Strategic voting when participation is costly

doi:10.1016/j.geb.2019.04.010

Games and Economic Behavior

Volume 116, July 2019, Pages 122-127

https://doi.org/10.1016/j.geb.2019.04.010 Get rights and content

Abstract

We study a general multiparty model of plurality rule elections with costly participation, and prove that strategic voting – that is, situations in which some voters abandon their most preferred alternative and vote strategically for the serious contender they dislike less – may emerge in equilibrium; just like when participation is costless/compulsory (Palfrey, 1989). This qualifies opposite claims made in more confined setups (e.g. Arzumanyan and Polborn, 2017), and establishes that Duverger's psychological effect is present in a much larger set of cases than currently believed.

Introduction

Duverger's (1951) law postulates that in plurality rule elections it is highly unlikely that all voters will vote sincerely for their top-ranked alternative. Indeed, when one is endowed with a unique vote, one should want to make the most of it. Since one's vote is relevant in the determination of the outcome only if this vote breaks or generates a tie for the first place, there is often a question between voting for the alternative one likes most and voting for the serious contender one dislikes less. This tension will lead supporters of less popular alternatives to behave “strategically.” That is, to abandon their top choices and back, among the serious contenders (i.e. among the alternatives that have substantial chances of winning the election), the one that gives them the highest utility. Duverger concluded that this line of reasoning – usually referred to as Duverger's “psychological” effect or factor (Cox, 1997) – will lead to strategic voting equilibria in which a small set of serious contenders – typically, two – is supported sincerely by individuals that like them more than any other candidate, and strategically by individuals whose top choice does not enjoy serious election prospects.

Formal analysis of strategic voting under plurality rule has confirmed the existence of strategic voting equilibria only when voting is costless or compulsory (Riker, 1982; Palfrey, 1989; Myerson and Weber, 1993; Cox, 1997; Fey, 1997). That is, only when full participation is guaranteed. In many relevant real life contexts though (e.g. U.S. presidential elections) participation is voluntary and costly for the voters. Hence, it is of utmost importance to investigate whether the described psychological effect still exists in these settings. So far, the literature has mainly focused on studying costly voting in the framework of two-party elections (see, for example, Palfrey and Rosenthal, 1983; 1985; Ledyard, 1984; Levine and Palfrey, 2007; Krasa and Polborn, 2009; Goeree and Grosser, 2007; Herrera et al., 2014; Krishna and Morgan, 2015; Tyson, 2016) and, surprisingly, very little is known regarding the effect of introducing voting costs in multiparty settings on the shaping of strategic voting incentives: Do strategic voting equilibria exist even when participation is voluntary and costly? That is, when turnout is partial, are there voters who abandon their top-ranked alternatives and who vote for the serious contender they dislike less?

To our knowledge, the only paper that tries to give a first answer to these questions is Arzumanyan and Polborn (2017). This paper studies multiparty elections under plurality rule with costly participation, and finds that strategic voting cannot take place in equilibrium. In particular, it demonstrates that in equilibrium all voted parties tie (in expectation) and all voters vote sincerely (i.e. all individuals that decide not to abstain, vote for their top-ranked alternative) suggesting that Duverger's psychological effect is not present when voting is costly. To arrive to this conclusion, Arzumanyan and Polborn (2017) examine a setup with three alternatives and general ordinal preferences (i.e. individuals were allowed to have any strict preference ordering over the three alternatives) but a very special structure of cardinal preferences and voting costs (i.e. all individuals enjoy 1, $λ \in (0, 1)$ and 0 units of utility by the election of their top-ranked, their second-best and their bottom-ranked alternative respectively; and voting costs are homogeneous). It is true that when voting is costless or compulsory, cardinal utilities are not really central in the shaping of equilibrium behavior, since the probabilities of ties and their ratios are by far the most relevant determinant factor of voters' behavior (e.g. Palfrey, 1989). When voting is voluntary and costly, though, cardinal utilities and voting costs are as much relevant as the probabilities of ties in determining who turns out to vote and, subsequently, whether strategic voting takes place or not. Hence, it is imperative that we try to study multiparty elections with costly voting, in a more general framework: i.e. allowing for a larger variety in voters' cardinal characteristics (namely, utility levels and participation costs), in order to get new and robust insights regarding the persistence of strategic voting.

