Ideology as Opinion: A Spatial Model of Common-value Elections

Transcription

1 Ideology as Opinion: A Spatial Model of Common-value Elections Joseph McMurray February 2016 Abstract This paper considers voting behavior when ideological differences reflect differences of opinion regarding a common objective, rather than fundamental conflicts of interest. Contrary to standard preference models but consistent with empirical evidence, citizens who lack confidence in their private opinions or are skeptical of the information of extremists remain ideologically moderate, and are less inclined to participate in an election. Electoral margins can be large, even when candidates are positioned symmetrically, and a jury theorem leads citizens on both sides of an issue to expect their side to win. This model also sets a stage for several useful extensions. JEL Classification Number D72, D82 Keywords: Voting, Elections, Ideology, Information Aggregation, Jury Theorem, Public Opinion, Swing Voter s Curse, Turnout, Abstention, Epistemic Democracy Assistant Professor of Economics, Brigham Young University. 133 FOB, Provo, UT Office , Fax joseph.mcmurray@byu.edu. Some of the material in this paper was included in my PhD dissertation at the University of Rochester. Special thanks to my advisor, Mark Fey, for his patient encouragement and extremely helpful advice. Thanks also to Ken Shotts, John Duggan, Navin Kartik, Gábor Virág, Stephane Wolton, Odilon Câmara, Jay Goodliffe, Adam Meirowitz, Rainer Schwabe, Roger Myerson, Tim Feddersen, Jean Guillaume Forand, Faruk Gul, Gordon Dahl, Chris Ellis, and participants at the Wallis Conference on Political Economy and the Stonybrook Game Theory Festival, for their interest and suggestions. 1

2 1 Introduction When voters take opposite sides on a political issue, it may be out of private interest: policy benefits one group while policy benefits another. In that case, democracy can serve to aggregate preferences, resolving conflicts in favor of larger groups, which is desirable from a utilitarian perspective. Alternatively, it may be that voters actually share a common goal, and simply hold different beliefs regarding which policy will deliver the desired end. In that case, as Condorcet s (1785) classic jury theorem points out, democracy can serve to aggregate information and determine what is truly optimal: a majority is less likely to be mistaken about an issue than a minority. Implicitly, any theory of political behavior must take a stand regarding the fundamental nature of voters disagreement. Inevitably, political outcomes will be viewed in the light of this underlying assumption. There are good reasons to suspect that preferences should be more fundamental to politics than information is. For one thing, differences of opinion could in principle be resolved, but conflicts of interest would remain. Moreover, conflicts of interest seem pervasive, and indeed unavoidable: issues such as wealth redistribution create obvious conflicts between the rich and the poor, and even classic examples of public goods create winners and losers when logistical issues are considered. Citizens might unanimously agree on the need for a public bridge, for example, but decisions of where to locate and how to pay for it are zero-sum. Consistent with this logic, existing literature strongly emphasizes preferences over information, as Section 2 describes. Analysis of the jury theorem remains limited almost exclusively to primitive binary issues, for example, while preference models have long included a continuum of policies. In the preference aggregation literature, policy preferences are determined by voter ideology, modeled as a preference parameter ranging from the liberal left to the conservative right. The earliest models of Hotelling (1929) and Downs (1957) assume perfect information, and more recent papers add commonly-valued components, but in a decidedly secondary role. The thesis of this paper is that, in spite of the logic above and contrary to the assumptions of most theoretical literature, several aspects of their behavior suggest that voters themselves might implicitly believe political questions to have answers that are fundamentally right or 2

3 wrong, as supposed by Condorcet (1785). 1 This conclusion is reached by extending the binary model in a natural way to include a continuum of policy possibilities. One policy in this continuum is ultimately the best thing to do, and would be unanimously favored except that its location is unknown. Voters form private opinions of what is best, modeled as private signals that are correlated with the truth. Ideology is not a preference parameter at all, then, but a product of Bayesian updating: a liberal citizen believes that the optimal policy lies somewhere to the left of center while a conservative believes it is on the right. 2 fact, this standard ideological geometry arises quite naturally even if truth is binary, because heterogeneous expertise still leads voters to form a spectrum of opinions, ranging from one extreme to the other. Empirical observations that corroborate the view of ideology as opinion include the following. First, citizens who lack political knowledge tend to remain ideologically moderate, reluctant to embrace one side or the other; as explained below, preference models give ideological moderates the strongest incentive to invest in candidate-specific information, therefore predicting the opposite correlation. Related to this, uninformed voters and informed ideological moderates both suffer from a swing voter s curse as in Feddersen and Pesendorfer (1996), and so abstain from voting or cast incomplete ballots; preference models with costly voting can explain abstention but not incomplete ballots. In A third empirical phenomenon that is easily explained in an information model is that of large margins of victory, which can arise here even if the menu of policies is perfectly symmetric, because citizens votes are correlated with one another. This correlation can also explain the empirical consensus effect whereby citizens on both sides of an issue expect to belong to the majority. In modeling voting as a game of common interest, this paper is not meant to suggest that conflicts do not exist or are unimportant, but rather that voters already take others preferences into account when deciding how to vote, viewing the world as if through the eyes of social planners. In essence, preference aggregation occurs internally, before votes are 1 Grofman and Feld (1988) trace this philosophical view (which is sometimes called epistemic democracy, following Coleman and Ferejohn 1986) to Rousseau s (1762) writings on the general will. 2 According to Gerring (1997), this is consistent with the etymology of the word ideology, whichhas historically been associated with beliefs, ideas, and world views. 3

