OBSTACLES TO EMPIRICAL ANALYSIS

There are several widely recognized problems with using nonexperimental data and methods to test theories of issue voting. First, the positions of candidates are endogenous to the decision rules that voters employ. Rational candidates adopt positions that they regard as conducive to winning elections. The pressure to attract voters reduces variation in candidate locations, precluding researchers from studying the full range of candidate configurations that would be useful for testing theories of issue voting. Taken to an extreme, electoral pressures could drive candidates to converge, leaving voters to decide on nonissue criteria.

The standard procedure for measuring the issue positions of candidates—asking voters where they think candidates stand—introduces additional endogeneity. If voters project locations onto candidates they prefer for nonspatial reasons, researchers will overstate the importance of whatever issue voting rule the voter has in mind. Some authors confront this problem by assuming that each candidate has a single location, equal to the average of all voters' individual assessments. This, however, biases estimates of proximity voting downward by introducing measurement error in the distance voters perceive between themselves and candidates, and by reducing the sample variation in this distance (Lewis and King Reference Lewis and King1999; see also Merrill Reference Merrill1995).

A second shortcoming of existing studies is the lack of critical tests: configurations of candidates and voters in which the theories predict different choices. The most commonly used data, seven-point issue scales from the National Election Study (NES), provide almost no critical tests of directional versus proximity voting. With integer scales running from 1 to 7, we cannot test the hypothesis that voters choose candidates on their side of the issue over candidates who are closer but on the opposite side.Footnote ³ Moreover, only eight scenarios pit more intense candidates against less intense candidates that are closer to the voter. Six of these involve candidates at the most extreme points on the scale, arguably beyond the region of acceptability. The only two uncontroversial tests arise when respondents at 5 face candidates at 5 (favored by proximity theory) and 6 (favored by directional theory), and the mirror image in which respondents at 3 face candidates at 2 and 3. Thus, of the 196 permutations of individual and candidate positions on a seven-point scale, only two (approximately 1%) provide strong tests of directional versus proximity theory. In practice, the proportion of critical tests is even smaller because voters often perceive candidates as taking competing sides of an issue, and therefore being on opposite sides of the neutral point.

Of course, even when voters and candidates are arrayed such that all three theories favor the same candidate, the mix of voting rules is crucial for strategy and selection in a democracy. The electorally optimal position for candidates depends on the frequency of each rule, even though the resulting configuration of candidates may provide few opportunities for using individual-level survey data to determine which rules voters employ. Notwithstanding the paucity of critical tests in existing polls, the mix of voting rules affects how sensitive candidates are to changes in voter positions, how sensitive the electorate is to changes in candidate positions, how movements in the status quo shape elections, and the kinds of policies a democracy produces.

A third roadblock to inference is multidimensionality. In standard surveys such as the NES, respondents report an overall feeling score or vote choice for each candidate, rather than saying which candidate they prefer on each policy issue. Without strong assumptions about the weight respondents place on each issue in the survey, it is difficult to infer what decision rule they are using. Behavior that appears consistent with one voting rule could, in many contexts, be made consistent with other rules by changing the presumed importance of each dimension. Moreover, existing data offer little opportunity to discover whether voters apply different decision rules to different issues, and estimates about the “average” decision rule necessarily depend on the set of items in the survey.

A fourth obstacle to empirical analysis is the presence of nonissue considerations. In actual elections, voters weigh not only the policy platforms of candidates but also their charisma, competence, trustworthiness, and other intangibles. These extra considerations make it difficult to isolate the issue criteria that voters use. The problem of identification is particularly serious when nonspatial considerations correlate with candidate locations and are difficult to hold constant in statistical analyses. Recognizing these problems, Adams, Bishin, and Dow (Reference Adams, Bishin and Dow2004, 357) concluded that with existing data, “no crucial test distinguishing the models is possible on the basis of voting behavior in a single election.”

