ISSN (Print) 0473-453X

Discussion Paper No. 1199

ISSN (Online) 2435-0982

EMPIRICAL PROPERTIESOF AN EXTENDED CES UTILITY FUNCTION
IN REPRESENTING DISTRIBUTIONAL
PREFERENCES

Keigo Inukai
Yuta Shimodaira
Kohei Shiozawa

December 2022

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Empirical properties of an extended CES utility
function in representing distributional preferences∗
Keigo Inukai†

Yuta Shimodaira‡

Kohei Shiozawa§

December 23, 2022

Abstract
In previous work, we proposed a method to address mathematical inconvenience by extending the constant elasticity of substitution (CES) utility
function in Inukai, Shimodaira, and Shiozawa (2022, ISER DP No.1195).
However, the relationships between the extended CES parameters and theexternal measurements are yet unrevealed. To explore these empirical properties of the extended CES utility function, in this paper we construct anonline experiment of Amazon Mechanical Turk workers using a modified dictator game, a public goods game, and a questionnaire. We then compare theparameters of the utility function according to the modified dictator game tobehavior in the public goods game and the responses to the questionnaire.
This provides evidence that the distribution parameter of the extended CES
utility function measures the preference for equality or selfishness. However,
we do not find any positive evidence that the substitution parameter measures
the preference for efficiency.
Keywords: CES, Distributional preferences, Equality, Efficiency, Amazon
Mechanical Turk
JEL codes: C91, D63, D64, D90∗

The authors are grateful to Nobuyuki Hanaki and participants in the 2021 Economic Science
Association Global Online Around-the-Clock Meetings for helpful comments and suggestions. The
authors would also like to thank Hayato Shimodaira for his technical support. Inukai acknowledges
the financial support of a Grant-in-Aid for Young Scientists (17H04780), Inukai and Shiozawa thesame through a Grant-in-Aid for Transformative Research Areas (21H05070), and Shimodairathrough a Grant-in-Aid for JSPS Fellows (19J12097) from the Japan Society for the Promotion of
Science. The experiments reported in this paper were approved by the Institutional Review Boardof the Faculty of Economics, Meiji Gakuin University.
†
Department of Economics, Meiji Gakuin University. E-mail: inukai@eco.meijigakuin.ac.jp‡
Graduate School of Economics, Osaka University. E-mail: u064674i@ecs.osaka-u.ac.jp§
Department of Economics, Takasaki City University of Economics.
E-mail: shiozawa@tcue.ac.jp1

1

Introduction

Debate about the trade-off between societal efficiency and fairness has prevailed
throughout human history, from early hunter-gatherers to the modern day. Someargue that while inequality may indeed arise, as society becomes richer, the poor
will also become richer. Others contend that it is more desirable to achieve equality,
even at the expense of social efficiency. This efficiency–fairness trade-off continues
to divide society. However, the question of distributive justice is not only a corequestion in the social sciences. In recent years, the allocation problem among individuals has also been pointed out as an essential adaptive issue in other organisms.
For this reason, we need to know more about the mechanisms that create the valuesof efficiency and fairness among people.
The pivotal element in studying individual attitudes toward efficiency and fairness begins by modeling and measuring individual preferences for efficiency and fairness. The outcome-based model addresses the concerns of individual participants inthe distribution of rewards between the individual and others (Fehr and Schmidt,
2006, Section 3.1; Cooper and Kagel, 2016, Section II). A method was proposed byAndreoni and Miller (2002) to model the distributional preferences elicited using themodified dictator game (henceforth MDG) using the constant elasticity of substitution (CES) utility function (Arrow et al., 1961), which has a flexible and helpful
specification in modeling behaviors. Andreoni and Miller (2002) employed the CES
utility function to encompass the three typical preference models—Rawlsian inequality aversion, altruistic, and selfish preferences—continuously. Further, Fisman
et al. (2007) studied individual heterogeneity of preferences with a two-dimensional
measure—selfishness and the elasticity of substitution—supposing that the parameter controlling elasticity also measures the preference for a trade-off between equality
and efficiency. Subsequent studies following Fisman et al.’s (2007) paradigm havebeen ongoing to this day (Fisman et al., 2014, 2015a,b, 2022; Li et al., 2017, 2022).

2

Although this procedure has been adopted in several applications, the standard
CES utility function is also known to have a theoretical difficulty in that the limit ofparameters corresponding to the Rawlsian maximin form, the parameter quantifying
selfishness becomes meaningless for identifying variations, and consequently, we may
obtain biased results. We have already addressed this in Inukai et al. (2022) andproposed an extension of the CES utility function, which allows us to interpret theselfishness parameter without bias.
Given the mathematical properties, the two parameters—the distribution andthe substitution parameter—of our extended CES utility function were expected
to measure the preference for equality or selfishness and the preference for efficiency, respectively, at orthogonal scales to each other. However, the relationshipbetween the extended CES parameters and individual behavior or thought has not
yet been clarified. For this paper, we conducted an MDG experiment, an experimenton public goods games, and questionnaires on demographics, socioeconomics, andpolitical attitudes/behaviors among participants of the Amazon Mechanical Turk
(https://www.mturk.com/, henceforth MTurk).1 We conducted the survey within
the framework of previous studies—concerning elite (Fisman et al., 2015b), politicalbehavior (Fisman et al., 2017), and demographic measures (Fisman et al., 2014)—
among the US population. Note that our study is not a direct replication of thesestudies. However, the accumulation of findings from a variety of samples is useful for obtaining robust results in research on (experimental) economics as a whole(Mullinix et al., 2015; Kohler, 2019; Snowberg and Yariv, 2021).
We examined the correlation of the parameters of the extended CES utility
by performing regressions with variables obtained from the public goods game experiment and questionnaire responses. The results of an online experiment with1

Evidence already exists that it is possible to obtain reliable results from the experiments using
the sample of MTurk participants (Mason and Suri, 2012; Rand, 2012; Höglinger and Wehrli, 2017).

3

an MTurk sample showed that the parameter expected to measure selfishness isnegatively correlated with cooperative behavior in public goods games, supporting
the interpretation that the parameter measures selfishness. However, we could not
obtain positive results indicating that the parameter controlling the substitution
elasticity measures the preference for efficiency.
The remainder of the paper is organized as follows. Section 2 describes theextended CES utility model and the MDG experiment. Section 3 provides details
on the public goods game and the questionnaires conducted following the MDG.
Section 4 describes the statistical procedures for multiple comparisons and Section 5
describes the experimental implementation. Section 6 presents the results, andSection 7 concludes.

2

Modified dictator game

2.1

Utility model and parameter estimation

To elicit individual distributional preferences, we used an MDG. We now consider
the decision problems associated with distributing the initial endowment between
the decision makers themselves and their opponents. When an individual decides ona percentage of the endowment distributed to themselves as s and to the opponent
as 1−s, their own reward is xs = s/ps , and the opponent’s reward is xo = (1−s)/po .
The ratio ps /po represents the relative price of the distribution, which is manipulated
by experimenters to measure individual distributional preferences. Here, the budget
constraint is represented as ps xs + po xo = 1.2
In Inukai et al. (2022), we documented an extended CES utility model that addresses the mathematical inconvenience of the standard CES utility function used to2

For simplicity, the amount of endowment is normalized to one.

4

model behaviors in MDGs. In this analysis, we characterize individual distributional
preferences using our extended CES utility function, which has the form
 1
 1+ρ −ρ
1+ρ −ρ − ρ

+
(1
−
α)
x
α
x
if ρ > 0osU (xs , xo ) =α ln(xs ) + (1 − α) ln(xo )if ρ = 0



1



 αx−ρ + (1 − α)x−ρ − ρ
if − 1 ≤ ρ < 0
s
o

(1)

where α ∈ [0, 1], ρ ∈ [−1, ∞). Let α and ρ be the distribution parameter and thesubstitution parameter, respectively.3
By solving the utility maximization problem subject to the budget constraint, we
obtain the demand function, which determines a decision maker’s own expenditure
share,
α/(1 − α)if 0 ≤ σ ≤ 1 α/(1 − α) + (ps /po )−(1−σ)
s(ps /po | α, σ) =
σ

α/(1
−
α)



σ
if σ > 1α/(1 − α) + (ps /po )−(1−σ)(2)

where σ = 1/(ρ + 1) ∈ (0, ∞) is the elasticity of substitution. Note that the value ofthe function s(•) corresponds to a percentage of the budget, that is, s = ps xs , andthus s(•) ∈ [0, 1].
In this study, as in Inukai et al. (2022), a Bayesian statistical model was formulated to estimate the two parameters of the extended CES utility function, α andρ. Each observation of the expenditure share s was assumed to be distributed on aunit interval in proportion to the normal distribution with location s(•) and scale
σs . For the error term scale, σs , the exponential distribution with the rate parameter λ = 0.5 was adopted as a prior distribution. For the distribution parameter
α, the uniform distribution on the unit interval is used as a noninformative prior

3

The naming of the parameters has been carefully selected to be neutral.

5

distribution. For the substitution parameter σ, the prior distribution was specified
in logarithmic form, ln σ, and set to Student’s t-distribution with degrees of freedomν = 4. To treat ln σ as a parameter with a unit interval scale, we converted it to therescaled substitution parameter τ ; that is, the value of the cumulative distributionfunction of the Student’s t-distribution with ν = 4, i.e.,
1 ln σ (ln σ)2 + 6
τ= +
3 .
2
2 (ln σ)2 + 4 2

2.2

(3)

Possible interpretations of the utility parameters

Based on the following mathematical discussion, the distribution parameter α is expected to represent a preference for (selfless–) equality–selfishness. Here, we discuss
the behavior of the function s at typical values of α. For α = 1 (α = 0), for any σ, we
obtain s = 1 (s = 0) and observe a completely selfish (selfless) distribution, independent of price. Alternatively, for α = 1/2, we obtain equation s/(1 − s) = (ps /po )1−σ
and find that the behavior changes depending on σ. If σ = 0, we obtain an equations/ps = (1 − s)/po or xs = xo which means that the payoffs received by both thedecision maker and the opponent are equal. If σ = 1, we obtain s = 1/2, whichcorresponds to the Nash bargaining solution (Nash, 1950) for an allocation problem
when the bargaining power is identical for the decision maker and the opponent. If
σ −→ ∞, s becomes a step function where ps /po = 1 is a switching point: s = 1
for ps < po , s = 0 otherwise. Given that the price ratio facing the decision makeris a random variable following a symmetric distribution where the mean is 1, theexpected s is 1/2 because s = 0 and s = 1 are realized with the same probability.4
In this way, the decision maker distributes equitably with respect to the opportunity

4

From this reasoning, the experimenter should instruct the participants in the MDG experimentto avoid forming the belief that the prices are drawn from a distribution that only one of themselvesor their opponent has advantages.

6

to obtain a payoff.5 In the three cases of α = 1/2, although the implementationdiffers depending on the elasticity of substitution σ, an equality-oriented distribution is observed in any case. Eventually, we expect that the parameter α measures
the preference for equality in the range between equality oriented and selfishness, or
selfless and equality oriented.
Next, let us discuss the interpretation of the substitution parameter ρ. A neutralinterpretation of the parameter ρ, which controls the elasticity of substitution σ, isa measure of the reactivity of the decision maker to prices, given the definition ofelasticity.
We consider three typical cases in general α ∈ (0, 1). Note that σ is interpretable
only if α is not 0 or 1. For σ = 0, we obtain an equation xs /xo = α/(1 − α): adecision maker determines the distribution so that the ratio of the payoffs matches
the odds of α. For σ = 1, we obtain an equation s/(1 − s) = α/(1 − α): a decision
maker determines the distribution according to the Nash bargaining solution, taking
the odds of α as the odds of bargaining power. For σ −→ ∞, s is a step function,
where ps /po = α/(1 − α) is a switching point: a decision maker more frequently gets
a whole payoff according to the odds of α—although the frequency depends on the
distribution of price the experimenter designs.
In the third case, s maximizes the α-weighted joint payoff among the decision
maker and the opponent αxs + (1 − α)xo . Given the weighted joint payoff for any
decision is always lower than the case of σ −→ ∞, it may (partially) be possible tointerpret that individuals with larger σ are efficiency oriented.65Note that the expected payoff amounts are not necessarily equal for both the decision makerand their opponent.
6Note that an individual with σ = 0 is not a minimizer of the weighted joint payoff, as they
do not choose corners unless α is 0 or 1. Furthermore, the magnitude relationship between theweighted joint payoff at σ = 0 and at σ = 1 depends on the parameter α and the offered price.
For example, we consider a situation where an individual whose α is 0.8 faces a budget constraint
4−1 xs + 8−1 xo = 1. The weighted joint payoff 0.8xs + 0.2xo is ∼ 3.02 if σ = 0 and is ∼ 2.88
if σ = 1. In this example, the weighted joint payoff decreases as σ increases from 0 to 1. This

