Boris Van Leeuwen
email: b.vanleeuwen@uvt.nl

Ingela Alger
email: .ingela.alger@tse-fr.eu

Gijs Van De Kuilen

Wieland M Üller

Arthur Schram

Sigrid Suetens

ESTIMATING SOCIAL PREFERENCES AND KANTIAN MORALITY IN STRATEGIC INTERACTIONS

Keywords: JEL codes: C49, C72, C9, C91, D03, D84 Social preferences, Kantian morality, other-regarding preferences, morality, experiment, structural estimation, finite mixture models

Theoretical work suggests that a form of Kantian morality has evolutionary foundations. To investigate the relative importance of Kantian morality and social preferences, we run a laboratory experiment on strategic interaction in social dilemmas.

We structurally estimate social preferences and Kantian morality at the individual and aggregate level. We observe considerable heterogeneity in preferences. A finite mixture analysis shows that the subject pool is well described as consisting of two types. One combines inequity aversion and Kantian morality, while the other combines spite and Kantian morality. The value of adding Kantian morality to well-established preference classes is also evaluated.

Behavioral and experimental economics has over the past decades provided a host of insights about the motivations that drive human behavior in social dilemmas. Notwithstanding the wealth of preference classes that have been considered-notably, altruism [START_REF] Becker | A theory of social interactions[END_REF], warm glow [START_REF] Andreoni | Impure altruism and donations to public goods: A theory of warmglow giving[END_REF], inequity aversion [START_REF] Fehr | A theory of fairness, competition, and cooperation[END_REF][START_REF] Bolton | Erc: A theory of equity, reciprocity, and competi-tion[END_REF], reciprocity [START_REF] Rabin | Incorporating fairness into game theory and economics[END_REF][START_REF] Charness | Understanding social preferences with simple tests[END_REF][START_REF] Dufwenberg | A theory of sequential reciprocity[END_REF][START_REF] Falk | A theory of reciprocity[END_REF], guilt aversion [START_REF] Charness | Promises and partnership[END_REF][START_REF] Battigalli | Guilt in games[END_REF], and image concerns [START_REF] Bénabou | Incentives and prosocial behavior[END_REF][START_REF] Ellingsen | Pride and prejudice: The human side of incentive theory[END_REF])-recent theoretical work has shown that yet another type of preferences is strongly favored by evolutionary forces. The novel element is a form of Kantian moral concern, so called Homo moralis preferences [START_REF] Alger | Homo moralis-preference evolution under incomplete information and assortative matching[END_REF][START_REF] Alger | Evolution of preferences in structured populations: genes, guns, and culture[END_REF]. The Kantian moral concern induces the individual to evaluate each course of action in the light of what material payoff (s)he would achieve, should others choose the same course of action. The purpose of this paper is to examine the explanatory power of such Kantian moral concerns, when these are assumed to be at work alongside consequentialistic concerns such as altruism and inequity aversion. We do this by way of conducting an experimental study.

The laboratory experiment consists of letting each subject choose strategies in three classes of two-player social dilemmas: sequential prisoners' dilemmas, mini trust games, and mini ultimatum bargaining games. In such sequential games one subject moves before the other, and it is this feature that allows us to distinguish consequentialistic motives from Kantian morality (à la Homo moralis, [START_REF] Alger | Homo moralis-preference evolution under incomplete information and assortative matching[END_REF]. Indeed, since each subject is told that he stands an equal chance of being a first-and a second-mover, Kantian morality would make him attach some value to the material payoff he would obtain if he played against himself. By contrast, a subject with purely consequentialistic preferences would make the subject attach value solely to the material payoff distribution that he expects to realize, given his beliefs about the opponent's strategy. 1 1 It is well known that the ability to control for subjects' beliefs when trying to identify their preferences is important [START_REF] Bellemare | Measuring inequity aversion in a heterogeneous population using experimental decisions and subjective probabilities[END_REF][START_REF] Miettinen | Revealed preferences in a sequential prisoners' dilemma: a horse-race between six utility functions[END_REF]. This is particularly true here, for Kantian morality reduces the sensitivity to beliefs. In the extreme case of an individual who would be driven entirely by the Kantian moral concern, the beliefs about the opponent's strategy would indeed be irrelevant, for such an individual would simply choose the "right thing to do." Hence, information about subjects' beliefs is crucial to distinguish Kantian moral concerns from consequentialistic ones. Accordingly, instead of hypothesizing subjects' beliefs about the behavior of their opponents (for example by some equilibrium hypothesis), we elicit each subject's belief in each strategic interaction. In further robustness checks, we 2 Positing a utility function with three parameters capturing attitudes towards unfavorable inequity, favorable inequity, and the Kantian moral concern, we use the observed individual choices and reported beliefs in 18 different games (six games in each game class) to structurally estimate the preference parameter values for each individual subject, using a standard random utility model. 2 The use of such structural models has become more commonplace in experimental and behavioral economics, including the estimation of social preferences [START_REF] Dellavigna | Structural Behavioral Economics[END_REF]. We also perform aggregate estimations, using a finite mixture approach, the same as that used by [START_REF] Bruhin | The many faces of human sociality: Uncovering the distribution and stability of social preferences[END_REF] in their statistical analysis of social preferences. 3 Not surprisingly, the estimations at the level of the individual subjects reveal a lot of heterogeneity in preferences. While many subjects appear to be averse to unfavorable inequity, some appear to be either indifferent or either like or dislike favorable inequity.

The behavior of most subjects is compatible with some concern for Kantian morality, and allowing for this motivational factor significantly improves the fit of the model to the data. Kantian morality further appears in all the aggregate estimations. The representative agent in the subject pool combines "behindness aversion" with Kantian morality. 4 Models with two or three types provide a much better fit than the representative agent model. Our finite mixture estimations thus capture the heterogeneity in a tractable way.

The two-types model has one type that combines inequity aversion with Kantian morality, while the other type combines "spite" or "competitiveness" -an aversion to being behind and taste for being ahead -with Kantian morality.

Importantly, allowing for Kantian morality substantially improves the fit of the model.

Model selection criteria and out-of-sample predictions indeed favor models with Kantian morality over those without. Comparing our main estimates to those based on a utility function with negative reciprocity as in [START_REF] Charness | Understanding social preferences with simple tests[END_REF] instead of the Kantian moral concern further shows that the value added of Kantian morality is in the same ballpark as such well-established motives as inequity aversion, altruism, and reciprocity.

Moreover, the out-of-sample predictions are more accurate with preferences that combine also impose rational expectations instead.

2 Social image concerns [START_REF] Bénabou | Incentives and prosocial behavior[END_REF] are muted because subjects are anonymously and randomly matched.

