Empirical foundation of valence using Aldrich–McKelvey scaling

Abstract

This paper uses data from the 2004 pre-election survey of the American National Election Study to test empirically different ways of incorporating a valence parameter into a Downsian utility function. We call particular attention to the problem of interpersonal incomparability of responses to the liberal-conservative scale, and use Aldrich–McKelvey’s pathbreaking method to obtain accurate distances between respondents and candidates, the key regressors. We find that the utility function the most supported by the empirical evidence, the intensity valence utility function, is the one which permits to make the better predictions for the 2004 presidential election. We also consider counterfactual analyses wherein we test if Bush, the candidate with the highest intensity valence, has dominant strategies which would have insured him to obtain a majority of the popular vote. According to the theory, it is known that the candidate with the highest intensity valence does not have such dominant strategies if the distribution of voters in the policy space is too heterogenous. Nevertheless, we show that the distribution of voters in 2004 is sufficiently homogenous for Bush to have dominant strategies.

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Notes

  1. 1.

    Since the presidential election of 2008, the ANES has asked respondents to rate only the two main Presidential candidates, i.e., the Democrat and the Republican. The ANES provides on its website a questionnaire utility which lists the years in which a question has appeared. The information concerning the thermometer questions for the candidates are lines 80, 81 and 82 (Part II “Candidate and incumbent evaluations”); see http://isr-anesweb.isr.umich.edu/_ANESweb/utilities/questutility/all.htm.

  2. 2.

    A voter votes sincerely when he votes for his most preferred candidate. Conversely, he votes strategically when he decides to vote for his second most preferred candidate because his most preferred one is unlikely to win.

  3. 3.

    For instance, using stated voting behaviors, Schofield et al. (2011, pp. 485–486) estimate mixed logit wherein the intercept term associated to one candidate measures his additive valence. Interestingly, they also add party-leader trait indices which reflect valence factors in some specifications.

  4. 4.

    Degan (2007) is a notable exception. Analyzing the 1968 and 1972 U.S. Presidential elections, she uses the first dimension of the DW-NOMINATE scores in the Senate as an accurate measure of candidates’ positions on a liberal-conservative scale. Concerning voters’ positions, she estimates a parametric distribution of their positions (rather than point estimates of individual positions) using stated choices and voters’ characteristics.

  5. 5.

    There are also some empirical studies which do not estimate the parameters of the utility function of voters to make predictions and/or counterfactual analyses as we do. They propose some estimations wherein divergence between candidates or from the median voter is explained by some proxies for valence advantage. Incumbency has been a standard (e.g., Burden 2004; Ansolabehere et al. 2001). More recently, Adams et al. (2011) and Stone and Simas (2010) use district expert informants in the 2006 House elections to distinguish between valence which reflects campaign skills or fundraising ability and valence that voters value for their own sake (competence, integrity). These papers do not consider the problem of interpersonal incomparability of responses. One exception is Zakharova and Warwick (2014). Using data from the Comparative Study of Electoral Systems, they show in particular that individuals’ valence judgements depend negatively on the distance between them and the parties.

  6. 6.

    Note that several papers also consider an additive valence, but this valence parameter is not fixed. They assume that it depends on campaign spending or a costly effort from the part of a candidate (see, e.g., Ashworth and Bueno de Mesquita 2009; Carrillo and Castanheira 2008).

  7. 7.

    If both candidates propose the same policy x, but voter i prefers one candidate because of his characteristics (which may include valence), then voter i has UCR preferences if he also prefers the same candidate when both propose the alternative policy \(x'\). Contrary to the additive valence, the intensity valence violates the UCR-property, as can be shown in Fig. 1.

  8. 8.

    Note that the intensity valence model is not the sole model which predicts that the set of voters who support a candidate with less ability is a non-convex set. For example, Miller (2011) combines an additive valence and the candidate’s ability of changing policy from an exogenous status quo. If the candidate the less able to change the status quo is additive-valence-advantaged, the set of voters who prefer this candidate is non-convex.

  9. 9.

