y (θ i ) = E[θ θ i, θ i = θ i ] = 2β α + 2β θ i. (3) In this equilibrium, each candidate is elected with probability 1/2 regardless of the signal realizations θ A and θ B. Furthermore, this is the unique symmetric pure-strategy equilibrium in which candidates use fully-revealing and continuous strategies. 20 Observe that the strategy given by (3) requires either candidate i to choose his platform to be the Bayesian estimate of the state assuming his opponent has received the same signal. This is an overreaction because he anticipates that, in expectation, his opponent s signal will be more moderate than his own, as the expectation of the opponent s signal equals his unbiased estimate of the state, β α+β θ i. When the voter believes that both candidates overreact to this degree, platforms do not affect winning probabilities because whenever candidate i increases his platform by ε > 0, equation (2) their average); this is always the case when θ A and θ B are sufficiently close, since the posterior mean is continuous in signals and for any ˆθ, E[θ θ A = θ B = ˆθ] = 2β ˆθ > E[θ θ α+2β A = ˆθ] = β ˆθ. 19 α+β A different way of seeing this is to note that when the voter conjectures that candidates are using unbiased strategies, then (2) implies that for any ( platform ) of candidate i, an increase in candidate i s platform by ε > 0 would raise the 2β voter s posterior by α+β ε > ε. Thus, increasing (resp. decreasing) his platform would benefit candidate i α+2β β 2 2 when he is located to the right (resp. the left) of i. Since under unbiased strategies a candidate s expectation of his opponent s platform is in between 0 and his own unbiased platform, pandering would reduce the probability of winning while overreacting would increase it. 20 In fact, the Appendix establishes a significantly stronger uniqueness result based on the analogous properties holding locally around some signal for each candidate; see Proposition B.1. 11

13 implies that the voter s posterior increases by between the two platforms (cf. fn. 19). ( 2β α+2β α+2β 2β ) ε 2 = ε/2, and thus she remains indifferent Although the linear coefficient in (3) is increasing in β and decreasing in α, the same is true for the unbiased strategy s coefficient, measured by 2β α+2β β α+β. The degree of overreaction in the equilibrium of Proposition 2, as β α+β, is non-monotonic in the parameters: it is increasing in β (resp. decreasing in α) when β 2 < α and decreasing in β (resp. increasing in α) when β 2 > α. The degree of overreaction vanishes as either α or β tend to either 0 or. 3.2 Equilibrium Welfare Even though the equilibrium of Proposition 2 fully reveals all available information to the voter, it does not use either politician s information efficiently because of their overreaction. This prompts the question of what other equilibria exist and whether there are any with higher voter welfare. Fix an arbitrary equilibrium. Given the candidates (possibly mixed) strategies, denote by v i the voter s welfare (i.e. ex-ante expected utility) from electing candidate i no matter which policy pair is actually proposed. Plainly, since this is a feasible strategy for the voter, the voter s equilibrium welfare cannot be smaller than max{v A, v B }. Our key result, Theorem 1 below, proves that this is in fact exactly the voter s equilibrium welfare even though, in equilibrium, the voter may be selecting both candidates with ex-ante positive probability. Since a candidate s policy can only depend on his own information, it follows that it is as if the voter s welfare in any equilibrium is influenced by just one candidate s signal and not by the other s. In this sense, electoral competition between office-motivated candidates leads to an inescapable inefficiency. Theorem 1. In any equilibrium, there is some candidate i {A, B} such that the voter s equilibrium welfare is equal to the welfare obtained by electing candidate i no matter the proposed policies. As the proof is somewhat involved but the result is central, we will sketch the main steps of the argument. The key insight is Lemma B.1 in the Appendix: any equilibrium must have the property 12