4 ever cast, and differences that remain on election day are differences of opinion regarding which policies will be best for society. A wealthy citizen supports income redistribution, for example, to help the poor, and to preserve democracy from the undue influence of other wealthy elites. Similarly, a poor citizen opposes redistribution that would benefit himself, fearing that it takes unfair advantage of a wealthy minority, squelches work incentives, or delivers too much power to political elites. When both sides of an issue employ the language of public goods, it is natural to worry that this is merely a rhetorical mask for self-interest. For most people, altruistic impulses are relatively subdued. Unlike other choices that citizens make, however, political decisions effect large numbers of people simultaneously. This can substantially amplify altruism: a citizen may prefer $100 to a policy that gives a $100 benefit to his neighbor, for example, but not to a policy that gives a $100 benefit toonehundred (or one hundred million) of his neighbors. More formally, consider a standard model of altruism, in which voter s utility function = + P 6= places positive weight on the well-being of each of his peers, and so can be rewritten as a weighted average =(1 ) + (1) of his own well-being and the average well-being = 1 P =1 of the population. When the number of peers is large, the second term (which is common across altruists) dominates, even if is close to zero. 3 In other words, even a citizen who is almost purely selfish should base political decisions on his perception of the common good, virtually ignoring his private interest. 4 3 The following thought experiment further supports the notion that citizens view political questions as if through the eyes of social planners: suppose that a citizen were asked to choose policy for a remote constituency, of which he himself is not a part. Presumably, such a citizen would be no more liberal or conservative on behalf of this other constituency than he is on behalf of his own, suggesting that his ideology depends little on his own interests. Furthermore, if given the option of choosing policy for a remote constituency consisting of individuals or a different remote constituency consisting of +1 individuals, the individual would presumably opt for the latter, consistent with the formulation in (1). 4 In principle, conflicts of interest could remain even between social planners, who prioritize common objectives differently, or aggregate preferences according to different social welfare functions. One might favor efficiency, for example, while another gives higher priority to equity. If neither is merely pursuing his 4

5 Empirically, voters indeed seem willing to subjugate their private interests to their perception of the common good. Summarizing earlier statistical studies, Caplan (2007, p. 149) writes that the [U.S.] elderly are not more in favor of Social Security and Medicare than the rest of the population... the unemployed are at most a little more in favor of government-guaranteed jobs, and the uninsured at most a little more supportive of national health insurance... Males vulnerable to the draft support it at normal levels... Fong (2001) reports Gallup poll evidence that preferences for redistribution correlate with voters beliefs regarding the prevalence of poverty, and about the relative importance of luck and effort for economic success, more strongly than with their own incomes or other demographic variables. 5 This translates strongly into voting behavior: according to American National Election Study (ANES) data, 36% of citizens with below-median incomes voted Republican in the 2012 U.S. presidential election while 52% of wealthier citizens voted Democrat. Thus, as Figure 1 illustrates, vote choice is correlated with income, but only weakly: a regression on all 28 income categories produces an 2 of only Preferences for redistribution behave similarly: asked whether they prefer that federal spending on aid to the poor be increased or decreased, 25% of respondents with below-median incomes favored reducing aid, while 44% of those with above-median incomes favored higher aid. 7 Moreover, a public spirited voter might presume that what is best for himself is likely good for society, as well; if so, even votes that appear selfish need not be. A focus on information is also consistent with the fact that policy decisions are exceedingly complex, so information is likely to be a first-order problem. Do financial instability and rising health care costs result from insufficient regulation, or excessive regulation? increased military activity deter terrorism, or provoke it? Does Are the poor helped or harmed own private interests, however, it is not clear that such differences should be robust: proceeding to discuss reasons why one welfare function or the other more accurately reflects the best interests of society, for example, these citizen-planners might eventually reach a consensus. 5 See also Corneo and Gruner (2002), Alesina and Angeletos (2005), Piketty (1995), Buera, Monge- Naranjo, and Primiceri (2011), and Giuliano and Spilimbergo (2014). 6 Piketty (1995) and Gelman et al. (2007) document similar trends in previous years and internationally, and Mueller (2003, ch. 14.4) cites several additional examples of groups who vote against their own interests. 7 These numbers exclude respondents who favor neither an increase nor a decrease in aid spending. 5

6 Figure 1 by minimum wage laws and workers unions? Preferences aside, experts and non-experts alike predict widely different outcomes for each of these policies, so even among voters with common objectives, policy disagreements are unsurprising. Field experiments by Gilens (2001), Luskin, Fishkin, and Jowell (2002), Banerjee et al. (2010), and Elias, Lacetera, and Macis (2015) confirm that changing voters information causes changes in their policy opinions and voting behavior. Page and Shapiro (1992) document numerous episodes of shifting public opinion. These changes are sufficiently pronounced that, as Magleby (1984) documents, ballot propositions that enjoy majority support only weeks before an election are often defeated on election day (and, similarly, underdog propositions often prevail). If preferences are immutable then changes in political support can only be attributed to changes in voters opinion. 8 Individual opinions oscillate much more than the aggregate: between the June, September, and November waives of a 1999 panel survey of Swiss voters, for example, Sciarini and Kriesi (2003) document how, on each of seven major policy issues, 25-40% of 8 Political swings are too rapid to be explained by demographic transitions (e.g. older generations being replaced by younger ones). 6

7 people reversed positions at least once, and 10% changed twice. 9 In addition to voters who reverse their stated policy positions, many remain officially undecided even until the day of an election. Broader ideological positions change much less frequently, but even these evolve over time, as Jennings and Markus (1984) document, especially while voters are young. Occasionally, such as after market crashes, business or government scandals, or wars or acts of terrorism, ideological shifts can also be quite abrupt. 10 Of course, evidence of changing opinions does not directly corroborate the common values assumption. Information about which of two policies transfers greater wealth from the rich to the poor, for example, could well lead rich and poor voters to update their policy stances in opposite directions. For the most part, however, Page and Shapiro (1992) find that new information pushes demographic groups in the same direction. Occasionally, disagreement has even given way to consensus, as when racial desegregation in U.S. schools was highly controversial in the 1950s but had few opponents by the 1980s. Berelson, Lazarsfeld, and McPhee (1954, ch. 7) also report that political conversions are most frequent among voters who interact and discuss politics with those of opposing views. In 2012, 43% of ANES respondents reported trying to convince others why they should vote for or against one of the parties or candidates, and 18% endorsed a candidate by displaying campaign paraphernalia. If political decisions were based primarily on idiosyncratic preferences then such persuasive efforts would be futile; that voters spend such time and energy attempting to educate and persuade one another (e.g. through debates, endorsements, and policy research) suggests an optimism that others, once properly informed, will share their own interest. Consistent with this, DellaVigna and Gentzkow (2010) conclude in a review of empirical literature that patterns of political persuasion are broadly consistent with informative communication and Bayesian updating. Unless all voters finally came to a consensus on all issues, it would be 9 Similar reversals are documented by Hill and Kriesi (2001). 10 Schuller (2015) documents a slight conservative shift after the terrorist attacks on September 11, In the wake of the 2008 financial crisis, former U.S. Federal Reserve chairman Alan Greenspan testified before Congress of being suddenly convinced that the ideology motivating his earlier efforts at financial deregulation had been wrong (Andrews, Edmund L. 10/23/2008. Greenspan Concedes Error on Regulation, New York Times. emphasis added). 7