    Indeed, and without loss of generality, consider that there are two candidates, \(j=1,2\). If the utility function of voter i if candidate j is elected is the Downsian utility function \(U(a_i,x_j)=-|x_j-a_i|\), then voter i strictly prefers candidate 1 if \(-|x_1-a_i|>-|x_2-a_i|\Leftrightarrow |x_1-a_i|<|x_2-a_i|\). Now if the utility function of voter i if candidate j is elected is \(U(a_{i},x_{j})=\delta -\lambda \left| x_{j}-a_{i}\right| \), with \(\lambda >0\), then voter i strictly prefers candidate 1 if \(\delta -\lambda \left| x_{1}-a_{i}\right| >\delta -\lambda \left| x_{2}-a_{i}\right| \Leftrightarrow |x_1-a_i|<|x_2-a_i|\).

  10. 10.

    If \(\lambda _{1}=\lambda _{2}\), the intensity valence utility function has no candidate-specific parameter, so it becomes equivalent to the Downsian utility function, as one can see in Eqs. (1) and (3); hence, it makes no sense to try to recover the other intensity valence parameters K and c when \(\lambda _{1}=\lambda _{2}\). Furthermore, if \(\lambda _{1}=\lambda _{2}\), it is in fact impossible to recover K and c in terms of the parameters of the unconstrained model as one can easily see below.

  11. 11.

    The second equality \(\delta _1-\frac{\lambda _1(\delta _1-\delta _2)}{\lambda _1-\lambda _2}=\delta _2-\frac{\lambda _2(\delta _1-\delta _2)}{\lambda _1-\lambda _2}\) is easy to verify: by multiplying each side by \((\lambda _1-\lambda _2)\), we obtain \(-\delta _1\lambda _2+\delta _2\lambda _1=\delta _2\lambda _1-\delta _1\lambda _2\).

  12. 12.

    The interviewer first said:

    I’d like to get your feelings toward some of our political leaders and other people who are in the news these days. I’ll read the name of a person and I’d like you to rate that person using something we call the feeling thermometer. Ratings between 50 degrees and 100 degrees mean that you feel favorable and warm toward the person. Ratings between 0 degrees and 50 degrees mean that you don’t feel favorable toward the person and that you don’t care too much for that person. You would rate the person at the 50 degree mark if you don’t feel particularly warm or cold toward the person. If we come to a person whose name you don’t recognize, you don’t need to rate that person. Just tell me and we’ll move on to the next one.

    At the same time, the survey also made use of a respondent booklet and showed a 0–100 degree scale indicating in addition the meaning of 15, 30, 40, 60, 70 and 85 degrees. Then, the interviewer asked respondent i to rate his affect toward the three main Presidential candidates. For instance, for John Kerry, the question was:

    How would you rate JOHN KERRY?

  13. 13.

    The wording of the question was as follows:

    We hear a lot of talk these days about liberals and conservatives. Here is a seven-point scale on which the political views that people might hold are arranged from extremely liberal to extremely conservative. Where would you place YOURSELF on this scale, or haven’t you thought much about this? [1] Extremely liberal, [2] Liberal, [3] Slightly liberal, [4] Moderate/middle of the road, [5] Slightly conservative, [6] Conservative, [7] Extremely conservative, [80] Haven’t thought much, [88] Don’t know, [89] Refused.

  14. 14.

    The questions concerning the locations of the candidates followed the self-placement question. As an example, the wording for Bush was as follows:

    Where would you place GEORGE W. BUSH on this scale?

  15. 15.

    We say that the missing observations concern nonvoters as well as few missing at random because they cannot concern only nonvoters. 50.3% of the observations are missing. If the missing observations only concerned nonvoters, the probability that someone of the electorate does not vote would be in the 99% confidence interval [0.466, 0.540]. However, 43.3 percent of the electorate did not vote according to the 2004 United States presidential election Wikipedia webpage (https://en.wikipedia.org/wiki/2004_United_States_presidential_election); 0.433 is slightly outside this confidence interval. That’s why the missing observations cannot concern only nonvoters. However, if they concern nonvoters as well as few missing at random, the Final sample remains representative of the electorate who goes to vote. Note also that one cannot use the 2004 pre-election ANES survey alone to know who is nonvoter. As it is well known, the reported turnout data from surveys are unreliable, due to heavy overreporting of turnout. The ANES survey is no exception: of the 1212 respondents who were interviewed in this pre-election survey, 15 answered that they will not vote, so 1.23% \(\left( \simeq \frac{15}{1212}\right) \). 55 respondents answered that they did not know, 16 refused to answer, and one observation is missing. It is possible that some of these 72 \((=55+16+1)\) additional respondents will not vote. If all of them do not vote, 7.17% \(\left( \simeq \frac{15+72}{1212}\right) \) of the respondents will not vote. Hence, according to the turnout data of the survey, the estimated bound for the probability of being a nonvoter is in the interval [0.0123, 0.0717]. Whatever the probability in this interval, we are very far below 0.433.