Finally, standard data sets typically do not contain the quantities necessary to distinguish discounting from the leading alternatives. The problem arises either because surveys imply that the neutral point is also the status quo (e.g., decrease taxes, keep taxes the same, increase taxes), or because they fail to ask about the location of the status quo. Merrill and Grofman (Reference Merrill and Grofman1997, 45), for example, could not test discounting vis-à-vis the alternatives because they “did not have a good operationalization of . . . the status quo point.” Likewise, Adams, Bishin, and Dow (Reference Calvert2004, 351) were “unable to distinguish between the directional and discounting models,” and therefore could “make no claims about the individual-level decision-making” that gave rise to their results. On occasion, researchers have asked what survey respondents expect candidates to deliver (Lacy and Paolino Reference Lacy and Paolino1998) or have exploited variation in political institutions that affect the ability of candidates to shape policy (Kedar Reference Kedar2005). These valuable studies have found support for discounting theory but have not estimated the prevalence of all three voting rules in the electorate.

In summary, previous research has been hampered by problems of endogeneity, the paucity of critical tests, the curse of high dimensionality, the presence of nonissue considerations, and inadequate measures of the status quo. As Lewis and King (Reference Lewis and King1999, 31) noted, “until survey researchers or experimentalists produce better measurement devices, or political methodologists generate better methodological approaches, the impasse will remain.” In the next section, we offer a theoretical and empirical way forward.Footnote ⁴

DERIVATION OF CRITICAL TESTS

We now derive the conditions under which the three decision rules lead to distinct voting patterns. Consider a one-dimensional policy space: a number line that increases in value from left to right. The voter, whose position is denoted by v, faces a choice between candidates c ₁ and c ₂, with c ₁ to the left of c ₂. We use the functions Prox(v), Disc(v), and Dir(v) to indicate which candidate—c ₁ or c ₂—the voter at v would prefer if he or she were a proximilist, a discounter, or a directionalist.

Figure 1(a) displays how voters would behave if they were following the proximity rule. Recall that proximity voters prefer closer candidates to more distant ones. To express this idea formally, define the candidate mean, c =(c ₁+c ₂)/2, as the midpoint between the two candidates. Voters to the left of c are closer to c ₁ than to c ₂, and therefore prefer c ₁ on proximity grounds. Voters to the right of c, however, are closer to c ₂, making that candidate their proximity favorite. Finally, voters who stand exactly on c are equidistant between the candidates, and therefore have no preference for one over the other.

FIGURE 1.

Proximity, Discounting, and Directional Rules

We use opposing arrows to summarize these predictions. Proximity voters who stand in the range covered by the left-pointing arrow, labeled Prox(v)=c ₁, prefer c ₁. Likewise, proximity voters with ideal points in the range covered by the right-pointing arrow, labeled Prox(v)=c ₂, prefer c ₂. The open circle above c means that, when a voter is equidistant between the candidates, proximity theory makes no prediction.

In the version of discounting theory that we examine, the policy that results from electing a candidate is a weighted average of the candidate's position and the status quo. Formally, p ₁=αQ+(1−α)c ₁ and p ₂=αQ+(1−α)c ₂, where Q is the location of the status quo and α ∈ (0,1) is the weight discounters assign to it. Let p =(p ₁+p ₂)/2 represent the midpoint between the policy outcomes that would result from choosing candidates 1 and 2, respectively. Voters to the left of p are closer to p ₁ (the policy outcome associated with candidate c ₁) than to p ₂ (the policy outcome associated with candidate c ₂), and would therefore favor c ₁ under a discounting rule. Voters to the right of p would favor c ₂, and voters at p would be indifferent between the two candidates.

Figure 1(b) illustrates these predictions for a scenario in which the status quo is to the left of the candidates. In such a configuration, the status quo pulls p ₁ to the left of c ₁ and pulls p ₂ to the left of c ₂, as indicated by the curved arrows below the number line. The straight arrows above the number line show how voters in different locations would behave if they were applying the discounting rule. Discounters in the region covered by the left-pointing arrow labeled Disc(v)=c ₁ would opt for c ₁, whereas discounters in the region spanned by the right-pointing arrow labeled Disc(v)=c ₂ would choose c ₂.