14 that for any on-path platform of a candidate i, say y i, the probability with which i would win when playing y i cannot depend on what signal i has received. To establish this, we first note that given any strategy for the voter, p(y A, y B ), the two candidates are engaged in a constant-sum Bayesian games where their payoffs depends only on the pair policy platforms and not (directly) on the signals. We prove in Appendix A a general result about any two-player constant-sum Bayesian-game whose payoffs depend only on the players actions and not on their types: every equilibrium has the property that the distribution of actions played by some type of a player would also be a best response for any other type of that player, even though the two types will generally hold different beliefs about the opponent s distribution of actions when types are correlated. Building on Lemma B.1, we show in Lemma B.2 that if an equilibrium is informative, it must be an ex-post equilibrium in the sense that the voter s strategy, p(y A, y B ) must be constant across all on-path platform pairs. Hence, a candidate would have no incentive to deviate even if he observed his opponent s platform before making his choice. Note, in particular, that this property is satisfied by the equilibria of Proposition 2. An intuition for this step is as follows. Consider any informative equilibrium. Since at least one candidate s platform is correlated with his signal, and since each candidate s signal provides him with information about his opponent s signal, it follows that at least one candidate s signal is informative about his opponent s platform. But then, the interim probability of winning for a candidate can be independent of his signal (as required by Lemma B.1) only if the winning probability is independent of which platforms are played on the equilibrium path. 21 This ex-post property for informative equilibria implies that in any equilibrium, informative or uninformative, one of two cases holds: (a) the equilibrium is non-competitive and there is a candidate i who is always elected regardless of the signal profile; or (b) the equilibrium is competitive and the 21 It is worth clarifying that this argument for informative equilibria does not use the assumption that the voter must randomize with equal probability when indifferent between candidates platforms. The ex-post property also applies to uninformative equilibria, but only under the randomization assumption. If the voter were not required to randomize uniformly when indifferent, there are uninformative equilibria with the flavor of matching pennies : for example, both candidates randomize uniformly over { x, x} for some x > 0, and the voter elects candidate A if y A = y B while she elects candidate B if y A = y B (and randomizes with equal probability off the equilibrium path). Nevertheless, the conclusion of Theorem 1 applies to uninformative equilibria as well even without the randomization assumption. 13

15 voter is indifferent between both candidates for all pairs of on-path platforms. In the latter case, the voter s expected utility given any on-path platform pair is, of course, the same regardless whom she elects. It follows that in either case, the voter s ex-ante expected utility can be evaluated by assuming that she always elects one of the two candidates, which is what Theorem 1 states. Theorem 1 determines an upper bound on the voter s welfare in any equilibrium of the Downsian election because despite there being two informed candidates, the voter s welfare might as well be determined by only one of their signals and the corresponding policy platform. It follows that there cannot be any equilibrium which delivers a higher welfare to the voter than she would obtain by always electing one candidate who plays the unbiased strategy, i.e. who choses policy optimally based on his signal. Our next result confirms that this outcome can be supported as a non-competitive equilibrium, and furthermore, that any competitive equilibrium yields strictly lower welfare. Theorem 2. There is a non-competitive equilibrium where the candidate who wins with probability one plays the unbiased strategy. There is no equilibrium that yields the voter a higher ex-ante expected utility, and any competitive equilibrium yields the voter strictly lower ex-ante expected expected utility. There are multiple constructions to verify the first statement of Theorem 2. Interestingly, the outcome can be supported even when the candidate who loses for sure, say i, plays the fullyrevealing strategy y i (θ i ) = θ i. Candidate i is overreacting to his information here to such an extent, choosing what would be an unbiased estimate only under a Laplacian or improper prior, that the voter never finds it optimal to elect him despite correctly inferring his information. 22 The second statement of Theorem 2 says that any real competition between candidates necessarily reduces the voter s welfare relative to a non-competitive equilibrium in which one candidate plays the unbiased strategy. 23 Clearly, the latter outcome can also be achieved if there were only 22 A feature of this equilibrium is that it is in pure strategies and does not entail any off-path platforms. This obviates concerns one might have with alternative constructions, such as the losing candidate playing a constant pure strategy, say y i(θ i) = 0, or a mixed strategy where the distribution of platforms has support R but is independent of his signal. It is worth noting that in the class of full-revealing strategies for the losing candidate, i, y i(θ i) = θ i is the only strategy that can be used to support a non-competitive equilibrium when the winner uses the unbiased strategy. 23 For example, one can explicitly compute the voter s welfare difference between a non-competitive election in which 14

16 one candidate contesting the election in the first place. This is a novel rationale, based on information aggregation of politicians policy-relevant information, for why non-contested elections may be beneficial to citizens. 3.3 Pandering and Welfare Having established that office-motivated candidates have incentives to overreact to information and that electoral competition does not promote efficiency, we now ask, from a normative perspective, what candidates strategies would maximize the voter s welfare in the Downsian game form. To address this question, it is analytically convenient to introduce an auxiliary game that we call the benevolent-candidates game. This has the same game form as we have studied so far and the same payoff function for the voter; the only difference is that candidates are not office-motivated but rather maximize the voter s utility. This auxiliary game is a team problem in the sense of Marschak and Radner (1972) and its Pareto-dominant equilibria identify the strategies which maximize the voter s welfare in our Downsian game form. The next result shows that an appropriate degree of pandering would actually be beneficial for voter welfare. Proposition 3. In the benevolent candidates game: 1. Unbiased strategies are not an equilibrium because candidates would deviate by pandering; 2. There is a symmetric fully revealing (non-linear) equilibrium with pandering, in which each candidate plays y(θ i ) = E[θ θ i, θ i < θ i ], (4) the winner plays the unbiased strategy and the competitive equilibrium of Proposition 2. The voter s ex-ante expected utility in the former is 1, while it is 4β2 +α 2 +5βα in the latter. Both expressions converge to 0 as either β α+β (α+2β) 2 (α+β) or α 0; hence, both equilibria are welfare equivalent in these limiting cases. However, the welfare difference between the two equilibria is not monotonic in the parameters. 15