8 difficult to guarantee that preferences do not also matter, but the observations above at least establish that voter preferences share a substantial common component. List and Goodin (2001) also point out that a common value paradigm can explain the empirical rareness of majority preference cycles, which social choice literature suggests should be ubiquitous, given the general intransitivity of the majority preference relation. 2 Related Literature Condorcet s (1785) original jury theorem has been generalized in a number of ways. Nitzan and Paroush (1982) accommodate heterogeneity in voters expertise. Ladha (1992) and Dietrich and Spiekermann (2013) analyze the impact of correlated judgment errors. De Clippel and Eliaz (2012) and List and Goodin (2001) analyze information aggregation under various voting rules. The latter allows a finite menu of policies, but otherwise these papers all maintain the binary structure of the original jury model. The same is true of extensions that assume strategic voting, beginning with Austen-Smith and Banks (1996). In contrast, the model below considers a truly spatial model, with a continuum of policies and, for one specification, a continuum of truth states. Bernhardt, Duggan, and Squintani (2009) and Ortoleva and Snowberg (2015) consider models with common-value components, but these are formulated merely as shocks that shift the distribution of policy preferences to the left or the right. A more common approach is to treat common-value valence issues as separable from a standard private-value spectrum of ideologies. In that case, Feddersen and Pesendorfer (1997, 1999) show that citizens infer commonly-valued information in equilibrium, so that voting behavior is determined almost exclusively by voters private preferences, rather than by their private signals. Because of this, the swing voter s curse actually vanishes in large elections. Most spatial voting models include a spectrum of preferences, but omit issues of common value. Typically, the geometry of ideological preferences is imposed exogenously, without explanation. Exceptions to this derive ideology from income. In Goodman (1973) income determines the demand for public good provision; in Romer (1975) and Meltzer and 8

9 Richard (1981) it determines the demand for income redistribution. Attributing political differences to income is problematic empirically, however, as Section 1 discusses. Moreover, the monotonicity of preferences in these settings hinges on the implicit restriction of redistribution and tax schemes to those that are monotonic in income. In principle, other tax schemes are also feasible, and as Tullock (1971) points out, these need not induce the familiar geometry: instead of transferring wealth from the rich to the poor, for example, the middle class could vote to expropriate wealth from both ends of the income distribution (or the rich and poor could expropriate from the middle class). In contrast, monotonicity in the model below follows naturally from the informativeness of private signals. 3 Model An electorate consists of citizens where, for mathematical convenience, is drawn from a Poisson distribution with mean, as in Myerson (1998). Together, these citizens must choose and implement a policy from the interval X =[ 1 1] of feasible alternatives, whichwillprovideacommonbenefit to every citizen. Let Z X and let Z denote an unknown state of the world, which designates the policy that in truth is best for society. For the most part, the analysis below assumes that Z = X, meaning that any policy that is feasible might also be optimal. For simplicity, the prior density of is uniform. 1 if Z 2 ( ) =. (2) 0 otherwise If policy X is implemented in state then each citizen receives utility ( ) = ( ) 2, which declines quadratically with the distance between and. This specification is convenient because, conditional on information Ω, expected utility is similarly quadratic in the policy choice. 11 [ ( ) Ω] = [ ( Ω)] 2 ( Ω) (3) That is, the optimal policy choice is the updated expectation of, and preferences are single-peaked, as in standard spatial voting models. 11 The final term in (3) does not depend on political outcomes, and so is suppressed in subsequent notation. 9

10 The concavity of ( ) also implies that voters are risk averse, and thus favor moderate policies,whichareguaranteednottobetoofarfromwhateveristrulyoptimal. 12 Ex ante, the optimal policy lies exactly at the center, ( ) =0. For some applications, it is more reasonable to assume that Z = { 1 1}, meaning that truth is binary. To end an economic recession, for example, Keynesian economic theory recommends a large policy of economic stimulus, while more classical economic theory views such stimulus efforts as wasteful; thus, the optimal level of stimulus is known ex ante to be either very large or very small. To accommodate binary issues, (2) can simply be reinterpreted as a mass function, meaning that = 1 and =1are equally likely. The densities below can be reinterpreted similarly, and proofs of formal results apply almost verbatim in that case, merely summing over Z instead of integrating. Accordingly, the discussion below moves freely between examples of continuous and binary uncertainty. To be clear, binary uncertainty makes no restriction on the set X of feasible policies: a moderatesized stimulus is still feasible even if it is not optimal per se in any state of the world. This is important because, in the face of uncertainty, risk-averse citizens might actually prefer a moderate stimulus so as to avoid the catastrophe of getting macroeconomic policy exactly wrong. Each citizen develops a private hunch regarding the location of the optimal policy, represented by a private signal S which is positively correlated with, and has the same domain S = Z. Conditional on, these signals are independent. Because citizens differ in expertise, their signals vary in quality. Specifically, a citizen s expertise is first drawn independently (from other citizens, and from ) from the set Q =[0 1] according to a common distribution, which is differentiable and has a strictly positive density. The density of = depends on and on. For convenience, this density is linear: ( ) = 1 (1 + ). 2 (4) The distributions,, and are common knowledge, but and are observed only privately. 12 The single-peakedness of ( ) would make it nearly concave even if the utility loss function were linear or convex, so results would be similar to those below. 10