  16. 16.

    One can test formally the null hypothesis that the Final sample is representative of the electorate who goes to vote, i.e., \(\text{ H}_0: S_b=0.5073\) versus \(\text{ H}_1: S_b\ne 0.5073\). The 90% confidence interval for Bush is [0.469, 0.537], so the null is not rejected.

  17. 17.

    As noticed in the Introduction, the Aldrich–McKelvey method permits to recover an accurate location of the candidates. Two additional advantages of the method have to be mentioned. First, the fact that the common dimension is the real line fits well with the theoretical framework in Sect. 2.1 which assumes that the policy space is \({\mathbb {R}}\). Second, and most importantly, the method permits to obtain an empirical distribution of the location of the respondents which is relatively continuous. In contrast, considering the self-placement \({\widetilde{a}}_i\) and the mean placement of each candidate Mean(\({\widetilde{x}}_{i,j}\)) to compute the distances will generate only 7 possible distances. In such a case, the lack of variability may be a problem for the estimations.

  18. 18.

    The assumption that the response process is multi-steps is in line with the current literature on the psychology of survey responses. For instance, the state-of-the-art book of Tourangeau et al. (2000) considers that a survey response process involves four steps: understanding the question, retrieving relevant information, using this information to make a judgment, and selecting and reporting of an answer.

  19. 19.

    In practice, the maximum likelihood estimates of the different SUR models considered are computed using an iterated Zellner scheme (see, e.g., Ruud 2000, p. 706).

  20. 20.

    A likelihood ratio statistic is twice the difference between the unconstrained maximum value of the log-likelihood function and the maximum subject to the restrictions: \(2\left( {\widehat{\ell }}_u-{\widehat{\ell }}_c\right) \), where \({\widehat{\ell }}_u\) and \({\widehat{\ell }}_c\) denote, respectively, the unconstrained and constrained maximum log-likelihood values. It is asymptotically chi-square distributed with degrees of freedom equal to the number of restrictions imposed.

  21. 21.

    Some readers may be curious of the results if we use instead of the Aldrich–McKelvey solution the mean placement of each candidate to compute the distance, as Adams et al. (2005) do. As already noticed, and as Armstrong II et al. (2014, p. 43) highlight, this is a naive solution. However, for the interested reader, “Appendix C” provides the results. The intensity valence model is the sole model which is not rejected, but as all the other models it is unable to make good predictions. It predicts that 55.64% of the voters will vote for Kerry under the assumption of sincere voting and under the one of strategic voting.

  22. 22.

    For moderate heterogeneity, no candidate has a dominant strategy which insures him to obtain the majority, and, more generally, no pure strategy Nash equilibrium exists; only mixed strategy equilibria exist in this intermediate case.

  23. 23.

    This counterfactual experiment should not be misinterpreted. It assesses the impact of candidates’ policy positions and their intensity valence, in particular the one of Bush, on the popular vote. By doing so, we follow, e.g., Adams et al. (2005, Chapter 12) who study if Dukakis would have been able to obtain a majority of the popular vote in 1988 under an alternative policy, or Degan (2007, p. 478) who show that Nixon would have won the popular vote in 1968 against any of the Democratic candidates. However, we do not claim that Bush should have played in practice a dominant strategy which would have insured him to obtain a majority of the popular vote, given that the President of the United States is not elected by the popular vote. It is elected through the Electoral College (EC) system. For a description of the EC system, see, e.g., Barthélémy et al. (2014).

  24. 24.

    We have chosen values for \(x_k\) ranging from −3.7 to 4.3 because the respondents are located between −3.704 and 4.233; see Table 1.

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Acknowledgements

Funding was provided by Labex MME-DII (Grant No. ANR-11-LBX-0023-01).

