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«Divided Majority and Information Aggregation: Theory and Experiment∗ Laurent Bouton Micael Castanheira Aniol Llorente-Saguer Boston University ...»

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Divided Majority and Information

Aggregation: Theory and Experiment∗

Laurent Bouton Micael Castanheira Aniol Llorente-Saguer

Boston University Université Libre de Bruxelles Max Planck Institute for

ECARES Research on Collective Goods

January 16, 2013


This paper studies theoretically and experimentally the properties of plurality and

approval voting when a majority gets divided by information imperfections. The majority faces two challenges: aggregating information to select the best majority candidate and coordinating to defeat the minority candidate. Under plurality, the majority cannot achieve both goals at once. Under approval voting, it can: welfare is strictly higher because some voters approve of both majority alternatives. In the laboratory, we find (i) strong evidence of strategic voting, and (ii) superiority of approval voting over plurality.

Finally, subject behavior suggests the need to study equilibria in asymmetric strategies.

JEL Classification: C72, C92, D70 Keywords: Multicandidate Elections, Plurality, Approval Voting, Experiments ∗ We thank participants to the ESA 2012 meetings in Tucson, the Political Economy Workshop at the Erasmus University Rotterdam, the Workshop on Social Protests and Political Influence and seminars at Boston University, Columbia, CREED, IMT Lucca, London School of Economics, Massachussets Institute of Technology, New York University, Oxford, Pittsburgh, Queen Mary, Royal Holloway, Tilburg and Warwick.

We particularly thank Alessandra Casella, Eric Van Damme, Christoph Engel, Olga Gorelkina, Ðura-Georg Grani´, Kristoffel Grechenig, Alessandro Lizzeri, Roger Myerson, Tom Palfrey and Jean-Benoit Pilet. We c would also like to thank Erika Gross and Nicolas Meier for excellent assistance at running the experiments.

We gratefully acknowledge financial support from the Max Planck Society. Micael Castanheira is a senior research fellow of the Fonds National de la Recherche Scientifique and is grateful for their support.

1 Introduction Elections are typically expected to achieve better-informed decisions than what an individual could achieve alone.1 The rationale is that if each voter can convey her privately-held information through her ballot, voting results will reveal the aggregate information dispersed in the electorate. However, this is a big “if”: in plurality elections, for instance, rational voters are typically expected to coordinate their ballots on only two alternatives, independently of the number of competing alternatives (Duverger’s Law). Therefore, unless the number of candidates is exactly two, information aggregation is dubious.

This limitation resonates with centuries of scholarly research on how to design an electoral system that can aggregate heterogeneous preferences and information in an efficacious way (see e.g. Condorcet 1785, Borda 1781, Myerson and Weber 1993, Myerson 1999, Piketty 2000, Bouton 2012). Frustration with plurality is also apparent in civil society: a large number of activists lobby in favor of reforming the electoral system2 and many official proposals have been introduced.3 One of the most popular alternatives to plurality is approval voting (AV).4 Yet a major hurdle stands in the way of reform: the substantial lack of knowledge surrounding the capacity of AV (or other systems) to outperform plurality. We need a better understanding of the properties of new electoral systems to identify and implement meaningful reforms.

With this purpose in mind, we study the properties of plurality and AV when voters are strategic but imperfectly informed. We focus on the case in which a majority both

needs to aggregate information and to coordinate ballots to defeat a minority alternative:

the Condorcet loser. Our analysis features two main novelties: first, we study these systems both theoretically and experimentally. Second, instead of focusing on the limiting properties of these systems when the electorate is arbitrarily large, we study them for any electorate size. This means that our conclusions are equally valid for committees and general elections.

A first theoretical finding is that, in plurality, the need to aggregate information proSee a.o. Condorcet 1785, Austen-Smith and Banks 1996, Feddersen and Pesendorfer 1996, 1997, Myerson 1998, Krishna and Morgan 2012, and the references therein. For limitations, see e.g. Bhattacharya 2012, Mandler 2012, and Morgan and Várdy 2012.

See e.g. the Electoral Reform Society (www.electoral-reform.org.uk) and the Fair Vote Reforms initiative (www.fairvote.org).

Two examples are North Dakota in 1987, where a bill to enact approval voting in some statewide elections passed the Senate but not the House and, more recently, the U.K., which held a national referendum in 2011 on whether to replace plurality voting with alternative voting.

Under approval voting, voters can “approve of” as many candidates as they want. Each approval counts as one vote and the candidate that obtains the largest number of votes wins (Weber 1977, 1995, Brams and Fishburn 1978, 1983, Laslier 2010, Nuñez 2010, Bouton and Castanheira 2012).

duces an equilibrium in which voters vote informatively (that is, their ballot conveys their private information), despite the need to coordinate against the minority. This equilibrium is not “knife edge”, and may rationalize the oft-observed pattern that strictly more than two candidates receive positive but different vote shares, despite the predictions of Duverger’s Law. When the minority is small, this equilibrium supports information aggregation, in the sense that the alternative with the largest expected vote share is the full information Condorcet winner. In contrast, when the minority is large, the alternative with the largest vote share is the Condorcet loser, in which case this equilibrium is highly inefficient. This equilibrium exists even when majority voters would benefit from collectively deviating towards a Duverger’s Law equilibrium.

In the same setup, we show that AV can always produce strictly higher welfare than plurality. Having the opportunity to approve of multiple alternatives allows the electorate to achieve both better coordination and information aggregation. While we cannot establish a general proof fully characterizing the equilibrium in approval voting,5 we are able to formulate two substantiated conjectures: (i) the symmetric equilibrium is unique, and (ii) the equilibrium strategy is such that voters approve of the candidate they deem best and sometimes also approve of the other majority candidate. These conjectures find support in one formal result and many numerical simulations.