11 The particular specification of (4) is not crucial to the logic of the results below, but is convenient for two reasons. First, for the case of binary and, (4) can be reinterpreted as a mass function. More importantly, this parameterization tractably separates a citizen s substantive information from his expertise. 13 With binary uncertainty, gives the correlation coefficient between and, meaning that a citizen with =1observes perfectly. With continuous uncertainty, the correlation is only 1 3, so even the highest quality signals include substantial noise, but with either specification is uniform on S and its precision increases with expertise. The lowest quality signal reveals nothing: if =0then and are independent. Citizens do not vote for policies directly. Instead, there is a menu of two policy alternatives,, representing the policy platforms of two candidates or parties, and. In order to maintain the focus on voting behavior, this paper treats these platforms as being endogenous. Taking the voting behavior derived in this paper as given, McMurray (2016a) then analyzes candidates location decisions, under a variety of assumptions regarding candidate motivations. Observing these platforms, citizens then vote (at no cost) for either candidate. 14 Avotingstrategy : Q S { } specifies a candidate choice { } for every realization ( ) Q S of private information. Let V denote the set of such strategies. Votes are cast simultaneously, and a winning candidate { } is determined by majority rule, breaking a tie if necessary by coin toss. The policy outcome is that candidate s policy platform. Expected utility can therefore be rewritten from (3) as follows, for any X. Z " # X [ ( ) Ω] = ( )Pr( = ) ( Ω). (5) Z = 13 Ottaviani and Sørensen (2006) use this multiplicatively linear structure to model expertise in a cheap talk setting, and further discuss its properties. For example, (4) can be interpreted as the weighted average ( ) = (1 ) 1 2 of the densities of an informative and an uninformative signal, with weights determined by a citizen s expertise. 14 Costly voting and costly information acquisition are important directions for future work, but beyond the scope of this paper. It is worth noting that ethical motivations such as altruism, which may be the source of voters common interest, have also been offered as explanations for costly voting (e.g. Edlin, Gelman, and Kaplan, 2007; Faravelli, Man, and Walsh, 2013). 11

12 Implicitly, Pr ( = ) in (5) depends on the strategies used by every voter. If his peers all voteaccordingtothestrategy V, a citizen s best response is the strategy V that maximizes (5) for every realization ( ) Q S of private information. A (symmetric) Bayesian Nash equilibrium is a strategy that is its own best response. 15 In defense of the controversial assumption that voters share a fundamentally common interest, Section 1 proposes that large elections might substantially amplify small levels of voter altruism. Since this is the key assumption of the model of this section, it is worth pausing to formalize this claim somewhat, before proceeding to analyze voters. To begin, suppose that preferences are arbitrarily heterogeneous, and consider a standard model of altruism, in which voter s utility function = + P 6= places positive weight on the well-being of each of his peers, and so can be rewritten as a weighted average =(1 ) + P of his own well-being and the average well-being = 1 =1 of the population. When the number of peers is large, the second term (which is common across altruists) dominates, even if is close to zero. In other words, even a citizen who is almost purely selfish should base political decisions almost entirely on his perception of the common good, rather than his private interest. In principle, conflicts of interest could remain even between social planners, who prioritize common objectives differently, or aggregate preferences according to different social welfare functions. One might favor efficiency, for example, while another gives higher priority to equity. If neither is merely pursuing his own private interests, however, it is not clear that such differences should be robust: proceeding to discuss reasons why one welfare function or the other more accurately reflects the true interests of society, for example, these citizen-planners might eventually reach a consensus. 15 In games of Poisson population uncertainty, symmetry is inevitable in equilibrium because the distribution of opponent behavior is the same for any two individuals within the game (unlike a game between a finite set of players), implying that a best response for one citizen is a best response for all. 12

13 4 Analysis This section analyzes equilibrium behavior. Section 4.1 begins by defining voter ideology and discussing empirical support for this informational formulation. Sections 4.2 and Section 4.3 then discuss how this translates into voting behavior and can produce large margins of victory. Section 4.4 considers the incentives for voter participation and abstention. Section 4.5 then analyzes voter welfare and Section 4.6 reinterprets the original model to accommodate the possibility of aggregate uncertainty. 4.1 Information and Ideology After observing his private information ( ) Q S, an individual uses Bayes rule to update his beliefs to the following, which, like the densities above, can be reinterpreted for binary and as a probability mass function. ( ) = 1 2 (1 + ) 1 (1 + ) = ( ). (6) 2 This posterior depends on and only through the product =. The density and conditional density of can be derived from ( ), ( ), and ( ) as Z Z Z Z Z ( ) = 1 = ( ) ( ) ( ) = 1 = ( ) 1 2 and Z Z ( ) = Q S Z Q S Q S 1 = ( ) 1 (1 + ) =(1+ ) ( ), 2 where 1 = is an indicator function that equals 1 when =. Analogous expressions can be derived for the case of binary uncertainty, by summing instead of integrating over S. As Section 3 discusses, a citizen s preferred policy is his expectation of. Using (6) and summing or integrating over Z, thisisgivensimplyby ( ) = for the case of binary uncertainty and by ( ) = 1 for the case of continuous uncertainty. Either 3 way, the preferred policy is simply proportional to, which can therefore be interpreted as a citizen s ideology. Citizens with negative values of are liberal, meaning that they believe the optimal policy is left of center, while citizens with positive values of are conservative, 13