7

2.3

Experimental design

We follow Fisman et al.’s (2007) protocol for the MDG experiment. The instructions
presented to the participants are provided in Appendix A.
Participants are required to distribute their initial endowment between themselves and an anonymous opponent so that they exhaust their budget. Following
the protocol, participants are then presented with a screen displaying budget constraint lines in a two-dimensional plane and make decisions by clicking on the lines.
The graphical interface allows participants to avoid the burden of calculating the
rewards from the prices.
The participants were instructed that the axes of the graph measure the payoffs.
Participants were also provided with a numerical value that indicated the amountthat corresponded to the coordinates on the graph on which they clicked. Note thatthe payoffs were presented in experimental tokens and that the maximum payoff for
one side in a single decision problem was less than 100 tokens.
The experiment consisted of 50 independent decision problems (i.e., budget constraints) corresponding to Fisman et al. (2007). However, rather than randomly
generating problems for each participant as their procedure, a predetermined problem set was commonly offered to all participants. Table 1 shows the set of problems
used. The columns “Max self” and “Max opp” are the intercepts of the budget
constraint line, which correspond to ps −1 and po −1 , respectively. Because of thevisual presentation of the budget constraint lines to the participants, we generated
the problems so that the angles of the lines were uniformly distributed to ensurethat the lines were visually unbiased. Therefore, the price ratios were distributed
in symmetry with a center of 1.7 The “Angle” column in Table 1 represents thereversal is more likely to occur on average when the price ratios that comprise a problem set are
distributed unevenly in favorable regions for the opponent.
7
Efficient methods of generating decision problems for parameter estimation are open to discussion. Baader et al. (2021) are working on simplifying the methodology and have suggested that8

angle ϕ between each line and the axis that measures the payoff to the decision
maker. Note that tan ϕ = ps /po . Figure 1 shows the budget constraint line in a
two-dimensional plane, where a decision maker’s own payoff in experimental tokens
is measured on the horizontal axis and the opponent’s payoff is measured on thevertical axis.8 Figure 2 shows demand curves s = s(ps /po | α = 1/2, σ) for several
values of σ and price ratios associated with budget constraints.
One unit of the experimental token was 0.02 USD. Each participant was paid
for realizing 5 of the 50 decision problems as dictator and 5 as recipient, while
Fisman et al. (2007) realized one each as dictator and recipient. Given our decision
problem set included some budget constraints with intercepts less than 50 tokens,
if we followed their method, we expected too much volatility in the participants’rewards. Therefore, to control the volatility of rewards, the number of decisions to
be realized was increased. To assign participant roles, either dictator or recipient,the “ring” matching algorithm was used, as documented in Crosetto et al. (2019).
Grech and Nax (2020) reported that when participants believe the dictator game
to be interactive, reciprocal motivation also drives them to transfer to their opponent
in the dictator game. To account for this, we did not explicitly instruct participantsthat the frequency of their becoming dictators is half of the 10 decisions to berealized. Thus, we expect to mitigate their reciprocal motives.

parameter estimation can be done with sufficient accuracy with fewer decision problems by devising price sampling. Imai and Camerer (2018) proposed an optimal adaptive method to generate
decision problems for the convex time budget experiment, which has a structure similar to the
MDG experiment.
8
In the experiment, the correspondence between both axes of the graphs presented to theparticipants and self/opponent was randomly determined for each participant.

9

10

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

70.00
75.00
80.00
85.00
40.00
96.26
41.00
75.92
42.00
61.84
95.00
68.67
96.00
57.01
97.00
47.80
98.00
40.34
99.00
34.18
40.00
99.12
42.00
86.39
44.00

Max self
7.36
10.01
12.96
16.21
8.79
24.00
11.45
23.50
14.30
23.00
38.38
30.00
45.17
28.80
52.45
27.60
60.29
26.40
68.81
25.20
31.25
82.00
36.77
80.00
43.09

Max opp
0.105
0.133
0.162
0.191
0.220
0.249
0.279
0.310
0.340
0.372
0.404
0.437
0.471
0.505
0.541
0.577
0.615
0.654
0.695
0.737
0.781
0.827
0.875
0.926
0.979

Price
6.0
7.6
9.2
10.8
12.4
14.0
15.6
17.2
18.8
20.4
22.0
23.6
25.2
26.8
28.4
30.0
31.6
33.2
34.8
36.4
38.0
39.6
41.2
42.8
44.4

Angle
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

76.38
46.00
66.53
48.00
57.81
25.20
68.81
26.40
60.29
27.60
52.45
28.80
45.17
30.00
38.38
23.00
14.30
23.50
11.45
24.00
8.79
16.21
12.96
10.01
7.36

Max self

Table 1: Set of decision problems

78.00
49.67
76.00
58.02
74.00
34.18
99.00
40.34
98.00
47.80
97.00
57.01
96.00
68.67
95.00
61.84
42.00
75.92
41.00
96.26
40.00
85.00
80.00
75.00
70.00

Max opp
1.021
1.080
1.142
1.209
1.280
1.356
1.439
1.528
1.625
1.732
1.849
1.980
2.125
2.289
2.475
2.689
2.937
3.230
3.582
4.011
4.548
5.242
6.174
7.495
9.515

Price
45.6
47.2
48.8
50.4
52.0
53.6
55.2
56.8
58.4
60.0
61.6
63.2
64.8
66.4
68.0
69.6
71.2
72.8
74.4
76.0
77.6
79.2
80.8
82.4
84.0

Angle

100
80

PASS

60
40
20
0
0

20

40

HOLD

60

80

100

Figure 1: Budget constraint lines offered to participants

���

σ=�

���
������ ��� ������������� ��� �������σ = ��

σ=�

���
���

���
���
σ = ����

���

���

���
��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ���������� ��� ��� ��� ��� ��� ��� ��� ��� ��� ������ / � �

�� / � �

σ = ���
σ = ���

Figure 2: Demand curves and offered prices
Note: The vertical axis measures the expenditure share s for the decision makers themselves, andthe horizontal axis measures the price ratio ps /po on a logarithmic scale. For all demand curves,
we fixed α = 1/2. The vertical lines represent the price ratios of each decision problem.

11

3

Measures and hypotheses

We implemented the public goods game experiment and the questionnaire survey
following the MDG and obtained the 26 variables listed in the following subsection.
We performed ordinal least squares regression to examine the relationship between
the collected variables (serving as independent variables) and the estimated utility
parameters (bα and τb) and the logarithmic scale of error terms (ln σbs ).9

3.1

Public goods gameThe extended CES parameters were compared with behavior in a public goods game.
As with dictator games, public goods games are experimental games used in many
economics and social psychology studies. There are many observations of cooperative behaviors in public goods game experiments, which are not supported by
the Nash equilibrium. Unlike the dictator game, the public goods game does notnecessarily require pure altruistic preferences as a motive for cooperative behavior
because they may also receive some payoffs from the opponents.10 However, Galizzi
and Navarro-Martinez (2019) have reported a correlation between altruistic behavior
in dictator games and cooperative behavior in public goods games. Given the distribution parameter α of the extended CES utility is believed to represent a preference
for equality–selfishness (the larger α, the more selfishness), we conjecture that α isnegatively correlated with cooperative behavior in the public goods game. Following
Galizzi and Navarro-Martinez’s (2019) findings, we conduct statistical analysis andtest the following hypotheses.

9

We only attempt to find a relationship between the variables and parameters, with no intentionof obtaining causality. Following the analysis in Fisman et al. (2014), we performed a regression
with the estimated utility parameters as the dependent variable.
10
Public goods games are frequently used to investigate the nature of reciprocally motivatedbehavior rather than to elicit altruistic preferences (Dhami, 2016; Drouvelis, 2021).

12

Hypothesis 1. The distribution parameter α of the extended CES utility is negatively correlated with the contribution in the public goods game.
The public goods game was played in groups of four players. The initial endowment was 20 tokens, each of which was exchanged for 0.06 USD. The marginal
per capita return was 0.4. The participants responded with their contribution to
the group as an integer from 0 to 20. We standardized each participant’s decisionand used the scores R public obtained in the regression analysis. Randomly chosenparticipants with a 10% chance were paid for the dictator game.11

3.2

Socioeconomic status of elitenessTo reveal the relationship between eliteness and distributional preferences, we obtained measures predicted to be correlated with eliteness and compared them withthe extended CES utility parameters. Fisman et al.’s (2015b) main finding is thatthe elite—authors defined as Yale Law School students—valued efficiency over equality, which was measured by the curvature parameter of the standard CES utility
function. As noted above, by the extended CES utility function, the preference
for efficiency can be measured as a component independent of the preference for
equality. Following Fisman et al. (2015b), we focus on the rescaled substitution parameter τ , which is expected to represent a preference for efficiency, and hypothesize
the following.
Hypothesis 2. The rescaled substitution parameter τ of the extended CES utility
is higher when socioeconomic status, which represents high eliteness, is high.
We collected the seven variables listed below as socioeconomic status measures to

11

The experiment followed the protocol of Fischbacher et al. (2001), where the experiment isdesigned to elicit the propensity for conditional cooperation. Therefore, the payoffs were calculatedusing the decisions of three unconditional contributions and one conditional contribution for eachgroup.

13

test the hypothesis. See Appendix B for the original survey questions and detailed
variable definitions.
R educational∗: education level
R educational1: dummy equal to 1 if respondent received a bachelor’s or
associate degree in college
R educational2: dummy equal to 1 if respondent received a master’s, doctoral, or professional degree
R income∗: income level
R income1: dummy equal to 1 if respondent’s annual income is less than
$30,000
R income2: dummy equal to 1 if respondent’s annual income is not less than$90,000
R employment: dummy equal to 1 if respondent is employee or business owner
R occupational∗: Occupational prestige
R occupational1: dummy equal to 1 if R employment = 1 and the respondent
belongs to professional and technical occupations
R occupational2: dummy equal to 1 if R employment = 1 and the respondent
belongs to higher administrative occupations

3.3

Political attitudes and behaviors

We reveal the relationship between political attitudes or behaviors and distributional preferences. Fisman et al. (2017) reported that the curvature parameterof the standard CES utility function could predict Americans’ political decisions:equality-oriented individuals were more likely to vote for Barack Obama in the 2012
presidential election or be affiliated with the Democratic Party. For equality-oriented
individuals, the distribution parameter α of the extended CES utility function is believed to be close to 0.5, but close to 1 for selfish individuals. Following Fisman
et al.’s (2017) conclusion, we focus on the distribution parameter α and hypothesize
the following.

14

Hypothesis 3. The distribution parameter α of the extended CES utility is lower for
those who support the Democratic Party or the redistributive policies that Democrats
mostly favor.
We collected the six variables listed below as measures of political attitudes andbehaviors to test the hypothesis. See Appendix B for the original survey questions
and detailed variable definitions.
R political: dummy equal to 1 if respondent thinks of self as liberal
R partisanship: dummy equal to 1 if respondent thinks of self as closer to the
Democratic party
R trump: dummy equal to 1 if respondent approves Trump’s job as President
R voting: dummy equal to 1 if respondent voted for Biden in the 2020 presidential
election
R redistribution: dummy equal to 1 if respondent agrees for redistributive policies
R basicincome: dummy equal to 1 if respondent agrees for basic income policies

3.4

Demographic characteristics

We collected the 12 variables listed below as demographic measures. See Appendix
B for the original survey questions and detailed variable definitions.
R age: standardized respondent’s age
R gender: dummy equal to 1 if respondent is female
R race∗: ethno-racial identity
R race1: dummy equal to 1 if respondent is Black or African American
R race2: dummy equal to 1 if respondent is Hispanic or Latino
R race3: dummy equal to 1 if respondent is Asian
R religious∗: religious identity
R religious1: dummy equal to 1 if respondent is ProtestantR religious2: dummy equal to 1 if respondent is Roman Catholic15

R religious3: dummy equal to 1 if respondent does not belong to a religion
R marital∗: marital status
R marital1: dummy equal to 1 if respondent is married
R marital2: dummy equal to 1 if respondent is widowed, divorced, or separated
R children: dummy equal to 1 if respondent has children
R metro: dummy equal to 1 if respondent lives in the metro area

4

Multiple comparison procedure

We collected 26 variables R ∗ and performed 26 statistical tests of the null hypothesis
for the regression coefficients for each parameter of the extended CES utility α andτ , and the scale of error terms ln σs . Each test is a two-sided t test. To account forthe multiple comparison problem over these 26 tests, we adjusted the p values using
the Benjamini–Hochberg procedure (Benjamini and Hochberg, 1995) and controlled
for the false discovery rate.
Furthermore, we conducted 26 equivalence tests to explicitly clarify the absenceof effects using the two-sided test procedure (TOST) (Lakens et al., 2018). We
detected ineffective variables using one-sided t tests that were smaller than thesymmetric smallest effect size of interest (SESOI). Following Lakens et al.’s (2018)suggestion, we set the SESOI based on a priori statistical power simulation for thenull hypothesis multiple testing.12 We searched for SESOI with a mean individual
power (see Porter, 2018) of about 0.8 and determined for αb, τb, and ln σbs to be
0.105, 0.62, and 0.39, respectively. The p values for each TOST were also adjusted
according to the Benjamini–Hochberg procedure.