In this paper we undertake this task and employ a rather general model regarding cardinal preferences and voting costs. We consider any arbitrary finite class of voters' cardinal preference types and variable voting costs – in the tradition of Palfrey and Rosenthal (1985) – and we prove that strategic voting equilibria indeed exist in multiparty elections, even when voting is voluntary and costly. That is, we show that Duverger's psychological effect survives in additional settings of applied interest by establishing that, when voting costs are heterogeneous and the space of cardinal preferences is rich, we always have equilibria in large elections such that the top-ranked alternative of a significant share of the non-abstaining voters does not coincide with any of the alternatives that are expected to receive a positive vote-share. To this end, we prove existence of Duvergerian equilibria, that is, two-candidate equilibria in which a voter either abstains or votes – sincerely or strategically – for one of the two candidates that are expected to be voted by the rest of the voters.

But why does strategic voting arise in this general framework and not when voting costs and utility levels are homogeneous? When, for example, there are three alternatives – say A, B and C – but only two of them are expected to receive a positive vote-share – say A and B – and voting costs are homogeneous, then, in large elections, only voters who have the highest stakes may turn out to vote (that is, voters whose utility difference between the two alternatives is at least as high as that of any other voter). Moreover, if one assumes that all voters with ordinal preferences $A ≻ C ≻ B$ and $B ≻ C ≻ A$ care necessarily more about the election's outcome, than every voter with ordinal preferences $C ≻ A ≻ B$ and $C ≻ B ≻ A$ (as do Arzumanyan and Polborn, 2017), then one, essentially, rules out that voters whose top-ranked alternative is C will turn out to vote in large elections. On the contrary, when one allows for richer spaces of cardinal preferences and/or heterogeneous participation costs, then this arguably knife-edge reasoning breaks down and, in large elections, a substantial share of the voting population may vote strategically for the serious contender they dislike less.

In what follows we first present the model (section 2) and then we proceed with the formal analysis (section 3).

Section snippets

The model

Let us assume that a society, $K = {1, 2, . . ., k}$ , that is composed of $k \in N^{0}$ individuals, has to make a policy choice from the set $M = {1, 2, . . ., m}$ with $m > 2$ . The preferences of each individual are given by a vector of real numbers $v_{i} = (v_{i}^{1}, v_{i}^{2}, . . ., v_{i}^{m}) \in V \subset {[0, 1]}^{m}$ which is interpreted in the following way: The utility that individual $i \in K$ derives from the implementation of policy $h \in M$ is $v_{i}^{h}$ . The type-space, V, is a finite subset of ${[0, 1]}^{m}$ with following properties: a) ordinal preferences are strict (i.e. for

Duvergerian equilibria

In a Duvergerian equilibrium there are exactly two policies that are expected to receive positive vote-shares, and a substantial share of voters engage in strategic voting. Our main contribution is to argue that such equilibria exist in multiparty elections, even when voting is costly. The structure of our argument is as follows: a) first we show that, in a restriction of our game to two policy alternatives and abstention, an equilibrium with partial participation, always exists (Subsection 3.1

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There are more references available in the full text version of this article.

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Citation Excerpt :
The literature on costly voting that initiated with Palfrey and Rosenthal (1983), Ledyard (1984), and Palfrey and Rosenthal (1985) has substantially grown since, especially from a theoretical viewpoint (see e.g. Börgers, 2004; Campbell, 1999; Flamand and Troumpounis, 2014; Gersbach et al., 2021; Goeree and Grosser, 2007; Herrera et al., 2014; Hughes, 2016; Krishna and Morgan, 2012; Myatt, 2015; Taylor and Yildirim, 2010a, 2010b), but also from an empirical viewpoint (see e.g. Agranov et al., 2018; Faravelli et al., 2019; Levine and Palfrey, 2007). Many papers have examined different aspects of one-round voting, such as which voting threshold should be used (Grüner and Tröger, 2019), whether voting should be voluntary or compulsory (Krasa and Polborn, 2009), what happens with more than two alternatives (Arzumanyan and Polborn, 2017; Xefteris, 2019), or they have explored the common-value case (Ghosal and Lockwood, 2009) and the case of risk-averse preferences (Grillo, 2017). From a technical perspective, many of these papers consider Poisson games (Meroni and Pimienta, 2017) due to their focus on large electorates.
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^☆: I would like to thank the editor, Marco Battaglini, an advisory editor and three anonymous referees for their helpful comments and suggestions.

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NoteStrategic voting when participation is costly☆

Abstract

Introduction

Section snippets

The model

Duvergerian equilibria

Games Econ. Behav.

Games Econ. Behav.

J. Econ. Theory

One person, many votes: divided majority and information aggregation

Econometrica

Making Votes Count: Strategic Coordination in the World's Electoral Systems

Political Parties

Stability and coordination in Duverger's law: a formal model of preelection polls and strategic voting

Am. Polit. Sci. Rev.

Welfare reducing polls

Econ. Theory

Turnout and power sharing

Econ. J.

Note
Strategic voting when participation is costly☆