14 and believe that the optimal policy is somewhere on the right. When is close to zero (on either side), a citizen can be said to be ideologically moderate. Whether uncertainty is continuous or binary, expertise is sufficiently heterogeneous that the set Θ of ideologies ranges from 1 to 1. Thesignofavoter sideology isthesameasthesignofhisprivatesignal. Recognizing that signals are noisy, however, an individual discounts his own information in proportion to his uncertainty. Even if his hunch is that the optimal policy is quite extreme, therefore, a citizen who knows that he lacks information will tend to remain ideologically moderate. Remark 1 formalizes this insight, by stating that the magnitude of increases both with the magnitude of and with information quality. The functional forms in (2) and (4) make this result especially easy to see, but a similar comparative static result would hold quite generally: for typical information structures, posterior beliefs are some weighted average of the prior and the signal (see Chambers and Healy, 2012), and citizens with noisier signals must place greater weight on their prior beliefs. Remark 1 Ideological intensity increases both in and in. The logic of Remark 1 underscores how naturally the familiar left-right geometry of ideology arises in this setting in contrast with a preference formulation of ideology, which, as Section 2 explains, requires restrictions on the set of feasible policies. Here, individuals sort themselves into philosophical camps : just as macroeconomists self-identify as Keynesian or classical, voters self-identify as liberal or conservative, according to their opinions regarding the appropriate role of government. With continuous uncertainty, there is a range of policies that might be optimal, so a range of opinions is automatic. Even if truth is fundamentally binary, though, opinions will inevitably range from full support for one extreme to full support for the other. The prediction that citizens who lack confidence in their private opinions remain ideologically moderate can be investigated empirically. Using four decades of ANES data, Figure 2 shows the distribution of self-reported ideology among citizens with different levels of information. 16 Voters with the highest level of information hold a range of political opinions, but 16 This 9-point ideological scale distinguishes survey respondents who originally placed themselves at the 14

15 Figure 2 consistent with Remark 1, the least informed group are noticeably more moderate. 17 Regressing ideological polarization on information yields a coefficient of (standard error 0 029), meaning that each level of information (out of five) makes a citizen more than one quarter-level more extreme (on a 9-point scale), on average. Treating ideology as a taste parameter (or heterogeneous prior, which has similar consequences) and allowing voter abstention, Larcinese (2009) and Oliveros (2013) offer an alternative explanation for why ideological moderates tend to have lower information levels: since moderates are close to indifferent between candidates policy positions, they lack an incentive to vote, and, since they won t be voting, also lack an incentive to invest in political information. This reasoning is compelling, except that the pattern above does not seem to be sensitive to voter abstention. Figure 2 is virtually identical, for example, when restricted to voters or to nonvoters, and empirical studies by Loewen, Milner, and Hicks (2008) and Selb and Lachat (2009) also find that voter information does not appreciably improve when center of a 7-point scale, but later admitted leaning either liberal or conservative. The information variable reflects citizens general level of information about politics and public affairs, as assessed subjectively by ANES interviewers on a 5-point scale. Other measures of information are more objective but less comprehensive. 17 Palfrey and Poole (1987), Abramowitz and Saunders (2008), and Berelson, Lazarsfeld, and McPhee (1954) note similar patterns. 15

16 voting is compulsory. 18 Another explanation for the pattern highlighted in Figure 2 is that the demand for news and political information is higher among extreme citizens than among moderates. On the surface, this may seem to be the natural prediction of a preference model, as ideological extremists have more at stake in an election than moderates, and so have stronger incentives to pay attention. This logic would only be valid, however, if the informational difficulty for voters were that of determining which candidate is on the left and which is on the right. 19 In general elections with established parties whose policy positions are far apart from one another, this is likely to be obvious. In that case, the voting decision of extreme voters is clear-cut: there is no need to invest time and energy in determining candidates (or their own) precise ideological positions or non-policy valence characteristics such as leadership skill, honesty, or intelligence, because these will be overwhelmed by ideological considerations, and will therefore not change their voting decisions. Instead, it is ideological moderates who should pay the most attention, as their decisions may well hinge on these details. 20 Thus, in a preference model, the most plausible sources of uncertainty should produce correlations opposite thoseobservedinfigure2. 21 If some citizens do have a larger appetite for news and political information, these should come to develop greater confidence in their opinions, and the logic of Remark 4.1 makes clear how this will translate into more extreme policy preferences. Mullainathan and Washington (2009) present what could be interpreted as evidence of a causal influence of information on 18 Moreover, this logic does not require preference heterogeneity: ideological moderates in this model, too, have an incentive to abstain, as Section 4.4 shows below, and if given the option, these would also have the lowest incentive to invest in additional private information. 19 This produces the observed correlation, for example, in the private-value model of Matejka and Tabellini (2015). 20 See Larcinese (2009) and Oliveros (2013). 21 Larcinese (2009) presents empirical evidence that the very most extreme citizens are slightly less well informed than citizens who are slightly less extreme. This could be explained by expanding the present model so that, after formulating their general policy opinions, voters could pay to invest in more specific information about candidates non-policy characeristics. Assuming that policy considerations are generally more important, citizens with strong policy views would need not make such investments, and so would have high levels of policy-relevant information, but low levels of candidate-specific information. 16

17 extremism: voters who were barely old enough to vote in the previous election hold more extreme views than those who were barely too young, perhaps because voting eligibility induced them to learn more about politics, thereby developing stronger opinions. Consistent with the result that voting behavior is determined by the product of and, Goren (1997) also reports that voters with extreme policy opinions but limited political knowledge vote similarly to moderates who are more knowledgeable. 4.2 Voting If citizens vote according to the strategy V then, in state Z, eachvotesfor candidate { } with probability Z Z ( ) = Q S 1 ( )= ( ) ( ), (7) where 1 ( )= is an indicator function that equals 1 when ( ) =. By the decomposition property of Poisson random variables (Myerson, 1998), the numbers and of and votes are independent Poisson random variables with means ( ) and ( ), sothe joint probability of vote totals = and = is given by ( ) = ( ) ( )!! [ ( )] [ ( )]. (8) In terms of (8), candidate wins the election by a margin of exactly 0 votes (alternatively, wins by votes) with probability ( ) = ( ) = X ( + ) (9) and wins by 0 votes (or wins by votes) with probability ( ) = ( ) = Candidate therefore takes office with probability Pr ( = ) = =0 X ( + ). (10) =0 X ( )+ 1 2 (0 ), (11) where the second term reflects the event of winning the tie-breaking coin toss. =1 17

