Idiosyncrasy, Information and the Impact of Strategic Voting

Strategic voting is a familiar phenomenon in plurality rule elections. Existing formal theories (Palfrey 1989, Cox 1994) predict the complete coordination of strategic voting, and hence strictly Duvergerian bipartism as a stable equilibrium outcome. I have argued elsewhere (Myatt 2002) that these theories are flawed, since they assume that voters have perfect common knowledge of the constituency situation. Modelling the information sources on which voters base their decisions, I have shown that the uniquely stable equilibrium outcome involves only partial coordination of strategic voting and hence multi-candidate support. In this paper, I expand my model in three ways. First, I show that (contrary to intuition) the incentive to vote strategically is lower in relatively marginal constituencies, after controlling for the distance from contention of a trailing preferred candidate. Second, I develop appropriate calibrations for the idiosyncrasy of voter preferences and the accuracy of their information sources. Third, I show that a calibration of the model is consistent with the impact of strategic voting in English parliamentary constituencies and the reported accuracy of voters’ understanding of the constituency situation. The model suggests that, in the British General Election of 1997, nearly 50 seats may have been lost by the Conservative party due to strategic vote switching. 1. PLURALITY RULE ELECTIONS AND STRATEGIC VOTING Plurality rule elections, where the winner is the candidate who receives the largest number of votes, are vulnerable to the phenomenon of strategic voting: A voter might well switch her vote away from her preferred candidate and toward a perceived leader, in the hope of exerting a greater influence over the outcome of the election. This phenomenon has long been the focus of political scientific study. Indeed, Farquharson (1969, pp.57–60) noted interest dating back to Pliny the Younger while Riker (1982b) cited the eloquent exposition of Droop (1871) in describing the strategic voting problem: “As success depends upon obtaining a majority of the aggregate votes of all the electors, an election is usually reduced to a contest between the twomost popular candidates . . . even if other candidates go to the poll, the electors THE IMPACT OF STRATEGIC VOTING 2 usually find out that their votes will be thrown away, unless given in favor of one or other of the parties between whom the election really lies.” Droop’s observation incorporates two interesting features. First, it expresses the idea that voters are instrumentally rational in their decision making. Specifically, they may choose to vote in a way that best influences who wins the election. Second, it is an early version of Duverger’s (1954) Law, which suggests that plurality rule elections will tend to lead to a two-party system. Of course, Duverger’s thinking moved beyond the phenomenon of strategic voting, as he envisaged a process in which political parties would react to likely outcomes of plurality elections. Nevertheless his “psychological effect” of strategic voting was a key component of this wider process, and one which he expected to generate only a tendency toward bipartism. Drawing upon these classic contributions, therefore, plurality elections might be expected to exhibit some strategic voting, and the relative dominance of two candidates within a voting district, but this local bipartism will not necessarily be complete. It remains, therefore, to ask how large the impact of strategic voting should be. Unfortunately, modern formal theories of strategic voting fail to answer this question in a convincing way. Palfrey (1989) and Myerson and Weber (1993) described formal models of strategic voting. The equilibria of these models are strictly Duvergerian, in the sense that strategic voting is complete: All voters fully coordinate on only two candidates. Cox (1994) highlighted the existence of a second category of non-Duvergerian equilibria. They involve vote switching away from a leading candidate and toward a trailing candidate resulting (in the context of a plurality election) in a tie for second place. Such a tie attenuates incentives to vote strategically, and hence is (technically) an equilibrium outcome. Perhaps unsurprisingly, these predictions are not supported by the data. In Section 7 I argue that this is the case in English parliamentary constituencies. Here, however, I consider the famous example of the 1970 New York senatorial election. This was highlighted by Riker (1982a) and others as a failure of coordination of strategic voting. In a “three 1A succint modern definition (Fisher 2002) is that: “A tactical voter is someone who votes for a party they believe is more likely to win than their preferred party, to best influence who wins in the constituency.” 