3 See also [START_REF] Bardsley | The Experimetrics of Public Goods: Inferring Motivations from Contributions[END_REF], [START_REF] Iriberri | Elicited beliefs and social information in modified dictator games: What do dictators believe other dictators do?[END_REF] and [START_REF] Breitmoser | Estimation of social preferences in generalized dictator games[END_REF], who use related mixture models to capture heterogeneity in social preferences. 4 Interestingly, this is in line with the theoretical prediction of [START_REF] Alger | Evolution of preferences in structured populations: genes, guns, and culture[END_REF], who show in a general model that preferences that combine material self-interest, a Kantian moral concern and other-regard at the material payoff level is what should be expected in most human populations.

3

Kantian morality with attitudes towards the realized payoff distribution, than any of the preferences without the Kantian moral concern.

Our paper fits in the large literature that estimates or tests models of social preferences. 5 In relation to this literature, our main contribution is that we allow for the possibility of Kantian morality as part of the motivation behind subjects' choices, in addition to social preferences. Closest to our work is the paper by [START_REF] Miettinen | Revealed preferences in a sequential prisoners' dilemma: a horse-race between six utility functions[END_REF], who also allow for this possibility. 6 Our study is similar to theirs in two respects. First, both experiments rely on sequential games (our experimental design was indeed inspired by theirs in this respect). Second, in both experiments the subjects' beliefs about opponents' choices are elicited and used as controls in the empirical estimations. The key difference between ours and their study is that our data set is much richer: we collect data on individual choices in 18 strategic interactions while in their study each subject faces one single sequential prisoners' dilemma. Our rich data set gives us access to a rich set of empirical tools. In particular, while [START_REF] Miettinen | Revealed preferences in a sequential prisoners' dilemma: a horse-race between six utility functions[END_REF] compare the explanatory power of six alternative utility functions, which involve either a consequentialistic, a reciprocity, or a Kantian concern, our data set allows us to estimate preference parameters at the individual level, and to apply finite mixture methods in order to detect the presence of common preference types that combine social preferences and Kantian morality. As indicated by our results, most subjects indeed appear to have such complex preferences. Furthermore, our data enables use of out-of-sample predictions to evaluate the explanatory power of the estimated preference types.

The remainder of this paper is organized as follows. Section 2 describes the experimental design and introduces the class of preferences we estimate, and Section 3 presents our econometric approach. The results are presented in Section 4, wherein we also report robustness checks and several measures of the value added of Kantian morality in our experiment. Section 5 concludes. 5 See, for example, [START_REF] Palfrey | Anomalous behavior in public goods experiments: how much and why?[END_REF]; [START_REF] Andreoni | Giving according to garp: An experimental test of the consistency of preferences for altruism[END_REF]; [START_REF] Charness | Understanding social preferences with simple tests[END_REF]; [START_REF] Engelmann | Inequality aversion, efficiency, and maximin preferences in simple distribution experiments[END_REF]; [START_REF] Bardsley | The Experimetrics of Public Goods: Inferring Motivations from Contributions[END_REF]; [START_REF] Fisman | Individual preferences for giving[END_REF]; [START_REF] Bellemare | Measuring inequity aversion in a heterogeneous population using experimental decisions and subjective probabilities[END_REF]; [START_REF] Blanco | A within-subject analysis of otherregarding preferences[END_REF][START_REF] Dellavigna | Testing for altruism and social pressure in charitable giving[END_REF]; [START_REF] Breitmoser | Estimation of social preferences in generalized dictator games[END_REF]; [START_REF] Iriberri | Elicited beliefs and social information in modified dictator games: What do dictators believe other dictators do?[END_REF][START_REF] Ottoni-Wilhelm | Why do people give? testing pure and impure altruism[END_REF] and, for a recent survey, see [START_REF] Cooper | Other-regarding preferences: A selective survey of experimental results[END_REF]. Closest to our work is the recent study by [START_REF] Bruhin | The many faces of human sociality: Uncovering the distribution and stability of social preferences[END_REF], who use the same finite mixture approach as we do, but who do not consider Kantian morality. 6 See also [START_REF] Capraro | Do the Right Thing: Preferences for Moral Behavior, Rather Than Equity or Efficiency per se, Drive Human Prosociality[END_REF], who evaluate the explanatory power of Homo moralis preferences in standard games; however, and by contrast to our experiment and that by [START_REF] Miettinen | Revealed preferences in a sequential prisoners' dilemma: a horse-race between six utility functions[END_REF], they rely on framing. More generally, economists are increasingly seeking to evaluate the explanatory power of non-consequentialistic motives; see, e.g., [START_REF] Bénabou | Eliciting moral preferences: theory and experiment[END_REF].

2 The experiment: game protocols, preferences, and procedures

Game protocols

In the experiment, subjects play three types of well-known game protocols, illustrated in Figure 1: the Sequential Prisoner's Dilemma protocol (SPD), shown in Figure 1a, the mini Trust Game protocol (TG), shown in Figure 1b, and the mini Ultimatum Game protocol (UG), shown in Figure 1c. 7 We use the standard notation for prisoners' dilemmas, where R stands for "reward", S for "sucker's payoff", T for "temptation", and P for "punishment", and we throughout assume T > R > P > S.

The objective of the experiment is to test whether Kantian morality (à la Homo moralis, [START_REF] Alger | Homo moralis-preference evolution under incomplete information and assortative matching[END_REF] can help explain the choices subjects make in these game protocols. A subject with such Kantian morality evaluates each strategy in the light of what his/her material payoff would be if, hypothetically, the opponent were to choose the same strategy. This requires that the interaction is symmetric. To symmetrize the game protocols in Figure 1-which are asymmetric with one first-mover and one second-mover-we make it clear to the subjects that they are equally likely to be drawn to play in each player role. This defines a symmetric (meta) game protocol, in which "nature" first draws the role assignment, with equal probability for both assignments, and then the players learn their respective roles. The game tree corresponding to this game protocol for the SPD is shown in Figure 2. A behavior strategy consists of specifying (potentially randomized) choices at all decision nodes in this game protocol. Let x = (x 1 , x 2 , x 3 ) denote the behavior strategy of subject i in this game tree: x 1 is the probability that i plays C as a first mover,

x 2 the probability that i plays C as a second mover following play C by the opponent, and

x 3 the probability that i plays C as a second mover following play D by the opponent.