12

Simulations were performed using data collected in a pilot experiment conducted in September
2020. Because some questionnaires were added after conducting the pilot experiment, we ran thesimulation using known data to supplement the missing variables.

16

We set the upper limit of the false discovery rate at 0.005. Setting the false
discovery rate at 0.005, even if all null hypotheses were rejected, less than one
hypothesis would be falsely detected because of a Type I error. We rejected thenull hypothesis when the adjusted p value was less than 0.005. This conservative
criterion results in decisive results.13
We preregistered the procedure for multiple comparisons and the SESOI values
in Open Science Framework (https://osf.io/68rn2).14

5

Implementation

The experiments were conducted in April 2021. The experimental software for the
MDG was developed from scratch using JavaScript and hosted on Firebase (https://firebase.google.com), and the public goods game and other questionnaires were
implemented using the platform provided by Qualtrics (https://www.qualtrics.
com).
We recruited participants via MTurk using the CloudResearch MTurk Toolkit(https://www.cloudresearch.com, Litman et al., 2017). By using CloudResearch’s
tool, we restricted the participants to US residents and tried to block workers basedon suspicious locations or duplicate IP addresses.
Arechar et al. (2017) have reported that participants’ decisions vary dependingon the time at which they respond. To avoid biasing the results over time, we divided
the experiment into the following seven waves starting at 1:00 AM, 7:00 AM, 9:00
AM, 1:00 PM, 3:00 PM, 7:00 PM, and 9:00 PM EDT. In each wave, we stopped

13

As commented by Grech and Nax (2020), the relationships between distributional preferencesand demographic measures delicate matters that warrant careful consideration.
14
After starting the first experimental wave, we found a mistake in the simulation code, so wereran the simulation and modified the SESOI in the registration. This modification was done independently of the data collected in the first wave. See https://osf.io/pn6ke for the registration
before modification.

17

recruiting when the number of participants that had completed all tasks reached 75(50 in the first wave).
A total of 500 MTurk workers participated in our study. The average age of theparticipants was 38.0 years (SD: 10.2, range: 21–74) and 33.7% of the participants
were female. Participants spent an average of 45.8 minutes (SD: 26.5, range: 3.6–
119.4) completing all tasks. They received a fixed completion fee of 2.50 USD atthe end of each wave and a variable bonus from the MDG for all participants andthe public goods game for participants that won a lottery with a 10% chance after
the completion of all waves. The average payment in the MDG was 4.99 USD (SD:
1.45, range: 1.22–9.81). In the public goods game, 52 lottery winners were paid
an additional 1.55 USD on average (SD: 0.36, range: 0.77–2.49). The average total
payment was 7.65 USD (SD: 1.52, range: 3.72–12.31) including the fixed fee.

18

6

Results

Figure 3 depicts a scatter plot of the parameter estimates α–τ for each participant.
Inspection of Figure 3 reveals that there are three principal clusters: the first is acluster with αb = 1 and τb that is widely distributed; the second is a cluster centeredat αb = 0.5 and τb = 0, and the third is a more widely distributed cluster centered atαb = 0.5 and τb = 0.2.
As we discussed in Inukai et al. (2022), if the α estimate is close to 0 or 1, theτ estimate has little credibility. In the analysis of the τ estimate we report in thefollowing sections, we omit participants that are purely selfless (5 participants) or
purely selfish (58 participants) using a one-sided test at the 10% level as in Li et al.
(2022). Participants excluded in the analysis of τ estimate are plotted with a crossin Figure 3.
Table 2 provides summary statistics for the measures of the public goods gameand the questionnaires. For some variables, statistics for the raw data before coding
to dummy variables or standardization are also provided.

19

1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
0, 1

Not

0, 1

Figure 3: Scatter plot of the extended CES utility parameter estimates α and τ
Note: Individuals for whom the null hypothesis that their α estimate is equal to 0 or 1 cannot berejected by a one-tailed test at the 10% significance level are identified by a cross. The contour
lines represent kernel density estimates.

20

Table 2: Summary statistics
Variables

Type

Mean

Std

Obs

R age (raw)R genderR race1
R race2
R race3
R religious1
R religious2
R religious3
R marital1
R marital2
R children
R metroR metro (raw)R public (raw)R educational1
R educational2
R income1
R income2
R employment
R occupational1
R occupational2
R political
R political (raw)
R partisanshipR trumpR trump (raw)
R voting
R redistribution
R redistribution (raw)R basicincome
R basicincome (raw)Integer
BooleanBooleanBooleanBooleanBooleanBooleanBooleanBooleanBooleanBooleanBoolean
1, 2, . . . , 90, 1, . . . , 20
BooleanBooleanBooleanBooleanBooleanBooleanBooleanBoolean
1, 2, . . . , 7
BooleanBoolean
1, 2, . . . , 7
BooleanBoolean
1, 2, . . . , 7
Boolean
1, 2, . . . , 7

38.018 10.182
0.337 0.473
0.130 0.337
0.040 0.196
0.076 0.265
0.128 0.335
0.468 0.500
0.238 0.426
0.660 0.474
0.056 0.230
0.556 0.497
0.862 0.346
1.986 1.738
8.794 6.517
0.584 0.493
0.224 0.417
0.176 0.381
0.176 0.381
0.948 0.222
0.290 0.454
0.082 0.275
0.374 0.484
4.298 2.032
0.590 0.492
0.490 0.500
3.828 2.161
0.724 0.447
0.742 0.438
5.046 1.589
0.752 0.432
5.236 1.538

500
498
500
500
500
491
491
491
500
500
500
484
484
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500

21

6.1

Relationships with the demographic characteristics

Figure 4 illustrates the scatter plots for each continuum variable versus each parameter estimate. Figures 5 and 6 show the cumulative distribution of parameter
estimates by category. Tables 3–5 report the regressions that analyze the relationships between the demographic variables and the estimated parameters.
b, is a positive correFor age (R age), see the upper row of Figure 4. Only for α
lation observed at the 5% significance level. The results are consistent with Fisman
et al.’s (2014) reported larger α for the group of participants over 60 years of age.
For gender (R gender), see the first row of Figure 5. For the distribution of αb,females stochastically dominate males. Taking α as a measure of selfishness, females
are more selfish, which is inconsistent with the reports of previous studies (Fisman
et al., 2014; Andreoni and Vesterlund, 2001; Kamas and Preston, 2015; Balafoutas
et al., 2012).
For ethno-racial identity (R race1–R race3), see the third row of Figure 5. Only
for ln σbs , is there a 5% significant difference between the four groups, with Asian
respondents displaying less noise. Fisman et al. (2014) reported African Americans
to have a smaller αb, and we obtained results consistent with this at the 10% significance level in a two-sided t test between groups using the dummy variable R race1.
However, a regression considering the three variables did not reveal this effect (seecolumn (3) in Table 3).
For religious identity (R religious1–R religious3), see the third row of Figure
5. For respondents identified as Catholic, αb is small, while ln σbs is large among thethree groups. The τb is large for respondents indicating no religion. These results
are inconsistent because Fisman et al. (2014) reported that religion does not have astrong effect.
For marital status (R marital1 and R marital2), see the first row of Figure
6. For respondents identified as married, both αb and τb are small among the three22

groups. The result for α
b lies opposite to the results reported by Fisman et al. (2014),
whereas the result for τb is consistent.
For parenthood (R children), see the second row of Figure 6. Respondents thatlive with their children have lower αb and τb.
For metro and non-metro residents (R metro), see the lower row of Figure 4 andthe third row of Figure 6. There is no difference between metro and non-metro
residents. In the questionnaire on the place of residence, 417 (86.2%) participants
responded that they lived in a county classified as a metro area, suggesting that thevariable R metro may be less informative.
Tables 3–5 provide the results of the regression analysis of the 12 variables observed in this subsection. Whereas the regression of the estimated α in Fisman et al.
(2014, Table 4) resulted in a coefficient of determination R2 of 0.041, in our resultsthe (adjusted) R2 is 0.116, indicating that the regression is not worsened. In the regression analysis, we rejected the hypothesis that R gender (female), R religious1(Protestant), and R religious3 (no religion) have no effect on the distribution parameter αb, respectively, at a level higher than 5% (see column (8) in Table 3). For
the rescaled substitution parameter τb, only R children was found to be effective atthe 5% significant level (see column (8) in Table 4). On the error term scale ln σbs ,R race3 (Hispanic or Latino), R religious1 (Protestant), and R religious3 (noreligion) were found to be effective at more than the 5% significant level (see column(8) in Table 5).
In the regressions of behavior in the public goods game experiment, the socioeconomic measures, and the political attitudes/behaviors presented in the following
subsections, we also detail the results controlling for the 12 variables listed in thissubsection.

23

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

20

30

40

50

60

Age

70

0.0

p = .042

ln s

R age

20

30

40

50

Age

60

70

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

p = .689

20

30

40

50

Age

60

70

p = .333

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

1 2 3 4 5 6 7 8 9Rural urban continuum

p = .635

0.0

ln s

R metro (raw)

1 2 3 4 5 6 7 8 9Rural urban continuum

p = .832

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

1 2 3 4 5 6 7 8 9Rural urban continuum

p = .437

Figure 4: Scatter plots for R age and raw R metro
Note: Left: against estimated α. Center: against estimated τ . Right: against estimated ln σs .
Simple regression lines are drawn. p values are for each slope of regression lines. Each plotted
point is jittered in the horizontal direction for raw R metro.

24

R gender
Male
Female0.6

Proportion

Proportion

0.8

0.4
0.2
0.0

1.0

1.0

0.8

0.8

0.6

0.6

Proportion

1.0

0.4
0.2

0.0

0.2

0.4

0.6

0.8

0.0

1.0

p = .005

0.0 0.2 0.4 0.6

0.4
0.2

Male
Female0.8 1.0

0.0

4.0

p = .088

3.0

2.0

ln s

Male
Female1.0 0.0

p = .756

R race∗Afr-Am
Hispanic
Asian
White

0.6

Proportion

Proportion

0.8

0.4
0.2
0.0

1.0

1.0

0.8

0.8

0.6

0.6

Proportion

1.0

0.4
0.2

0.0

0.2

0.4

0.6

0.8

0.0

1.0

p = .458

0.0 0.2 0.4

Afr-Am
Hispanic
Asian
White
0.6 0.8 1.0

0.4
0.2
0.0

4.0

p = .945

3.0

2.0

ln s

Afr-Am
Hispanic
Asian
White
1.0 0.0

p = .026

R religious∗ProtestantCatholic
Irreligion

0.6

Proportion

Proportion

0.8

0.4
0.2
0.0

1.0

1.0

0.8

0.8

0.6

0.6

Proportion

1.0

0.4
0.2

0.0

0.2

0.4

0.6

p < .001

0.8

1.0

0.0

0.0 0.2 0.4

ProtestantCatholic
Irreligion
0.6 0.8 1.0

p < .001

0.4
0.2
0.0

4.0

3.0

2.0

ln s

ProtestantCatholic
Irreligion
1.0 0.0

p = .001

Figure 5: Distribution of estimates grouped by R gender, R race∗, and R religious∗
Note: Left: CDF of estimated α. Center: CDF of estimated τ . Right: CDF of estimated ln σs .
p values are based on the one-way ANOVA.

25

R marital∗
MarriedWidow/Divorced
Never married

0.6

Proportion

Proportion

0.8

0.4
0.2
0.0

1.0

1.0

0.8

0.8

0.6

0.6

Proportion

1.0

0.4
0.2

0.0

0.2

0.4

0.6

0.8

0.0

1.0

p < .001

0.0 0.2

MarriedWidow/Divorced
Never married
0.4 0.6 0.8 1.0

0.4
0.2
0.0

4.0

p < .001

3.0

MarriedWidow/Divorced
Never married
2.0
1.0 0.0

ln s

p = .081

R children
With children
Without

0.6

Proportion

Proportion

0.8

0.4

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Proportion

1.0

p < .001

0.0 0.2 0.4

0.4
0.2

With children
Without
0.6 0.8 1.0

0.0

4.0

p < .001

3.0

With children
Without
2.0
1.0 0.0

ln s

p = .070

R metro
Urban
Rural

0.6

Proportion

Proportion

0.8

0.4

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

p = .982

0.8

1.0

Proportion

1.0

0.0 0.2 0.4 0.6

p = .767

Urban
Rural
0.8 1.0

0.4
0.2
0.0

4.0

3.0

2.0

ln s

Urban
Rural
1.0 0.0

p = .593

Figure 6: Distribution of estimates grouped by R marital∗, R children, and R metro
Note: Left: CDF of estimated α. Center: CDF of estimated τ . Right: CDF of estimated ln σs .
p values are based on the one-way ANOVA.