2For instance, this traditional example is used effectively in the undergraduate text of Morton (2001). THE IMPACT OF STRATEGIC VOTING 3 Candidate Votes Share James Buckley 2,288,190 39% Charles Goodell 1,434,472 24% Richard Ottinger 2,171,232 37% Total 5,893,894 100% TABLE 1. The 1970 New York Senatorial Election horse race” two liberal candidates, Richard L. Ottinger and Charles E. Goodell, competed against the conservative James R. Buckley. More specifically, Goodell was an incumbent Republican who had taken a liberal stance on the Vietnam War, and hence received the nomination of the Liberal Party. The New York Conservative Party, however, rather than nominating Goodell as a “fusion” candidate instead supported the conservative Buckley. I present the outcome of this election in Table 1. A widely held belief was that the liberal vote was split between Goodell and Ottinger, allowing the win for Buckley. The outcome was certainly not Duvergerian. Neither was it non-Duvergerian in the sense of Cox (1994), since such equilibria require the challengers to tie for second place. The failure of formal theories to accommodate such electoral outcomes is not necessarily a function of the instrumental rationality postulated by such theories. Rather, it is due to their informational underpinnings. The Cox-Palfrey model assumes that voters have perfect common knowledge of the constituency situation. Similarly, it is interesting to note that Droop’s (1871) logic can apply only when the identities of the parties “between whom the election really lies” are known to voters. Uncertainty over such identities is the important feature that is missing from existing theories. In my companion paper (Myatt 2002), I address this issue. I develop a model in which voters base their decisions on informative signals of the constituency situation. The uniquely stable voting equilibrium entails only partial coordination of strategic voting, and thus is consistent with the multi-candidate support observed in reality. In other words, I predict some, but not complete, strategic voting. Whereas my model offers an interesting range of comparative static results, further analysis is necessary to identify exactly how much strategic voting is consistent with a formal theory. This is provided in the remainder of the present paper. THE IMPACT OF STRATEGIC VOTING 4 2. QUALIFIED MAJORITY VOTING Consider a stylized interpretation of the 1970 New York senatorial election. A first simplification is to suppose that all conservatives voted for Buckley, and that all liberals ranked him as their least preferred candidate. Liberal voters faced a potential coordination problem. Inspecting Table 1, a fraction γ = 39%/61% ≈ 0.635 of liberals needed to coordinate behind either Goodell or Ottinger in order to defeat Buckley. They were playing a qualified majority voting game: A qualified majority γ needed to coordinate in order to stop the disliked Conservative. This is exactly the scenario consider in my companion paper (Myatt 2002), and it proves to be a convenient representation of the strategic voting problem. I offer a simplified exposition here. Proofs are omitted, and the arguments are heuristic — the reader is referred to the companion paper for a formal treatment. I suppose that there are group of voters who wish to avoid a disliked third option j = 0, and to do so must vote for one of two challenging candidates j ∈ {1, 2}. A win by the disliked candidate generates a zero payoff, whereas a win by challenger j yields a payoff of uij > 0 to voter i. A fraction γ > 12 need to coordinate in order to defeat j = 0. I write p for the proportion of the electorate (where the electorate in this case refers to those who dislike candidate 0) who vote for candidate 1. Candidate 1 (e.g. Ottinger) wins if p > γ, candidate 2 (e.g. Goodell) wins if 1 − p > γ ⇔ p < 1 − γ and candidate 0 (e.g. Buckley) wins otherwise. The outcome of the New York case satisfied p = 37%/61% ≈ 0.602. How should an instrumental voter act in such a scenario? She can only influence the outcome of the election when there is a tie for the lead. In the scenario described here, this happens whenever p = γ or p = 1−γ. It is important, therefore, to assess what such an instrumental voter knows about p. The models of Palfrey (1989), Myerson andWeber (1993) and Cox (1994) all assume that voters’ preferences (and hence their voting decisions) are drawn independently from a commonly known distribution. In a large electorate, this means that p is effectively known. If p = γ, then an instrumental voter will find it optimal to vote for candidate 1, yielding a payoff of ui1. All others will do the same, and a Duvergerian THE IMPACT OF STRATEGIC VOTING 5 outcome obtains. A similar analysis applies when p = 1− γ. When p is not equal to either of these values, then instrumental concerns cannot guide a voter’s choice. Clearly, the assumed common knowledge of p is both unrealistic and drives the pathological results of existing formal voting theories. Suppose, then, that p were unknown. I use a density function f(p) to model a voter’s beliefs over