Costly information acquisition. Is it better to toss a coin?

Highlights

• We study a strategic model of common value elections with endogenous information acquisition and heterogeneous voters.

• Elections can fail to aggregate information.

• We find necessary and sufficient conditions for information aggregation.

• Inefficiencies can be corrected by limiting participation to the most competent citizens.

Abstract

This paper presents a strategic model of common value elections with endogenous information acquisition. It proves that majoritarian elections can fail to aggregate information when voters have heterogeneous skills and provides necessary and sufficient conditions for information aggregation. Inefficiencies can be partially corrected by limiting participation to the most competent citizens, a result which provides a rational foundation for epistocratic government.

Introduction

A growing literature in Political Economy has analyzed the ability of large elections to aggregate information. According to the Condorcetʼs Jury Theorem, as long as each voter is more likely to be right than wrong, the probability that an election will result in the best possible outcome approaches one as the number of voters increases. Scholars have proved this result both under naive voting (see Berg, 1993, Berend and Paroush, 1998; Ladha, 1992, Ladha, 1993) and under strategic voting (see Austen-Smith and Banks, 1996, Feddersen and Pesendorfer, 1997, Myerson, 1998; and Wit, 1998). However, most of this work has assumed that information is exogenous and independent on the size of the electorate, while acquiring information, in the real word, costs time and effort.¹ Schumpeter (1950) and Downs (1957) pointed out that voters have little incentive to collect costly information since they are very unlikely to change the election outcome. Thus, voters free-ride, under-producing information. This is the so-called “rational ignorant hypothesis” which is consistent with empirical evidence (see, for instance, Delli Carpini and Keeter, 1996, Blendon et al., 1997, Nannestad and Paldam, 2000, Caplan, 2001) and could have important implications for the quality of democratic deliberations. Furthermore, scholars have found that the degree of political knowledge is uneven and positively correlated with education. In other words, the more educated a citizen, the more likely they are to have correct information (see Blendon et al., 1997, Caplan, 2001, Nannestad and Paldam, 2000; Lau and Redlawsk, 2006; Nordin, 2009). Then, the observed heterogeneity in political knowledge would reflect an underlying heterogeneity of votersʼ skills. This evidence also suggests that people possess limited political knowledge because processing the available information requires a significant effort (see Aidt, 2000).

The objective of this paper is to understand whether large elections aggregate information when the electorate has different information acquisition skills and can acquire better or worse information. We introduce a model of common value elections with two candidates. Citizens agree as to which of the two candidates is better for each state of the world, but when it comes time to vote they do not know the state of the world. They do not have access to a free and reliable source of information, and need to make an effort to process the information they are exposed to. Thus, obtaining more precise information incurs an increasingly higher cost. Voters have different abilities to process information; less skilled voters must invest more effort to extract the same amount of information. It follows that the quantity of information that each voter acquires will be determined endogenously at equilibrium.

We begin the analysis by characterizing optimal strategies and provide an intuitive characterization: voters who acquire information equate marginal gains to marginal costs. All equilibria are characterized by a cutoff type, which is the highest type acquiring information. We go on to prove the validity of the rational ignorance hypothesis. Information is a public good, and a large population amplifies the free riding problem by reducing the probability that any single vote will be decisive. Although the framework is simple, these preliminary findings account for three empirically relevant facts: (i) the average voter is poorly informed; (ii) only a small fraction of the electorate is informed; (iii) the less skilled the voters, the less information they acquire, resulting in an uneven distribution of information.

Equilibria with information acquisition in large electorates exist only if the marginal cost of acquiring no information for the most skilled voters are zero (Corollary 1). Even when this condition holds and under generic conditions, any bias in favor of one candidate aggravates the free riding problem: an equilibrium with information acquisition exists if and only if the expected gains from electing the better candidate in the two states of the world are equal. This implies that uninformed voters vote randomly, introducing noise in the process of information acquisition and aggregation. In equilibria with information acquisition, every elector is, on average, more likely to be right than wrong. However, the average quality of information acquired by voters decreases so fast that the probability that the better candidate will actually be elected approaches one-half in a large electorate. In short, large elections might perform no better than a fair coin toss (Theorem 1).

It is natural to ask whether less expensive access to information could alleviate this shortcoming. Reducing the marginal cost of acquiring information for the most skilled voters encourages voters to acquire information of better quality. If the marginal costs of acquiring an infinitesimal amount of information grow negligibly for the most skilled voters, equilibria with information acquisition exist when priors are not too biased in favor of one of the two candidates (Theorem 2).² In this cases, elections aggregate a positive amount of information. However, full information aggregation requires at least the non-convexity of the marginal costs functions (Theorem 2). Intuitively, the conditions for information aggregation requires that some voters receive some relevant information, for free. This requirement is mathematically non-generic, but it does not seem excessively strong. However, whether this condition is satisfied in the real world, is an empirical question which is beyond the objective of this paper.

We continue by considering the overall efficiency of elections, and prove that they are efficient if and only if they perfectly aggregate information. Indeed, it is always possible to find a strategy profile such that the probability of electing the better candidate approaches one and the aggregate costs approach zero when there are many electors. However, such strategies are not equilibria. In a strategic setup, voters do not internalize the informational externality.