17 and the voter elects i when y i > y i, or equivalently from (4), i wins when θ i > θ i The above pandering equilibrium provides strictly higher voter welfare than the profile in which both candidates play unbiased strategies (and the voter responds optimally), and hence than any equilibrium of the game with office-motivated candidates. The first part of Proposition 3 reaches the same conclusion for benevolent candidates as Proposition 1 does for office-motivated candidates: unbiased strategies do not form an equilibrium. The logic, however, is now precisely opposite to the one presented for the game with office-motivated candidates. Recall from the discussion following Proposition 1 that if candidates were to play unbiased strategies, the voter would elect the candidate whose platform (and hence signal) is more extreme. While this induces office-motivated candidates to overreact to their information in order to increase their winning probabilities, it has the opposite effect on benevolent candidates because of a winner s curse. Since a benevolent candidate cares about his platform choice only in the event of winning, conditioning on winning informs him that his opponent s signal is more moderate this own, and hence that he should moderate his policy platform, i.e. he should pander. This logic suggests why pandering is desirable. The second part of Proposition 3 shows that there is indeed a symmetric and fully-revealing equilibrium with pandering in which the candidate with the more extreme platform (or signal) would win. It is intuitively clear that the strategy (4) is optimal for a benevolent candidate if that candidate wins when his opponent s signal is more moderate than his own. What requires some work is showing that it would be optimal for the voter to elect the candidate with the more extreme platform (and hence signal) given these strategies Using the closed-form expression for truncated Normal distributions, equation (4) can be expressed as ( ) ( ) 1 α σ α+β θi φ 1 α+2β σ α+β θi y (θ i) = β α + β θi σ β φ α + 2β Φ ( 1 σ α α+β θi ) Φ ( ), 1 α+2β σ α+β θi where σ = α+2β, and φ( ) and Φ( ) are respectively the density and cumulative distributions of the standard normal (α+β)β distribution, N (0, 1). To see that this strategy has pandering, consider any θ i > 0 (with a symmetric argument for ( ) ( ) ( ) ( ) θ i < 0). Then 0 < y (θ i) < β α+β θi because φ 1 α σ α+β θi 1 α+2β > φ σ α+β θi 1 α > 0 and Φ σ α+β θi > Φ 1 α+2β σ α+β θi > Note that as θ i {, + }, the strategy in (4) becomes approximately the unbiased strategy, The 16 β α+β θi.

18 Finally, the third part of Proposition 3 confirms that the pandering equilibrium in part two of the Proposition yields higher voter welfare than the profile of unbiased strategies, and hence than any equilibrium of the game with office-motivated candidates. 26 While this can be checked through a direct computation, we use the following more powerful and instructive argument. Theorem 2 tells us that voter welfare in any equilibrium with office-motivated candidates is no higher than in a non-competitive equilibrium where the winner plays the unbiased strategy. In turn, that welfare is lower than if both candidates were to play unbiased strategies and the voter responded optimally (since in this case it would be suboptimal for the voter to always elect one candidate). It suffices, therefore, to show that both candidates playing unbiased strategies yields lower voter welfare than both pandering according to (4). Now recall that as we reasoned in proving Proposition 1, if both politicians played unbiased strategies then the voter s optimal response would lead to the candidate with the more extreme signal winning. But among candidates strategies with this property, we have: Proposition 4. The strategy profile where both candidates play (4) maximizes voter welfare among all candidates strategy profiles in which the voter s optimal response would lead to candidate i winning whenever θ i > θ i. Consequently, the pandering equilibrium of Proposition 3 is Pareto dominant among all equilibria of the benevolent-candidates game in which a candidate wins when he has the more extreme signal. Thus, at least under the requirement that a candidate must win when he has the more extreme signal, the optimal way to aggregate information in the Downsian game form is for candidates to pander according to (4); in particular, this dominates both candidates playing unbiased strategies and hence any equilibrium of the office-motivated game. While a complete proof has been elusive, we conjecture that both candidates pandering according reason is that the distribution of θ i θ i is Normal with mean E[θ θ i] = θi, and hence as θi (resp. ), α+β conditioning on winning becomes uninformative as θ i E[θ θ i] = α θi (resp. ). 26 α+β Note that while it would not be an equilibrium for candidates to play unbiased strategies either when they are office motivated (Proposition 1) or benevolent (Proposition 3, part one), we are only discussing here what the voter s welfare would be if both candidates played unbiased strategies and the voter responded optimally. β 17