The location of the status quo is crucial to expectations about p ₁ and p ₂, and therefore to the choices of discounters. Had the status quo in our example been between the candidates, it would have played a moderating role by moving p ₁ to the right of c ₁ and moving p ₂ to the left of c ₂. If the status quo had been on the far right, in contrast, it would have tugged both policy outcomes to the right of the candidates' announced positions, thereby shifting p to the right and expanding the set of discounters who would have chosen c ₁.

Directional theory emphasizes how candidates and voters are arrayed with respect to the neutral point, N, which may differ from the location of the status quo. When candidates are on opposite sides of the neutral point, as in Figure 1(c), directional voters prefer the candidate who advocates their side. Rabinowitz and Macdonald (Reference Rabinowitz and Macdonald1989) formalized this intuition by stipulating that each voter favors the candidate whose product (v−N)(c−N) is largest. If a voter takes the left side of the issue, then (v−N)(c ₁−N) > (v−N)(c ₂−N), and therefore c ₁ is the better choice. If a voter takes the right side of the issue, the inequality reverses, and directional theory favors c ₂. Neutral voters, who stand exactly on N, have no directional basis for choosing one candidate over the other. The opposing arrows mark the regions in which directional voters would prefer one of the candidates.

What happens when both candidates take the same side of an issue? Directional voters on that side prefer the more intense candidate, whereas directional voters on the opposite side prefer the less intense candidate. We illustrate this pattern in Figure 1(d), where both candidates endorse the left side of the issue. Applying the directional criterion, voters to the left of N would have (v−N)(c ₁−N) > (v−N)(c ₂−N) and choose c ₁. Voters to the right of N, however, would have the reverse inequality and select c ₂.

Having introduced our notation, we offer two propositions, which are proven in Appendix A.

Proposition 1. Proximity and discounting theories make distinct predictions if and only if the voter is between the status quo and the midpoint of the two candidates, and the discounter places a sufficiently high weight on the status quo. In symbols, let α* = (v − c)/(Q−c. Then Prox(v) ≠ Disc(v) iff α > α* and either Q < v < cor its reflection, c < v < Q.

Proposition 2. Proximity and directional theories make distinct predictions if and only if the voter is between the neutral point and the midpoint of the two candidates. In symbols, Prox(v) ≠ Dir(v) iff either N < v < cor c < v < N.

Proposition 1 distinguishes proximity from discounting theory. According to proximity theory, voters to the left of c prefer c ₁, whereas voters to the right of c prefer c ₂. The policy midpoint p plays an analogous role in discounting theory: voters to the left of p support c ₁, but voters to the right of p support c ₂. Consequently, the proximity and discounting rules imply different voting behavior only when the voter is between c and p. Proposition 1 establishes the necessary and sufficient conditions for “sandwiching” the voter between c and p. A key element in Proposition 1 is the weight of the status quo. If the voter placed no weight on the status quo, then p would coincide with c and the theories would make identical predictions. As the weight of the status quo increases, p shifts away from c in the direction of Q.Footnote ⁵ The value α* marks the transition point at which p crosses to the opposite side of v, causing discounters and proximilists to part ways.

As Proposition 2 shows, proximity and directional theories diverge under a different sandwich condition, in which the voter stands between c and N. Consider, for example, the configuration in which c < v < N. A directional voter would prefer c ₁, the more intense candidate on the voter's side, whereas a proximity voter would prefer c ₂, who is closer to the voter's ideal point.

From these two propositions, we derive predictions for all possible arrangements of candidates, voters, the neutral point, and the status quo, for all possible weights that discounters could assign to current policy (see Appendix A for proofs). Table 1 lists the six key scenarios.Footnote ⁶ For each scenario, we give the implied sandwich conditions and indicate the choices a proximilist, a discounter, and a directionalist would make. Each scenario has an unlisted reflection—reversing the arrangement of candidates, the voter, the neutral point, and the status quo—that would lead to the opposite voting behavior. For example, the first row shows that with arrangement N < v < c and v ≤ Q, proximilists and discounters would prefer c ₁, whereas directionalists would prefer c ₂. In the mirror image arrangement, c < v < N and Q ≤ v, proximilists and discounters would favor c ₂, but directionalists would incline toward c ₁.