26

27
500
0.006
4.141∗∗

Observations
Adjusted R2
F Statistic

498
0.014
7.887∗∗∗

0.615∗∗∗(0.012)0.059∗∗∗(0.021)(2)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

0.635∗∗∗(0.010)0.041∗∗(0.020)Const.

R metroR children

R marital2

R marital1

R religious3

R religious2

R religious1

R race3

R race2

R race1

R gender

R age

(1)

500
−0.0010.888

0.642∗∗∗(0.011)−0.049(0.030)0.007(0.051)−0.002(0.038)

(3)

491
0.088
16.739∗∗∗0.615∗∗∗(0.024)0.106∗∗∗(0.036)−0.043(0.028)0.108∗∗∗(0.031)

(4)

500
0.047
13.355∗∗∗0.712∗∗∗(0.018)−0.112∗∗∗(0.022)−0.045(0.045)

(5)

500
0.039
21.289∗∗∗0.686∗∗∗(0.015)−0.091∗∗∗(0.020)(6)484−0.0020.001

0.001(0.030)0.635∗∗∗(0.027)(7)

Table 3: Regressions of estimated α on demographic characteristics

474
0.116
6.157∗∗∗

0.039∗(0.022)0.055∗∗(0.021)−0.045(0.030)0.028(0.049)0.017(0.038)0.103∗∗∗(0.036)−0.024(0.029)0.081∗∗(0.033)−0.059∗(0.032)−0.063(0.051)−0.030(0.029)0.019(0.029)0.641∗∗∗(0.041)

(8)

28
437
−0.0020.160

Observations
Adjusted R2
F Statistic

435
0.004
2.928∗

0.407∗∗∗(0.014)−0.041∗(0.024)(2)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

0.395∗∗∗(0.011)−0.009(0.023)

Const.

R metroR children

R marital2

R marital1

R religious3

R religious2

R religious1

R race3

R race2

R race1

R gender

R age

(1)

437
−0.0040.444

0.396∗∗∗(0.013)−0.030(0.033)0.012(0.056)0.028(0.042)

(3)

429
0.045
7.775∗∗∗

0.406∗∗∗(0.026)−0.046(0.041)−0.054∗(0.031)0.077∗∗(0.035)

(4)

437
0.030
7.783∗∗∗

0.458∗∗∗(0.021)−0.094∗∗∗(0.025)−0.009(0.050)

(5)

437
0.045
21.374∗∗∗0.451∗∗∗(0.016)−0.101∗∗∗(0.022)(6)423−0.0020.088−0.010(0.032)0.404∗∗∗(0.030)(7)

Table 4: Regressions of estimated τ on demographic characteristics

414
0.051
2.848∗∗∗−0.012(0.026)−0.030(0.025)−0.006(0.033)0.017(0.055)0.023(0.044)−0.055(0.042)−0.045(0.033)0.043(0.038)0.001(0.037)0.073(0.058)−0.065∗∗(0.033)−0.007(0.032)0.458∗∗∗(0.046)

(8)

29

500
−0.0000.940−2.158∗∗∗(0.036)−0.070(0.072)498−0.0020.096−2.150∗∗∗(0.044)−0.024(0.076)(2)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

Observations
Adjusted R2
F Statistic

Const.

R metroR children

R marital2

R marital1

R religious3

R religious2

R religious1

R race3

R race2

R race1

R gender

R age

(1)

500
0.014
3.355∗∗−2.175∗∗∗(0.040)0.194∗(0.107)0.287(0.183)−0.256∗(0.135)

(3)

491
0.033
6.594∗∗∗−1.968∗∗∗(0.087)−0.341∗∗∗(0.132)−0.107(0.102)−0.422∗∗∗(0.114)

(4)

500
0.006
2.532∗−2.286∗∗∗(0.067)0.176∗∗(0.080)0.209(0.166)

(5)

500
0.005
3.290∗−2.231∗∗∗(0.054)0.131∗(0.072)(6)484−0.001
0.287−0.057(0.107)−2.103∗∗∗(0.099)(7)

Table 5: Regressions of estimated ln σs on demographic characteristics

474
0.043
2.753∗∗∗−0.102(0.081)−0.013(0.080)0.164(0.110)0.237(0.182)−0.293∗∗(0.141)−0.387∗∗∗(0.135)−0.133(0.109)−0.415∗∗∗(0.122)0.094(0.119)0.254(0.190)−0.042(0.106)−0.043(0.107)−1.963∗∗∗(0.151)(8)

6.2

Relationships with the public goods game

Figure 7 presents the scatter plots for R public against each parameter estimate.
Table 6 reports regressions that analyze the relationship between R public and theestimated parameters. The results of the regression analysis also include a model inwhich the demographic variables are added to the independent variables.
We find a negative correlation between the contributions in the public goods
game and the distribution parameter αb. Our results support Hypothesis 1.
Furthermore, we find a positive correlation between contributions and the error
term scale ln σbs .

30

31

0

5

10

15

Contribution in PGG

205

p = .77010

15

Contribution in PGG

20

Figure 7: Scatter plots for contributions in the public goods game

0

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0

5

p = .002

10

15

Contribution in PGG

Note: Left: against estimated α. Center: against estimated τ . Right: against estimated ln σs . Simple regression lines are drawn. p
values are for each slope of the regression lines.

p < .001

0.0

0.4

0.4

0.0

0.6

0.6

0.2

0.8

0.8

0.2

1.0

1.0

ln s

20

Table 6: Regressions on behavior in the public goods gameα estimate

R public
Const.
Demogr. Controls
Observations
Adjusted R2
F Statistic

(1)

(2)

−0.100∗∗∗(0.020)0.635∗∗∗(0.010)−0.117∗∗∗(0.019)0.643∗∗∗(0.039)

No
500
0.047
25.816∗∗∗

Yes
474
0.179
8.924∗∗∗

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

τ estimate

R public
Const.
Demogr. Controls
Observations
Adjusted R2
F Statistic

(1)

(2)

0.007
(0.023)0.394∗∗∗(0.011)−0.010(0.024)0.459∗∗∗(0.046)No437−0.002
0.086

Yes
414
0.049
2.636∗∗∗

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

ln σs estimate

R public
Const.
Demogr. Controls
Observations
Adjusted R2
F Statistic

(1)

(2)

0.220∗∗∗(0.071)−2.158∗∗∗(0.036)

0.257∗∗∗(0.074)−1.969∗∗∗(0.149)

No
500
0.017
9.513∗∗∗

Yes
474
0.065
3.537∗∗∗

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

32

6.3

Relationships with the socioeconomic status of eliteness

Figure 8 shows the cumulative distribution of the parameter estimates by category.
Tables 7–9 report regressions that analyze the relationship between the socioeconomic status variables and estimated parameters.
For education level (R educational1 and R educational2), see the first row of
Figure 8. We observe that the higher the level of education, the lower the value of
αb, i.e., the more equality oriented they tend to be. This may be inconsistent withFisman et al.’s (2015b) finding that Yale law students defined as the elite have agreater αb and are not equality oriented.
For income level (R income1 and R income2), see the second row of Figure 8.
There are no differences among the three groups for α and τ , which is inconsistent
with Fisman et al.’s (2014) finding that being in a low-income group had a negative
effect on αb and a positive effect on τb. We find that ln σbs for respondents classified
in the high-income group is smaller at the 5% significance level.
For employment status (R employment), see the third row of Figure 8. For αbalone, there is a significant difference at the 5% level. However, the result that α
b islower for the employed than for the unemployed is in the opposite direction of theeffect reported by Fisman et al. (2014).
For occupation (R occupationnal1 and R occupationnal2), see the fourth row
of Figure 8. There are no differences among the three groups. The absence ofdifferences between respondents belonging to privileged occupations and others may
not be consistent with Fisman et al.’s (2015b) findings.
In the regression analysis, for the distribution parameter αb, we find that only
R educational2 (master and above) has a negative effect at the 5% significant level(see column (10) in Table 7). For the rescaled substitution parameter τb, we find that
R educational1 (bachelor) has an effect at the 5% significance level (see column(10) in Table 8). Given the sign of the coefficient is negative, it appears that at least

33

those participants that completed their bachelor’s degree are not efficiency oriented.
Ultimately, we obtain no evidence to positively support Hypothesis 2.
Only the effect of R income2 is significant based on the regression analysis (seecolumn (10) in Table 9).

34

R educational∗
Bachelor
Master & Above
Other

0.6

Proportion

Proportion

0.8

0.4
0.2
0.0

1.0

1.0

0.8

0.8

0.6

0.6

Proportion

1.0

0.4
0.2

0.0

0.2

0.4

0.6

0.8

0.0

1.0

p < .001

0.0 0.2

Bachelor
Master & Above
Other
0.4 0.6 0.8 1.0

0.4
0.2
0.0

4.0

p < .001

3.0

Bachelor
Master & Above
Other
2.0
1.0 0.0

ln s

p = .636

R income∗< $30, 000

Middle$90, 000

0.6

Proportion

Proportion

0.8

0.4
0.2
0.0

1.0

1.0

0.8

0.8

0.6

0.6

Proportion

1.0

0.4

< $30, 000

0.2
0.0

0.2

0.4

0.6

0.8

0.0

1.0

p = .711

0.4< $30, 000

0.2

Middle$90, 0000.0

0.0 0.2 0.4 0.6 0.8 1.0

Middle$90, 000

4.0

p = .825

3.0

2.0

ln s

1.0

0.0

p = .026

R employment
Unemployed
Employed

0.6

Proportion

Proportion

0.8

0.4

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Proportion

1.0

p = .036

0.0 0.2 0.4

0.4
0.2

Unemployed
Employed
0.6 0.8 1.0

0.0

4.0

p = .360

3.0

Unemployed
Employed
2.0
1.0 0.0

ln s

p = .406

R occupational∗
Pro/Tech
Admin
Other

0.6

Proportion

Proportion

0.8

0.4
0.2
0.0

1.0

1.0

0.8

0.8

0.6

0.6

Proportion

1.0

0.4
0.2

0.0

0.2

0.4

0.6

p = .279

0.8

1.0

0.0

0.0 0.2 0.4

Pro/Tech
Admin
Other
0.6 0.8 1.0

p = .605

0.4
0.2
0.0

4.0

3.0

2.0

ln s

Pro/Tech
Admin
Other
1.0 0.0

p = .277

Figure 8: Distribution of estimates grouped by R educational∗, R income∗, R employment, and R occupational∗
Note: Left: CDF of estimated α. Center: CDF of estimated τ . Right: CDF of estimated ln σs .
p values are based on the one-way ANOVA.

35

36

No
500
0.038
10.730∗∗∗

Demogr. Controls
Observations
Adjusted R2
F Statistic

Yes
474
0.121
5.635∗∗∗

0.659∗∗∗(0.044)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

0.718∗∗∗(0.022)−0.022(0.028)−0.065∗∗(0.033)−0.087∗∗∗(0.026)−0.141∗∗∗(0.031)

Const.

R occupational2

R occupational1

R employment

R income2

R income1

R educational2

R educational1

(2)

(1)

No
500
−0.0030.341

0.632∗∗∗(0.012)0.022(0.027)−0.000(0.027)

(3)

Yes
474
0.112
5.281∗∗∗

0.648∗∗∗(0.042)−0.015(0.027)0.001(0.028)

(4)

No
500
0.007
4.404∗∗

0.725∗∗∗(0.044)−0.094∗∗(0.045)

(5)

Yes
474
0.114
5.676∗∗∗

0.650∗∗∗(0.057)−0.011(0.047)

(6)

No
500
0.001
1.280

−0.010(0.022)−0.059(0.037)0.643∗∗∗(0.013)(7)

Table 7: Regressions of estimated α on socioeconomic status

Yes
474
0.112
5.278∗∗∗−0.004(0.022)−0.019(0.037)0.643∗∗∗(0.041)

(8)

No
500
0.035
3.589∗∗∗−0.081∗∗∗(0.027)−0.140∗∗∗(0.033)0.001(0.027)0.012(0.027)−0.055(0.047)0.020(0.023)−0.035(0.037)0.762∗∗∗(0.047)

(9)

Yes
474
0.113
4.173∗∗∗−0.023(0.030)−0.070∗∗(0.035)−0.021(0.028)0.010(0.028)−0.006(0.049)0.005(0.023)−0.015(0.037)0.674∗∗∗(0.061)

(10)

37

No
437
0.031
8.006∗∗∗

Demogr. Controls
Observations
Adjusted R2
F Statistic

Yes
414
0.059
2.845∗∗∗

0.500∗∗∗(0.050)Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

0.486∗∗∗(0.025)−0.072∗∗(0.033)−0.037(0.038)−0.116∗∗∗(0.029)−0.101∗∗∗(0.034)

Const.