19 to (4) maximizes voter welfare even without the assumption that a candidate wins when he has the more extreme signal. To interpret better this requirement and see some intuition for why it is unlikely to be restrictive, consider any symmetric strategy profile where both candidates play y( ) that is symmetric around 0. For the unbiased strategy, y(θ i ) = E[θ θ i ], we have y ( ) = the overreaction strategy identified in Proposition 2, y(θ i ) = E[θ θ i, θ i = θ i ], we have y ( ) = Presuming differentiability, one can verify that whenever y ( ) [0, β β+α ; for 2β α+2β. 2β α+2β ], it would be optimal for the voter to elect the candidate with the more extreme platform and hence the more extreme signal. 27 Thus, roughly speaking, the requirement that a candidate wins when he has the more extreme signal is satisfied so long as neither candidate overreacts by more than he would when conditioning on the opponent having received the same signal as he did. It appears unlikely that such a degree of overreaction could be socially desirable. 4 Discussion Having established our main results, we now turn to three issues. Subsection 4.1 explains how our results extend to other voter preferences and information structures; Subsection 4.2 relaxes the assumption that candidates are committed to their platforms; and Subsection 4.3 shows robustness to a broader set of candidate motivations. 4.1 Preferences and Information Structures Our fundamental results which identify the welfare loss from electoral competition by office-motivated candidates, Theorem 1 and Theorem 2, hold very generally. An inspection of the proof of Theorem 1 shows that the only juncture at which the statistical information structure plays any role is to ensure that if a candidate i s strategy is informative, then the distribution of i s platforms from the point of view of his opponent, i, is not linear in θ i. Plainly, this is a property that will typically hold 27 In fact, one can establish that the only equilibrium of the benevolent-candidates game in which both candidates use the same differentiable strategy y( ) that is symmetric around 0 and satisfies y 2β ( ) [0, ] is the pandering α+2β equilibrium identified in Proposition 3. 18

20 in any information structure in which θ A and θ B are correlated with each other through the true state θ. As long as this property is satisfied, Theorem 1 would hold regardless of what the statistical structure is, whether and how candidates wish to deviate from the unbiased strategy profile, what the voter s utility function U(y, θ) is, how the voter randomizes if indifferent (cf. fn. 21), and what the policy space is. For example, the result applies to the models of Heidhues and Lagerlof (2003) and Loertscher (2012), thereby generalizing some of those authors observations in their models. The results on pandering do require more structure. However, they extend directly to a broad family of well-known statistical structures in which the distribution of the candidates signals is in an Exponential family and the prior is a conjugate prior. This includes a variety of widely-used distributions (see fn. 7); indeed, in Appendix D, we provide a Beta-Bernoulli example whose structure is quite different from the Normal-Normal structure and yet has similar forces at work. The important property within the Exponential family is that the posterior expectation of the state θ given a prior mean parameter, say θ 0, and any number of signal realizations, θ 1,..., θ n, takes a linear form: E[θ θ 1,..., θ n ] = n i=0 θ iw i n i=0 w i, for some positive coeffcients w 0,..., w n (Jewel, 1974; Kass et al., 1997). When the distribution of each θ i θ is identical for i = 1,..., n, 28 one can take w i = w 1 for all i = 1,..., n, and hence E[θ θ 1,..., θ n ] = w 0 w 0 +nw 1 θ 0 + nw 1 n i=1 θ i w 0 +nw 1 n ( nw n 1 i=1 E[θ θ 1,..., θ n ] θ 0 = θ i w 0 + nw 1 n. Therefore, θ 0 ), (5) while n i=1 E[θ θ i] θ 0 = n n i=1 ( ) w0 w 0 +w 1 θ 0 + w 1 w 0 +w 1 θ i n θ 0 = w 1 ( n i=1 θ i w 0 + w 1 n θ 0 ). (6) It is immediate from (5) and (6) that for any n > 1 and any vector of signal realizations, when one compares the average of the individual posterior expectations with the posterior expectation given the average signal, both shift in the same direction relative to the prior mean, but the latter 28 The points made below hold even when this is not the case, but this simplification makes the argument transparent. 19