Our theoretical analysis poses an interesting trade-off between these two electoral systems. On the one hand, one could claim that AV is more complex than plurality because it extends the set of actions that each voter can take.6 Hence, there is a risk that actual voters make more mistakes under AV, which could wash out its favorable theoretical properties.

On the other hand, our theoretical findings are that AV reduces the number of equilibria and therefore simplifies strategic interactions amongst voters. In other words, AV should facilitate the voters’ two-pronged goal of aggregating information and coordinating ballots to avoid a victory of the Condorcet loser.

We ran controlled laboratory experiments to assess the validity of these theoretical In contrast, Bouton and Castanheira (2012) fully characterize the equilibrium for arbitrarily large electorate sizes. In the presence of “doubt”, the equilibrium proves to be unique and implies full information and coordination equivalence. That is, the full information Condorcet winner always has the largest expected vote share. In contrast, Goertz and Maniquet (2011) provide an example in which aggregate information does not obtain if sufficiently many voters assign a probability zero to some states of nature.

With three alternatives, plurality offers four possible actions: abstain, and vote for either one of the three alternatives. AV adds another four possible actions: three double approvals, and approving of all alternatives. Saari and Newenhizen (1988) argue that this may produce indeterminate outcomes, and Niemi (1984) argues that AV “begs voters to behave strategically”, in a highly elaborate manner. In contrast, Brams and Fishburn (1983, p28) show that the number of undominated strategies can be smaller under AV than under plurality.

findings. They reveal interesting patterns and support most predictions. We first study setups in which information is symmetric across states of nature. Under plurality, we observe the emergence of both types of equilibria: when the minority is sufficiently small, all groups stick to playing the informative equilibrium. By contrast, when the minority is “large”, in the sense that the informative equilibrium leads to the Condorcet loser winning with a high probability, all groups gave up aggregating information and coordinated their ballots on a same alternative, as predicted by Duverger’s Law. Under AV, some subjects double vote to increase the vote shares of both majority candidates. As predicted, the amount of double voting increases with the size of the minority. However, the absolute level of double voting is lower than predicted.

Comparing the two systems, we observe that subjects make fewer strategic mistakes under AV than under plurality. Moreover, when the minority is large, subjects need more time to reach equilibrium play in plurality than in AV. This suggests that voters handle more easily the larger set of voting possibilities offered by AV than the need to select an equilibrium under plurality.

Next, and in contrast with theory (which focuses on symmetric equilibria), individual behavior in AV displays substantial heterogeneity among subjects: many subjects always double vote, whereas many other subjects always single vote their signal. The observation that double-voting increases with the size of the minority is mainly driven by a switch in the relative number of subjects in each cluster. This pattern points to the need to extend the theory and consider equilibria in asymmetric strategies.7 Extending the model in this direction, we find that this type of behavior is indeed an equilibrium which performs particularly well in explaining the level of double-voting observed in the laboratory.

We then turn to those treatments in which the quality of information varies across states, and find that subjects adjust their behavior in line with theoretical predictions. In the case of plurality, the data provides further evidence that three-candidate equilibria are a natural focal point when majority voters have common values. In the case of AV, the results are even stronger, in the sense that voters converge faster to the theoretical prediction.

Last, we analyze the welfare properties of both electoral systems. A valuable feature of a common value setup is that it allows us to make clear welfare predictions: in equilibrium, the majority voters’ payoff should be strictly higher with AV than with plurality. This is exactly what we observe in all different treatments. Actually, information aggregation In two-candidate elections, Ladha et al. (1996) have identified situations in which there exists an asymmetric equilibrium in which voters who receive the same signal behave differently.

becomes so efficient with AV that realized payoffs become very close to what a social planner who observes all signals could achieve.

Beyond testing the very predictions of the model, these experiments also shed new light on voter rationality: determining whether voters behave strategically and respond to incentives is a central issue in the quest for better political institutions.8 The advantage of our setup is two pronged. First, the need to aggregate information produces different —often opposite— voting incentives from the need to coordinate ballots. Therefore, we can test whether and in which proportion subjects react to a change in incentives when we modify the relative value of coordination versus information aggregation. Second, studying multicandidate rather than two-candidate elections widens the set of electoral systems (and thus of voter incentives) that can be analyzed. In our case, the predicted behavior of voters is substantially different between plurality and AV. To the best of our knowledge, our paper is the first laboratory experiment which explores multi-alternative elections with common value voters.9 As should be clear from the above description of the results, it offers overwhelming support to voters behaving strategically in this context.

2 A common value model

We consider a voting game with an electorate of fixed and finite size who must elect one

policy  out of three possible alternatives,   and . The electorate is split in two groups:

 active voters who constitute a majority, and  voters who constitute a minority. There are two states of nature:  = { }, which materialize with probabilities  ()  0. While these probabilities are common knowledge, the actual state of nature is not observable before the election.

Active voters’ utility depends both on the policy outcome and on the state of nature:

utility is high ( =  ) if  is elected and the state is , or if  is elected and the state is . It is intermediate ( =  ∈ (0  )) if  wins and the state is  or if  wins and the state For evidence of strategic behavior in experimental settings with information aggregation, see Guarnaschelli, McKelvey and Palfrey (2000), Battaglini et al. (2008, 2012), Goeree and Yariv (2010), and Bhattacharya, Duffy and Kim (2012).

Surprisingly, the experimental literature on multicandidate elections with private value voters is also quite slim. The seminal papers of Forsythe et al (1993, 1996) are closest to our paper. See also Rietz (2008) or Palfrey (2012) for detailed reviews of that literature. Van der Straeten et al. (2010) also study AV experimentally although in substantially different settings.

is . Finally, utility is low (normalized to zero) if  is elected:

–  –  –

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