The experiment consists of two treatments—the BDM treatment and the VA treatment—implemented in a between-subject design and paid under random lottery incentives. Each treatment consists of two parts, and subjects know that only one part is payoff-relevant:Footnote ³

1. 1. In part 1, we elicit the distributional preferences of subjects with the Equality Equivalence Test (EET) by Kerschbamer (Reference Kerschbamer2015), and then, we expose them to an incentivised survey.

2. 2. In part 2, a lottery ticket (giving either $w=12$ EUR or 0 EUR with equal probabilities) is auctioned off under either the BDM or the VA. Subjects’ initial endowment in part 2 is $e=10$ EUR.Footnote ⁴

EET: This procedure exposes each subject to a number (in our case 10) choices between two allocations, each specifying a payoff for the subject and a payoff for a randomly assigned anonymous second subject. In half of the choices, there is advantageous inequality (the deciding subject is ahead in monetary terms), in the other half, there is disadvantageous inequality (the deciding subject is behind in monetary terms). From the choices of the subject, x- and y-scores measuring the benevolence in the two domains of inequality are calculated. A higher x-score (y-score) means more benevolence in the domain of disadvantageous (advantageous) inequality. These scores jointly determine the distributional type of the subject.

BDM: In the BDM, each subject i is asked to submit a bid $b_i$ , then a random price $p_i$ is drawn. The allocation rule is:

1. (i) if $b_i>p_i$ : subject i buys the lottery ticket at price $p_i$ ;

2. (ii) if $b_i<p_i$ : subject i keeps the endowment; and

3. (iii) if $b_i=p_i$ : either (i) or (ii) is implemented with equal probability.

VA: In the VA, subjects are randomly assigned to pairs. In each pair, the subjects—denoted i and j with $i\ne j$ —are asked to submit bids $b_i$ and $b_j$ . The allocation rule is:

1. (i) if $b_i>b_j$ : subject i buys the lottery ticket at price $b_j$ , while subject j keeps the endowment;

2. (ii) if $b_i<b_j$ : subject j buys the lottery ticket at price $b_i$ , while subject i keeps the endowment; and

3. (iii) if $b_i=b_j$ : either (i) or (ii) is implemented with equal probability.

Consider a subject i of distributional type $d\in \left(\text{self},,,\text{alt},,,\text{spite}\right)$ , where self stands for selfish, alt for altruistic, and spite for spiteful. Let $f_i^d\left(·\right)$ be strictly increasing and normalise $f_i^d\left(0\right)\equiv 0$ . In the BDM, the unique admissible bid for subject i of type d, $b_i^d$ , is implicitly defined by

(2.1) $\begin{array}{c}\frac{1}{2}{f}_i^d\left(e,+,w,-,b_i^d\right)+\frac{1}{2}{f}_i^d\left(e,-,b_i^d\right)=f_i^d\left(e\right).\end{array}$

Let $(x_i,x_j)$ denote the final monetary allocation and $u_i^d\left(x_i,,,x_j\right)$ the associated utility of subject i of distributional type d. In the VA, the utility for a selfish subject is—as in the BDM—of the form $u_i^{\text{self}}\left(x_i,,,x_j\right)=f_i^{\text{self}}\left(x_i\right)$ . Thus, $b_i=b_i^{\text{self}}$ is a weakly dominant strategy for selfish in the VA. Sincere bidding is not necessarily a weakly dominant strategy for non-selfish subjects in the VA. An altruist i’s utility increases in the monetary payoff of j, i.e., $u_i^{\text{alt}}\left(x_i,,,x_j\right)=f_i^{\text{alt}}\left(x_i\right)+g_i^{\text{alt}}\left(x_j\right)$ , where $g_i^{\text{alt}}(·)$ is strictly increasing. Conversely, under spite, we have $u_i^{\text{spite}}\left(x_i,,,x_j\right)=f_i^{\text{spite}}\left(x_i\right)-g_i^{\text{spite}}\left(x_j\right)$ , where $g_i^{\text{spite}}(·)$ is again strictly increasing. Note that our formulation allows for a possible dependence of a subject’s risk preferences on her distributional type. That is, we explicitly allow the $f_i^d\left(·\right)$ function to depend on the distributional archetype of the subject. We assume that if $x_j$ is good (bad), i.e., altruism (spite), i is more (less) risk-averse in j’s monetary gains than in her own. Empirical evidence includes, e.g., Chakravarty et al. (Reference Chakravarty, Harrison, Haruvy and Rutstrom2011) or Mengarelli et al. (Reference Mengarelli, Moretti, Faralla, Vindras and Sirigu2014). In extending the distributional preferences (as defined here for certain outcomes) to a lottery framework, we follow the expected utility approach. As pointed out, among others, by Fudenberg and Levine (Reference Fudenberg and Levine2012) and Saito (Reference Saito2013), an alternative procedure for extending distributional preferences to lotteries is to replace the certain income in the utility function with its expected value. This would allow for ex-ante fairness, while the expected utility extension does not. While the distinction between ex-ante and ex-post fairness is interesting and important, in general, it seems less relevant in our context. The reason is that our focus is on the change in the behaviour of distributional archetypes who have preferences that are monotonic in the material payoff of the other subject (i.e., selfish, altruists, and spiteful), while the ex-ante vs ex-post fairness discussion is most relevant for distributional archetypes who have preferences that are non-monotonic in the material payoff of the other subject (i.e., fairness-minded or inequality averse subjects).Footnote ⁵ See Fudenberg and Levine (Reference Fudenberg and Levine2012) and Saito (Reference Saito2013) for a discussion.

Restricting ourselves to admissible bids, we can prove (see Appendix A for details):

Proposition 1.

In the VA,

1. (i) a selfish subject never has an incentive to deviate from $b_i^{\text{self}}$ ;

2. (ii) an altruistic subject never has an incentive to overbid relative to $b_i^{\text{alt}}$ , and might have an incentive to underbid;

3. (iii) a spiteful subject never has an incentive to underbid relative to $b_i^{\text{spite}}$ , and might have an incentive to overbid.

From Proposition Footnote 1 and the fact that selfish and altruistic subjects combined comprise the majority in experimental data (see Kerschbamer Reference Kerschbamer2015), we derive:

Theoretical prediction 1. Comparing bids across treatments,

1. (i) selfish subjects will not change their bids, while altruistic (spiteful) types will underbid (overbid) in VA relative to the BDM;

2. (ii) in the aggregate, we expect underbidding in the VA relative to the BDM.