Likewise, let y = (y 1 , y 2 , y 3 ) denote the behavior strategy used by the opponent (subject j). Each strategy pair (x, y) determines the realization probability η (x,y) (γ) of each play γ of the game protocol, where a play is a sequence of moves through the game tree, from its "root" to one of its end nodes (see Figure 2). For example:

η (x,y) ((1, C, C)) = x 1 •y 2 2 and η (x,y) ((2, D, C)) = (1-y 1 )•x 3 2 .
Turning to the two other game protocols, when the trust game protocol is symmetrically randomized, a behavior strategy is a vector, x = (x 1 , x 2 ) ∈ [0, 1] 2 , where x 1 is the

i = 1 j = 2 (R, R) C y 2 (S, T ) D C x 1 j = 2 (T , S) C y 3 (P , P ) D D 1 2 j = 1 i = 2 (R, R) C x 2 (T , S) D C y 1 i = 2 (S, T ) C x 3 (P , P ) D D 1 2
probability with which i invests (selects (I)) and x 2 the probability with which i gives back something (selects G) if the first-mover invested. When the ultimatum game protocol is symmetrically randomized, a behavior strategy is a vector,

x = (x 1 , x 2 ) ∈ [0, 1] 2 ,
where x 1 is the probability with which i proposes an equal sharing (selects (E)), and x 2 the probability with which i accepts an unequal sharing (selects (A)). Like in the SPD game protocol, for both the TG and the UG protocols we denote by y = (y 1 , y 2 ) the strategy of i's opponent j, and write η (x,y) (γ) to denote the probability of each play γ of the game protocol at hand.

Having formally defined the game protocols, we are in a position to define the utility function that we posit.

Social preferences and Kantian morality

Let the expected utility of a subject i playing against a subject j be

u i (x, y) = (1 -κ i ) • γ η (x,y) (γ) • π i (γ) (1)
-α i • γ η (x,y) (γ) • max 0, π ij (γ) -π i (γ) -β i • γ η (x,y) (γ) • max 0, π i (γ) -π ij (γ) +κ i • γ η (x,x) (γ) • π i (γ) ,
where x and y are i's and j's behavior strategy, respectively, π i is i's material utility following play γ and π ij is j's material utility following play γ. The utility function in (1) nests many familiar utility functions in the literature. Clearly, setting all three parameters to zero, α i = β i = κ i = 0, represents pure self-interest and thus amounts to the classical Homo oeconomicus. The [START_REF] Fehr | A theory of fairness, competition, and cooperation[END_REF] model of inequity aversion is obtained by setting α i ≥ β i > 0 and κ i = 0. One obtains [START_REF] Becker | A theory of social interactions[END_REF] model of pure altruism by setting κ i = 0 and α i = -β i , for some β i ∈ (0, 1). 8 Here β i is the individual's "degree of altruism", the weight placed on the other subject's material utility, while the weight 1β i is placed on own material utility. Pure Homo moralis preferences are obtained by setting α i = β i = 0 and κ i ∈ (0, 1). Here κ i is the individual's "degree of Kantian morality", the weight placed on the material utility that would be obtained if both subjects in the interaction at hand played x, the strategy used by individual i, while the weight 1κ i is placed on own material utility, given the strategy profile (x, y) effectively played. The utility function in (1) also nests the [START_REF] Charness | Understanding social preferences with simple tests[END_REF] model without reciprocity. In Section 4.4 we extend the utility function to also accommodate reciprocity as formalized in [START_REF] Charness | Understanding social preferences with simple tests[END_REF].

Because each subject in our experiment faces risky decisions (the monetary payoff depends on the decision of the opponent, which the subject does not know when making the decisions), we allow for risk aversion. Thus, the term π i (γ) in equation ( 1) is the Bernoulli function value that the individual attaches to his or her monetary payoff under 8 See also the note by [START_REF] Engelmann | How not to extend models of inequality aversion[END_REF] on extending inequity aversion models to incorporate altruism.

play γ. We will call π i (γ) the individual's material utility under play γ. If the monetary payoff allocation after a play γ is m i (γ) , m j (γ) , we assume that the individual's own material utility is of the CRRA form

π i (γ) = m i (γ) 1-r i -1 1 -r i , ( 2
)
where r i is the (constant) degree of relative risk aversion of subject i. We further assume that each subject evaluates his or her opponent's monetary payoff in terms of own risk attitude. 9 Hence, subject i evaluates the opponent j's monetary payoff as follows:

π ij (γ) = m j (γ) 1-r i -1 1 -r i . ( 3
)
Risk neutrality is the special case when r i = 0, and we identify the special case r i = 1 with logarithmic utility for money: then π i (γ) = ln m i (γ) and π ij (γ) = ln m j (γ).

Distinguishing Kantian morality from social preferences

Many experimental studies use dictator game protocols to estimate social preferences.

An advantage of such protocols is that they contain no strategic element, and hence there is no need to elicit subjects' beliefs about other subjects' behaviors. However, this class of game protocols would not allow us to distinguish between social preferences and Kantian morality à la Homo moralis. To see why, consider a dictator game in which the donor may transfer any part of his endowment w to the recipient, and the amount transferred will be multiplied by a factor m > 1. Suppose that both players face an equal probability of being the donor, and denote by x ∈ [0, w] and y ∈ [0, w] their respective strategies (how much to give in the donor role). Consider first a risk-neutral pure altruist i, with

β i = -α i ≥ κ i = 0,
and thus a utility function of the form (the factor 1/2 represents nature's draw of roles):

u i (x, y) = 1 2 [(1 -β i )(w -x + my) + β i (mx + w -y)] . ( 4
)
If instead i is a risk-neutral pure Homo moralis, with κ i ≥ α i = β i = 0, then his or her expected utility is:

u i (x, y) = 1 2 [(1 -κ i )(w -x + my) + κ i (mx + w -x)] . ( 5
)
9 There is experimental evidence that both students and financial professionals exhibit such false consensus [START_REF] Roth | Stereotypes and false consensus: How financial professionals predict risk preferences[END_REF]. Moreover, there is experimental evidence that people make the same decisions under risk (in the gain domain) for themselves and others [START_REF] Andersson | Deciding for others reduces loss aversion[END_REF].

Comparison of the second terms in these utility functions reveals that while an altruist cares about the other individual's monetary payoff (mx + wy)/2 (which depends on the other's strategy y), an individual driven by Kantian morality instead cares about the monetary payoff (mx + wx)/2, which would result if both players were to use i's strategy

x. Nonetheless, as shown by the derivatives with respect to own strategy x, the trade-off for altruists and Kantian moralists is the same here:

du i (x, y) dx = 1 2 [β i m -(1 -β i )] , (6)
and

du i (x, y) dx = 1 2 (κ i m -1) . ( 7
)
Whether an altruist or a Kantian moralist, the individual either gives the whole endowment or nothing at all: indeed, dividing the right-hand side of (6) by 1β i , and letting

σ i ≡ β i
1-β i , we see that the altruist gives everything if σ i exceeds 1/m while the Kantian moralist gives everything if κ i exceeds 1/m. 10 Therefore, we would be unable to separate altruism from a Kantian concern using dictator games. 11 By instead using game protocols that contain strategic elements and collecting data on decisions at all nodes in the game tree as well as beliefs about opponent's play, our experimental design allows us to discriminate between social and Kantian moral preferences.