18 By the environmental equivalence property of Poisson games (Myerson, 1998), an individual from within the game reinterprets and as the numbers of and votes cast by his peers; by voting himself, he can add one to either total. If he votes for candidate, for example, that candidate will win with probability P =0 ( )+ 1 2 ( 1 ) instead of (11). Thisisanincreaseof Pr (P ) = 1 2 (0 )+ 1 2 ( 1 ), (12) which is the probability that a vote for candidate reverses the election outcome, or is pivotal (event P ). The first term on the right-hand side of (12) corresponds to the scenario in which the candidates tie and loses the tie-breaking coin toss; the second, the scenario in which wins the coin toss but loses the election by exactly one vote. In terms of (12), the difference ( ) in expected utility between voting and voting is given by ( ) = {[ ( ) ( )] Pr (P ) } {[ ( ) ( )] Pr (P ) } = {[ ( ) ( )] Pr (P ) } = [2 ( )( )Pr(P ) ] = 2( )[ ( P ) ]Pr(P ), (13) where = 1 2 ( + ) is the midpoint between the two candidates platforms and Pr (P ) =Pr(P )+Pr(P ) (14) is the probability that either an vote or a vote is pivotal (event P). 22 Since, (13) is positive if and only if ( P ) is sufficiently high. This expectation is increasing in a voter s ideology, as the proof of Lemma 1 shows, below; accordingly, Lemma 1 characterizes best-response voting as an ideological strategy, defined in 22 Comparing electoral outcomes with and without his own vote is clearly the rational thing for a voter to do, but pivotal voting models remain somewhat controversial because, empirically, voters do not seem to be cognizant or even capable of such a calculus (Esponda and Vespa, 2014). For most of the results below, the issue is moot, since equilibrium voting turns out also to be sincere. 18

19 Definition 1. Such behavior is simple: a citizen votes ifhisideologyistotheleftofsome threshold and votes if his ideology is to the right. An ideological strategy can be notated as a function of a citizen s ideology, rather than of and separately. Definition 1 A voting strategy V is ideological, with ideology threshold [ 1 1], if if ( ) =. 23 if Lemma 1 For any voting strategy V, there is a unique best response V. Moreover, is ideological. When citizens follow an ideological strategy, the expected vote shares ( ) for each candidate reduce from (7) to the following. ( ) = ( ) = Z 1 Z 1 (1 + ) ( ) (15) (1 + ) ( ). (16) The best response to an ideological strategy is another ideological strategy, so using a fixed point argument on the space of ideology thresholds, Proposition 1 builds on Lemma 1 to state the existence of an equilibrium. In fact, this equilibrium turns out to be unique. Proposition 1 A unique Bayesian Nash equilibrium V exists, and is ideological Electoral Margins In preference models with perfect information, the candidate whose policy position is closest to the median voter s ideal point attracts a majority of voters, and wins the election. If candidates are equidistant from this policy whether they converge or remain polarized they tie (in expectation). Empirically, many elections are quite close, but many are not: 23 The specification of voting behavior right at the threshold here and in Definition 2 below is unimportant, as this is realized with zero probability. 24 Uniqueness here is only up to the specification of behavior for citizens right at the ideological threshold, who in equilibrium are indifferent between voting and. 19

20 according to Mueller (2003, ch. 11), the historic average margin of victory in U.S. gubernatorial elections is nearly 23%; as Krehbiel (1997) points out, large bipartisan majorities are also common in legislative votes. One possibility, of course, is that candidates are asymmetric with respect to the median position. In the common value setting of this paper, however, large margins can also arise even when candidate positions are symmetric, because of the correlation between citizens opinions. To see this, suppose that = or, equivalently, that =0. In that case, the symmetry of the model is such that the unique equilibrium ideological voting strategy has =0, meaning that voting is symmetric: citizens with negative signals all vote and citizens with positive signals all vote. Expected vote shares then reduce from (7) to ( ) = ( ) = Z 1 Z (1 + ) ( ) 2 = ( ), 4 producing a margin of victory ( ) = ( ) ( ) = 1 ( ) 2 in state. Across states of the world, the average margin [ ( )] = Z 1 is positive, as Proposition 2 states formally ( ) ( ) = 1 ( ) (17) 4 Proposition 2 If = then, in the unique Bayesian Nash equilibrium, the expected margin of victory is positive, and increases in ( ). The last part of Proposition 2 points out that (17) increases in ( ), meaningthatlarge margins of victory are especially likely when the general level of information in the electorate is high or, equivalently, when a policy question is obvious. This is consistent with empirical observation: in electoral decisions such as retaining public officials with clear track records 25 In the case of binary uncertainty, ( ) = ( ) = R 1 0 [ ( )] = 1 2 ( ). 1 2 (1 + ) ( ) = ( ), so 20

21 of quality governance or revising archaic government procedures or constitutional language, landslide victories are commonplace. Interpreting a large margin of victory as evidence that a decision was obvious is also natural in settings such as jury verdicts and constitutional amendments, where supermajority rules are explicitly utilized to avoid taking action until the benefits of doing so are unambiguous. 26 Intuitively, the reason for Proposition 2 is that, since individuals opinions are each correlated with the truth, they are also (indirectly) correlated with one another. This insight is simple, but seems not to have been documented in existing literature. A similar result could be derived in a more standard probabilistic voting model, such as those of Wittman (1977) orcalvert(1985),wherecandidateslocatearoundthemedianvoter sperceivedlocation,but his actual location is unknown. How plausible this is, however, depends on the presumed source of uncertainty. If uncertainty merely stems from the sampling error in opinion polls, for example, it should attenuate as polls grow large. With modern sample sizes as large as they are, the sample median should be close to the true median. In an information setting, by contrast, early poll results may be quite unreliable because individuals (rationally) wait to research candidates and policy issues until shortly before voting takes place. In light of (17), the fact that most elections do not result in landslides but instead are quite close indicates that, for these political questions, ( ) must be close to zero. This is consistent with well-known empirical evidence of egregious voter ignorance, as well as the observation in Section 1 that most political questions are exceedingly complicated. In fact, Section 3 assumes for simplicity that ( ) has full support, but in that case the most expert citizens should largely be in agreement with one another. For many political issues, there is no such consensus even among intellectual elites, suggesting that is close to zero even for the most highly informed. 26 Building on this paper, McMurray (2016b) considers a model in which candidates interpret large margins of victory as mandates from voters to move in one policy direction or the other. 21