p. Given this uncertainty, she can always envisage the possibility that p = γ so long as f(γ) > 0, and so instrumental concerns are always relevant. The relative likelihood of p = γ and p = 1−γ is determined by f(γ) and f(1 − γ). A vote for candidate 1 will yield a payoff gain of ui1 in the former case, and a vote for candidate 2 will yield a payoff gain of ui2 in the latter. It is optimal for her to vote for candidate 1 whenever: ui1f(γ) ≥ ui2f(1− γ) ⇔ log [ ui1 ui2 ] } {{ } ũi + log [ f(γ) f(1− γ) ] } {{ } λ ≥ 0 If it were not for the term λ = log[f(γ)/f(1 − γ)] then individual i would vote for her first choice candidate. λ, therefore, is the incentive to vote strategically, and is determined by the relative likelihood of the two “pivotal events” p = γ and p = 1 − γ. The voter balances this incentive against her relative preference ũi = log[ui1/ui2]. Importantly, it is uncertainty over p (represented by f(p)) that is the determinant of strategic voting — and this uncertainty is missing from the Cox-Palfrey framework. This analysis is decision-theoretic, characterizing optimal voting conditional on a voter’s beliefs. Taking a game-theoretic approach allows me to consider the generation of such a beliefs. A voter will consider the preferences, beliefs and strategies of others in order to evaluate f(p). In Myatt (2002) I construct a game-theoretic model in the following way. A voter’s preferences, summarized by ũi, are broken down into two components: ũi = η + εi where εi | η ∼ N(0, ξ) 3The Cox-Palfrey framework supposes that each individual casts a voter for candidate 1 with independent probability p. As the electorate grows large, the actual fraction who vote for candidate 1 becomes arbitrarily close to p with near certainty. However, there is a remove possibility that realized support will be equal to γ or 1− γ. For p > 1/2, the former event is infinitely more likely. THE IMPACT OF STRATEGIC VOTING 6 The component η is common to all voters. In fact, it is the preference of the median voter in the electorate. The element εi is idiosyncratic to voter i, and varies across the electorate. The variance term ξ represents the extent of this variation, and hence is a measure of voters’ idiosyncrasy. A voter’s true first preference is for candidate 1 whenever ũi ≥ 0. This means that the true support π for candidate 1 is: π = Pr[ũi ≥ 0 | η] = Φ(η/ξ) ⇒ η = ξΦ−1(π) where Φ is the cumulative distribution function of the standard normal distribution. Importantly, I suppose that a voter does not know η, and hence neither does she know the true support for candidates 1 and 2. She is initially ignorant (a diffuse prior over η) but accumulates information on the relative support for the two candidates. This information includes her own relative preference ũi, which is a signal of η. I summarize this information in an informative signal δi. This satisfies: δi | η ∼ N ( η, ξ m ) ⇒ η | δi ∼ N ( δi, ξ m ) (1) A justification for this specification is relatively straightforward. If a voter observes the preferences ofm−1 randomly selected voters, plus her own, then the mean of this sample δi will have an expectation of η and a variance of ξ2/m. It follows that m indexes the precision of a voter’s information source. A voting decision, then, is contingent on a voter’s preferences (summarized by ũi) and her information (summarized by the signal δi). It is natural to seek a monotonic equilibrium that is increasing in both of these inputs — so that a voter is more likely to vote for candidate 1 when she prefers this candidate (ũi ↑) or this candidate is thought to be more popular among the electorate (δi ↑). Adopting the convention that a voter chooses her true first preference when f(γ) = f(1− γ) = 0, I show (Myatt 2002) the following. 4Given the normality assumption, the signal δi generated in this way is a sufficient statistic for η, and hence captures entirely the information and beliefs of a voter. 5This justification for the signal specification also results in correlation between a voters signal δi and her preferences ũi — in fact (Myatt 2002) the correlation coefficient satisfies ρ = 1/ √ m. This correlation is accounted for in all of the formulae that follow. See Appendix A.1 for more details. THE IMPACT OF STRATEGIC VOTING 7 Proposition 1 (Myatt (2002)). There is a unique symmetric and monotonic voting equilibrium. The equilibrium strategy is linear in its inputs: Individual i votes for candidate 1 if and only if ũi + bδi ≥ 0 for some finite b > 0. b is increasing in the required qualified majority γ and the precision of the information sourcem, but decreasing in the idiosyncrasy parameter ξ. It satisfies: 2Φ−1(γ) ξ ≤ b √ m ≤ Φ −1(γ) ξ √√√2 + 2√(Φ−1(γ))2 + ξ2 (Φ−1(γ))2 On average, a voter faces a strategic incentive λ = bη = bξΦ−1(π). Some implications are immediate. The equilibrium involves partial, but not complete strategic voting since the average strategic incentive is finite. It incorporates a voter’s information sources. It allows for