Those scholars opposing egalitarian and unmediated democracy have often asserted that citizens do not have enough knowledge or skill to govern, an argument which is apparently supported by Theorem 1. These arguments were used by Plato (2000) to defend the concept of government by philosophers, and by John Stuart Mill (1890, Ch. 8) to propose a form of democracy where the educated class enjoys more voting power. Estlund (2008) calls this types of government “epistocracy of the educated thesis”.³ We test such arguments and consider a simple model of epistocracy where only wiser (more skilled) citizens are allowed to vote. We prove that if the skill threshold is properly selected, elections partially aggregate information. We then characterize the conditions that allow perfect information aggregation under an epistocratic government. Restricting voting rights improves information aggregation because it reduces the noise introduced by the random vote of uninformed citizens. The result suggests that a model with voluntary participation might be superior to a model with compulsory participation, because the former reduces the noise introduced by poorly informed citizens (see also Krishna and Morgan, 2012 and McMurray, 2013). However, in an epistocratic model this is reached by denying voting rights to a considerable amount of citizens (actually, to most citizens), while, under voluntary voting, less informed voters prefer to abstain rather than vote for either candidate. This the so-called “swing voterʼs curse” (see Feddersen and Pesendorfer, 1996).

One of the first paper to deal with endogenous information acquisition in elections is Martinelli (2006), who investigates whether the effects of the rational ignorance hypothesis are severe enough to undermine information aggregation. He considers a model with costly information acquisition where voters can acquire information of varying quality and have the same cost function. As a result, voters acquire information of decreasing quality but every elector acquires the same amount. When there are many electors the precision of the signal declines to zero, consistently with the rational ignorance hypothesis. If the marginal costs of acquiring an infinitesimal amount of information are zero, large elections partially aggregate information, despite rational ignorance (Theorem 2, p. 232). Furthermore, if the marginal costs of acquiring an infinitesimal amount of information grow slowly then elections perfectly aggregate information. However, the paper also shows that if the electorate is ideologically heterogeneous the conditions for information aggregation are more stringent (Theorem 6, p. 240). Martinelli (2007) presents an alternative model with heterogeneous costs but only one type of information, whose quality does not depend on the size of the electorate. As a result, a decreasing segment of the electorate acquires information but the quality is always the same. When there are many electors only the electors with the lowest costs acquire information. If zero cost types have positive density, large elections aggregate some information despite rational ignorance.

Differently than this paper, Martinelli, 2006, Martinelli, 2007 cannot simultaneously account for the low and uneven level of information in the electorate usually found by empirical literature. In Martinelli (2006) there is no heterogeneity so only the quality of information acquired by each voter converges to zero. In Martinelli (2007), the quality of information each voter can acquire is fixed so only the cutoff type converges to zero. In our paper, both effects appear simultaneously. The result is that the speed of convergence to zero of the information acquired by voters in this model “doubles” the speed of convergence to zero of the information acquired by voters in Martinelli, 2006, Martinelli, 2007. This finding explains why this simple change in the setup alters the conditions for information aggregation and make them more restrictive (see Section 4).

Oliveros (2011) introduces a models which is similar to the one presented by Martinelli (2006). In his model all agents have the same cost functions but they have two dimensions of heterogeneity in their preferences. It follows that electors endogenously acquire information of different qualities. Differently than in this paper and in Martinelli, 2006, Martinelli, 2007, in his model equilibria with information acquisition exist for every prior, because he assumes that there is a fraction of electors who vote irrationally for one candidate or for the other one. This assumptions guarantees that pivotal probabilities are bounded away from zero which allows to prove the existence of equilibria with information acquisition through Browerʼs fixed point theorem. However, information aggregation is possible under conditions that are similar to the ones found by Martinelli (2006).

Other papers have considered information acquisition in elections in models where the quality of the signal is constant and independent of the size of the electorate, and agents bear a fixed cost to acquire information. Thus, in these models it is not possible to have an arbitrarily large number of citizens acquiring information. In particular, Persico (2004) studies the relationship between optimal decision rules and the accuracy of the signal. He proves that qualified majority rules are optimal only if the signal is sufficiently accurate. Koriyama and Szentes (2009) show that undersized committees may generate large welfare losses, while oversized committees generate only small inefficiencies.⁴ Information acquisition in elections has also been considered by Feddersen and Sandroni (2006a). They use their own ethical voter model (Feddersen and Sandroni, 2006b) to predict that a significant fraction of the electorate will acquire independent information, and that the fraction of informed voters may decrease with the quality of information. This behavior contradicts the results of Martinelli (2006).

The structure of this article is as follows. Section 2 introduces the model, and Section 3 evaluates the validity of the rational ignorance hypothesis. Section 4 studies the existence of equilibria with information acquisition and their information aggregation properties. Section 5 considers the asymptotic efficiency of the equilibria. Section 6 investigates a simple epistocracy model, and the main conclusions are presented in Section 7. Proofs are provided in Appendix A.