R occupational2

R occupational1

R employment

R income2

R income1

R educational2

R educational1

(2)

(1)

No
437
−0.0040.193

0.393∗∗∗(0.014)0.015(0.030)−0.008(0.030)

(3)

Yes
414
0.047
2.452∗∗∗

0.463∗∗∗(0.048)−0.011(0.031)
0.011(0.032)

(4)

No
437
−0.0000.841

0.438∗∗∗(0.049)−0.046(0.050)

(5)

Yes
414
0.050
2.668∗∗∗

0.425∗∗∗(0.064)0.040(0.053)

(6)

No
437
−0.0020.503−0.011(0.025)−0.040(0.041)0.401∗∗∗(0.014)(7)

Table 8: Regressions of estimated τ on socioeconomic status

Yes
414
0.046
2.437∗∗∗−0.003(0.025)−0.013(0.042)0.460∗∗∗(0.046)

(8)

No
437
0.021
2.338∗∗−0.118∗∗∗(0.031)−0.104∗∗∗(0.037)0.004(0.030)0.002(0.031)
0.011(0.053)0.008(0.026)−0.021(0.041)0.475∗∗∗(0.052)

(9)

Yes
414
0.051
2.177∗∗∗−0.082∗∗(0.035)−0.048(0.040)−0.010(0.031)
0.013(0.033)0.065(0.056)−0.004(0.027)−0.012(0.042)0.457∗∗∗(0.069)

(10)

38

No
500
−0.0020.453−2.169∗∗∗(0.082)
Yes
474
0.042
2.469∗∗∗−1.907∗∗∗(0.163)−0.107(0.106)−0.021(0.123)−0.010(0.095)0.074(0.112)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

Demogr. Controls
Observations
Adjusted R2
F Statistic

Const.

R occupational2

R occupational1

R employment

R income2

R income1

R educational2

R educational1

(2)

(1)

No
500
0.011
3.695∗∗−2.102∗∗∗(0.044)−0.058(0.096)−0.261∗∗∗(0.096)

(3)

Yes
474
0.050
2.783∗∗∗−1.977∗∗∗(0.157)0.005(0.101)−0.238∗∗(0.102)

(4)

No
500
−0.0010.691−2.286∗∗∗(0.158)0.135(0.162)

(5)

Yes
474
0.041
2.550∗∗∗−2.025∗∗∗(0.212)0.073(0.175)

(6)

No
500
0.001
1.287

−0.103(0.081)
0.094(0.133)−2.136∗∗∗(0.045)(7)

Table 9: Regressions of estimated ln σs on socioeconomic status

Yes
474
0.042
2.473∗∗∗−0.088(0.082)0.062(0.137)−1.948∗∗∗(0.152)

(8)

No
500
0.011
1.798∗−0.007(0.098)0.131(0.119)−0.037(0.099)−0.268∗∗∗(0.098)0.166(0.170)−0.113(0.085)0.075(0.135)−2.260∗∗∗(0.170)

(9)

Yes
474
0.048
2.252∗∗∗−0.098(0.109)0.035(0.129)0.013(0.103)−0.244∗∗(0.104)0.140(0.181)−0.080(0.086)0.061(0.138)−2.039∗∗∗(0.224)

(10)

6.4

Relationships with political attitudes and behaviors

Figure 9 depicts scatter plots for each continuum variable against each parameter
estimate. Figures 10 and 11 plot the cumulative distribution of parameter estimatesby category. Tables 10–12 report regressions that analyze the relationship between
the political attitudes and behaviors variables and the estimated parameters.
For ideological identity (R political), see the first row of Figure 9 and the firstrow of Figure 10. The finding that respondents that self-identify as liberal have alarge αb, indicating selfishness, may not be consistent with Fisman et al.’s (2017)
finding that Democrats have a small αb and value equality.15 We also find that ln σbs
is larger for respondents self-identifying as conservative.
For partisanship (R partisanship), see the second row of Figure 10. For the
distribution of αb, Democrats stochastically dominate Republicans at the 5% significance level. This result is inconsistent with Fisman et al.’s (2017) finding that
Democrats have a small αb and value equality. We also find that ln σbs is larger for
Republican respondents at the 5% significance level.
Regarding approval of Trump’s job (R trump), see the second row of Figure 9 andthe third row of Figure 10. We observe effects in the same direction as the resultsof ideological self-identification and partisanship for αb, τb, and ln σbs . In contrast
to the observation of no significant differences in τb in ideological self-identificationand partisanship, τb is significantly smaller—but not stochastically dominated—for
Trump approvers.
For voting behavior for Biden, see the fourth row of Figure 10. There are nodifferences between Biden and Trump voters for αb, τb, and ln σbs . Our results may
then be inconsistent with Fisman et al.’s (2017) finding that voters for Democratic
candidate Obama in the 2012 election were equality oriented. Interestingly, however,15

According to the Pew Research Center, Democrats are more likely to self-identify as liberal(Oliphant, 2019).

39

44.9% of Republican respondents responded that they voted for Biden, although
95.0% of the Democrats responded that they voted for Biden.16 It seems impossible
to compare behavior in the 2012 and 2020 elections according to partisanship.
For agreement with redistributive policies, see the third row of Figure 9 and theupper row of Figure 11. We observe that the higher the degree of agreement forredistributive policy, the lower the value of αb and τb at the 5% significance level withuncoded data. Under the assumption that α measures the preference for equality–
selfishness, it is consistent that supporters of redistributive policies have smaller αb,indicating an equality orientation.
For agreement with basic income policies, see the fourth row of Figure 9 and thelower row of Figure 11. Compared with the figures for the approval of redistributive
policies, the approval of basic income policies has an effect in the same direction,
but the results are not significant.
In the regression analysis, we find that only R political (liberal) has a positiveeffect on the distribution parameter α
b at the 1% significance level (see column (14) inTable 10). For the rescaled substitution parameter τb and the error term scale ln σbs ,we find that no variable has a significant effect (see column (14) in Tables 11 and
12). Given the only variable with a significant effect, R political, did not display
the expected direction of the effect, we obtain no evidence to support Hypothesis 3.

16

The Pew Research Center reported that some Republicans voted for Biden (Igielnik et al.,2021).

40

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

1

2

3

4

5

6

Ideological identification

7

0.0

ln s

R political (raw)

1

2

3

4

5

6

Ideological identificationp < .001

7

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

1

2

3

4

5

6

7

6

7

6

7

6

7

Ideological identification

p = .062

p < .001

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

1

2

3

4

5

6

Approval for Trump

7

0.0

ln s

R trump (raw)

1

2

3

4

5

6

Approval for Trump

p < .001

7

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

1

2

3

4

5

Approval for Trump

p = .001

p < .001

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

1

2

3

4

5

6

7

0.0

ln s

R redistribution (raw)

1

2

3

4

5

6

7

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

1

2

3

4

5

Agreement for redistribution

Agreement for redistribution

Agreement for redistribution

p = .033

p = .041

p = .664

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

1

2

3

4

5

6

7

0.0

ln s

R basicincome (raw)

1

2

3

4

5

6

7

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

1

2

3

4

5

Agreement for basic income

Agreement for basic income

Agreement for basic income

p = .435

p = .206

p = .478

Figure 9: Scatter plots for raw R political, R trump, R redistribution, and R basicincome
Note: Left: against estimated α. Center: against estimated τ . Right: against estimated ln σs .
Simple regression lines are drawn. p values are for each slope of regression lines. Each plotted
point is jittered in the horizontal direction.

41

R political
Left
NeutralRight

0.6

Proportion

Proportion

0.8

0.4
0.2
0.0

1.0

1.0

0.8

0.8

0.6

0.6

Proportion

1.0

0.4
0.2

0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.0 0.2 0.4 0.6

p < .001

Left
NeutralRight
0.8 1.0

0.4
0.2
0.0

4.0

p = .263

3.0

2.0

ln s

Left
NeutralRight
1.0 0.0

p < .001

R partisanship
Republicans
Democrats

0.6

Proportion

Proportion

0.8

0.4

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Proportion

1.0

0.0 0.2 0.4

p = .017

0.4
0.2

Republicans
Democrats
0.6 0.8 1.0

0.0

4.0

p = .986

3.0

Republicans
Democrats
2.0
1.0 0.0

ln s

p = .024

R trump
Trump apvl
Disapprove

0.6

Proportion

Proportion

0.8

0.4

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Proportion

1.0

0.0 0.2 0.4

p < .001

0.4
0.2

Trump apvl
Disapprove
0.6 0.8 1.0

0.0

4.0

p = .016

3.0

Trump apvl
Disapprove
2.0
1.0 0.0

ln s

p < .001

R voting
Trump
Biden

0.6

Proportion

Proportion

0.8

0.4

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

p = .558

0.8

1.0

Proportion

1.0

0.0

0.2 0.4 0.6

p = .522

Trump
Biden
0.8 1.0

0.4
0.2
0.0

4.0

3.0

2.0

ln s

Trump
Biden
1.0 0.0

p = .087

Figure 10: Distribution of estimates grouped by R political, R partisanship, R trump, and R voting
Note: Left: CDF of estimated α. Center: CDF of estimated τ . Right: CDF of estimated ln σs .
p values are based on the one-way ANOVA.

42

R redistribution
Agree w/ redist
Disagree

0.6

Proportion

Proportion

0.8

0.4

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Proportion

1.0

p = .029

0.0 0.2

0.4
0.2

Agree w/ redist
Disagree0.4 0.6 0.8 1.0

0.0

4.0

p = .126

3.0

Agree w/ redist
Disagree2.0
1.0 0.0

ln s

p = .710

R basicincome
Agree w/ BI
Disagree

0.6

Proportion

Proportion

0.8

0.4

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.2

0.2

0.0

0.0

0.0

0.2

0.4

0.6

p = .2670.8

1.0

Proportion

1.0

0.0 0.2 0.4

Agree w/ BI
Disagree
0.6 0.8 1.0

p = .759

0.4
0.2
0.0

4.0

3.0

Agree w/ BI
Disagree2.0
1.0 0.0

ln s

p = .308

Figure 11: Distribution of estimates grouped by R redistribution and R basicincome
Note: Left: CDF of estimated α. Center: CDF of estimated τ . Right: CDF of estimated ln σs .
p values are based on the one-way ANOVA.

43

44
No
500
0.087
48.794∗∗∗

Demogr. Controls
Observations
Adjusted R2
F Statistic

Yes
474
0.146
7.240∗∗∗

0.623∗∗∗(0.040)Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

0.584∗∗∗(0.012)

0.098∗∗∗(0.023)0.138∗∗∗(0.020)Const.

R basicincome

R redistribution

R voting

R trump

R partisanshipR political

(2)

(1)

No
500
0.001
1.280

0.622∗∗∗(0.016)0.023(0.020)

(3)

Yes
474
0.114
5.688∗∗∗

0.637∗∗∗(0.041)

0.009(0.021)

(4)

No
500
0.048
26.350∗∗∗0.685∗∗∗(0.014)−0.100∗∗∗(0.020)

(5)

Yes
474
0.123
6.120∗∗∗

0.672∗∗∗(0.043)−0.052∗∗(0.023)

(6)

No
500
−0.0000.824

0.650∗∗∗(0.019)−0.020(0.022)(7)

Table 10: Regressions of estimated α on political attitudes and behaviors

Yes
474
0.115
5.705∗∗∗

0.647∗∗∗(0.042)−0.014(0.023)

(8)

45
No
500
0.008
4.771∗∗

Demogr. Controls
Observations
Adjusted R2
F Statistic

Yes
474
0.117
5.815∗∗∗

0.656∗∗∗(0.042)−0.030(0.024)(10)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

0.672∗∗∗(0.020)−0.050∗∗(0.023)

Const.

R basicincome

R redistribution

R voting

R trump

R partisanshipR political

(9)

No
500
0.000
1.236

−0.026(0.023)0.655∗∗∗(0.020)

(11)

Yes
474
0.114
5.679∗∗∗−0.007(0.024)0.645∗∗∗(0.043)

(12)

No
500
0.119
12.229∗∗∗0.148(0.025)−0.034(0.025)−0.052∗∗(0.023)−0.056∗∗(0.026)−0.035(0.026)−0.005(0.027)0.696∗∗∗(0.028)

∗∗∗

(13)

Yes
474
0.155
5.833∗∗∗

0.116∗∗∗(0.027)−0.018(0.026)−0.037(0.025)−0.048∗(0.027)−0.034(0.027)0.008(0.028)0.684∗∗∗(0.046)(14)

Table 10: Regressions of estimated α on political attitudes and behaviors (cont’d)

46
No
437
0.002
1.821

0.384∗∗∗(0.014)
Yes
414
0.051
2.694∗∗∗

0.464∗∗∗(0.046)−0.026(0.028)0.032(0.023)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

Demogr. Controls
Observations
Adjusted R2
F Statistic

Const.