bi-directional strategic switching, since some realizations of δi will point to the wrong challenging candidate. It also offers extensive comparative statics. All of these features are highlighted, explored and explained in Myatt (2002). A number of of issues demand further analysis, however. The first is the impact of strategic voting. The model predicts only partial coordination, but is the degree of strategic voting generated by the model consistent with that in the data? The second is the pattern of strategic voting. Whereas Proposition 1 offers comparative statics in terms of γ and π, these are indirect functions of the expected vote shares in a three horse race. The third is the importance of strategic voting. In particular, which election results are likely to have been affected by strategic voting? I respond in the subsequent sections of the paper. 3. THE IMPACT OF STRATEGIC VOTING A first step in measuring the impact of strategic voting is to consider the probability that an individual in an appropriate risk population votes strategically By “risk population” I mean those for whom a strategic vote makes sense. Suppose (without loss of generality) that π > 1/2 ⇔ η > 0, so that candidate 1 is the leading challenger. A voter who prefers candidate 2 (so that ũi < 0) but believes that candidate 1 is in front (so that δi > 0 ⇒ λ = bδi = 0) faces a positive strategic incentive and is “at risk” of voting strategically. THE IMPACT OF STRATEGIC VOTING 8 An easy way to consider an “at risk” individual is to equip her with a signal satisfying δi = η (so that has an accurate expectation of the constituency situation) and then examine her behavior. She votes for candidate 1 whenever bδi + ũi ≥ 0, or equivalently (given that δi = η) when (1 + b)η + εi ≥ 0. Hence the realized support for candidate 1 is: p = Φ ( (1 + b)η ξ ) = Φ((1 + b)Φ−1(π)) (2) Of course, this voter actually prefers candidate 1 with probability π, and so a strategic vote is observed with probability p− π. Furthermore, she is at risk of voting strategically when she prefers candidate 2 (with probability 1 − π) and hence the impact of strategic voting, measured as a proportion of the risk population, is: p− π 1− π = Φ((1 + b)Φ−1(π))− π 1− π (3) Equation 3 measures the impact of strategic voting. In a similar way, I may assess the importance of strategic voting. I can invert Equation 2 to yield π as a function of p: π = Φ(Φ−1(p)/(1 + b)). This tells me what the election outcome would have been in the absence of strategic voting. If p > γ > π, then strategic voting changes the identity of the winning candidate. For a more accurate assessment of this effect, I must also account for the variation in voter’s signals. Doing so (Appendix A.1) generates the following: p = Φ ( (1 + b)η √ var[bδi + ũi | η] ) = Φ ( (1 + b)η ξ √ m m+ b2 + 2b ) (4) = Φ ( (1 + b)Φ−1(π) √ m (m− 1) + (1 + b)2 ) (5) ⇒ π = Φ ( Φ−1(p) 1 + b √ (m− 1) + (1 + b)2 m ) (6) Equation 3 measures the impact of strategic voting, requiring as input the parameters γ, π,m and ξ2. Equation 6 allows me to “invert” an election outcome (i.e. remove the effect 6Here I have actually given her a signal δi = η, and so εi ∼ N(0, ξ). If I had considered her behavior conditional on her actually receiving a signal δi, then the distribution of εi changes slightly. The reason is that ũi and hence εi is a component of the signal δi, and so knowledge of δi influences the distribution of εi. Correcting for this has a minimal effect on the analysis — see Appendix A.1. 7A voter’s response to her signal b is contingent on γ,m and ξ. THE IMPACT OF STRATEGIC VOTING 9 of strategic voting, and requires as input the parameters γ, p and m. The parameters γ, π and p summarize the constituency situation, and I consider them in more detail in Section 4. In the calibration exercises of Sections 5–7 I examine the response of strategic voting to the information parameter m. Before doing this, however, I must consider the specification of the idiosyncrasy parameter ξ. ξ measures the idiosyncrasy of preferences across the electorate. An increase in ξ results in a larger number of voters who are more heavily committed to their preferred candidate (for instance, when ui1/ui2 is large). As I show in Appendix A.2, a value for ξ may be pinned down by the preferences of the median supporter of a particular candidate. To understand this idea, consider the median supporter of candidate 1, and suppose that she prefers candidate 1 twice as much as candidate 2, so that u1 = 2u2. Since she is the median supporter, it follows that Pr[u1 ≥ 2u2] = π/2. Together with π, this equation is sufficient solve for ξ. In fact, for this example and π = 1/2, it solves for ξ = 1.056 (see the Appendix). I thus fix ξ at this value for the calibration exercises of Sections 5–7. 