The key effect is that an individual with a Kantian moral concern is not only influenced by his belief about the opponent's actual play, but also by what he would himself have done had the player roles been reversed (information that we collect in the experiment). Put differently, an important consequence of Kantian morality is that a subject's preferences over moves off the equilibrium path associated with a strategy pair (x, y) may influence his or her decisions on its path. This differs sharply from altruism, inequity aversion or spite, which induce consequentialistic reasoning. 10 This observation is in line with a more general comparison of behavioral predictions for altruists and Kantian moralists in [START_REF] Alger | Homo moralis-preference evolution under incomplete information and assortative matching[END_REF], see also Alger and Weibull (2017). 11 We would face the same identification problem with allocation tasks. Consider a subject i who faces the choice between the allocations (S, T ) and (P , P ), where the first entry is monetary payoff to self and the second entry is monetary payoff to the other subject, with T > P > S. A risk-neutral subject i with a utility function of the form in (1) strictly prefers (S, T ) to (P , P ) if and only if κ i (T -P )α i (T -S) > P -S. Hence, a subject who selects (S, T ) can be driven either by pure altruism (-α i > 0 = κ i ), by pure Kantian morality

(κ i > 0 = α i )
, by a combination of these, or by a combination of behindness aversion and Kantian morality

(α i • κ i > 0).
Concretely, consider first a (symmetrically randomized) Trust Game protocol (see Fig-

ure 1b) with 2R > T + S, and suppose that an individual i believes that the opponent will play K ("keep") as second-mover. If this individual i has no Kantian morality and is either selfish or driven by behindness aversion (α i > 0), he will choose N ("not invest") as first-mover. By contrast, if he has Kantian morality of a sufficiently large degree κ i , then he will, as first-mover, choose I ("invest"), because he would himself play G ("give back") as second mover. Likewise, in the (symmetrically randomized) Sequential Prisoner's Dilemma protocol (Figure 1a), suppose that 2R > T + S > 2P and consider a subject who believes that the other will choose D both as first-mover and as second-mover. Despite this belief, a subject i with a large enough degree of Kantian morality would nevertheless evaluate the play C followed by C, because this is the play he would choose if he met himself.

Turning finally to the Ultimatum Game protocol, as in Figure 1c, we use it to conduct a formal analysis of the effect of Kantian morality (a formal analysis of the other two game protocols is provided in Appendix A1). A risk-neutral subject i obtains the following expected utility from using behavior strategy x = (x 1 , x 2 ) when he believes that the opponent will use behavior strategy ŷ = ( ŷ1 , ŷ2 ) (the randomization factor 1/2 has been omitted):

u i (x, ŷ) = (1 -κ i )[x 1 R + (1 -x 1 ) ŷ2 T + (1 -x 1 ) (1 -ŷ2 ) S (8) + ŷ1 R + (1 -ŷ1 ) x 2 P + (1 -ŷ1 ) (1 -x 2 ) S] -[α i (1 -ŷ1 ) x 2 + β i (1 -x 1 ) ŷ2 ] (T -P ) + κ i [x 1 R + (1 -x 1 ) x 2 T + (1 -x 1 ) (1 -x 2 ) S + x 1 R + (1 -x 1 ) x 2 P + (1 -x 1 ) (1 -x 2 ) S].
The partial derivatives with respect to x 1 and x 2 are thus:

∂u i (x, ŷ) ∂x 1 = (1 -κ i )[R -ŷ2 T -(1 -ŷ2 ) S] + β i • ŷ2 (T -P ) (9) + κ i • [2 (R -S) -x 2 (T + P -2S)] ∂u i (x, ŷ) ∂x 2 = (1 -κ i ) (1 -ŷ1 ) (P -S) -α i • (1 -ŷ1 ) (T -P ) + κ i • (1 -x 1 ) (T + P -2S) . ( 10
)
To see the two key effects of Kantian morality mentioned above, we compare an individual who is inequity averse but does not have a Kantian concern (κ i = 0) to one who has a Kantian concern but is not inequity averse (α i = β i = 0). First, when considering the effect of his choice as a first-mover, x 1 , the inequity-averse individual pays no attention to his choice as a second-mover, while the Kantian moralist does (i.e., x 2 shows up in the derivative if and only if κ i 0). Likewise, when considering the effect of his choice as a second-mover, x 2 , the inequity-averse individual pays no attention to his choice as a first-mover, while the Kantian moralist does (i.e., x 1 shows up in (10) if κ i 0). Second, the expressions ( 9) and ( 10) show that beliefs about the opponent's play (information that we elicit from the subjects) matter less for a pure Kantian moralist than for a purely inequity averse individual. In the extreme case where κ = 1 > α = β = 0, the Kantian moralist chooses the strategy that would maximize the expected material payoff should both players choose it, irrespective of what (s)he believes the opponent will play.

Clearly, disentangling an individual's social preferences from his or her Kantian moral preferences requires controlling for his or her beliefs about the opponent's play. We therefore elicit subjects' such beliefs (by way of the quadratic scoring rule). We describe the experimental procedures, including the belief elicitation procedure, in the next subsection.

Procedures

In total, 136 subjects (69 men, 67 women) participated in the experiment. We conducted 8 sessions at the CentERlab of Tilburg University, with between 12 and 22 subjects per session. Using the strategy method, each subject made decisions both as a first mover and a second mover for 18 game protocols (6 SPDs, 6 TGs and 6 UGs), for different monetary payoff assignments T , R, P and S, listed in Table 1. 12All payoffs are denoted in 'points', where one point is equivalent to 17 eurocents. The order of the game protocols was randomly determined at the beginning of each session.