R basicincome

R redistribution

R voting

R trump

R partisanshipR political

(2)

(1)

No
437
−0.0020.004

0.395∗∗∗(0.017)−0.001(0.023)

(3)

Yes
414
0.054
2.807∗∗∗

0.474∗∗∗(0.047)−0.036(0.024)

(4)

No
437
0.011
5.816∗∗

0.422∗∗∗(0.016)−0.053∗∗(0.022)

(5)

Yes
414
0.049
2.624∗∗∗

0.461∗∗∗(0.048)−0.004(0.027)

(6)

No
437
0.005
3.057∗

0.427∗∗∗(0.021)−0.044∗(0.025)(7)

Table 11: Regressions of estimated τ on political attitudes and behaviors

Yes
414
0.053
2.764∗∗∗

0.475∗∗∗(0.047)−0.034(0.026)

(8)

47
No
437
0.003
2.356

0.423∗∗∗(0.022)−0.039(0.025)

Yes
414
0.050
2.682∗∗∗

0.469∗∗∗(0.048)−0.023(0.027)(10)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

Demogr. Controls
Observations
Adjusted R2
F Statistic

Const.

R basicincome

R redistribution

R voting

R trump

R partisanshipR political

(9)

No
437
−0.0020.095−0.008(0.026)0.400∗∗∗(0.022)

(11)

Yes
414
0.049
2.623∗∗∗

0.004(0.027)0.456∗∗∗(0.048)

(12)

Yes
414
0.047
2.139∗∗∗−0.013(0.032)−0.030(0.030)−0.025(0.030)−0.028(0.032)−0.017(0.032)0.033(0.032)
0.493∗∗∗(0.053)0.017(0.029)−0.003(0.029)−0.061∗∗(0.026)−0.067∗∗(0.030)−0.026(0.031)
0.028(0.031)
0.468∗∗∗(0.032)
No
437
0.017
2.281∗∗

(14)

(13)

Table 11: Regressions of estimated τ on political attitudes and behaviors (cont’d)

48

No
500
0.026
14.428∗∗∗Demogr. Controls
Observations
Adjusted R2
F Statistic

Yes
474
0.046
2.742∗∗∗−1.939∗∗∗(0.151)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

−2.054∗∗∗(0.045)−0.139(0.088)−0.279∗∗∗(0.073)

Const.

R basicincome

R redistribution

R voting

R trump

R partisanshipR political

(2)

(1)

No
500
0.006
3.801∗−2.074∗∗∗(0.056)−0.142∗(0.073)

(3)

Yes
474
0.042
2.578∗∗∗−1.941∗∗∗(0.154)−0.056(0.078)

(4)

No
500
0.027
15.104∗∗∗−2.293∗∗∗(0.050)0.276∗∗∗(0.071)

(5)

Yes
474
0.047
2.784∗∗∗−2.053∗∗∗(0.159)0.149∗(0.086)

(6)

No
500
−0.0010.726−2.108∗∗∗(0.068)−0.069(0.080)(7)

Table 12: Regressions of estimated ln σs on political attitudes and behaviors

Yes
474
0.041
2.563∗∗∗−1.941∗∗∗(0.155)−0.048(0.084)

(8)

49
No
500
−0.0020.138−2.181∗∗∗(0.071)

0.031(0.082)

Yes
474
0.041
2.536∗∗∗−1.966∗∗∗(0.157)0.005(0.088)(10)

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01.

Demogr. Controls
Observations
Adjusted R2
F Statistic

Const.

R basicincome

R redistribution

R voting

R trump

R partisanshipR political

(9)

No
500
0.000
1.042

0.085
(0.083)−2.222∗∗∗(0.072)

(11)

Yes
474
0.041
2.564∗∗∗

0.052(0.089)−1.991∗∗∗(0.158)

(12)
∗∗

No
500
0.031
3.703∗∗∗−0.197(0.093)−0.023(0.093)0.183∗∗(0.085)0.032(0.096)−0.027(0.099)0.142(0.100)−2.270∗∗∗(0.104)

(13)

Yes
474
0.040
2.104∗∗∗−0.111(0.103)−0.002(0.098)0.121(0.096)0.011(0.104)−0.015(0.103)0.088(0.106)−2.060∗∗∗(0.176)(14)

Table 12: Regressions of estimated ln σs on political attitudes and behaviors (cont’d)6.5

Effective and ineffective variables

We conducted multiple comparisons according to the procedure described in Section
4 and conservatively identified variables with and without effect.
Tables 13–15 provide the regression results for all variables on the distribution
parameter αb, the rescaled substitution parameter τb, and the error term scale ln σbs .
The adjusted R2 of the regression is 0.208 for αb, 0.045 for τb, and 0.064 for ln σbs .
Column “p” in each table provides the p values and column “q” the adjusted pvalues (q values) using the Benjamini–Hochberg procedure. Column “TOST p” ineach table reports the p values of the equivalence test and column “TOST q” adjusts
these p values (q values) using the Benjamini–Hochberg procedure. The rows forvariables for which the q value of the t test is below 0.005, the upper limit of thefalse discovery rate we set are highlighted in red, and the rows for variables for which
the q value of the equivalence test are less than this value are highlighted in blue.
Figures 12–14 provide a visual comparison of the effect size of each variable. The
thick and thin lines indicate confidence intervals of 95% and 99.5%, respectively.
The dot-dashed vertical line represents the SESOI set so that the mean individual
power is 0.8 for the upper limit of the false discovery rate is 0.005, and the dashedvertical line represents the SESOI set for the upper limit of the false discovery rate
is 0.05.17
After selecting the variables, only R public and R political on the distribution parameter αb were found to be effective. On αb, R religious2, R children, R metro, R educational1, R income2, R occupational1, R partisanship, and R basicincome were found to be ineffective. On the rescaled substitution parameter τb, R age, R gender, R race1, R marital1, R metro, R public, R income1,
R income2, R occupational1, R occupational2, R political, R partisanship,

17

Determined for αb, τb, and ln σbs to be 0.08, 0.1, and 0.3, respectively.

50

R voting, R trump, R redistribution, and R basicincome were found to be ineffective. On the error term scale ln σbs , R age, R gender, R children, R metro, R income1, R occupational1, R political, R partisanship, R voting, R redistribution,
and R basicincome were found to be ineffective.
It is noteworthy that for τb, all measures of political attitude and behavior are
shown to be ineffectual. While conclusive findings cannot be made on the rescaled
substitution parameter τ , particularly regarding whether it truly reflects preferences
for efficiency, it appears reasonable to conjecture that the correlations between parameter τ and political attitudes/behavior are not robust. Further precision inmeasurement or additional data is necessary to scrutinize relationships between parameters and other variables in greater depth.

51

52

0.02190.02090.02880.04730.03730.03540.0286
0.03470.0316
0.04880.02740.02790.01940.02870.03380.02640.02680.04640.02230.0356
0.02650.02520.02690.02500.02670.02700.0611

0.05290.0475−0.05480.0290−0.00190.0825−0.00440.0345−0.0434−0.0581−0.01790.0019−0.1115−0.0095−0.0405−0.0296−0.0027
0.0028−0.0053−0.0324
0.0999−0.0146−0.0545−0.0433−0.03450.02000.7120

[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[

0.00990.0063−0.1115−0.0640−0.0752
0.0130−0.0607−0.0337−0.1056−0.1539−0.0717−0.0530−0.1496−0.0659−0.1069−0.0815−0.0553−0.0884−0.0492−0.10240.0478−0.0640−0.1074−0.0925−0.0870−0.0330
0.5919pq,0.0959 ] 0.0160 0.1239,
0.0886 ] 0.0238 0.1239,
0.0018 ] 0.0578 0.2147,
0.1220 ] 0.5407 0.7698,
0.0715 ] 0.9602 0.9602,
0.1520 ] 0.0201 0.1239,
0.0518 ] 0.8766 0.9602,
0.1027 ] 0.3208 0.5957,
0.0187 ] 0.1705 0.4926,
0.0378 ] 0.2345 0.5082,
0.0360 ] 0.5148 0.7698,
0.0567 ] 0.9465 0.9602, −0.0733 ] 0.0000 0.0000,
0.0470 ] 0.7413 0.9602,
0.0259 ] 0.2314 0.5082,
0.0223 ] 0.2634 0.5268,
0.0500 ] 0.9207 0.9602,
0.0941 ] 0.9511 0.9602,
0.0385 ] 0.8111 0.9602,
0.0377 ] 0.3641 0.6310,
0.1520 ] 0.0002 0.0024,
0.0349 ] 0.5625 0.7698, −0.0016 ] 0.0434 0.1882,
0.0059 ] 0.0844 0.2743,
0.0180 ] 0.1969 0.5082,
0.0731 ] 0.4582 0.7446,
0.8322 ] 0.0000

95% Conf. Interval

Note: Observations: 474, Adjusted R2 = 0.208, F (26, 447) = 5.777 (p < .001).

R age
R genderR race1
R race2
R race3
R religious1
R religious2
R religious3
R marital1
R marital2
R children
R metroR public
R educational1
R educational2
R income1
R income2
R employment
R occupational1
R occupational2
R political
R partisanship
R voting
R trump
R redistribution
R basicincome
Const.

Std. Err.

Coef

Table 13: Regressions of estimated α

0.00880.0031
0.04130.05440.00300.26220.00020.02140.02600.16820.00080.00010.63030.00050.02850.00220.00010.01420.00000.02110.42380.00020.03070.00700.00430.0009TOST p0.01640.00730.05120.06430.00730.28400.00130.03270.03760.19010.00280.00110.63030.00210.03900.00650.00100.02460.00010.03270.44070.00120.03980.01410.00930.0028TOST q53

0.02700.02600.03440.05690.04590.04380.0341
0.04310.03830.05830.03330.03300.02450.03600.04170.03180.03330.05610.02700.04270.03220.03030.03240.03030.03270.0326
0.0737−0.0133−0.0257−0.0227−0.0009
0.0358−0.0623−0.0460
0.03590.01270.0705−0.0683−0.0058−0.0207−0.0800−0.0459−0.0171
0.00330.0667−0.0007−0.0122−0.0190−0.0279−0.0267−0.0274−0.00420.03790.4850
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
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[
[
[
[
[

−0.0664−0.0769−0.0904−0.1127−0.0545−0.1484−0.1130−0.0489−0.0625−0.0441−0.1337−0.0707−0.0689−0.1508−0.1278−0.0796−0.0622−0.0436−0.0538−0.0963−0.0823−0.0876−0.0905−0.0871−0.0686−0.02610.3400pq,0.0398 ] 0.6229 0.8524,
0.0255 ] 0.3248 0.7565,
0.0450 ] 0.5097 0.8283,
0.1109 ] 0.9870 0.9870,
0.1261 ] 0.4364 0.7565,
0.0238 ] 0.1557 0.7565,
0.0209 ] 0.1774 0.7565,
0.1206 ] 0.4058 0.7565,
0.0879 ] 0.7399 0.9597,
0.1850 ] 0.2273 0.7565, −0.0028 ] 0.0409 0.5319,
0.0591 ] 0.8601 0.9870,
0.0275 ] 0.3992 0.7565, −0.0091 ] 0.0271 0.5319,
0.0360 ] 0.2713 0.7565,
0.0454 ] 0.5905 0.8524,
0.0689 ] 0.9209 0.9870,
0.1769 ] 0.2351 0.7565,
0.0523 ] 0.9779 0.9870,
0.0718 ] 0.7751 0.9597,
0.0442 ] 0.5543 0.8477,
0.0317 ] 0.3580 0.7565,
0.0370 ] 0.4102 0.7565,
0.0322 ] 0.3668 0.7565,
0.0601 ] 0.8971 0.9870,
0.1019 ] 0.2452 0.7565,
0.6300 ] 0.0000

95% Conf. Interval

Note: Observations: 414, Adjusted R2 = 0.045, F (26, 387) = 1.750 (p = .014).

R age
R genderR race1
R race2
R race3
R religious1
R religious2
R religious3
R marital1
R marital2
R children
R metroR public
R educational1
R educational2
R income1
R income2
R employment
R occupational1
R occupational2
R political
R partisanship
R voting
R trump
R redistribution
R basicincome
Const.

Std. Err.