4. DISTANCE FROM CONTENTION AND MARGINALITY Intuition suggests that strategic voting should be greater in marginal constituencies when the preferred candidate is far from contention. In the context of the current model, when the preferred candidate is far from contention any pivotal event will almost always involve the leading challenger, and so formal theory agrees with this aspect of informal intuition. The marginality hypothesis is more problematic, however. The idea is that pivotal events are more likely in more marginal constituencies. This idea has no role in a theory of purely instrumental voting, however, since it is the relative probability of different pivotal events that matters and not the absolute probability of a pivotal event. 8Empirically the distance from contention is well known as a strong predictor of strategic voting, allowing the measure to become a basis to construct validity tests for different measures of strategic voting (Franklin, Niemi and Whitten 1992, 1993, 1994 and Evans and Heath 1993, 1994). 9Cain (1978, p. 644) provides a classic example of this hypothesis in his analysis of strategic voting in Britain. He expects the pressure to defect (i.e. vote strategically) to be lower in “noncompetitive” constituencies. THE IMPACT OF STRATEGIC VOTING 10 In the present model, the constituency situation is summarized by γ and π. Unfortunately, these parameters are not an ideal way to separate the effects of marginality and distance from contention. Suppose, for instance, that γ > π > 1/2. An increase in π (associated with higher strategic voting incentives) will increase the gap between the two challenging candidates and hence the distance from contention of candidate 2. Unfortunately this parameter change also closes the gap between candidate 1 and the required qualified majority, making the constituency more marginal. It is unclear whether increased strategic voting is due to marginality or the distance from contention. A different representation of the constituency situation is required. To do this, I expand consideration to the all members of the constituency, including those who vote (exogenously) for the disliked j = 0, and write the true support for these three candidates as ψ0, ψ1 and ψ2 where ψ0 + ψ1 + ψ2 = 1. This scenario may be easily mapped back into the qualified majority voting game. The qualified majority of dissatisfied voters required to defeat j = 0 is γ = ψ0/(1 − ψ0). The true support for candidate 1 among the electorate is π = ψ2/(1− ψ0). Using the notation ψj , the ideas of marginality and distance from contention may be formalized. Consider a constituency in which at least some strategic voting is needed to defeat j = 0, so that ψ0 > ψ1 > ψ2, or equivalently γ > π. Then I may define: Winning Margin = w = ψ0 − ψ1 and Distance from Contention = d = ψ1 − ψ2 These parameters (together with ψ0 + ψ1 + ψ2 = 1) are sufficient to determine the constituency situation. Solving linearly to obtain ψ0, ψ1 and ψ2 in terms of w and d: ψ0 = (1 + 2w + d)/3 ψ1 = (1− w + d)/3 ψ2 = (1− w − 2d)/3  ⇒ γ = 1 + 2w + d 2− 2w − d and π = 1− w + d 2− 2w − d Using this formulation, I may change w and d separately, and determine the effect on γ and π. By inspection, both γ and π are increasing in d, and so (as expected) an increase in THE IMPACT OF STRATEGIC VOTING 11 the distance form contention results in greater strategic incentives. γ is clearly increasing in the winning margin w. Simple algebra confirms that this π is also increasing in w. Fixing the distance from contention, an increase in the size of the winning margin w (making the constituency lessmarginal) actually increases the incentives to vote strategically. Proposition 2. When coordination is required to defeat j = 0 (ψ0 > ψ1 > ψ2) the incentive to vote strategically increases with both the winning margin and the distance from contention. Proof. See Appendix A.3. This prediction runs against established intuition. After controlling for the distance from contention, strategic voting should be lower in more marginal constituencies. An appeal to the data is required. In recent work, Fisher (2000) finds that strategic voting increases with the winning margin in recent British General Elections. The effect is small, but nevertheless statistically significant when estimated across the three elections of 1987, 1992 and 1997. This analysis is extended by Myatt and Fisher (2002) and Fisher (2001). In the former paper, a version of the strategic incentive λ is shown to explain the pattern of strategic voting across all three elections, and its inclusion eliminates the significance of the winning margin and distance from contention measures. In the latter paper, Fisher (2001) includes a range of other explanatory variables, including political interest, education, party identification and local campaign spending. A version of λ continues to have strong explanatory power. Thus there is strong empirical support for the present theory relative to informal intuition. Interestingly, this provides a partial response to the critique of rational choice theory offered by Green and Shapiro (1994): The formal theory offers explanations that differ from intuition, and yet better