For each game protocol, subjects first indicated what they would do at each decision node

and second what they believed others would do at each decision node. In all game protocols, we used neutral labels. Two of the 18 game protocols were randomly selected for payment. For one game protocol, subjects were paid based on their actions and for the second game protocol they were paid based on the accuracy of their beliefs. For the payment based on actions, subjects were randomly matched in pairs and randomly assigned the role of first-mover or second-mover. Based on the actions in a pair, earnings for both subjects in the pair were calculated. For the payment based on beliefs, one decision node was randomly selected and subjects were paid using a quadratic scoring rule. At the beginning of each session, subjects were randomly assigned a cubicle and read the instructions on-screen at their own pace. Subjects also received a printed summary of the instructions. At the end of the instructions subjects had to successfully complete a quiz to test their understanding of the instructions before they could continue. After completing the game protocols, we elicited risk attitudes using an incentivized method similar to the method of [START_REF] Eckel | Sex differences and statistical stereotyping in attitudes toward financial risk[END_REF]. Self-reported demographic data was gathered by way of asking the subjects to complete a short questionnaire at the end of the session. The instructions, quiz questions and risk elicitation task are reproduced in Appendix A4. Sessions took around 1 hour and subjects earned between e10.50 and e26.90 with an average of e18.80. Key features of the experimental design and main analyses were pre-registered. 13Prior to describing how we will analyze the data, we present some descriptive statistics.

Descriptive statistics

In Table 1, we present an overview of the average actions and beliefs for each game protocol. On average, observed behavior follows patterns that accord well with other experiments. For example, in the SPDs, on average subjects display conditional cooperation (x 2 > x 3 ). In the TGs, increasing the temptation payoff T and decreasing the sucker payoff S (compare game protocols 7 vs 8, 9 vs 10, 11 vs 12) reduces both trust (x 1 ) and trustworthiness (x 2 ). In the UGs, lower offers (P ) are accepted less frequently (x 2 ). Moreover, on average actions (x) and beliefs (y) are highly correlated (see also Figure A.1 in Appendix A3). Table A.2 in Appendix A3 presents all decisions in the risk elicitation task. Based on their lottery choice, most subjects (83%) are classified as being risk-averse.

Statistical analysis

The econometric strategy consists in producing both individual and aggregate estimates of the parameters in the utility function specified in (1) using a random utility model. In the main specification we control for the subjects' stated beliefs (note that this implies that no equilibrium assumption is needed). We will then conduct several robustness checks and propose ways to evaluate the value-added of including Kantian morality.

Individual preferences

For each subject i, we estimate the individual's social and moral preference parameters α i , β i , and κ i as specified in (1), using a standard additive error specification. We refer to these preference parameters using the vector θ i = (α i , β i , κ i ). For each individual, we infer the risk parameter r i from the lottery choices in the [START_REF] Eckel | Sex differences and statistical stereotyping in attitudes toward financial risk[END_REF] task (see Table A.2 in Appendix A3). As robustness checks, we also estimate r i alongside the other parameters and we carry out the analysis under the alternative assumption that all subjects are risk neutral (all r i = 0), see Section 4.3. We consider pure strategies (that is, assigning a unique action at each decision node), and assume that subject i's true (expected) utility from using pure strategy x i when ŷi is i's expectation about his opponents behavior, is a random variable of the additive form

ũi (x i , ŷi , θ i ) = u i (x i , ŷi , θ i ) + ε ix i ,
where u i (x i , ŷi , θ i ) is the expected utility of using strategy x i given beliefs ŷi following from the utility function in (1), and ε ix i is a random variable representing idiosyncratic tastes not picked up by the hypothesized utility u i (x i , ŷi , θ i ). Such a random utility specification sometimes induces choice of actions that do not maximize the deterministic component u i (x i , ŷi , θ i ). Assuming that the noise terms ε ix i are statistically independent (between subjects and across pure behavior strategies x i for each subject) and Gumbel distributed with the same variance, the probability that subject i will use strategy x i , given his probabilistic belief ŷi about the opponent's play is given by the familiar logit formula [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behavior[END_REF]:

p i (x i , ŷi , θ i , λ i ) = exp [(u i (x i , ŷi , θ i )) /λ i ] x ∈X g exp [(u i (x , ŷi , θ i )) /λ i ] , ( 11
)
where λ i > 0 is a "noise" parameter, which is estimated alongside the preference parameters in θ i , and X g denotes the set of pure strategies in game protocol g ∈ G, where G is the set of game protocols. The smaller the parameter λ i is, the higher is the probability that individual i makes his or her choices according to the hypothesized utility function u i (x i , ŷi , θ i ). We use maximum likelihood to estimate the preference parameter vector θ i = (α i , β i , κ i ) and the "noise" parameter λ i for each individual i. 14 Then, the probability density function can be written as: ,g,x) , (12) where x i is the vector of the observed pure strategies of individual i, ŷi is the vector of stated beliefs of individual i about opponent's strategy in all the game protocols, and

f (x i , ŷi , θ i , λ i ) = g∈G x∈X g p i (x, ŷi , θ i , λ i ) I(i
I(i, g, x)
is an indicator function that equals 1 if i played strategy x in game protocol g and 0 otherwise.

Aggregate estimations

We estimate preference parameters both for a representative agent and a given number of "preference types". For the representative agent, we simply aggregate all individual decisions and treat them as if they come from a single decision-maker. For the types estimations, we use finite mixture models, similar to the approach used by Bruhin et al.

(2019). The finite mixture estimations allow us to capture heterogeneity in the population in a tractable way. For these estimations, we assume that there is a given number of types K in the population. For each type k = {1, ..., K}, we estimate the parameter vector

θ k = (α k , β k , κ k
) and the noise parameter λ k .

In a recent paper, [START_REF] Apesteguia | Monotone stochastic choice models: The case of risk and time preferences[END_REF] show that estimating CRRA parameters using a random utility model may be problematic. To avoid this, we estimate the social preference and Kantian morality parameters under the assumption that all subjects have logarithmic utility over monetary outcomes (i.e. we impose r k = 1 for all types k). Given that most subjects in our experiment are risk averse according to the lottery task, assuming homogeneous risk aversion seems a better approximation of the data than assuming homogeneous risk neutrality. In subsection 4.3 we relax this assumption and also run estimations where we estimate the CRRA parameter r k alongside the social preference and morality parameters. As an additional robustness check, we also run the estimations imposing risk-neutrality (i.e. r k = 0 for all types k).

The log-likelihood is then given by:

ln L = N i=1 ln        K k=1 φ k • f (x i , ŷi , θ k , λ k )        , ( 13
)
where φ k is the population share of type k in the population. To maximize the loglikelihood in (13), we use an Expectation-Maximization (EM) algorithm (see for instance

McLachlan, Lee, & Rathnayake, 2019). 15 As part of the EM algorithm, we estimate the posterior probabilities τ i,k that individual i belongs to type k by:

τ i,k = φ k • f (x i , ŷi , θ k , λ k ) K m=1 φ m • f (x i , ŷi , θ m , λ m ) . ( 14
)
4 Results

Individual preferences

Figure 3 shows the marginal distributions of the estimated individual preference parameters α i , β i , and κ i . 16 For all three parameters, we observe considerable heterogeneity.

Most estimates of α i and κ i are positive and signed-ranks tests confirm that the parameter distributions are located to the right of zero (p < 0.001 for both α i and κ i estimates).

By contrast, most estimates of β i are negative, and this is again confirmed by a signedrank test (p = 0.003). Hence, we find that most subjects are motivated by a combination of Kantian morality (κ i > 0) and spite (α i > 0, β i < 0).

Table 2, which shows summary statistics for the parameter estimates, provides further support for the pattern observed in Figure 3. Median and mean estimates are positive for α i and κ i , but negative for β i . Moreover, the relatively large standard deviations indicate that there is considerable heterogeneity in social preferences and Kantian morality. 17

Figure 4 illustrates the pairwise correlations between the three preference parameter estimates. The left panel of Figure 4 shows that the estimates for α i and β i are negatively correlated (Spearman's ρ = -0.295, p = 0.002, n = 109), and again that there is substantial heterogeneity. For many individuals we observe a combination of α i > 0 and β i < 0, 15 We use 24 sets of starting values. 16 In the estimations, we do not restrict the size or the sign of the parameter estimates. For most subjects, the parameter estimates are of reasonable size. However, for some subjects we obtain very large estimates of α i , β i , and/or κ i (in absolute value), suggesting that our utility function (1) does not explain the decisions of these subjects well, either because they use a decision rule not nested in (1), or because their decisions are simply too noisy to be generated by any utility function. In the remainder of this section, we report results for our 'core sample', which consists of the 109 subjects for whom all three preference parameter estimates lie between -2 and 2. The fraction that we leave out in the main text (19,6%) is comparable in size to the fraction of 26.3% for whom [START_REF] Fisman | Individual preferences for giving[END_REF] conclude that their decisions are too noisy to be utility-generated. In Appendix A3 we report results based on data for all 136 subjects. While the latter results are more noisy, they are qualitatively quite similar to those for the core sample. 17 For these estimates we used the risk elicitation task to determine r i . In subsection 4.3 we provide robustness tests where we estimate r i alongside the preference parameter, or impose risk neutrality. Figure 4: Correlations between estimated preference parameters.

-2-1.5-1 -.5 0 .5 1 1.5 2 estimated β i -2-1.5-1 -.5 0 .5 1 1.5 2 estimated α i -2-1.5-1 -.5 0 .5 1 1.5 2 estimated κ i -2-1.5-1 -.5 0 .5 1 1.5 2 estimated α i -2-1.5-1 -.5 0 .5 1 1.5 2 estimated 2019) call it, "behindness aversion," with Kantian morality. For the estimates of β i and κ i we find a negative correlation (Spearman's ρ = -0.173, p = 0.071, n = 109). We also use copula methods to describe the joint parameter distributions for the individual estimates of α i , β i and κ i . As for the pairwise correlations reported above, we observe that the individual estimates of α i , β i and κ i are not statistically independent.

Appendix A2 provides more details.

Aggregate estimations

We now turn to estimation of preferences at the aggregate level (see section 3.2 for details). To distinguish these estimates from the individual ones, we use an index k to designate the type. Table 3 presents the estimates of the finite mixture models for one, two and three types.

The representative agent

When assuming only one type, that is, a representative agent, we obtain the estimates α 0 = 0.14, β 0 = 0.00, and κ 0 = 0.21, where the index 0 stands for the representative agent.

In other words, the representative agent dislikes disadvantageous inequity, is indifferent with respect to advantageous inequality, and has a positive degree of Kantian morality.

20

The representative agent thus exhibits Kantian morality and behindness aversion.

The two-and three-type models

As can be seen in Table 3, in both multi-type models all types exhibit Kantian morality (κ k > 0), roughly of the same order of magnitude as the representative agent. There is much stronger heterogeneity in terms of the inequity aversion parameters α k and β k : some types exhibit behindness aversion (α k > 0) while other types are (close to) indifferent to behindness (α k ≈ 0); and some types disliking behind ahead (β k > 0) while other types like it (β k < 0).

More specifically, when assuming two types, the most common type (Type 1) exhibits (mild) inequity aversion, with parameter estimates α 1 = 0.06 and β 1 = 0.09, combined with a degree of Kantian morality κ 1 = 0.23. This type represents about 62% of the subjects. The other type, Type 2, exhibits a combination of strong spite ("negative altruism")

and Kantian morality, with α 2 = 0.27, β 2 = -0.31, and κ 2 = 0.18. 18For each subject i, we estimate the posterior probability τ i,k that i belongs to type k (as defined in ( 14)). By taking the largest value τ i,k for each subject i, we can assign each of the subjects to one of the types. Table A.6 in Appendix A3 lists the chosen strategies per game protocol type based on this classification. "Type 2 subjects", who combine spite and Kantian morality, mostly choose to always defect (D, D, D) in the SPDs (in 87%) of the cases, while "Type 1 subjects", who combine inequity aversion and Kantian morality, choose (D, D, D) less frequently (38%) and often conditionally cooperate (C, C, D) instead (32%). Similarly, in the TGs, Type 2 subjects most frequently choose not to invest as first mover and to "keep" as second mover (N , K) (85%), while Type 1 subjects most frequently invest as first mover and "give" as a second mover (I, G) (43%). In the UGs, Type 2 subjects mostly choose the unequal option as a first mover (74%) and accept unfair offers as a second mover (97%). Instead, Type 1 subjects most frequently propose an equal payoff (68%) and accept fewer unequal offers (88%).

When assuming three types, for all types we again estimate a positive Kantian morality parameter κ k . In comparison with the results under the two-types approach, Type 3 is very close to the previous Type 2. This type is again characterized as combining spite with Kantian morality, and represents a similar fraction of the population (37%). 19 The new Type 2 combines (relatively strong) inequity aversion with Kantian morality. It represents around 17% of the population. Type 1 is very close to Homo moralis. The inequity aversion parameters α 1 and β 1 are not significantly different from zero (at the 5% level), while the Kantian morality κ 1 is positive and significant. This type represents 33% of the population. In sum: under the three-types approach, Type 1 displays Kantian morality, Type 2 is inequity averse and moral, and Type 3 is spiteful and moral.

In terms of chosen strategies, Type 3 behaves almost identical as Type 2 in the twotypes model. The new Types 1 and Type 2 differ in some respects. In the SPDs, the new Type 2 acts conditionally cooperative more often than Type 1. Similarly, Type 2 chooses to "give" more often than Type 1 in the TGs. In the UGs, Type 2 refuses unequal offers more frequently than Type 1.

