Susceptibility to strategic voting: a comparison of plurality and instant-runoff elections1 Andrew C. Eggersy Tobias Nowackiy Abstract Reformers and researchers often speculate that some voting systems are more sus- ceptible to strategic voting than others, but existing measures are unsatisfying: they essentially ask how likely a voter is to regret a sincere vote, not how likely a voter is to anticipate that an insincere vote is beneficial. We propose a new approach to assessing susceptibility to strategic voting (and other voting system properties) under more realistic assumptions about uncertainty and voter behavior, and we use it to compare three-candidate plurality and instant-runoff (IRV) elections. Drawing on preference data from 160 surveys, we estimate that being a strategic voter (vs. always voting sincerely) is between 5 and 35 times more beneficial in plurality than in IRV. IRV is less susceptible to strategic voting both because insincere votes are more risky in IRV and because strategic behavior by other voters tends to reduce the benefit of being strategic. 1This version: January 17, 2020. Thanks to Steve Callander, Gary Cox, Justin Grimmer, Matias Iaryczower, Paul Klemperer, Dimitri Landa, Gregory Martin, David Myatt, Karine van der Straeten, and seminar audiences at Trinity College Dublin, the Toulouse School of Economics, the Democracy and Policy Lab at Stanford University, and the ECPR Standing Group on Strategic Interactions for valuable feedback. Thanks to Mats Ahrenshop, Tak Huen Chau, and Tom Robinson for research assistance. Earlier versions were circulated starting June 2019 under the title \Comparing strategic voting incentives in plurality and instant-runoff elections". yNuffield College and Department of Politics and International Relations, University of Oxford, UK. email: [email protected] yStanford University. email: [email protected] 1 1 Introduction What voting method should we use to choose a leader, a representative, or a policy from a set of multiple alternatives? This question is relevant not just to constitutional designers and voting theorists but also to voters who have in recent years been asked to evaluate electoral reforms in large-scale referendums in the U.K., British Columbia, and U.S. states and cities.2 While there are certainly other factors to consider, many voting theorists argue that we should prefer voting methods that are less susceptible to strategic voting (e.g. Dummett, 1984; Chamberlin and Featherston, 1986; Brams and Fishburn, 2002; Tideman, 2018), because (among other reasons) strategic voting can make election results difficult to interpret (Satterthwaite, 1973; Green-Armytage, Tideman and Cosman, 2016) and requires some voters to forego the evident expressive benefits of a sincere vote (Hamlin and Jennings, 2011; Pons and Tricaud, 2018). Accordingly, a large body of research has attempted to measure the susceptibility of various voting methods to strategic voting. Following Gibbard(1973) and Satterthwaite(1975)'s proofs that any reasonably fair voting system might produce a manipulable election result (i.e. a result where at least one voter could get a better outcome by casting an insincere vote), researchers have sought to estimate how often each voting system would produce manipulable election results (e.g. Chamberlin and Featherston, 1986; Saari, 1990; Favardin and Lepelley, 2006; Ornstein and Norman, 2014; Green-Armytage, Tideman and Cosman, 2016; Tideman, 2018). Studies in this manipulability literature suggest, for example, that plurality elections (as used in the U.K. or U.S.) are far more susceptible to strategic voting than elections held under instant-runoff rules (also known as the Hare system, the alternative vote, or ranked-choice voting, as used in Australia). One key problem with previous efforts to measure susceptibility to strategic voting is that they ignore the uncertainty voters face when they make their voting decision. In essence, the manipulability literature asks how often a voting result could be manipulated by a voter (or group of voters) who knows exactly how other voters have voted. Ignoring uncertainty simplifies the analysis, of course, and it is appropriate for estimating how often voters would 2The UK held a national referendum on electoral reform in 2011, British Columbia held three electoral reform referendums between 2005 and 2018, the U.S. state of Maine adopted instant-runoff voting for federal elections by referendum in 2016, and New York City did so for municipal primaries and special elections in 2019. 