TABLE 1.

Predicted Choices in Six Scenarios

Note: α* =(v−c)/(Q−c).

Figure 2 illustrates our predictions. Each panel in Figure 2 can be interpreted in light of our sandwich conditions: whether the voter is sandwiched between p and c (such that proximity and discounting theories disagree), and whether the voter is sandwiched between N and c (such that proximity and directional theories disagree). To isolate directional voting, for example, one must find scenarios in which the second sandwich condition holds but the first one does not. Figure 2(a) shows examples of the two scenarios that achieve this effect. To isolate proximity voting, both sandwich conditions must hold, as in Figure 2(b). To isolate discounting, the first sandwich condition must be true while the second must be false, as in Figure 2(c). Finally, when neither sandwich condition holds, all three decision rules agree, per Figure 2(d).

FIGURE 2.

Scenarios That Isolate Each Decision Rule

EXPERIMENTAL DESIGN

We designed an experiment that placed voters in the various scenarios and recorded the choices they made. The experiment, administered to a representative sample of U.S. adults over the Internet, began by measuring respondents' views about federal health care policy (Figure 3(a)). We chose health care because it is central to national policy debates,Footnote ⁷ is used to illustrate the differences between directional and proximity theories (e.g., Rabinowitz and Macdonald Reference Rabinowitz and Macdonald1989, 97), and does not require the status quo to be located at the neutral point.

FIGURE 3.

Measuring the Voter's Position and Preferences About Candidates

The question format in Figure 3(a) was especially suitable for testing all three theories. Following the NES, we labeled only the end points of the scale. Assigning verbal labels to every point might have reduced ambiguity and increased reliability (Krosnick Reference Krosnick1999), but Rabinowitz and Macdonald (Reference Rabinowitz and Macdonald1989, 95) argued that labeling only the end points is best for comparing the theories. Such a scale not only “corresponds very nicely to the requirements of the traditional spatial model,” but also “permits directional thinking. Respondents thinking in directional terms would have little difficulty responding to this question. They would simply choose their preferred policy direction and how intensely they favored it.” In summary, NES-type scales “allow for either a directional or proximity interpretation, which is important for . . . empirical work” (Macdonald, Listhaug, and Rabinowitz Reference Macdonald, Listhaug and Rabinowitz1991, 1114).Footnote ⁸

We modified the NES format in two ways that improved our ability to draw inferences. First, we increased the number of scale points from seven to eleven to obtain a larger number of critical tests.Footnote ⁹ Second, we adopted different numeric labels. NES scales typically run from 1 to 7, leaving some ambiguity about the location of the neutral point. Our scale, in contrast, was centered on 0, a value respondents associate with neutrality (Schwarz et al. Reference Schwarz1991).

After recording the respondent's view, we displayed two candidates on the same scale and asked, based on the issue at hand, which candidate the respondent preferred (Figure 3(b)). To test proximity, discounting, and directional theories in their purest form, we described the candidates as “A” and “B.” Our design thereby avoided two obstacles described previously: the endogeneity of candidate locations and the presence of nonissue considerations.Footnote ¹⁰

Finally, we measured respondents' perceptions about the location of the status quo. “To the best of your knowledge,” we asked, “which point on the scale most closely represents the policy of the current government in Washington?” We placed this question last to avoid priming respondents to consider current policy when choosing between candidates. Our estimates of discounting may, therefore, be conservative, compared to what one would obtain by encouraging people to include the status quo in their voting calculus.