Coef

Table 14: Regressions of estimated τ

0.00000.00000.00060.00940.01570.04870.00460.01090.00070.13430.02280.00010.00000.06370.01650.00010.00000.11180.00000.00210.00020.00020.00050.00020.00000.0015TOST p0.00000.00010.00120.01360.02040.05510.00710.01500.00140.13430.02700.00020.00000.06900.02040.00040.00020.11620.00000.00350.00050.00050.00100.00050.00020.0026TOST q54

0.08490.08120.11190.18360.14480.13720.11110.13470.12270.18930.10630.10830.07530.11150.13120.10250.10390.18020.08650.13830.10300.09760.10450.09720.10370.10470.2373−0.1271−0.0023
0.14670.2296−0.1778−0.3387−0.1807−0.35780.13750.2523−0.0732
0.01620.2406−0.1069−0.02230.0287−0.2156
0.1387−0.0634
0.1016−0.0810−0.02480.04970.1279−0.0202
0.0753−2.1505
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
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[

−0.2939−0.1620−0.0732−0.1313−0.4623−0.6083−0.3991−0.6224−0.1037−0.1197−0.2822−0.1966
0.0927−0.3259−0.2801−0.1727−0.4199−0.2155−0.2334−0.1703−0.2834−0.2167−0.1556−0.0631−0.2240−0.1305−2.6169
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,

0.03970.15730.36670.59050.1068−0.0691
0.0377−0.09310.37860.62430.13570.22890.38850.11220.23550.2302−0.01130.49290.10670.37350.12130.16700.25500.31880.18350.2810−1.6842

95% Conf. Interval
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]

0.13500.97730.19050.21180.22020.01390.10460.00820.26330.18330.49140.88130.00150.33820.86510.77920.03860.44200.4646
0.46320.43170.7994
0.63450.18880.84540.47260.0000pNote: Observations: 474, Adjusted R2 = 0.064, F (26, 447) = 2.244 (p = .001).

R age
R genderR race1
R race2
R race3
R religious1
R religious2
R religious3
R marital1
R marital2
R children
R metroR public
R educational1
R educational2
R income1
R income2
R employment
R occupational1
R occupational2
R political
R partisanship
R voting
R trump
R redistribution
R basicincome
Const.

Std. Err.

Coefq
0.52050.97730.52050.52050.52050.12080.52050.10630.57040.52050.67250.9166
0.03870.67250.91660.91660.25090.67250.67250.67250.67250.91660.82480.52050.9166
0.6725

Table 15: Regressions of estimated ln σs0.00100.00000.01510.19150.07170.35430.03010.40550.02010.23360.00150.00030.02380.00570.00260.00020.04700.08200.00010.01880.00140.00010.00060.00360.00020.0014

TOST p0.00340.00000.02620.21640.08870.36850.04120.40550.03070.25310.00360.00130.03440.01060.00570.0012
0.0611
0.09690.00090.03060.00360.00090.00220.00720.00120.0036TOST qR_ageR_genderR_race1
R_race2
R_race3
R_religious1
R_religious2
R_religious3
R_marital1
R_marital2
R_children
R_metroR_public
R_educational1
R_educational2
R_income1
R_income2
R_employment
R_occupational1
R_occupational2
R_political
R_partisanship
R_voting
R_trumpR_redistribution
R_basicincome
−0.2

−0.1

0.0

0.1

Coefficient estimate

Figure 12: Confidence intervals for α estimate

55

0.2

R_ageR_genderR_race1
R_race2
R_race3
R_religious1
R_religious2
R_religious3
R_marital1
R_marital2
R_children
R_metroR_public
R_educational1
R_educational2
R_income1
R_income2
R_employment
R_occupational1
R_occupational2
R_political
R_partisanship
R_voting
R_trumpR_redistribution
R_basicincome
−0.2

−0.1

0.0

0.1

Coefficient estimate

Figure 13: Confidence intervals for τ estimate

56

0.2

R_ageR_genderR_race1
R_race2
R_race3
R_religious1
R_religious2
R_religious3
R_marital1
R_marital2
R_children
R_metroR_public
R_educational1
R_educational2
R_income1
R_income2
R_employment
R_occupational1
R_occupational2
R_political
R_partisanship
R_voting
R_trumpR_redistribution
R_basicincome
−0.8

−0.4

0.0

0.4

Coefficient estimate

Figure 14: Confidence intervals for ln σs estimate

57

0.8

6.6

Correlations between error term scale and utility parameters

Table 16 shows the correlation coefficients for the full sample and for the sample
excluding participants classified as purely selfless or selfish. We obtain a negative
correlation between the error term scale ln σbs and α
b in the population, which includes purely selfless/selfish participants. Furthermore, in the population excluding
participants purely selfless/selfish, we find a positive correlation between ln σbs andτb, while the negative correlation against α
b disappears. A careful inspection of thescatter plot between αb and ln σbs shown in the left panel of Figure 15 reveals that
there is a cluster around αb = 1 and ln σbs = −3.5, and that this cluster brings anegative correlation between αb and ln σbs .
These observations suggest that it is necessary to control the estimates of utility
parameters and the existence of purely selfless/selfish participants to analyze therelationship between ln σbs and the variables. Comparing columns (1) and (2) inTable 17, we can see that controlling for αb improves the explanatory power of themodel in terms of adjusted R2 . Comparing columns (3) and (4) in Table 17, which
are the analyses for the sample excluding purely selfless/selfish participants, we cansee that controlling for τb improves the adjusted R2 . Comparing columns (1) and(5) in Table 17, we can see that controlling for the indicator of participants purely
selfless (Iselfless ) and selfish (Iselfish ) also improves the adjusted R2 .
The negative correlation between R public and ln σbs is reduced by controllingfor αb or Iselfish , or by excluding purely selfless/selfish participants. Similarly, thenegative correlation between R religious1 and ln σbs weakens (but is not wholly
diminished) through adding control variables. Conversely, for R religious3 andR income2, the coefficients increase in magnitude through the addition of control
variables. These two variables then provide relatively more explanatory power forthe error term scale ln σbs than the other variables.
58

Table 16: Correlation coefficients between estimates
α
b
α
b
τb
ln σbs

.36
−.21

τb

ln σbs

.19

−.02
.31

−.04

0.0

0.0

1.0

1.0

ln s

ln s

Note: The lower triangular part of the matrix is for all participants,
and the upper triangular part (italicized) is for only participants that
reject the null hypothesis αb = 0, 1.

2.0
3.0
4.0

2.0
3.0
4.0

0.0 0.2 0.4 0.6 0.8 1.0
0, 1

Not

0, 1

0.0 0.2 0.4 0.6 0.8 1.0
0, 1

Not

0, 1

Figure 15: Scatter plots between error term scale and estimated utility parameters
Note: Left: against estimated α. Right: against estimated τ . Individuals for whom the null
hypothesis that their α estimate is equal to 0 or 1 cannot be rejected by a one-tailed test at the
10% significance level are identified by a cross. The contour lines represent kernel density
estimates.

59

Table 17: Regression of estimated ln σs controlling for utility parameter estimates
(1)

(2)

(3)

(4)

(5)

(6)
0.015
(0.206)0.687∗∗∗(0.162)−0.638∗(0.360)−1.024∗∗∗(0.138)−0.083(0.080)0.016(0.077)0.119(0.106)0.218(0.172)−0.206(0.136)−0.234∗(0.130)−0.177∗(0.104)−0.337∗∗∗(0.127)0.076(0.115)0.111(0.178)0.022(0.101)
0.044(0.102)0.129∗(0.074)−0.068(0.105)−0.026(0.123)0.005(0.096)−0.204∗∗(0.098)0.128(0.169)−0.077(0.081)
0.092(0.130)−0.024(0.098)−0.037(0.092)0.009(0.099)0.134(0.091)
0.003(0.098)0.101(0.099)−2.418∗∗∗(0.270)
No
474
0.177
4.396∗∗∗∗∗−0.388(0.183)αb0.941∗∗∗(0.144)τb−0.127(0.085)−0.002(0.081)0.147(0.112)0.230(0.184)−0.178(0.145)−0.339∗∗(0.137)−0.181(0.111)−0.358∗∗∗(0.135)0.137(0.123)0.252(0.189)−0.073(0.106)0.016(0.108)0.241∗∗∗(0.075)−0.107(0.111)−0.022(0.131)
0.029(0.102)−0.216∗∗(0.104)0.139(0.180)−0.063(0.087)0.102(0.138)−0.081(0.103)−0.025(0.098)0.050(0.104)0.128(0.097)−0.020(0.104)0.075(0.105)−2.151∗∗∗(0.237)−0.107(0.085)0.016(0.081)
0.125(0.112)0.241(0.183)−0.178(0.144)−0.307∗∗(0.137)−0.182(0.111)−0.344∗∗(0.134)0.121(0.122)0.230(0.189)−0.080(0.106)0.017(0.108)0.197∗∗(0.078)−0.111(0.111)−0.038(0.131)
0.017(0.102)−0.217∗∗(0.104)0.140(0.180)−0.065(0.086)0.089(0.138)−0.042(0.104)−0.030(0.097)0.029(0.105)0.111(0.097)−0.034(0.103)0.083(0.104)−1.874∗∗∗(0.270)−0.079(0.080)0.006(0.078)0.131(0.102)0.103(0.169)−0.205(0.137)−0.300∗∗(0.130)−0.190∗(0.101)−0.338∗∗∗(0.128)0.068(0.114)0.095(0.173)−0.054(0.099)0.012(0.098)0.073(0.073)−0.061(0.107)−0.024(0.124)0.073(0.095)−0.207∗∗(0.099)0.243(0.167)−0.064(0.080)0.140(0.127)0.014(0.096)−0.061(0.090)−0.048(0.097)−0.044(0.090)−0.041(0.097)0.138(0.097)−2.033∗∗∗(0.219)−0.067(0.076)0.030(0.074)0.152(0.097)0.104(0.161)−0.239∗(0.130)−0.242∗(0.124)−0.146(0.097)−0.372∗∗∗(0.122)0.056(0.108)0.029(0.165)0.010(0.095)0.018(0.093)0.093(0.069)0.014(0.103)0.019(0.118)0.089(0.090)−0.210∗∗(0.094)0.180(0.159)−0.063(0.076)0.152(0.121)
0.032(0.091)−0.034(0.086)−0.023(0.092)−0.018(0.086)−0.037(0.093)0.102(0.092)−2.490∗∗∗(0.220)−0.408(0.343)−0.759∗∗∗(0.114)−0.098(0.081)
0.002(0.078)0.106(0.107)0.215(0.176)−0.180(0.138)−0.269∗∗(0.131)−0.199∗(0.106)−0.312∗∗(0.129)0.085(0.117)0.163(0.181)−0.021(0.102)0.037(0.104)0.116(0.074)−0.106(0.106)−0.045(0.125)0.002(0.098)−0.198∗∗(0.099)0.164(0.172)−0.074(0.083)0.088(0.132)−0.025(0.099)−0.050(0.094)−0.011(0.100)0.113(0.093)−0.010(0.099)0.121(0.101)−2.083∗∗∗(0.227)

No
474
0.064
2.244∗∗∗

No
474
0.071
2.345∗∗∗

Yes
414
0.018
1.296

Yes
414
0.114
2.969∗∗∗

No
474
0.147
3.901∗∗∗

Iselfless
Iselfish
R age
R genderR race1
R race2
R race3
R religious1
R religious2
R religious3
R marital1
R marital2
R children
R metroR public
R educational1
R educational2
R income1
R income2
R employment
R occupational1
R occupational2
R political
R partisanship
R voting
R trump
R redistribution
R basicincome
Const.
Excluded
Observations
Adjusted R2
F Statistic

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01. Participants who classified purely selfless or selfish are
excluded if “Excluded” is Yes.

60

7

Concluding remarks

Using theoretical arguments, we have previously believed that the extended CES
utility function, proposed by Inukai et al. (2022), measures preferences for equality–
selfishness in the distribution parameter α, and efficiency in the substitution parameter ρ (or τ ). In this paper, we studied empirical implications of the parameters byconducting a modified dictator game followed by a public goods game and a survey
among participants in Amazon Mechanical Turk.
Upon comparing the results for 26 variables with conservative criteria, we foundthat only two variables, behavior in the public goods game, R public, and ideologicalself-identity, R political, exhibit a statistically significant effect on the distribution
parameter αb. Based on the observation that a smaller the value of αb corresponds
to a larger contribution to the public goods game, we conclude that the supposition
that α reflects selfishness is reasonable. For the self-identification of ideology, R political, we observed a greater αb for the liberal respondents. However, thisobservation contradicts the findings of Fisman et al. (2017). This suggests that theresults of previous studies on the relationship between the parameters of the utility
function and other variables may not be generalizable, even if this discrepancy is
attributed to differences between subject pools (Kohler, 2019).
Furthermore, the equivalence tests revealed that the measures of political attitudes and behavior are irrelevant for the rescaled substitution parameter τb. To
conclude from the results of our regression analysis, it would be premature to suggest that τ has nothing to do with behavior in public goods games, socioeconomic
measures, and political attitudes/behavior. As noted in Inukai et al. (2022), τ estimation tends to involve greater inferential uncertainty than α, even though bothparameters are measured on the same scale. This property inevitably makes it difficult to identify effects in the regression analysis of τb. To more thoroughly examine
the relationship between τb and various variables, it is necessary to attempt to reduce61the inferential uncertainty by increasing the sample size. Additionally, for the MDG
experimental tasks that each participant takes, it may be beneficial to increase thenumber of tasks or to redesign them with a more efficient composition for estimation.