In sum, the aggregate estimates lead to two observations. First, we observe relatively little heterogeneity in estimates of the morality parameter κ k . In most cases, κ k is around 0.2, showing that most people are well described by having Kantian morality concerns.

Second, we note that in both multi-type models, we do not observe types who are best described by pure self-interest (α k = β k = κ k = 0). 20 Nonetheless, self-interest is still an important driver for all the types.

Comparing the one-, two-, and three-types models

Clearly, adding more types improves the fit of the model, but this comes at the cost of parsimony as well as precision of allocating individuals to types. Information criteria like the Bayesian information criterion (BIC) are not well suited to select the number of clusters (or in our case, 'types') in finite mixture models. In a recent overview paper on the use of finite mixture models, [START_REF] Mclachlan | Finite mixture models[END_REF] recommend using the 'integrated completed likelihood' (or 'integrated classification', ICL, [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF]. This criterion is approximated by

ICL = -2 ln L + d ln N + EN (τ ), ( 15
)
where the log-likelihood function ln L is defined as in (13), d is the number of estimated parameters, and N is the number of individuals in our sample. The last term in ( 15) is types models. All but one subject who is classified as Type 2 in the two-types model, are classified as Type the entropy

EN (τ ) = - K k=1 N i=1 τ i,k ln τ i,k , ( 16
)
where τ i,k is the estimated posterior probability of individual i belonging to type k, as defined in ( 14). This implies that the stronger individuals are assigned to types (i.e. all τ i,k 's close to zero or one), the lower the entropy will be. In other words, the ICL extends the BIC by adding an additional penalty if individuals are assigned imprecisely to types. 1996), which is defined as:

N EC = EN (τ ) ln L(K) -ln L(1) , ( 17
)
where ln L( 1) is the log-likelihood of the representative agent model and ln L(K) the loglikelihood of the model with K types. Hence, the NEC weighs the precision of the type classifications τ i,k by the increase in the log-likelihood compared to the representative agent model.

Table 3 shows statistics for both the ICL and the NEC. For both metrics, a lower score indicates a more preferred model. The NEC selects the two-types model and the ICL selects the three-types model. Table A.5 in Appendix A3 shows estimates and goodness-offit metrics for a four-types model. The four-types model performs worse on both criteria than the two-types and three-types models in Table 3. Note that marginal improvement in the ICL score is largest when going from the representative agent to the two-types model. In sum, assuming two types instead of a representative agent brings us a long way in capturing the heterogeneity in the population.

Robustness

Estimating risk attitudes

In the main analysis, we imposed values for the CRRA parameter r. In the individual estimations, we based r i on the decision in the lottery task while in the aggregate esti-

3-types model

Notes: Distributions of the estimated posterior probability τ i,k of individual i belonging to type k for the two-types and three-types finite mixture models reported in Table 3.

mations we assumed that everyone has logarithmic utility (i.e. r k = 1). We also estimate the CRRA parameter r alongside the social preference and Kantian morality parameters.

Doing so does not affect our estimates by much.

First, at the individual level, estimating r i alongside the preference parameters (α i , β i , κ i ) does not affect the estimates by much. The estimated preference parameters are strongly correlated (Spearman rank correlations: ρ = 0.639, p < 0.001, n = 109 for α i , ρ = 0.566, p < 0.001, n = 109 for β i , and ρ = 0.606, p < 0.001 for κ i ) although the correlation between the imposed and estimated r i values is weak (ρ = 0.069, p = 0.478). The estimates of α i , β i and κ i are not systematically smaller or larger using either method (signed-rank tests, p = 0.198, n = 109 for α i , p = 0.228, n = 109 for β i , and p = 0.388, n = 109 for κ i ).

Second, estimations of the finite mixture models that include the estimation of a CRRA parameter r k for each type k lead to a value for r k close to 1 in most cases (see

Risk neutrality

In yet another robustness check we estimate the social preference and Kantian morality parameters under the alternative assumption that all subjects are risk neutral (i.e., r i = 0 for all subjects i). Figure 6 shows scatter plots of individual parameter estimates under both assumptions, with estimates under risk neutrality on the horizontal axis and estimates under constant (individual specific) relative risk aversion (CRRA) on the vertical axis. Each dot represents an individual subject. The diagrams suggest that the riskneutral and CRRA estimates are strongly correlated. Indeed, for the inequity parameter α i (when behind) the Spearman rank correlation is ρ = 0.802. For the inequity parameter β i (when ahead) it is ρ = 0.774, and for the Kantian morality parameter κ i it is ρ = 0.627 (all three rank correlations hold for p < 0.001, n = 109).

The middle panel in Figure 6 also shows that the β i estimates are much higher under risk neutrality than under CRRA. 22 Indeed, for 94 out of 109 subjects, the risk-neutral estimate is higher than the CRRA estimate (signed-rank test, p < 0.001). 23 By contrast, 21 Table A.7 shows that for both two-types (panel B) and three-types (panel C) models, subjects are separated in almost the same groups as when we impose r k = 1. 22 One can easily see how assuming risk neutrality would bias estimates of β k . Take for example the UG protocol. Both risk aversion and 'aheadness aversion' (β i > 0) would induce one to choose E over U . 23 Moreover, for most subjects (80 out of 109), β i is positive under risk neutrality (signed-rank test, p < 0.001).

Figure 6: Correlations between risk neutral and CRRA estimates -2-1.5-1 -.5 0 .5 1 1.5 2 estimated α i (crra) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated α i (risk-neutral) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated β i (crra) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated β i (risk-neutral) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated κ i (crra) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated κ i (risk-neutral)

Notes: Figures shows estimates smaller the risk-neutral estimates of κ i (80 out of 109, signed-rank test: p < 0.001) and α i (64 out of 109, signed-rank test: p = 0.068) are lower for most subjects than under CRRA. 24 For the majority of subjects (72 out of 109), assuming CRRA preferences instead of risk neutrality leads to a higher log-likelihood, indeed indicating a better fit under CRRA preferences.