Our design was unique not only because it exogenously placed candidates on the scale and measured Q and vote choice, but also because it employed an efficient algorithm for locating candidates. Complete randomization of candidate positions would have been inefficient for two reasons. First, it would have generated many situations in which at least one theory failed to make a clear prediction. Second, full randomization would have resulted in an undesirably high proportion of ties, situations in which all three theories made the same prediction.

We gained efficiency by taking v into account when assigning candidates. In particular, we ensured that the theories made clear predictions (avoided situations in which voters would have been indifferent) by arraying the candidates such that c ≠ v, and by not treating respondents who located themselves at the neutral point. We also screened-out respondents at −5 and +5 because all three theories would have made the same prediction for those voters. A more efficient algorithm would have conditioned on Q as well, but asking about the status quo before displaying the candidates would have run the risk of priming bias.

To quantify the inefficiency of full randomization, we simulated the patterns that would have arisen if we had presented the respondents in our sample with a random array of candidates. Approximately 23% of the scenarios would have been indeterminate (at least one theory would not have made a prediction) and an additional 65% would have failed to discriminate (all three theories would have favored the same candidate). A further 9% would not have been informative, according to criteria discussed in the next section of the article. Only 3% of the cases would have provided critical tests necessary to estimate the frequency of proximilism, discounting, and directionalism in the electorate. Thus, simple random assignment—the standard approach in survey experiments—would have yielded remarkably little usable data. Informative cases were 13 times more common with our stratified randomization algorithm than with pure random assignment.Footnote ¹¹

Thus, by letting formal theory guide the design of our experiment, we greatly increased the opportunity to estimate the relative importance of the three leading accounts of issue voting. Moreover, by assigning candidate locations that made optimal use of each voter's location, rather than distributing candidates completely at random, we conducted the experiment at far lower cost than otherwise would have been required.

STATISTICAL MODEL

From Table 1, which provided a basis for the experiment, we also developed estimators for the rates of proximity, discounting, and directional voting in the electorate. Suppose that all electors intend to follow one of the three canonical voting rules. Let π_dir be the proportion of people who intend to follow a directional rule, let π_disc be the proportion who intend to discount, and let π_prox=1 − π_dir − π_disc represent the remaining share that intend to apply the proximity rule. To allow for the possibility of implementation error, assume that each voter makes the wrong choice—mistakenly selects a candidate not favored by that voter's decision rule—with probability e. (In Appendix B, we explore alternative assumptions about error.)

We can then model the probability that a randomly selected voter would choose candidate 2 (c ₂). When confronted with scenario I, proximilists and discounters will mistakenly choose c ₂ with probability e, whereas directionalists will correctly choose c ₂ with probability 1−e. Consequently, the overall probability of choosing c ₂ under scenario I can be decomposed as Pr(c ₂|scenario I) ≡ π_I = π_proxe + π_disce + π_dir(1−e), or π_dir(1−2e) + e. Scenarios II and III imply a different pattern, in which all three types of voters will select c ₂ only by mistake, such that Pr(c ₂|scenario II) ≡ π_II = e and Pr(c ₂|scenario III) ≡ π_III = e. By similar logic, Pr(c ₂|scenario IV) ≡ π_IV = π_proxe + π_disc(1−e) + π_dire, or π_disc(1−2e) + e; Pr(c ₂|scenario V) ≡ π_V = π_proxe + π_disce + πdir(1−e), or π_dir(1−2e) + e; and Pr(c ₂|scenario VI) ≡ π_VI = π_proxe + π_disc(1−e) + π_dir(1−e), or (π_dir + π_disc)(1−2e) + e.