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Correspondence to Fabian Gouret.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

I am grateful to the Editor Huseyin Yildirim and two referees for valuable criticisms. I also thank William Kengne and Alfonso Valdesogo for discussions on bootstrap, and Renaud Foucart for useful comments on a preliminary version, as well as Vladimir Apostolski, François Belot, Thomas Brodaty, Frédéric Chantreuil, Jean Lainé, Mathieu Martin, Agustín Pérez-Barahona, Donald Saari, Julien Vauday and participants at the workshops “Voting, Democracy and Inequality” in Caen and “Advances in Economic Design: Games, Voting and Measurement” in Paris. All remaining errors are mine, and I acknowledge the financial support of Labex MME-DII (ANR-11-LBX-0023-01) and a “Chaire d’Excellence CNRS”

Appendices

Appendix A: Estimation results taking issue scale responses at face value

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4. However, the responses to the liberal-conservative scale are taken at face value to obtain the distances, i.e., \(d_{i,j}=|{\widetilde{a}}_i-{\widetilde{x}}_{i,j}|\).

Table 7 provides the results. The first result is that the Downsian and the additive valence models are rejected according to the likelihood ratio tests, while the intensity valence model is not. These findings are similar to those obtained in Sect. 4. However, one should clearly have in mind that the fact that they are identical to the one with an Aldrich–McKelvey correction cannot permit to claim or suggest that an Aldrich–McKelvey correction is not necessary. The findings in Table 7 would be credible under the assumption that taking the data at face value is not a problem. This assumption is refutable and has been refuted, i.e., statistically tested and rejected in Sect. 3.1. Indeed, we have shown that the null hypothesis of interpersonal comparability of responses is rejected. Given that there is clear problem of interpersonal incomparability of responses, any econometrics using these data at face value cannot be considered as credible.

The main reason of providing these results is mainly to show that when the data are taken at face value the utility function is linear in distance as it is sometimes mentioned (see, e.g., Adams et al. 2005, p. 17). Indeed, given the estimated coefficient \({\widehat{\gamma }}=1.038\) and its estimated standard error \({\widehat{se}}({\widehat{\gamma }})=0.066\), the 95% confidence interval of \(\gamma \) is [0.909, 1.168], so the null hypothesis \(\text{ H}_0: \gamma =1\) is not rejected. As the results with the Aldrich–McKelvey correction suggest (Table 4), and as Adams et al. aptly explain, the fact that the utility is linear in distance in Table 7 is due to the fact that “the policy scales from which distance is measured [...] are constrained to specified finite intervals.”

Table 7 Maximum likelihood estimates of the 4 SUR models relaxing the assumption that the utility functions are linear in distance and taking liberal-conservative scale responses at face value

Appendix B: Support for each candidate according to the Downsian and additive valence models

B.1 Support for each candidate according to the Downsian model

Using the locations of Kerry, Nader and Bush (\(x_k=-0.422\), \(x_n=-0.394\) and \(x_b=0.816\)) and the Downsian utility function parameters obtained in Column [2] of Table 4, Fig. 7 describes the estimated Downsian utilities given by the three candidates. As shown in Table 1, note that voters are located between −3.704 and 4.233 on the liberal-conservative space. However, Fig. 7 only considers the interval of the liberal-conservative space ranging from -1 to 1. We do so in order to clearly visualize when the estimated utilities intersect. Indeed, Kerry and Nader are so close (\(x_k=-0.422\) and \(x_n=-0.394\)) that it would have been difficult to visualize these intersections if we had shown all the possible values taken by the respondents in the policy space. The main objective of this Appendix is to fully understand why when the voters vote sincerely, the Downsian model predicts unrealistically that Nader will obtain 31.56% of the votes. Thus, we focus on the situation wherein all the voters vote sincerely.

It is easy to see from Fig. 7, that:

  1. (i.)

    The voters whose bliss points are strictly higher than \(a_2\simeq 0.211\) obtain the highest level of utility with Bush. If so, these voters will vote for Bush. Then, according to the Downsian model, the relative frequency of voters who will vote for Bush is \({\widehat{S}}_b=\frac{\sharp \{i=1,\ldots , N ;~ a_i>a_2\}}{N}\simeq 0.4468\).

  2. (ii.)

    The voters whose bliss points are strictly between \(a_1\simeq -0.408\) and \(a_2\simeq 0.211\) obtain the highest level of utility with Nader. If so, these voters will vote for Nader. The relative frequency of voters who will vote for Nader is then \({\widehat{S}}_n=\frac{\sharp \{i=1,\ldots , N ;~ a_1<a_i<a_2\}}{N}\simeq 0.3156\).