2 regret a sincere vote in a given voting system, which may be an important consideration. But there is good reason to care about the extent to which voters might anticipate the need to cast an insincere vote in a given voting system, not only whether they might regret a sincere vote. If (like Riker, 1982) we are concerned that strategic voting might make election results difficult to interpret, for example, we want to know whether voters (or the elites who mobilize them) would be likely to perceive ex ante an opportunity to benefit from an insincere vote; answering that question requires us to consider the strategic voting problem that voters face before the election, when results are uncertain. (As we discuss further below, there are circumstances when a voting system could appear highly manipulable ex post but not ex ante, so the two approaches may yield different answers.) For this and other reasons the manipulability literature offers an incomplete measure of voting systems' susceptibility to strategic voting. What is missing is a measure of how much the typical voter would benefit ex ante from an insincere vote given reasonable assumptions about voter uncertainty and voter behavior. In this paper we introduce a new way of assessing susceptibility to strategic voting that improves on previous work in three principal ways. First, to characterize typical voter prefer- ences we draw on 160 election surveys across 56 democracies; like Green-Armytage(2014) and Green-Armytage, Tideman and Cosman(2016) we use voters' numerical evaluations of parties as utility measures, but we draw on a larger set of preference distributions in order to produce more broadly representative results. Second, we consider the uncertainty voters face when they decide to vote in advance of elections. Instead of asking whether our survey respondents could benefit from an insincere vote given perfect knowledge of others' votes, we compute for each respondent the expected utility of each possible vote given a reasonable model of beliefs over election outcomes; crucially, these beliefs are realistically uncertain, with the precision cali- brated from empirical work on voters and forecasters in recent UK elections (Fisher and Myatt, 2017; Eggers and Vivyan, 2020). The susceptibility of a voting system to strategic voting can then be measured by the average gain in expected utility each voter would receive by switch- ing from always voting sincerely to always casting the optimal strategic (i.e. highest expected utility) vote. Third, we consider how voters' incentive to vote strategically depends on other voters' strategic behavior. We begin by assuming (as does the manipulability literature) that 3 voters expect other voters to vote sincerely; we then use a novel iterative polling algorithm to compute strategic voting incentives when voters expect other voters to vote strategically to varying degrees. This allows us to check how strategic voting incentives are shaped by (beliefs about) strategic voting by other voters, which we argue may be an important determinant of a voting system's susceptibility to strategic voting. We use this approach to compare strategic voting incentives in elections held under plurality (the most common single-winner voting system in the world) and instant-runoff voting (IRV), a comparison that is particularly relevant because of recent referendums in both the U.K. and U.S. in which voters were asked to choose between the two systems (see footnote2). In contrast to plurality elections, in which each voter casts a ballot for a single candidate and the winner is the candidate with the most votes, voters in IRV elections rank the candidates and the winner is determined by successively eliminating less-popular candidates.3 IRV variants are used to elect legislators at the state and federal level in Australia, presidents of Sri Lanka, India, and Ireland, mayors of Sydney, London, San Francisco and several other American cities, and (as of 2018) all federal offices in the U.S. state of Maine. IRV's advocates sometimes argue that it is immune to strategic voting,4 but critics are quick to point out circumstances in which a savvy voter could benefit from an insincere vote in an IRV election (e.g. Fishburn and Brams, 1983; Dummett, 1984). The manipulability literature finds IRV to be among the least ex post manipulable voting methods and plurality to be among the most ex post manipulable (e.g. Chamberlin and Featherston, 1986; Green-Armytage, Tideman and Cosman, 2016), but no previous study systematically compares the two voting methods in a way that takes into account uncertainty or that considers how other voters' strategic behavior shapes strategic voting incentives. 3Most commonly, in each round the candidate receiving the fewest top rankings is eliminated from all ballots until one candidate remains (or, equivalently, until one candidate has a majority of top rankings). In the system used to elect the mayor of London and and the president of Sri Lanka (sometimes called the \contingent vote" or \supplementary vote" system), all candidates but two are eliminated immediately. The other variation within the family of instant-runoff systems is how many candidates voters may rank (up to two in the London mayoral election, up to three in San Francisco, as many as desired in some Australian states, and all in other Australian states). In the three-candidate case (our focus), the elimination procedures are identical and the number of candidates voters rank makes little or no difference.