To estimate the parameters in this model (π_prox, π_disc, π_dir, and e), we need a procedure for determining which scenario each of our survey respondents faced. It was easy to ascertain who confronted scenario I or scenario II. Those scenarios depended solely on the arrangement of the candidates, the voter, the neutral point, and the status quo, all of which we either measured or experimentally manipulated. It was more difficult to allocate the remaining respondents into scenarios III through VI. As is clear from Table 1, these scenarios depend on an unobserved quantity: the weight that each discounter, in his or her own mind, attaches to the status quo. Scenarios III and IV, for example, involve identical arrangements of the observed quantities (c, v, N, and Q). They nonetheless predict different behavior by discounters, depending on whether the weight they assign to the status quo is above or below the cutoff value α*. A discounter who gives the status quo low weight (α < α*) will choose the same candidate as a proximilist, but a discounter who gives the status quo high weight (α > α*) will select the other candidate. The remaining two scenarios in Table 1 (scenarios V and VI) are also isomorphic for all types except discounters, whose preferences depend on whether their private value for α is greater than α*.

We therefore devised a method for inferring whether discounters—to the extent they exist—had been assigned to scenario III versus IV or to scenario V versus VI. Our approach relies on the concept of an informative subset. Of the cases in which voters face Q < v < c, N (the arrangement shared by scenarios III and IV), we regard the portion in which α* < .1 as informative. Within that subset, any discounter who puts even slight weight on the status quo would satisfy the condition that α > α*, and would, therefore, perceive themselves as facing scenario IV instead of scenario III. Likewise, of the cases in which voters confront N, Q < v < c (the arrangement common to scenarios V and VI), we treat ones with α* < .1 as informative. In those cases, all but the most trivial discounters would see themselves in scenario VI, not scenario V.

Subsets in which α* ≥ .1 are not as informative because they make it impossible to distinguish reliably between discounters and proximity voters. In arrangements where α* is high, even discounters who place substantial weight on the status quo may fail to satisfy the condition that α > α*. In our experiment, they would make the same choices as similarly situated proximilists, even though they reached a decision after discounting the announced positions of the candidates. To estimate whether and to what extent discounters exist in the electorate, we must focus on the subset of cases in which we can actually detect them and discard the remainder.

This framework implies a set of error-adjusted estimators for the prevalence of directional, discounting, and proximity rules, where IV* denotes the informative subset of scenarios III and IV, and where VI* denotes the informative subset of scenarios V and VI.

(1)

(2)

(3)

In our sample, there were no instances of scenario IV*, so Equation (2) simplified to π_disc = (π_VI* − π_I)/(1−2π_II). By computing the proportion of respondents in scenarios I, II, and VI* who chose candidate c ₂, and substituting the proportions into Equations (1) and (2) as values for π_I, π_II, and π_VI*, one can estimate the rates of proximity, discounting, and directional voting while taking voter error into account.Footnote ¹² We quantify the uncertainty associated with our estimates by taking bootstrap samples from the pruned data set that contains only scenarios I, II, and VI*.Footnote ¹³

ESTIMATED PREVALENCE OF THE DECISION RULES

The surveys discussed in this article were administered by Knowledge Networks, an Internet-based polling firm, with support from the National Science Foundation. By using random digit dialing to recruit participants and by providing Internet access to households, Knowledge Networks was able to administer our questionnaires to a nationally representative sample of U.S. adults.Footnote ¹⁴ The interviews took place between December 2005 and June 2006, and 3,315 people (72% of invitees) agreed to take the survey. We did not continue with respondents who located themselves at −5, 0, or +5 on the first item, the health care scale, because they would not have yielded critical tests. Of the remaining people, 1,564 completed our experimental protocol, yielding 1,045 informative cases.Footnote ¹⁵

Table 2 summarizes the behavior of respondents in each informative scenario. The table shows, for example, that 25.67% of the 261 respondents who faced scenario I chose candidate c ₂ rather than c ₁. The table also reports the decisions of citizens in scenarios II and VI*.Footnote ¹⁶ As noted previously, the sample contained no instances of the informative subset IV* because citizens who faced the arrangement shared by scenarios III and IV did not locate the status quo such that α* was less than .1.Footnote ¹⁷

TABLE 2.

Predicted versus Observed Choices

Note: When estimating the observed preference for c ₂, we classified a case as scenario I if N < v < c and v ≤ Q; as scenario II if v < c, N and v ≤ Q; and as scenario VI* if N, Q < v < c and α* < .1. The sample contained no examples of scenario IV*, in which Q < v < c, N and α* < .1.