22 4.4 Voter Participation The analysis above assumes that every citizen must vote, whereas in most elections, citizens are allowed to abstain, and many do so. Accordingly, this section redefines voting strategies : Q S { 0} to allow abstention, denoted as action 0. Let V 0 denote the set of such strategies. Definition 2 redefines an ideological strategy in this setting, using two thresholds instead of one. That is, a citizen whose ideology is sufficiently liberal votes for candidate, while one whose ideology is sufficiently conservative votes. If the two thresholds coincide then everyone votes; if the thresholds are distinct then citizens between the thresholds abstain from voting for either candidate. Definition 2 A voting strategy 1 2 V 0 is ideological, with ideology thresholds 1 2, if [ 1 1 ] if 1 2 ( ) = 0 if ( 1 2 ). if [ 2 1] With ideological strategies redefined, Lemma 2 reiterates Lemma 1 in stating that the unique best response to any voting strategy is ideological. Proposition 3 then reiterates Proposition 1 in stating that an equilibrium exists. Lemma 2 For any voting strategy V 0, the unique best response V 0 is ideological. With continuous distributions of and (for continuous uncertainty), the set of citizens whose signals are completely uninformative is of measure zero. If candidate platforms are distinct, therefore, then no citizen is indifferent between the two candidates. Intuitively, since voting in this model is costless, this may seem to suggest that every citizen will vote, meaning that the ideological thresholds 1 and 2 coincide in equilibrium. Contrary to this intuition, however, Proposition 3 states that 1 2, implying that a positive fraction of the electorate choose not to participate. An ideological strategy with abstention is illustrated in Figure 3: citizens with high and low ideology vote and, respectively, while ideological moderates abstain. 22

23 Figure 3: Voting behavior, by opinion and expertise, for a belief threshold strategy. Proposition 3 (Swing voter s curse) If V 0 is a Bayesian Nash equilibrium then is ideological, with 1 2. Moreover, such an equilibrium exists. 27 The reason why citizens abstain in equilibrium is the swing voter s curse identified by Feddersen and Pesendorfer s (1996): since private opinions are correlated with the truth, the candidate with the truly superior policy platform is more likely to win the election by one vote than to lose by one vote, which means that a vote for the inferior candidate is more likely to be pivotal than a vote for the superior candidate. A completely uninformed voter and, by continuity, a weakly informed voter is therefore more likely to do harm than good by voting, and so prefers to abstain. As noted above, the strategic complexity ofthepivotalvotingcalculusmakesitsomewhatcontroversial. Asnotedbelow,however, solving the conceptually simpler problem of a social planner produces the same behavior, thus making it more plausible. The basic idea of delegating to those with better information is quite straightforward, even if the mathematical derivation is not, and the swing voter s curse finds strong support in the laboratory experiments of Battaglini, Morton, and Palfrey (2010). Consistent with the prediction of Proposition 3, numerous studies document an empirical correlation between voter turnout and information variables such as voter education, political 27 Logic discussed in McMurray (2013) suggests that multiple equilibria are possible here, but only for exotic distributions of private information. 23

24 knowledge, age, access to news media, and contact from campaign workers. 28 Lassen (2005), Gentzkow (2006), Banerjee et al. (2010), and Larcinese (2007) also present evidence that the relationship between information and voter participation is causal. A more traditional explanation for voter abstention is the time cost of traveling to the polls, waiting in line, and so on, but as Feddersen and Pesendorfer (1996) point out, this cannot explain why citizens also frequently cast incomplete ballots, even after voting costs have already been paid. The swing voter s curse offers a natural explanation for this, if a voter has clear, strong opinions on some issues, but not others, and so delegates the latter to those who know more. support of this explanation, Magleby (1984, ch. 6) and Wattenberg et al. (2000) find that educated and politically knowledgeable voters are the most likely to cast completed ballots. 29 As Figure 3 makes clear, citizens who lack expertise are not the only ones subject to the swing voter s curse: since ideology is the product of and, those with high levels of expertise but moderate signals abstain as well. Such individuals are not poorly informed in an absolute sense but, believing the optimal policy to be moderate, have trouble choosing between one candidate who seems too far to the left and another who seems too far to the right. Even if were the same for every citizen, then, ideological moderates would abstain to avoid the swing voter s curse. This spatial implication of the swing voter s curse does not seem to have not been documented in the binary models of existing literature. The prediction that voter participation depends jointly on information and ideology has strong empirical support. Using ANES data, Figure 4 displays voter participation levels by information level and ideology. According to probit estimates in Table 1, each information level makes the average citizen 9% more likely to vote in the primary election, 9% more likely to register in the general election, 6% more likely to vote conditional on registering, and 4% 28 For a review of this empirical literature, see McMurray (2015). Voter turnout is also higher in national and general elections than in local races and primaries, perhaps because partisan labels and media exposure give voters greater confidence in their opinions. Similarly, participation tends to be low for policy initiatives, which are typically non-partisan, and which many find confusing (Magleby 1984, ch. 6). 29 Similarly, Gerber, Huber, Biggers, and Hendry (2015) find that general election voters are less likely to also participate in primary elections when they lack information or express a belief that other voters will do a good job on their behalf. In 24

25 Figure 4 more likely to cast a complete ballot conditional on voting. Similarly, moving one ideological category increases these participation levels by 2%, 1%, 1%, and1%, respectively. These estimates are consistent with the findings of Palfrey and Poole (1987), Keith et al. (1992, ch. 3), and Abramowitz and Saunders (2008), that voter participation and information are jointlycorrelatedwithideology. BadeandRice (2009) suggest an alternative theoretical link between information and ideology based on persuasion and the demand for information, but find the additional link with participation inexplicable. Sobbrio and Navarra (2010) find evidence that supports the strategic theory even more specifically, pointing out that participation is correlated with information, even conditional on ideology Those authors actually interpret their finding as evidence against the swing voter s curse, but this is because, following Feddersen and Pesendorfer (1996), they conceptualize ideology as a preference parameter, independent of voter beliefs. 25