explain the data. 5. APPLICATION: NEW YORK 1970 I am now in a position to calibrate the Myatt (2002) model and examine the impact and important of strategic voting in the context of reasonable parameter values. I return to THE IMPACT OF STRATEGIC VOTING 12 the case of the 1970 New York senatorial election (Table 1). Mapping this example into the model yields the parameters γ and p: γ = ψ0 ψ1 + ψ2 = 2, 288, 190 3, 605, 704 = 0.635 and p = ψ1 ψ1 + ψ2 = 2, 171, 232 3, 605, 704 = 0.602 Hence liberal voters needed to achieve a qualified majority of γ = 0.635 to defeat the disliked Buckley (see Section 1) and yet achieved only p = 0.602, and this split in the liberal vote allowed Buckley to win. Following the analysis of Section 3, I fix the idiosyncrasy parameter at ξ = 1.056. For any specified m, I may now calculate the equilibrium response b of a voter to her signal δi. Proposition 1 gives bounds to b. The exact value, however, is the solution to the following fixed point equation: b = 2mΦ−1(γ) √ var[bδi + ũi | η] ξ2(1 + b) = 2 √ mΦ−1(γ) ξ √ b2 + 2b+m b2 + 2b+ 1 (7) With this in hand, I am able to calculate b for various m. For a variety of values for π, I calculate the impact of strategic voting, by substituting into Equation 3. The results are displayed in Figure 1(a). As expected, the impact of strategic voting on the risk population increases with bothm and π. For smallm and π, strategic voting is limited. However, it takes only moderate values of m to increase the equilibrium impact of strategic voting to (when π is relatively large) higher levels. Are such larger values form reasonable? Adopting themicro-foundation for the informative signal δi described in Section 2, namely sampling of preferences of others, it would seem that larger values form are appropriate. An individual voter might be expected to interact with a large number of people during day-to-day interaction, yielding a largem. This may be misleading, however. Even if the number of individuals is observed is large, the effective value form will be rather lower. I offer two reasons here. First, the observation of the preferences of others is likely to occur with noise. Second, and most importantly, if a voter interacts with individuals whose preferences are correlated, then the observation of additional voters is likely to add little extra information. I address this second issue formally in Section 6. To cope with these issues, I can check the plausibility ofm by computing the accuracy of beliefs. 10This is an application of the equilibrium solution from Myatt (2002). See Appendix A.4 for details. THE IMPACT OF STRATEGIC VOTING 13 0 5 10 15 20 25 30 m = Precision of Information 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Im pa ct = p − π 1 − π .......................... π = 0.52 ......................... ............................................... ........................................................................... .................................................................................................. ................................ .... .... .... .... π = 0.54 ........ .... .... ... . .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... . .... .... .... .... .... .... .... .... .... .... .... .... .... .. .. . . . . . . . . π = 0.56 ... .... .... ..... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) The Impact of Strategic Voting 0 5 10 15 20 25 30 m = Precision of Information 0.5 0.6 0.7 0.8 0.9 1.0 A cc ur ac y = Φ ( √ m Φ − 1 (π )) .......................... π = 0.52 ................................................................................. ........................................................................................................................................................................... .................................... .... .... .... .... π = 0.54 .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . . . . . . . . π = 0.56 .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Accuracy of Voter Beliefs FIGURE 1. Strategic Voting in the New York Senatorial Election I define the accuracy of beliefs as the probability that the informative signal correctly identifies the leading challenger. Continue to assume (without loss of generality) that η > 0. A signal indicates the correct leader when δi. This occurs with probability α = Pr[δi ≥ 0] = Φ(( √ m/ξ)η) = Φ( √ mΦ−1(π)). This last formula allows me to cross-check m against the accuracy of beliefs. For instance: m = 25 ⇒ α = Φ( √ mΦ−1(π)) =  0.599 π = 0.52 0.692 π = 0.54 0.775 π = 0.56 THE IMPACT OF STRATEGIC VOTING 14 with further values illustrated in Figure 1(b). The calibration exercise demonstrates that if the outcome of the New York senatorial election is to be consistent with the present model then either the true value for π must be relatively close to 1/2 or the information available to voters must not identify the challenger with very high probability. In fact, I can calculate the parameter π that is consistent with the data by using Equation 6. Doing so with m = 25 yields π = 0.528. This suggests that, absent strategic voting, Ottinger’s true support may have been closer to 0.528 × 61% ≈ 32.2%. Of course, the value of m = 25 implies that voters’ beliefs were not particularly accurate, and hence this “inversion” procedure is subject to the critique that an instrumentally rational theory requires voters to have relatively “fuzzy” beliefs. I turn attention to this issue in the next section. 