From the 1,045 cases in Table 2, we inferred the prevalence of proximilism, discounting, and directionalism in the population. Our estimates imply that about 57.7% of citizens use a proximity decision rule, 27.6% choose after discounting the announced positions of candidates, and the remaining 14.7% apply directional logic.Footnote ¹⁸ It bears repeating that these estimates already account for voter error. We assumed that each voter mistakenly chose the wrong candidate—chose c ₁ when they preferred c ₂, or vice versa—with probability e. (Our conclusions are robust to alternative assumptions about error, as shown in Appendix B.) We then calculated e as the rate at which citizens in scenario II chose c ₂ and incorporated e into our estimates of the three voting rules.

Consider the case of directional voting. Had we ignored error, we would have classified 25.7% of voters as directionalists, because that share of citizens chose c ₂ when presented with scenario I. However, we estimated an implementation error rate of about 15.5%. In our model, this means that around 15.5% of voters who intended to follow proximity or discounting rules in scenario I inadvertently selected the directional favorite. At the same time, around 15.5% of directionalists who encountered scenario I mistakenly opted for the proximity/discounting favorite. Equation (1) takes these countervailing errors into account. The error-corrected estimate of directional voting is, therefore, (π_I − π_II)/(1−2π_II) = (0.2567-0.1552)/(1−2 × 0.1552) ≈ 14.7%.

Our estimates and associated measures of uncertainty are depicted in Figure 4. The figure, a ternary plot, is useful for displaying three estimates that must sum to 100 and are therefore interdependent. The height of any point in the plot (the perpendicular distance from the base to the top vertex) measures the frequency of proximity voting. If all voters were proximilists, we would place a point on the upper vertex. If only half were proximilists, the point would sit on a horizontal line parallel to the base and midway between the top and the bottom. And if proximilists were entirely absent from the sample, we would place a point somewhere on the triangle's base. Similarly, the rate of discounting is equal to the perpendicular distance from the left side of the triangle to the lower right vertex, with points on the left side implying no discounting and a dot on the lower right vertex implying that all voters are discounters. Finally, the rate of directional voting is given by the perpendicular distance from the right side of the triangle to the lower left vertex.

FIGURE 4.

Estimated Prevalence of Voting Rules

The black dot in the triangle's interior represents our best estimate of the shares of the population that adhere to each voting rule. Dotted lines emanate from the point to the relevant numeric scales on each side. The line that runs parallel to the base and intersects the left side of the triangle shows that the dot is 57.7% of the distance from the base to the top, representing the share of proximilists in the population. Dotted lines that parallel the left and right sides of the triangle reflect the proportions of discounters and directionalists, respectively. The solid line around the dot is a 95% confidence region.Footnote ¹⁹

The closer a point is to a given vertex (with closeness measured perpendicular to the base), the more pervasive the decision rule associated with that vertex. The dashed lines that divide the triangle into three equal areas establish which voting rule is most common. The dot and its associated confidence region fall entirely within the upper region. Thus, a plurality of voters in our experiment behaved consistently with the proximity rule.

TESTING ADDITIONAL IMPLICATIONS OF THE THEORIES

To further validate our experimental method and test additional implications of the theories, we examined whether the distribution of proximity, discounting, and directional voting differs for subgroups of the population. Some authors have speculated that less sophisticated and less informed citizens are especially likely to employ a directional decision rule because it is less cognitively and informationally demanding. Empirical studies have found little support for this hypothesis, however (Merrill and Grofman Reference Merrill and Grofman1999, 79). Along similar lines, one might hypothesize that discounting, which requires an understanding of the policy-making process, increases with the sophistication of voters. We used educational attainment as a proxy for political knowledge and sophistication.

To assess the effect of education, we estimated our model separately for people with and without a college degree. Estimates for the two subgroups appear in Figure 5(a). Directional voting was about twice as common among the less educated respondents (18.6%) as among the more educated ones (8.5%). In contrast, we found little evidence of a connection between discounting and education. The college graduates in our sample practiced discounting only 3.3 percentage points more often than those without college degrees.

