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Electronic supplementary material for article entitled “Equality, Efficiency, and Sufficiency: Responding to Multiple Parameters of Distributive Justice During Charitable Distribution”
1 Distributive justice task
1.1 Task details
In the distributive justice task, participants made choices about the hypothetical distribution of meal loss to economically disadvantaged American adults. It was described to participants that each recipient had been promised 28 meals but that some of these meals needed to be taken away. Participants were required to choose between two options that differed in how this loss of meals was distributed amongst the three recipients in each trial. Each trial began with a screen containing a box labelled with the text ‘Start’. When participants clicked on this box using the mouse, a screen was displayed like that shown in Figure S1. Participants made their choice by clicking in one of the boxes in the upper corners of the screen. Once participants made their choice, the number of meals taken away from each participant was displayed in text for four seconds before the next trial began. Participants received $10 for completing this task. In order to encourage a sense of realism, participants were made aware that the researchers would donate up to $15 to a charity for poverty relief in conjunction with their performance of the task.
Fig. S1 Screen display during a trial of the distributive justice task. Recipient images are omitted here to maintain privacy. Clicking in the box in the upper left corner would take 15 meals away from Melissa. Clicking in the box in the upper right corner would take 11 meals away from each of Naomi and Jan. The mouse cursor began in the bottom centre of the screen.
The current experiment employed a different range of dilemmas than that used in the original version of the task (Hsu et al. 2008). Thirty-two trials were analysed in total, shown in Table S1. Trials were presented in a unique random order for each participant. The images of the recipients in each trial were frontal shots of the head, matched within trials and between dilemma types on age, sex, posture, and other physical characteristics. Images were sourced from the Center for Vital Longevity database (Minear and Park 2004), the MUCT database (Milborrow et al. 2010), the Spacek database (Spacek), the Database of Faces (AT&T Laboratories Cambridge), and the Psychological Image Collection at Stirling (PICS).
Table S1 Trial list for the distributive justice task
Dilemma Type / Trial / Meals LostaRecipient 1 / Recipient 2 / Recipient 3
LowEff-LowEqu / 1 / 15 / 13 / 3
2 / 15 / 13 / 5
3 / 19 / 17 / 3
4 / 19 / 17 / 5
5 / 23 / 21 / 3
6 / 23 / 21 / 5
7 / 27 / 25 / 3
8 / 27 / 25 / 5
LowEff-HighEqu / 1 / 15 / 8 / 8
2 / 15 / 9 / 9
3 / 19 / 10 / 10
4 / 19 / 11 / 11
5 / 23 / 12 / 12
6 / 23 / 13 / 13
7 / 27 / 14 / 14
8 / 27 / 15 / 15
MedEff-HighEqu / 1 / 15 / 10 / 10
2 / 15 / 11 / 11
3 / 19 / 12 / 12
4 / 19 / 13 / 13
5 / 23 / 14 / 14
6 / 23 / 15 / 15
7 / 27 / 16 / 16
8 / 27 / 17 / 17
HighEff-HighEqu / 1 / 15 / 13 / 13
2 / 15 / 15 / 15
3 / 19 / 15 / 15
4 / 19 / 17 / 17
5 / 23 / 17 / 17
6 / 23 / 19 / 19
7 / 27 / 19 / 19
8 / 27 / 21 / 21
aIn each dilemma, participants chose to take meals away from either recipient 1 or both recipient 2 and recipient 3.
1.2 Calculation of the inequality-aversion parameter
Each trial of the distributive justice task involved a choice between two meal loss distributions, each of which was associated with a different degree of efficiency and a different degree of equality. A meal loss distribution is denoted as (x1, x2, x3), where x1 is the meal loss of recipient 1, x2 is the meal loss of recipient 2, and x3 is the meal loss of recipient 3. For example, a trial might involve choosing between either taking 15 meals away from recipient 1, denoted as (15, 0, 0), or taking 13 and 5 meals away from recipient 2 and recipient 3, respectively, denoted as (0, 13, 5).
The efficiency associated with each option was quantified in terms of the total number of meals lost. The marginal efficiency, M, of a choice between two options was obtained by the following formula, where Mc is the number of meals lost in the chosen option and Mu is the number of meals lost in the unchosen option.
The inequality associated with each option was quantified using the Gini coefficient (Hsu et al. 2008; Xu 2003). This coefficient is a common measure of the inequality of resource distribution amongst a population, and represents the discrepancy between the observed distribution and a perfectly even distribution (Xu 2003). Gini coefficient values range between 0 and 1, with greater values representing greater inequality. The formula for calculating the Gini coefficient of discrete data is as follows, where n is the number of data points in the distribution and x is the value of a data point.
The distribution associated with each option within the distributive justice task has three values, for example, (15, 0, 0), such that n = 3. Therefore, the Gini coefficient formula can be expanded to the following, where, as before, (x1, x2, x3) represents the number of meals taken away from each of the three recipients in a given trial.
The marginal inequality, G, associated with each choice was calculated using the following formula, where Gc is the inequality of the chosen option and Gu is the inequality of the unchosen option.
Perceived utility for a particular distribution was modelled as Ui = M – iG,where Ui is the perceived utility of the distribution for subject i, M is the efficiency of the distribution, G is the inequality of the distribution, and i is proportional to the extent to which equality is valued by subject i. Higher values of are considered to be indicative of a more inequality-averse preference during the distributive justice task, and was labelled as the inequality-aversion parameter. This model is designed to capture the trade-off between efficiency and equality required during the distributive justice task. A key assumption made by the model is that efficiency is valued linearly. See Hsu et al. (2008) for a discussion of this assumption of linearity. The marginal utility of a choice within a given trial is then represented by the following formula.
The less even distribution in each trial is denoted in vector form as x1, and the more even distribution is denoted as x2.The perceived utility of x1is denoted as u(x1), and the perceived utility of x2is denoted as u(x2).The probability that a participant would choose the x1 allocation rather than the x2 allocation in a given trial was calculated using the following formula (Hsu et al. 2008).
The parameter was evaluated via the following log likelihood function, where yi = 1 for trials in which the x1 distribution was selected, and yi = 0 for trials in which the x2 distribution was selected.
A value of was obtained for each participant by finding the maximum of this log likelihood function when the participant’s own values of M,G,and yi for each trial were inserted. The maximum was found by applying the Nelder-Mead simplex algorithm (Nelder and Mead 1965) using the fminsearch function in Matlab™ R2007a, Version 7.4.0.287 (The Mathworks, Inc., Natick, MA). Given that the fminsearch function finds the minimum of a function, the log likelihood function was multiplied by 1 before implementing the Nelder-Mead simplex algorithm. This meant that the value calculated described the maximum of the original log likelihood function. The algorithm was applied from 10 starting points selected at random within the range of 50 to 50, based broadly on the distribution of reported in (Hsu et al. 2008). The value of associated with the highest likelihood value was used as a participant’s final measure of behaviour on the task. Finding the maximum in this way gave an individual value of that best fit the model of inequality aversion to the data for each participant. It should be noted that there were no alternate solutions produced for any of the participants, as the values of estimated from different starting points converged.
Supplementary material references
AT&T Laboratories Cambridge. Database of faces. Available via AT&T archive. Retrieved 1 Nov 2010
Hsu, M., Anen, C., & Quartz, S. R. (2008). The right and the good: Distributive justice and neural encoding of equity and efficiency. Science, 320, 1092–1095.
Milborrow, S., Morkel, J., & Nicolls, F (2010) The MUCT Landmarked Face Database. Available via the Pattern Recognition Association of South Africa. Retrieved 1 Nov 2010
Minear, M., & Park, D. C. (2004). A lifespan database of adult facial stimuli. Behavior Research Methods, Instruments, & Computers, 36, 630–633.
Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308–313.
PICS. Face database. Available via PICS. Retrieved 1 Nov 2010
Spacek L Face database. Available via University of Essex. Retrieved 1 Nov 2010
Xu K (2003) How has the literature on Gini’s index evolved in the past 80 years? Available via Department of Economics at Dalhousie University Working Papers Archive. Retrieved 5 Dec 2011