A NEW THEORY OF STRATEGIC VOTING

Existing game-theoretic analysis of plurality rule elections predicts the complete coordination of strategic voting: A strict interpretation of Duverger’s Law. I reach a different conclusion. A group of voters must partially coordinate behind one of two challenging candidates in order to avoid the success of a disliked incumbent. Departing from existing models, the popular support for each challenger is uncertain. Individuals base their votes upon informative signals of candidate support levels. These represent either the social communication of political preferences throughout the electorate, or alternatively the imperfect observation of opinion poll information. The uniquely stable voting equilibrium entails only limited strategic voting and hence partial coordination. This is due to the surprising presence of negative feedback: An increase in strategic voting by others actually reduces the incentives for an individual to vote strategically. Hence stable multi-candidate support is perfectly consistent with instrumental rationality and fulfilled expectations. 1. DUVERGER’S LAW AND STRATEGIC VOTING Duverger (1954) introduced his Law by noting that the “simple-majority single-ballot system favors the two-party system. ” He envisaged an ongoing process involving both voters and political parties with bipartism as an eventual conclusion. Whereas this vision involved only a tendency toward bipartism under the plurality rule, the formal analyses of more recent contributors have generated a rather stricter version of Duverger’s Law. Cox (1994) and Myerson and Weber (1993) reinforce Palfrey’s (1989) claim that: “[W]ith instrumentally rational voters and fulfilled expectations, multicandidate contests under the plurality rule should result in only two candidates getting any votes. ” This paper is based on Myatt (1999). Elements have appeared previously under the title “Strategic Voting under the Qualified Majority Rule” (Myatt 2000). Stephen D. Fisher inspired this work with his extensive empirical research on tactical voting in Britain, and with many hours of conversation on the topic. I thank colleagues, seminar participants and reviewers for the many helpful comments that have (hopefully) improved the quality of this work since its inception. I remain responsible for any remaining errors. A NEW THEORY OF STRATEGIC VOTING 2 These authors considered plurality elections where each individual casts a single vote and the candidate with the largest number of votes wins. They found that the uniquely stable equilibrium outcome involves positive support for only two candidates. This is the result of strategic voting, where an individual may switch her vote away from her preferred candidate. Their Duvergerian prediction is that voters will fully coordinate their strategic behavior. Unfortunately, this strictly bipartite prediction is not borne out by the data. Both the United Kingdom and India provide examples of plurality voting systems with multi-candidate support at a constituency level. This might suggest a lack of instrumental rationality on the part of voters. Alternatively, it may point to weaknesses in the specifications of existing formal theories that drive their strictly Duvergerian conclusions. I will argue that partial coordination of strategic voting is perfectly consistent with stable equilibrium behavior on the part of instrumentally rational individuals. In other words, an appropriately specified formal model predicts only a tendency toward bipartism. This argument stems from the observation that existing formal theories assume the perfect common knowledge of the constituency-wide support for different candidates. My response is a model in which voters are uncertain of the preferences of others. They base their decisions on noisy signals of the distribution of preferences throughout the electorate. The analysis shows that strategic voting is a self-attenuating rather than selfreinforcing phenomenon: An increase in strategic voting by others reduces the incentives for an individual to vote strategically. This negative feedback pulls strategies away from full coordination and toward multi-candidate support as a stable voting equilibrium. 1Myerson and Weber (1993) succeed in characterizing equilibria for a wider variety of electoral systems. 2Such equilibria are Duvergerian. Cox (1994) highlighted the existence of non-Duvergerian equilibria. Such equilibria involve switching away from a more to less popular candidate, generating a tie for second place, and thus moderating the incentives for strategic voting. Unfortunately, non-Duvergerian equilibria are highly unstable (Fey 1997) and hence fall outside the class of stable equilibrium predictions. 3Analysis of strategic voting in Britain is provided by Johnston and Pattie (1991), Lanoue and Bowler (1992) and Niemi, Whitten, and Franklin (1992) inter alia. New research, based on British Election Study data and analyzing standard intuitive predictions, the bimodality hypothesis of Cox (1994) and the predictions of this paper is reported by Myatt and Fisher (2002). Riker (1976) offered an analysis of the Indian case, and hypothesized that reduced strategic voting was due to the presence of clear Condorcet winner, against which strategic voting is futile. This hypothesis is not supported by the formal analysis presented here. A NEW THEORY OF STRATEGIC VOTING 3 The argument begins with the following scenario: Imagine a parliamentary constituency or voting district in which a group of dissatisfied voters wish to dislodge a disliked incumbent office-holder. To do so, a proportion γ > 1 2 of the dissatisfied voters must vote in favor of one of two challenging candidates. This is a qualified majority voting game: The qualified majority γ must successfully coordinate if the incumbent is to be defeated. This scenario captures many elements of modern plurality elections. In the 1997 British General Election the incumbent (and unpopular) Conservative party polled between 1/3 and 1/2 of the votes cast for the three major parties in 270 out of 529 English constituencies. In more than half of England, therefore, anti-Conservative voters needed to successfully coordinate behind either the Labour or Liberal Democrat candidate to ensure a Tory defeat. What determines the behavior of an instrumental voter in this scenario? She may only influence the outcome of the election with a casting vote. This pivotal event occurs when the vote share of one of the challengers is just equal to the required qualified majority γ. An extra vote will then tip the balance away from the disliked incumbent. An instrumental voter then balances the relative probability of the two pivotal events (corresponding to a successful “challenge” by one of the preferred candidates) against her relative preference for the two candidates. This insight is clear from earlier decision-theoretic work by McKelvey and Ordeshook (1972) and Hoffman (1982), and was further explored in a game-theoretic context by Palfrey (1989), Myerson and Weber (1993) and Cox (1994). Such formal theories successfully highlight these important elements of an instrumental voter’s decision calculus. Unfortunately, the game-theoretic treatments share a common feature: Individual preferences and voting decisions are drawn independently from a commonly known distribution. It is this, and this alone, that leads to strictly Duvergerian predictions. To see why, consider the relative probability of pivotal events. Mathematically, as the electorate grows large, the probability of a pivotal event involving the leading challenger becomes infinitely larger than the probability of such an event involving the trailing challenger. Equivalently, and perhaps more intuitively, if a pivotal event occurs, then it almost always involves the leading challenger. Any instrumental voter will, therefore, switch her vote to the leader. Game-theoretic reasoning is unnecessary for a A NEW THEORY OF STRATEGIC VOTING 4 Duvergerian conclusion: As long as the electorate is large and voting decisions are drawn independently from a commonly known distribution, almost all instrumental voters will switch, even before they account for strategic switching by others. The fact that voting decisions are drawn independently from a commonly known distribution means that any uncertainty in the Cox (1994), Palfrey (1989) and Myerson and Weber (1993) models is only apparent, and not real. The assumed knowledge of the underlying vote generating process, ensures that such individual-level uncertainty has no affect in the aggregate: As the electorate grows large, the Law of Large Numbers begins to bite, allowing an observer to precisely predict the election’s outcome. My aim, therefore, is to remove this potentially undesirable feature and consider seriously the informational underpinnings of the Cox-Palfrey framework. In particular, I wish to allow voters to be uncertain of the levels of support enjoyed by the candidates. To achieve this objective, I separate voters’ preferences into common and idiosyncratic components. The common component (reflecting the preferences of the median voter) is shared by all individuals, whereas the idiosyncratic component is distributed independently throughout the electorate. Crucially — and in contrast to Cox-Palfrey — there is constituency uncertainty in that the common component is unknown to voters. As the electorate grows large, the idiosyncratic components average out, but constituency uncertainty over the common component remains. This means that only uncertainty over the common component matters when considering the realized aggregate preferences (and hence decisions) of a large electorate. Put simply, it is only uncertainty over the identity of the median voter that matters, and yet the Cox-Palfrey modelling paradigm permits only uncertainty over deviations from such a median. 4The accurate knowledge required by voters in the Cox-Palfrey framework i