FIGURE 5.

Estimates by Demographic Group

The political characteristics of respondents may also affect their incentives to use a discounting decision rule. Fiorina (Reference Fiorina1992), for example, hypothesized that when moderate citizens confront extreme politicians, they vote to put different branches of government under the control of rival camps that must compromise to produce policy. Others have found evidence that politically independent voters are especially likely to engage in this sort of calculus (Lacy and Paolino Reference Lacy and Paolino1998, 1181). Accordingly, we hypothesized that two groups—people without partisan affiliations and people with moderate political ideologies—would be more likely to engage in discounting behavior than more politically committed and extreme citizens.

To evaluate this hypothesis, we estimated our model separately for the most and least partisan thirds of our sample (strong partisans versus independents and leaners) and for the most and least ideological thirds of the sample (those at 1, 2, 6, or 7 versus those at 4 on the 7- point ideology scale). Figure 5(b) confirms that the three decision rules had different distributions among independents than among strong partisans. The estimated share of independents that discount (38.9%) was nearly twice as large as the share of strong partisans that discount (22.0%). More striking and less expected, directionalists were almost absent among political independents but comprised one-fourth of strong partisans. Similarly, Figure 5(c) shows that political moderates were nearly twice as likely as strong ideologues to employ a discounting decision rule (37.9% versus 18.6%). Thus, the political characteristics and commitments of voters affect patterns of issue voting.

THE POSSIBILITY OF MIXED MODELS

Our statistical model assumes that each voter follows one of the three canonical issue voting rules. An alternative perspective holds that all voters employ a common, mixed decision rule. Rabinowitz and Macdonald (Reference Rabinowitz and Macdonald1989, 110) raised the possibility that voters combine directional and proximity criteria, and Iversen (Reference Iversen1994) developed a theoretical justification for such a model. Merrill (Reference Merrill1995), too, offered a mixed directional-proximity model, which Merrill and Grofman (Reference Merrill and Grofman1999) extended by incorporating discounting to yield a “unified theory of voting.”

Although some authors have found support for mixed models, Merrill and Grofman (Reference Merrill and Grofman1999, 79) noted a problem of interpretation: “It is possible that some voters evaluate candidates according to the proximity model, whereas others use a directional measure. In other words, heterogeneity among voters is an alternative explanation to individual mixed utilities.” Unfortunately, neither standard surveys nor our experiments allow one to know definitively whether the voting population is composed of homogenous mixers, heterogeneous purists, or something in between.

Researchers may gain insight, however, by examining whether people in different demographic subgroups employ different rules. Merrill and Grofman (Reference Merrill and Grofman1999, 79) found little evidence that levels of directional voting vary with political sophistication and other demographic factors, and therefore concluded that people probably make compound utility calculations. “Still,” they noted, “it is possible that an alternative method of subgrouping would reveal heterogeneity for utility functions.”

Our experiments uncovered evidence that different subgroups judge the platforms of candidates in different ways. This conclusion was possible because our study, unlike previous ones, distinguished empirically among all three decision rules. More research is needed, but our findings support the hypothesis of voter heterogeneity—the claim that different voters employ different pure rules or different mixtures—rather than the hypothesis that all voters have homogeneous mixtures (for further evidence of heterogeneous decision criteria, see Rivers Reference Rivers1988).

If, however, one believes that voters mix directional, discounting, and proximity rules, our findings establish bounds on the shares of the population that include each element in their mix. Based on our data, at least 14.7% of citizens have a directional component to their utility function, and at least 27.6% consider the location of the status quo when choosing among candidates. It is more difficult to put lower bounds on the proportion of voters that assign at least some weight to pure proximity because the proximity-consistent choices in our sample could be due to discounters who place very low weight on the status quo. However, our data show that more than 85% of citizens have either a proximity or a discounting component in their issue voting calculus.