  3. (iii.)

    The voters whose bliss points are strictly less than \(a_1\simeq -0.408\) obtain the highest level of utility with Kerry. The relative frequency of voters who will vote for Kerry is then \({\widehat{S}}_k=\frac{\sharp \{i=1,\ldots , N ;~ a_i<a_1\}}{N}\simeq 0.2375\).

Fig. 7
figure7

Estimated Downsian utility functions. Note: This figure depicts the three estimated Downsian utilities in function of a obtained in Table 4. The (red) solid curve depicts the estimated utility if Bush is elected. The (blue) dashed curve depicts the estimated utility if Kerry is elected. The (green) dotdash curve depicts the estimated utility if Nader is elected. The locations of Kerry, Nader and Bush are \(x_k=-0.422\), \(x_n=-0.394\) and \(x_b=0.816\), respectively. Finally, \(a_1\simeq -0.408\) and \(a_2\simeq 0.211\)

Thus, the Downsian model predicts unrealistically that Nader will obtain 31.56% of the vote (under the assumption of sincere voting) because he is located on the right of Kerry (\(x_n=-0.394\) and \(x_k=-0.422\)) and attracts all the voters located between \(a_1\simeq -0.408\) and \(a_2\simeq 0.211\). This is an interval wherein a massive heap of voters are located as shown in Fig. 3.

B.2 Support for each candidate according to the additive valence model

Using the locations of Kerry, Nader and Bush (\(x_k=-0.422\), \(x_n=-0.394\) and \(x_b=0.816\)) and the estimated additive valence utility function parameters obtained in Column [3] of Table 4, Fig. 8 describes the estimated additive valence utilities given by the three candidates. The main objective is to understand why, even under the assumption of sincere voting, nobody will vote for Nader.

It is easy to see from Fig. 8 that:

  1. (i.)

    The voters whose bliss points are strictly higher than \(a_1\simeq 0.022\) obtain the highest level of utility with Bush. If so, these voters will vote for Bush. Then, according to the Downsian model, the relative frequency of voters who will vote for Bush is \({\widehat{S}}_b=\frac{\sharp \{i=1,\ldots , N ;~ a_i>a_1\}}{N}\simeq 0.5249\).

  2. (ii.)

    The voters whose bliss points are strictly less than \(a_1\simeq 0.022\) obtain the highest level of utility with Kerry. The relative frequency of voters who will vote for Kerry is then \({\widehat{S}}_k=\frac{\sharp \{i=1,\ldots , N ;~ a_i<a_1\}}{N}\simeq 0.4751\).

The results are similar under the assumptions of sincere voting and strategic voting because the locations of Kerry (\(x_k=-0.422\)) and Nader (\(x_n=-0.394\)) are very close, and Kerry has an additive-valence advantage over Nader (\({\widehat{\delta }}_k>{\widehat{\delta }}_n\)), as shown in Column [3] of Table 4. Consequently, the higher additive valence of Kerry implies a higher level of utility with Kerry than with Nader for all voters. So even under the assumption of sincere voting, nobody will vote for Nader according to the additive valence model.

Fig. 8
figure8

Estimated additive utility functions. Note: This figure depicts the three estimated additive utilities in function of a obtained in Table 4. The (red) solid curve depicts the estimated utility if Bush is elected. The (blue) dashed curve depicts the estimated utility if Kerry is elected. The (green) dotdash curve depicts the estimated utility if Nader is elected. The locations of Kerry, Nader and Bush are \(x_k=-0.422\), \(x_n=-0.394\) and \(x_b=0.816\), respectively. Finally, \(a_1\simeq 0.022\)

Appendix C: Estimation results considering the mean placement of each candidate to compute each distance

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4, as well as the ability of the different models to predict the 2004 presidential election, as in Table 5. However, the individual positions are taken at face value and we use the mean of the perceived locations of the candidates to obtain the distances, i.e., \(d_{i,j}=|{\widetilde{a}}_i-\text{ Mean }({\widetilde{x}}_{i,j})|\), where Mean(\({\widetilde{x}}_{i,j}\)) is the mean of the perceived locations of candidate j, i.e., Mean(\({\widetilde{x}}_{i,k}\))=2.692, Mean(\({\widetilde{x}}_{i,n}\)) = 2.745 and Mean(\({\widetilde{x}}_{i,b}\) ) =5.591; see Table 1.