62

A

Instruction for the modified dictator game

INSTRUCTION
Introduction
This HIT is an academic experiment about decision-making.
You will be paid a <xed fee for completing tasks and a bonus.
When you complete the HIT, you will receive 2.50 USD.
And you will receive a bonus after the other workers in our experiment
have completed the tasks.
The bonus you will earn depends partly on your decisions and the otherparticipants' decisions, and partly on chance.
The HIT consists of two experiments (Part I and II) and several
questionnaire surveys.
The entire HIT may be complete within 40 minutes.
At the end of the HIT, you will receive a completion SECRET KEY andinput it on your MTurk screen.
Details of how to make decisions and receive payments will be provided
below.

63

Part I
Decision Problems
In part I, you will repeatedly participate in 50 independent decision
problems in the same form.
You will decide how to allocate your tokens to others.
The tokens in part I will be exchanged for USD at the following rate:
1 token = 2¢.
In each decision problem, you will be in a pair with another person
chosen randomly from the other workers in this experiment.
You will then be asked to allocate tokens between yourself and the other.
Note that the person who will be paired with you may differ in eachproblem.

How to Operate
See the Dgure below.

64When the experiment begins, a graph with a thick black line is displayed,
as shown in the 8gure.

65

Clicking or tapping anywhere on the line makes an orange point (!)
appear.

66

This orange point on the line represents a token allocation.
The horizontal axis of the graph measures the amount you HOLD, and thevertical axis measures the amount you PASS to the other.
For example, as illustrated in the Cgure above, the point represents anallocation in which you HOLD 22.5 tokens and PASS 56.2 tokens.
Thus, if you choose this allocation, you will receive 22.5 tokens, and theother will receive 56.2 tokens.
You can see the actual numbers: the amount you HOLD and the amount
you PASS at the bottom of the graph.

67

Points A, B, and C in the 0gures below show an example.
Point A means you will receive nothing, and the other will receive 90
tokens.

68

Point B means you will receive 60 tokens, and the other will receivenothing.

69

Point C means you and the other will receive equally 36 tokens.

70In each problem, you will be asked to decide a HOLD/PASS combination
by dragging the orange point with a mouse.
You can adjust the allocation by pressing

+

and

−

buttons at theOKbutton.

bottom of the graph.
To move on to the next problem, press the

Next, a new graph with a different line will be displayed.
Then you will be asked to allocate with a new person in anotherindependent problem.
You will need to complete this process 50 times.

Payments for You
The following describes how the amount of your bonus in part I is
calculated.
The computer randomly chooses 10 of the 50 problems.
For each problem chosen, the computer randomly selects a worker to bepaired with you and randomly chooses whether to adopt your decision or
the paired person's decision.
If the decision you made is adopted, you receive tokens that you answer
to HOLD.
If, on the other hand, the paired person's decision is adopted, you receive
tokens that they answer to PASS.
As we have already explained, each token will be worth two cents, andless than a cent will be rounded up.

71Trying Operation
You can manipulate the graph in the box below.
Practice the operation here.
In the graph below, three points are displayed as examples:• ■: HOLD nothing and PASS 60 tokens;• ■: HOLD 36 tokens and PASS 36 tokens;• ■: HOLD 90 tokens and PASS nothing.
In the experiment, these points will not be displayed.

When you want to start the experiment after practice, move to the bottomof this page, enter your Amazon Mechanical Turk Worker ID, and press
the

BEGIN

button.

72

[Move to the bottom of this page]

CLICK or TAP on the thick line.

When you click or tap on the thick line,
the values are displayed HERE.

73

Begin the Part I
Please make sure that your Amazon Mechanical Turk Worker ID is 9lled
in the box below.
It is NOT your E-mail address.
If we do not have your correct Worker ID, we will not be able to pay you.

↓
Input your MTurk Worker ID

XXX

BEGIN

74

B

Questions and variable definitions

B.1

Socioeconomic status of elitenessEducation level Participants answered the question “What is the highest level ofschool you have completed or the highest degree you have received?” by choosing oneof these options: “Less than high school degree,” “High school graduate (high school
diploma or equivalent including GED),” “Some college but no degree,” “Associatedegree in college (2-year),” “Bachelor’s degree in college (4-year),” “Master’s degree,” “Doctoral degree,” and “Professional degree (JD, MD).” All participants wereforced to answer this question. We defined two dummy variables, R educational1
and R educational2, representing the choice of 1) “Associate degree in college”
or “Bachelor’s degree in college,” and 2) “Master’s degree,” “Doctoral degree,” or“Professional degree,” respectively.

Income level Which category represents the total combined income of all members of your family (living here) during the past 12 months before taxes?” bychoosing one of these options: “Less than $10,000,” “$10,000 to $19,999,” “$20,000
to $29,999,” . . ., “$90,000 to $99,999,” “$100,000 to $149,999,” and “$150,000 or
more.” All participants were forced to answer this question. We defined two dummy
variables, R income1 and R income2, representing 1) the annual income is less than
$30,000 and 2) not less than $90,000, respectively.
Employment status Participants answered the question “What is your current
employment status?” by choosing one of these options: “Full-time employee,” “Parttime employee,” “Self-employed or business owner,” “Unemployed and looking forwork,” “Student,” and “Not in labor force (for example: retired, or full-time parent).” All participants were forced to answer this question. We defined a dummy
variable R employment representing the choice of “Full-time employee,” “Part-time75employee,” or “Self-employed or business owner.”

Occupational prestige Participants who indicated that they were employed (i.e.,
R employment = 1) answered the “To which of the following occupational groups
do you belong?” by choosing one of these options: “Professional and technical(for example doctor, teacher, engineer, artist, accountant, nurse),” “Farm worker(for example farm laborer, tractor driver),” “Higher administrative (for example
banker, executive in big business, high government official, union official),” “Service(for example restaurant owner, police officer, waitress, barber, caretaker),” “Sales(for example sales manager, shop owner, shop assistant, insurance agent, buyer),”“Skilled worker (for example foreman, motor mechanic, printer, seamstress, tool anddie maker, electrician),” “Clerical (for example secretary, clerk, office manager, civil
servant, bookkeeper),” “Unskilled worker (for example laborer, porter, unskilled
factory worker, cleaner),” “Semi-skilled worker (for example bricklayer, bus driver,cannery worker, carpenter, sheet metal worker, baker),” and “Other.” These options presented to the participants were randomly ordered for each participant. All
participants were forced to answer this question. We defined two dummy variables,
R occupational1 and R occupational2, representing the choice of 1) “Professional
and technical,” and 2) “Higher administrative,” respectively.

B.2

Political attitudes and behaviors

Think of self as liberal Participants answered the question “Here is a 7-point
scale on which the political views that people might hold are arranged from very
liberal (left) to very conservative (right). Where would you place yourself on thisscale?” by choosing between 1 and 7. All participants were forced to answer thisquestion. We defined a dummy variable R political that is an indicator for liberalindividuals whose answer is less than 4 on the 7-point scale.

76

Democrats Participants answered the question “Do you think of yourself as closer
to the Republican or Democratic party?” by choosing one of these options: “Democrats,”“Democratic leaners,” “Equally close to Democrats and Republicans,” “Republicanleaners,” and “Republicans.” All participants were forced to answer this question.
We defined a dummy variable R partisanship that is an indicator for Democrats
whose answer is “Democrats” or “Democratic leaners.”

Trump job approval Participants answered the question “Looking back on Donald Trump’s four years (2017–2021) in office, in general, do you approve or disapproveof his job as President?” on a 7-point scale: “Strongly disapprove (1),” “Disapprove,” “Somewhat disapprove,” “Neutral,” “Somewhat approve,” “Approve,” and“Strongly approve (7).” All participants were forced to answer this question. We
defined a dummy variable R trump that is an indicator for individuals whose answer
is “Somewhat approve,” “Approve,” or “Strongly approve.”

Voting for Biden in 2020 Participants answered the question “In the 2020 presidential election, for which candidate would you have voted?” by choosing one ofthese options: “Donald Trump,” “Joe Biden,” and “Vote for neither/Other.” Options for Trump and Biden were randomly ordered by participants. All participants
were forced to answer this question. We defined a dummy variable R voting that is
an indicator for individuals whose answer is “Joe Biden.”Favor for redistributive policies Participants answered the question “Do you
agree or disagree that the government in Washington should redistribute wealthfrom the rich to the poor?” on a 7-point scale: “Strongly disagree (1),” “Disagree,”“Somewhat disagree,” “Neutral,” “Somewhat agree,” “Agree,” and “Strongly agree(7).” All participants were forced to answer this question. We defined a dummy
variable R redistribution that is an indicator for individuals whose answers are77

“Somewhat agree,” “Agree,” or “Strongly agree.”Favor for basic income policies Participants answered the question “Do you
agree or disagree about such a system that would automatically guarantee a certain
basic income to all permanent residents? (Andersson and Kangas, 2007)” on a 7point scale: “Strongly disagree (1),” “Disagree,” “Somewhat disagree,” “Neutral,”“Somewhat agree,” “Agree,” and “Strongly agree (7).” All participants were forced
to answer this question. We defined a dummy variable R basicincome that is anindicator for individuals whose answer is “Somewhat agree,” “Agree,” or “Strongly
agree.”

B.3

Demographic characteristics

Age Participants chose their year of birth using a dropdown menu. We calculated
age from the responses and conducted a regression analysis with standardized values
R age. All participants were forced to answer this question.
Gender Participants answered the question “With respect to gender, how do you
self-identify?” by choosing one of these options: “Female,” “Male,” “Other,” and“Choose not to answer.” All participants were forced to answer this question. We
defined a dummy variable R gender representing female identity for all participants
except those who chose not to answer.

Ethno-racial identity In response to the question “With respect to race or ethnicity, how do you self-identify?”, participants were asked to choose from the following options: “American Indian or Alaska Native,” “Native Hawaiian or PacificIslander,” “Black or African American,” “White,” “Hispanic or Latino,” “Asian,”and “Other.” These options presented to participants were randomly ordered foreach participant. Participants were allowed to choose more than one applicable op-78

tion. They could also move on to the next question without selecting one, i.e., without answering the question. We defined three dummy variables, R race1, R race2,
and R race3, representing the unique choice of 1) “Black or African American,” 2)“Hispanic or Latino,” and 3) “Asian,” respectively, for all participants except those
who did not choose any of the options.

Religious identity Participants answered the question “What religion do you belong to or identify yourself most close to?” by choosing one of these options: “Jew,”“Hindu,” “Protestant,” “Roman Catholic,” “Orthodox (Russian/Greek/etc.),” “Muslim,” “Buddhist,” “Other Christian (Evangelical/Pentecostal/Free church/etc.),”“Other,” “Do not belong to a religion,” and “Choose not to answer.” These options presented to the participants were randomly ordered for each participant. All
participants were forced to answer this question. We defined three dummy variables, R religious1, R religious2, and R religious3, representing the choice of
1) “Protestant,” 2) “Roman Catholic,” and 3) “Do not belong to a religion,” respectively, for all participants except those who chose not to answer.

Marital status Participants answered the question “With respect to gender, how
do you self-identify?” by choosing one of these options: “Married,” “Widowed,”“Divorced,” “Separated,” and “Never Married.” All participants were forced to answer this question. We defined three dummy variables, R marital1 and R marital2,
representing the choice of 1) “Married” and 2) “Widowed,” “Divorced,” or “Separated,” respectively.

Parenthood Participants answered the question “Do you have children living withyou?” by choosing yes or no. All participants were forced to answer this question.
We defined a dummy variable R children representing those who answered yes.

79

Metro residents We asked participants for their ZIP code or Federal Information Processing System (FIPS) county code to obtain information on whether they
lived in a metro area.18 Participants could choose not to answer both the ZIP and
FIPS codes. We converted the ZIP codes into FIPS codes and then obtained theUnited States Department of Agriculture (USDA) rural–urban continuum code corresponding to each participant’s FIPS code. We defined participants with a USDA
rural–urban continuum in the range of 1–3 as residents of the metro area, represented
by a dummy variable R metro.19

18

In the early wave of the experiment, we only asked for the ZIP code. We received a message
from one respondent that they did not want to answer the question because they believe that the
ZIP code is personally identifiable information. In response to this request, we decided to add theoption of answering with the FIPS code, which is more difficult to identify, instead of answeringwith the ZIP code.
19
We retrieved the USDA rural–urban continuum code at https://www.ers.usda.gov/
data-products/rural-urban-continuum-codes.aspx. Our procedure to identify metro residents was followed Huff and Tingley (2015).

80

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