26 Table 1 An alternative explanation for the correlation between ideology and voter participation is that voting is costly: as is well known, moderate voters are less willing to pay voting costs because they are close to indifferent between policies on the left and right. This explanation is problematic, however, for at least two reasons. First, as Figure 4 makes clear, the joint correlation between information, ideology, and participation is just as strong for the decision of whether to cast a complete or incomplete ballot where traditional voting costs are irrelevant as for the decision of whether to vote or to abstain. Second, costly voting leads to the well-known counterfactual prediction that citizens should not participate in large elections, because a vote is so unlikely to be pivotal. The most common resolution to this paradox is the hypothesis that voters are motivated by a sense of civic duty, but as I show in McMurray (2015), this explanation eliminates the predicted correlation between information and participation, because in large elections participation and abstention are determined entirely by duty (whether voters share a common interest or not), so information no longer matters. Moreover, in the context of a private-value model, it seems that 26

27 ideological moderates, if anyone, should have the strongest sense of civic duty, since these voters prefer policies that are closest to what the social planner prefers. The same is not true in the model above, where the greatest public good is provided by the most expert citizens, and ideological extremists often belong to this group. 4.5 Welfare This section analyzes the implications of equilibrium voting for social welfare. Let denote the unique equilibrium policy outcome of the game when the expected number of citizens is. With common preferences, it is uncontroversial to measure social welfare simply by the expected utility [ ( )] of an individual citizen, which averages over the various realizations both of and of each citizen s private information, but also depends implicitly on the equilibrium strategy adopted by voters. Like the equilibrium characterizations above, the welfare results of this section apply whether uncertainty is continuous or binary. Condorcet s (1785) classic jury theorem which Krishna and Morgan (2011) call the first welfare theorem of political economy states that public opinion in large elections almost surely favors the better of two alternatives. Proposition 4 merely affirmsthatthis holds for two arbitrary policies within this spatial environment, and notes further that the unique equilibrium voting strategy is also the voting strategy that uniquely maximizes social welfare. Proposition 4 (Jury Theorem) If then, for any, there exists a unique voting strategy V that maximizes [ ( )]. This strategy is also the unique Bayesian Nash equilibrium of the game, and is therefore ideological. Moreover, arg max { } ( ). The existence of an optimal voting strategy is not surprising, but is also not trivial, because the set of voting strategies is not compact under the standard topology, so standard fixed point theorems cannot be applied directly. The proof of Proposition 4 first points out that there is an optimum within the compact set of strategies that are ideological, and then shows by construction that every non-ideological voting strategy is inferior to another that is ideological. That a socially optimal strategy is also individually optimal, and therefore 27

28 constitutes an equilibrium, follows from the common value assumption, as in McLennan (1998). Uniqueness then follows from Proposition 1. Proposition 4 is a normative result, but also sheds light on empirical facets of voter behavior. For one thing, it gives a rationale for the popularity of democratic institutions: even citizens whose private opinions differ from the majority opinion are broadly supportive of following majority rule. It also explains why political rhetoric often cites public opinion for support especially when consensus is high, which, as noted above, is evidence that a decision was obvious. The result that private opinions and public opinion are both correlated with also means that they are correlated with each other, which could explain the consensus effect documented by Ross, Green, and House (1977), whereby individuals on both sides of an issue expect to belong to the majority. In essence, a citizen who favors one policy over another predicts that other reasonable citizens, after weighing the evidence, will come to the same conclusion. In the 2012 U.S. presidential election, for example, 96% of ANES survey respondents who planned to vote Democrat also predicted a Democratic victory, while 83% of those who planned to vote Republican predicted a Republican victory. As Figure 5 illustrates, predictions are strongly monotonic in ideology. That is, the voters who are most confident that their side is right are also the most confident that their side will win. According to a least-squares regression of forecasts on ideologies, moving one category to the right on a 9-point ideological scale makes a citizen 11% (standard error 0 3%) morelikelyto predict a Republican presidential victory. Proposition 4 restricts attention to strategies in V, which do not allow voter abstention. As Theorem 3 states, however, many citizens will abstain if allowed to do so. Since every citizen receives an informative signal, it may seem that this merely throws away valuable information. If so, this might vindicate efforts to penalize non-voters with stigma or fines. To the contrary, however, Proposition 5 states that the optimal voting strategy in V 0 constitutes an equilibrium as well. Since strategies with no abstention are still feasible when abstention is allowed, the optimal strategy in V 0 is even better than the optimal strategy in V, implying that social welfare is higher with equilibrium abstention than without. This is consistent with survey evidence noted by Gerber, Huber, Biggers, and Hendry (2015), that 28

29 Figure 5 many voters view not knowing enough about the candidates as an acceptable excuse with their friends and family for not voting. Proposition 5 If then, for any, there exists a voting strategy V 0 that maximizes [ ( )]. This strategy is also a Bayesian Nash equilibrium of the game, and is therefore ideological. Moreover, arg max { } ( ). As in McMurray (2013), one way to understand how aggregating additional signals could reduce the quality of the electoral decision is to note that the decision of whether to vote or abstain conveys private information beyond the content of the vote itself; when voting is made mandatory, this additional information is lost. Ideally, each citizen s opinion would be utilized, but would be weighted according to its precision, as in Nitzan and Paroush (1982) and Shapley and Grofman (1984). However, majority rule instead weights each vote equally. When voting is mandatory, the sign of is recorded for each citizen, but its magnitude is not utilized. In equilibrium, abstention provides a crude mechanism by which citizens can transfer weight from their own votes to those who are more confident (whether because of greater expertise or less ambiguous signals). As noted above, equilibrium predictions based on pivotal voting inferences are somewhat controversial, given the strategic sophistication that seems required. Proposition 5 sheds light on this issue because, by comparison, a social planner s choice of participation threshold 29