Table 8 Maximum likelihood estimates of the 4 SUR models considering the mean placement of each candidate to compute each distance
Table 9 Popular vote: Reality and point estimates according to the different models (based on the results considering the mean placement of each candidate to compute each distance)
Fig. 9
figure9

Estimated additive valence and intensity valence utility functions (based on the results considering the mean placement of each candidate to compute each distance). Notes: Panel (A) depicts the three estimated additive valence utilities in function of \({\widetilde{a}}\) obtained in Table 8 and \({\widetilde{a}}_1\simeq 4.005\). Panel (B) depicts the three estimated intensity valence utilities in function of \({\widetilde{a}}\) obtained in Table 8 and \({\widetilde{a}}_1\simeq 4.018\). In both panels, the (red) solid curve depicts the estimated utility if Bush is elected. The (blue) dashed curve depicts the estimated utility if Kerry is elected. The (green) dotdash curve depicts the estimated utility if Nader is elected. The locations of Kerry, Nader and Bush are Mean(\({\widetilde{x}}_{i,k}\))=2.692, Mean(\({\widetilde{x}}_{i,n}\)) = 2.745 and Mean(\({\widetilde{x}}_{i,b}\) ) =5.591, respectively

Table 8 provides the estimates. The first results are that the Downsian and the additive valence models are rejected according to the likelihood ratio tests, while the intensity valence model is not. These results are similar to those obtained in Sect. 4. However, one should clearly have in mind that the fact that these results are identical to the one with an Aldrich–McKelvey correction cannot permit to claim or suggest that an Aldrich–McKelvey correction is not necessary. As already noticed, using the mean placement of each candidate to compute the distances is problematic. As pointed out by Armstrong II et al. (2014, p. 43), it is prone to failure in case of heteroskedasticity. Furthermore, recall that each respondent i provides his placement \({\widetilde{a}}_i\) on a 7-point scale, ranging from 1 to 7. By considering the mean placement of each candidate to compute the distances, we have only 7 possible distances. The lack of variability may be a problem for the estimations.

Table 9 illustrates the inability of all the models to make good predictions when we use the mean of the perceived locations of the candidates to obtain the distances. Recall that in Table 5, when one uses an Aldrich–McKelvey correction, the point estimates of the fractions of vote obtained via the intensity valence model under the assumption of sincere voting (i.e., \({\widehat{S}}_b\simeq 0.5099\), \({\widehat{S}}_k\simeq 0.4652\), \({\widehat{S}}_n\simeq 0.0249\)) or under the assumption of strategic voting (i.e., \({\widehat{S}}_b\simeq 0.518\), \({\widehat{S}}_k\simeq 0.482\), \({\widehat{S}}_n=0\)) are very close to the percentages of vote obtained by the candidates in reality (i.e., \(S_b=0.5073\), \(S_k=0.4827\), \(S_n=0.0038\)). In Table 9, the intensity valence model implies that 55.64% of the voters will vote for Kerry under the assumption of sincere voting and under the one of strategic voting (Lines [5a] and [5b]). The point estimates obtained with the Downsian (Lines [3a] and [3b]) and the additive valence models (Lines [4a] and [4b]) are also unrealistic.

Finally, note that the intensity valence and the additive valence models provide exactly the same predictions. It may appear surprising given that the estimated coefficients of the intensity valence utility functions and the additive valence utility functions are very different, as shown in Table 8. In particular, the estimated slope coefficients are candidate-specific and very different in the case of the intensity valence while they are identical in the case of the additive valence. The reason of these similar predictions is easy to see from Fig. 9. The additive and the intensity valence utility functions are very different as shown in Figure C1, but the predictions are similar mainly because the policy space for the respondents is the 7-point scale. In the case of the additive valence, the voters whose bliss points are higher than \({\widetilde{a}}_1\simeq 4.005\) obtain the highest level of utility with Bush. If so, these voters will vote for Bush. The voters whose bliss points are less than \({\widetilde{a}}_1\simeq 4.005\) obtain the highest level of utility with Kerry and will vote for him. Concerning the intensity valence, the voters whose bliss points are higher than \({\widetilde{a}}_1\simeq 4.018\) obtain the highest level of utility with Bush and will vote for him. The voters whose bliss points are less than \({\widetilde{a}}_1\simeq 4.005\) obtain the highest level of utility with Kerry and will vote for him. Thus, in both cases, the respondents who place themselves at \({\widetilde{a}}\in \{1,2,3,4\}\) vote for Kerry and respondents who place themselves at \({\widetilde{a}}\in \{5,6,7\}\) vote for Bush.