6. COMMUNITY SAMPLING AND FUZZY BELIEFS The calibration exercise for the New York senatorial election suggests that the present theory is consistent with observed data only when there are small amounts of information available to voters. If I were to introduce the possibility of opinion polls, then the probability that a voter will be able to identify the lead challenger will be relatively large. In such situations, therefore, the theory may well predict too much strategic voting, leading me to question the assumption of instrumental rationality. Turning to other scenarios, however, this is not necessarily the case. The plurality rule is used in the parliamentary constituencies of the United Kingdom. At a constituency level, opinion polls are not typically available, and hence a voter must learn of the constituency situation via social communication. If I am to take seriously the sampling procedure described in Section 2, then I must also recognize that voters are likely to communicate with other individuals who are similar to them. I formalize this idea with a notion of community communication. Suppose that each individual belongs to a community, and that a fraction ω of her idiosyncratic component is shared with all individuals in the community. 11As highlighted in Myatt (2002), for 1997 UK General Election, Evans, Curtice and Norris (1998) note that 47 nationwide opinion polls were conducted during the election campaign. By contrast, only 29 polls were conducted in 26 different constituencies at a constituency level, out of a total of 659 constituencies. THE IMPACT OF STRATEGIC VOTING 15 This is formalized as follows: εi = θ + ε̃i ⇒ ũi = η + θ + ε̃i where var[θ | η] = ωξ var[ε̃i | η] = (1− ω)ξ Thus preferences are equal to a common component across the constituency, plus a community component and then a further individual component. The parameter ω represents the relative importance of the community effect. This specification may seem reasonable; after all, the political preferences of the electorate may well be affected by events at a community level. Now, suppose that a voter is only able to sample the preferences of individuals from her own community. Since θ is common to every sample member, it cannot be averaged out. This means that var[δi | η] ≥ var[θ | η]. In fact, the best that a voter can do is to identify perfectly the typical member of her community. This would be equivalent to the observation of η + θ, and hence a signal δi = η + θ. In this case: var[δi | η] = var[θ | η] = ωξ ⇒ m = var[δi | η] ξ2 = 1 ω It follows that there is an upper bound on the precision of information available to a voter. For instance, if ω = 0.2 so that 20% of individual preference variation is due to variation across communities, thenm ≤ 5. This analysis suggests that appropriate values for m might well be very low indeed. In Figure 2(a) I plot the impact of strategic voting against the importance of community variation ω. By inspection, even if a small element of community variation dramatically reduces the impact of strategic voting. The presence of community variation has additional implications for the pattern of correlation between a voter’s preferences and her opinions. Suppose that candidate 1 is the true leading challenger (i.e. η > 0) but that a particular voter i prefers candidate 2, so that ũi < 0. When ω is large, so that the individual variation is small relative to community variation, then it is highly likely that voter i is drawn from a community where η+ θ < 0. If her opinions are based on the observation of her community (so that δi = η+θ as above) then she will believe that her candidate 2 is the leading challenger. In other words, for large ω, voters are likely to live in communities where their opinions are shared, and THE IMPACT OF STRATEGIC VOTING 16 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ω = Importance of Community Variation 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Im pa ct = p − π 1 − π ........................... π = 0.55 .................................................................................................................................................................................................................................................................................................................. .... .... .... .... π = 0.60 ..................................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . . . . . . . . π = 0.65 ..................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .