Defining Lexicographic Preferences

Some theoretical models of voter preferences consider two types of candidate attributes: valence and political (or ideological) attributes (for example, Franchino and Zucchini Reference Franchino and Zucchini2015). Valence attributes, V _c, are valued identically by all voters, and more is always preferred to less of a given valence attribute. Examples could be the competence and integrity of the candidate. Political attributes, P _c, instead divide voters by their ideological stance, and here, the party of the candidate, c, may be the most important cue. They are defined by a utility function, U _i, for voter, i:

(1)$$U_i = \alpha V_c + \beta ( {\vert {P_c-P_i} \vert } ).$$

Equation 1 shows that political attributes depend on the voter's own policy position, whereas valence attributes have a positive (or negative) influence on utility for all candidates. The simple model also allows for the possibility that voters may assign different salience or weight to different attributes – indicated by the parameters α and β.

We propose a third type of attributes, lexicographic attributes, L. Drakopoulos (Reference Drakopoulos1994) defines strict lexicographic preferences in the following way:

Let x′ and x′′ be two alternatives defined as two bundles of attributes:

$${x}^{\prime} = ( {{{x}^{\prime}}_1, \;\,{{x}^{\prime}}_2; \;\ldots {{x}^{\prime}}_n} ) $$

$${x}^{\prime \prime} = ( {{{x}^{\prime \prime}}_1, \;{{x}^{\prime \prime}}_2; \;\ldots {{x}^{\prime \prime}}_n} ).$$

x′ is preferred to x″ if

$$\eqalign{& {\rm either}\;1) \;{{x}^{\prime}}_1 > {{x}^{\prime \prime}}_1 \cr & {\rm or}\;\;\;\;\;\,2) \;{{x}^{\prime}}_1 = {{x}^{\prime \prime}}_1; \;\;\;{{x}^{\prime}}_2 > {{x}^{\prime \prime}}_2 \cr & {\rm or}\;\;\;\;\;\,3) \;{{x}^{\prime}}_1 = {{x}^{\prime \prime}}_1; \;\;\;{{x}^{\prime}}_2 = {{x}^{\prime \prime}}_2; \;\;\;{{x}^{\prime}}_3 > {{x}^{\prime \prime}}_3.} $$

In other words, x ₂ is only considered if there is a tie between ${x}^{\prime}_1$ and ${x}^{\prime \prime}_1$, and x ₃ is only considered if there is also a tie between $x_2^{\prime}$ and $x_2^\prime\prime$ and so forth until ${x}^{\prime}_{n-1} = {x}^{\prime \prime}_{n-1}; \;\;{x}^{\prime}_n > {x}^{\prime \prime}_n.$

‘Strict lexicographic preferences' means that an individual would always place absolute priority on the first-order attribute, x ₁, and would never substitute any part of it to gain anything in terms of other, lower ordered attributes (that is, x ₂ and x ₃). In a sense, the other attributes are dominated by this first-order attribute. As the name implies, these are very strict assumptions that would probably rarely occur in practice. It may be that not all attributes are ordered lexicographically by the respondent, but only the first dominating attribute. Scott (Reference Scott2002) refers to in Lancaster's definition of dominating preference as a situation where condition 1) is satisfied, but the other attributes (x ₂ and x ₃) may behave as substitutable.Footnote ³

We define lexicographically dominated second-order attributes, L, as attributes with a salience that depends on the choice set:

(2)$$U_i = \alpha V_c + \beta ( {\vert {P_c-P_i} \vert } ) + \gamma ^dL_c.$$

The effect of second-order attributes, γ ^d, is a function of the choice set. Let c′ and c′′ be two candidates with attribute values:

$${c}^{\prime} = ( {{v}^{\prime}, \;{\,p}^{\prime}, \;{l}^{\prime}} ) $$

$${c}^{\prime \prime} = ( {{v}^{\prime \prime}, \;{\,p}^{\prime \prime}, \;{l}^{\prime \prime}} ).$$

The effect of second-order attributes, γ ^d, depends on whether the candidates, c′ and c′′, are from the same party (p′; = p′′) or not. We define

(3)$$\gamma ^d = \left\{{\matrix{ {\gamma^{dominated}, \;} & {{\rm if}\;{\,p}^{\prime}\ne {\,p}^{\prime \prime}} \cr {\gamma^{dominated}, \;} & {{\rm if}\;{\,p}^{\prime} = {\,p}^{\prime \prime}.} \cr } } \right.$$

When there is no tie between the two values on the lexicographically dominating first-order attribute, P (that is, p′ ≠ p′′), the second-order attribute is dominated, and its effect, γ ^dominated, should be smaller than the effect when there is a tie. When there is a tie on the first-order attribute, the second-order attribute is not dominated, γ ^dominated, and may play a larger role in determining the candidate choice:

$$\gamma ^{dominated} < \gamma ^{dominated}.$$

In the candidate literature, this influence of party affiliation on the effect of other attributes has been referred to as the way in which party affiliation ‘dampens’ (McGraw Reference McGraw, Druckman, Green, Kuklinski and Lupia2011, 190) or ‘attenuates’ (Kam Reference Kam2007, 348) the effects of other characteristics or traits of the candidate. We use this difference in the effect of dominated and non-dominated second-order attributes as an indication of lexicographic preferences.Footnote ⁴

Lexicographically dominated attributes may also be political in the sense that if they have an effect, γ ^dominated > 0, the effect depends on the ideal of the voter, L _i:

(4)$$U_i = \alpha V_c + \beta ( {\vert {P_c-P_i} \vert } ) + \gamma ^d( {\vert {L_c-L_i} \vert } ).$$

For example, Republicans may prefer male over female candidates, and Democrats may prefer the opposite. However, if party affiliation dominates the influence of gender, gender only has an effect when the voter's preference for party affiliation has been satisfied.

Using Conjoint Experiments to Test for Lexicographic Preferences

We suggest that conjoint experiments can be used to test for lexicographic preferences.Footnote ⁵

In conjoint experiments, respondents are often asked to choose between (or rate) two alternatives. The candidates are presented in profiles containing a number of attributes. The respondents are often asked to complete several of such choice tasks, thereby increasing statistical power (Hainmueller, Hopkins, and Yamamoto Reference Hainmueller, Hopkins and Yamamoto2014).

By randomly assigning the values (levels) of the attributes to the profiles (for example, woman or man for the attribute gender), the conjoint method makes it possible to estimate the average marginal component effect (AMCE) of each of the attributes on the choice of the candidates. Hainmueller, Hopkins, and Yamamoto (Reference Hainmueller, Hopkins and Yamamoto2014, 10) show that the AMCE represents the marginal effect of an attribute, π _l, averaged over the joint distribution of the remaining attributes:

(5)$$\eqalign{\overline {\pi _l} ( t_1, \;t_0, \;p( {\vector t}) ) = & \sum\limits_{( t, {\vector t}) \in \tilde{T}} {\rm {\opf E}} [ Y_i( t_1, \;t, \;{\vector t}) -Y_i( t_0, \;t, \;{\vector t}) \vert ( T_{ijk[ \neg l] }, \;{\vector T}_{i[ \neg j] k}) \in \tilde{T}] \cr & \quad \times p( T_{ijk[ \neg l] } = t, \;{\vector T}_{i[ \neg j] k} = {\vector t}\vert ( T_{ijk[ \neg l] }, \;{\vector T}_{i[ \neg j] k}) \in \tilde{T}).} $$

In relation to Equation 2, π _l represents α, β, or γ, that is, the effects of valence, political, or lexicographic attributes. t ₁ is the treatment level and t ₀ the reference level of the attribute. Y _i is the potential outcome for respondent i. T _ijk[¬l] is the vector of levels of the other attributes (not l) for respondent i, of the candidate profile, j, in choice task k. T_i[¬j]k is the levels of attributes in the other profile, not j. $\tilde{T}$ is the set of all possible values of the attributes.Footnote ⁶

Important to the study of lexicographic preferences, the AMCE of attribute l is not only identified by averaging the other attributes of the candidate, j, that is, T _ijk[¬l], but also all the attributes of the alternative candidate, j, in the same choice task, that is, T_i[¬j]k. If party affiliation is a dominating first-order lexicographic preference, respondents will (primarily) use the party attribute to decide which candidate to vote for. However, when both candidates in the choice task have the same party affiliation, the respondents look for other second-order attributes to reach a decision. In candidate conjoint experiments with only two party values (for example, Republican and Democrat), the respondents will choose between two candidates from the same party in 50 per cent of the tasks (on average). The AMCE is the effect of attributes such as race and gender estimated as an average of the effect in these choice tasks, with a tie on party affiliation and the effect in the other (50 per cent) tasks when the party affiliation differs. The average marginal component effect, AMCE, of second-order lexicographic preference, γ, is an average of the two types of tasks, weighted by the share of choice tasks in which the two candidates are from different parties, w _d:

(6)$$\gamma ^{AMCE} = w_d\gamma ^{dominated} + w_{\neg d}\gamma ^{dominated}, \;w_d + w_d = 1.$$

To test for lexicographically dominated second-order attributes, we propose (1) splitting the data into tasks with a tie on the hypothesized first-order attribute (in our case, party affiliation) and tasks with variation in the first-order attribute, (2) estimating the marginal component effect of the other attributes in each of these subsets, and (3) comparing the size of two sets of marginal component effects. Statistically significant differences in the marginal component effects will indicate that the other preferences are second-order lexicographic preferences dominated by the first-order preference.

Characteristics of the Voters

If a lexicographic second-order preference is also political, γ ^dominated represents an average of the conditional effect on voters for which the value of the attribute is congruent with their policy position, γ ^{dominated,congruent} , and voters for which it is not congruent, γ ^{dominated,congruent} . We should, therefore, expect this relationship:

(7)$$\gamma ^{dominated, congruent} < \gamma ^{dominated} < \gamma ^{dominated, congruent}$$

In other words, we can test for political second-order preferences by comparing the effects when the attributes are congruent and not congruent with the political positions of the voters. Similarly, we can compare respondents from the same party with weak and strong partisanship to examine whether lexicographic preferences are more prevalent among strong partisans.

Pre-Registered Candidate Choice Experiment

Since our analysis of the Kirkland and Coppock (Reference Kirkland and Coppock2018) data was obviously not pre-registered, and since our analysis involves a large number of statistical tests, there is a risk of false-positive tests. To test the results more formally, we pre-registered a studyFootnote ⁷ with the following hypothesis:

H₁: When two candidates share party affiliation, Democrats prefer (a) female and (b) black candidates more than Republicans do – and more so than when candidates have different party affiliations.

We recruited 662 MTurk respondents from the United States for the study presented here.Footnote ⁸ Only respondents with a HIT approval rate of 98 per cent or above and a HIT approval number above 5,000 were included. They received $1.21 when completing the survey, which lasted about 10 minutes (the survey also contained questions not used in this study).Footnote ⁹ The data were collected between 19 and 29 June 2020.

The survey presented respondents with five conjoint choice tasks in which they could choose between two candidates. The choice was not forced. The candidates had six attributes (party, gender, race, age, job, and experience). Table A1 lists the different attributes and values (levels) in the conjoint experiment. The order of the attributes was randomized between each rendition of the table.

Table A1. Attributes and levels in the conjoint experiment

We used a smaller number of levels per attribute than Kirkland and Coppock (Reference Kirkland and Coppock2018) to increase statistical power and make our study comparable to other conjoint studies. The choice and party affiliation variables in the pre-registered study were measured as follows:

Choice variable, Y : ‘Which of these two candidates would you choose? If in doubt, please choose the one that you like the most’ (single choice between candidates 1 and 2, not forced).

Party affiliation of the respondent: ‘Generally speaking, do you usually think of yourself as a Republican, a Democrat, or as an independent?’ (‘Strong Democrat’, ‘Weak Democrat’, ‘Independent Democrat’, ‘Independent’, ‘Independent Republican’, ‘Weak Republican’, ‘Strong Republican’.)

The first three response categories were coded as Democrat and the last three as Republican. Strong Democrats (Republicans) were coded as the first (last) category. Weak Democrats (Republicans) were coded as those who identified as ‘Independent Democrat[s]’ (‘Independent Republican[s]’).

Figure A1 shows that when respondents could choose between candidates from two different parties (first panel), the effect of Republican candidates was about −0.25. This is similar to the results based on the Kirkland and Coppock (Reference Kirkland and Coppock2018) data, which show that we can replicate this result and that Democratic respondents similarly dominate our MTurk sample.

More importantly, the figure indicates that gender and race are second-order lexicographic preferences. They tend to have larger effects when candidates are from the same party – even if the difference for gender is not statistically significant. Similarly, age and experience also seem to be lexicographic second-order preferences in our data.

However, if these attributes are also political attributes, their effect will depend on the share of Republican and Democratic respondents in the sample. Therefore, we turn to Figure A2 to examine the effects by respondent party and, thereby, test H₁. The second panel confirms the first part of the hypothesis: ‘When two candidates share party affiliation, Democrats prefer (a) female and (b) black candidates rather than Republicans do’. By contrast, the first panel shows that when the parties of the two candidates differ, there are minimal effects on gender and race, if any, for Republican and Democratic respondents. Consequently, the third panel confirms the last part of H₁, ‘and more so than when candidates have different party affiliations’. The difference in the effect of gender and race between Republicans and Democrats is significantly larger when they choose between two candidates from the same party rather than between candidates from different parties. In other words, these results support the theory that race and gender are second-order lexicographic and political preferences among US voters.

We note that the results in Figure A2 also indicate that age is a second-order lexicographic preference and that it is political in the sense that it is only the Democratic respondents who dislike older candidates if they choose between candidates from the same party. Experience seems to be a second-order valence attribute in the sense that both Republican and Democratic respondents tend to prefer more experienced candidates when they are from the same party, even if this is only statistically significant for the Democratic respondents (Figure A2).

Strong and Weak Partisans