Appendix D: Estimation results excluding the extreme voters

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4, as well as the ability of the different models to predict the 2004 presidential election, as in Table 5. However, the extreme voters, i.e., those whose bliss points are strictly less than \(a_1\) or higher than \(a_4\) in Figure 5 are excluded. The sample is now composed of 592 respondents. Note that according to the Aldrich–McKelvey correction, the location of Kerry is \(x_k=-0.420\), the one of Nader is \(x_n=-0.396\), and the one of Bush is \(x_b=0.816\) with this subsample; hence, the locations of the candidates are very similar to those obtained with the Final sample described in Table 1.

Table 10 provides the estimates of the different models. The first result is that, again, the intensity valence is the sole model which is not rejected by the data, considering the simple likelihood ratio tests or the bootstrapped likelihood ratio tests (the procedure is similar to the one presented in Sect. 4.2).

Table 10 Maximum likelihood estimates of the 4 SUR models excluding the extreme voters

Table 11 provides the point estimates of the fractions of vote obtained via the different models under the assumption of sincere voting and under the assumption of strategic voting. The results are very close to the percentages of vote obtained in Table 5.

Table 11 Popular vote: Reality and point estimates according to the different models (extreme voters excluded)

Appendix E: Estimation results excluding the respondents who rate Nader at zero

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4, as well as the ability of the different models to predict the 2004 presidential election, as in Table 5. However, those who rate Nader at zero at the thermometer questions are excluded. The sample is now composed of 536 respondents. Note that according to the Aldrich–McKelvey correction, the location of Kerry is \(x_k=-0.425\), the one of Nader is \(x_n=-0.391\), and the one of Bush is \(x_b=0.816\) with this subsample; hence, the locations of the candidates are very similar to those obtained with the Final sample described in Table 1.

Table 12 provides the estimates of the different models. The first result is that, again, the intensity valence is the sole model which is not rejected by the data, considering the simple likelihood ratio tests or the bootstrapped likelihood ratio tests (the procedure is similar to the one presented in Sect. 4.2).

Table 12 Maximum likelihood estimates of the 4 SUR models excluding the respondents who rate Nader at zero

Table 13 provides the point estimates of the fractions of vote obtained via the different models under the assumption of sincere voting and under the assumption of strategic voting. The results are very close to the percentages of vote obtained in Table 5.

Table 13 Popular vote: Reality and point estimates according to the different models (respondents who rate Nader at zero are excluded)

Appendix F: Estimation results excluding the liberal respondents who rate Nader at zero

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4, as well as the ability of the different models to predict the 2004 presidential election, as in Table 5. However, the liberal respondents who rate Nader at zero at the thermometer questions are excluded. By “liberal respondents” we mean those whose bliss points are strictly less than zero according to the Final sample. The sample is now composed of 571 respondents. Note that according to the Aldrich–McKelvey correction, the location of Kerry is \(x_k=-0.418\), the one of Nader is \(x_n=-0.398\), and the one of Bush is \(x_b=0.816\) with this subsample; hence, the locations of the candidates are very similar to those obtained with the Final sample described in Table 1.

Table 14 provides the estimates of the different models. The first result is that, again, the intensity valence is the sole model which is not rejected by the data, considering the simple likelihood ratio tests or the bootstrapped likelihood ratio tests (the procedure is similar to the one presented in Section 4.2).

Table 14 Maximum likelihood estimates of the 4 SUR models excluding the liberal respondents who rate Nader at zero

Table 15 provides the point estimates of the fractions of vote obtained via the different models under the assumption of sincere voting and under the assumption of strategic voting. The results are very close to the percentages of vote obtained in Table 5.

Table 15 Popular vote: Reality and point estimates according to the different models (liberal respondents who rate Nader at zero are excluded)

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Gouret, F. Empirical foundation of valence using Aldrich–McKelvey scaling. Rev Econ Design (2021). https://doi.org/10.1007/s10058-021-00243-w

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Keywords

  • Spatial models of voting
  • Valence
  • Survey
  • Aldrich–McKelvey scaling

JEL Classification

  • D72
  • C81