Table 4 shows the estimates of finite mixture models under risk neutrality. Comparing these results with those in Table 3, one sees that, qualitatively, estimates of the parameters α k and κ k are not much affected. For all types in Tables 3 and4, α k and κ k are positive, under both risk hypotheses, although the Kantian morality parameter values somewhat lower under risk neutrality than under CRRA. In line with the individual parameter estimates (see the middle panel in Figure 6), the finite mixture estimates of the parameters β tend to be much higher under risk neutrality than under CRRA. Moreover, under riskneutrality, all estimates of β k are non-negative, in contrast to the CRRA estimates, where we observed β k < 0 for some types k. 25 The ICL criterion allows comparison of the fit of the CRRA and risk-neutral models, respectively (see Tables 3 and4). For any given number of types, the CRRA model has 24 Most risk-neutral estimates of κ i (96 out of 109) and α i (92 out of 109) are positive (signed-rank tests, p < 0.001) 25 Table A.7 shows that the assignment of subjects to types for the risk-neutral two-types (panel D) model, is very similar to when we impose r k = 1. For the three-types models (panel E), the type classification is again similar under both assumptions, but some who are classified as "Type 2" with r k = 1 are classified as "Type 1" under risk-neutrality. Figure 7: Correlations between estimates using subjective and rational expectations -2-1.5-1 -.5 0 .5 1 1.5 2 estimated α i (subjective) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated α i (re) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated β i (subjective) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated β i (re) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated κ i (subjective) -2-1.5-1 -.5 0 .5 1 1.5 2 estimated κ i (re)

Notes: Figures shows estimates smaller

Rational expectations

So far, we assumed that people maximize expected utility given their (reported) subjective expectations. In this subsection we investigate what happens to the estimated preference parameters if we take rational expectations instead.

Figure 7 shows correlations between the individual estimates using subjective and rational expectations. For all three preference parameters, the estimates under the two assumptions are strongly correlated. For the inequity parameter α i (when behind) the Spearman rank correlation is ρ = 0.553. For the inequity parameter β i (when ahead) it is ρ = 0.635, and for the Kantian morality parameter κ i it is ρ = 0.344 (for all three rank correlations: p < 0.001, n = 109). For most subjects, the log-likelihood is larger when we assume rational expectations instead of subjective expectations (67 out of 109, signedrank test: p = 0.043), indicating that assuming rational expectations actually improves the fit for most subjects.

Table 5 shows the finite mixture estimates when we assume rational expectations. The representative agent with rational expectations is characterized by a combination of spite (α k > 0, β k < 0) and morality (κ k > 0). Compared to the model with subjective expectations (see Table 3), the estimates for α k and κ k are larger when we assume rational expectations.

The estimate for β k is negative when we assume rational expectations, where it was zero under subjective expectations. For the representative agent model, the log-likelihood is lower when assuming rational expectations. For the two-types model and three-types model, assuming rational expectations leads to qualitatively similar results as under subjective expectations. For the two-types model, Type 1 again displays a combination of (mild) inequity aversion and morality, Type 2 combines spite with morality. 26 The ICL scores of both multi-type models are somewhat lower under rational expectations, indicating a slightly better fit under rational expectations. Most importantly however, the estimated preference parameters for the multi-type models are very similar under both assumptions.

The value added of Kantian morality

In the preceding sections, we showed that estimated Kantian morality parameters tend to be positive, both at the individual and aggregate level. In this subsection, we benchmark the added value of the Kantian morality parameter against other parameters, and also against reciprocity.

Individual estimations

We conduct likelihood-ratio tests to see if adding the Kantian morality parameter κ i to a model with only the two social preference parameters α i and β i improves the fit. The likelihood-ratio tests reveal that adding κ i improves the fit for 21 individuals at the 5% level (and for 32 individuals at the 10% level). For comparison, likelihood ratio tests when adding either α i to (β i , κ i ), or β i to (α i , κ i ), improves the fit at the 5% level for 20 and 26 individuals, respectively (at the 10% level, for 25 (α i ) and 37 (β i ) individuals).

Hence, in terms of value added at the individual level, all three preference parameters are in roughly the same ballpark.

A more general approach is to consider all models that are nested in (1) and apply standard information criteria. We use both the Bayesian information criterion (BIC) and Akaike's Information Criterion (AIC), each of which is based on the log-likelihoods and adds a penalty for each parameter. The lower score, the better fit. More precisely, the criteria are:

BIC = -2 ln(L) + d ln (18), (18)
and

AIC = -2 ln(L) + 2d, (19)
where ln ( 18) in ( 18) comes from the 18 observations per subject. Since ln 18 ≈ 2.89 > 2, BIC gives a heavier penalty per parameter than AIC. subjects (32.0%), respectively. The right panel shows the results from the same exercise, but now applied to AIC. Then the best-fitting model at the individual level includes the parameter κ i for 48 subjects (or 44.0%). Again, a larger number of subjects than for α i (31 subjects, or 28.4%) and also slightly more subjects than β i (40 subjects, or 36.7%).

Aggregate estimations

We also evaluate the value added of Kantian morality for the finite mixture estimations.

Table A.9 in Appendix A3 shows estimates for finite mixture models with only α k and β k (i.e. where κ i = 0). For any given number of types, these fixed mixture estimates give substantially higher ICL scores than the model including Kantian morality, indicating that fixed mixture estimates that include the parameter κ i provide a better fit.

Reciprocity vs. Kantian morality

We can compare the value added of the Kantian morality parameter, to the value if one were to instead of Kantian morality add reciprocity. For this purpose, we modify the utility function in (1) to replace the Kantian morality term by a term that represents negative reciprocity as in [START_REF] Charness | Understanding social preferences with simple tests[END_REF], which leads to

u i (x, y) = γ η (x,y) (γ) • π i (γ) (20) -α i • γ η (x,y) (γ) • max 0, π ij (γ) -π i (γ) -β i • γ η (x,y) (γ) • max 0, π i (γ) -π ij (γ) -δ i • q • γ η (x,y) (γ) • max 0, π ij (γ) -π i (γ) ,
where q = 1 if the other player 'misbehaved' and q = 0 otherwise. Following [START_REF] Charness | Understanding social preferences with simple tests[END_REF], we label a first-mover action as misbehavior if it excludes an outcome that has maximal joint monetary payoffs. For our case this means that defecting as a first mover in a SPD protocol (if 2R > T +S, which holds for 5 out of 6 SPDs), and not investing in a TG protocol constitutes misbehavior (note, however, that the δ i term cancels in latter case, as not investing will lead to equal payoffs for both players). In addition, we also label not proposing an equal split in the UGs as misbehavior.

In Table A.10 in Appendix A3 we provide the results of finite mixture models based on (20). The three-types model has the lowest ICL score among the reciprocity models.

Based on the ICL score, the three-types reciprocity model performs better than the mixture models with only α k and β k (see A.9). This shows that, adding reciprocity improves the fit of the model. Importantly however, the three-types model that allows for Kantian morality instead of reciprocity has an even lower ICL score, suggesting that Kantian morality adds more than reciprocity in our setting.

Out-of-sample predictions

So far, we evaluated the performance of different models based on information criteria.

As an alternative, we consider the predictive accuracy of different models by conducting out-of-sample predictions. For each of the 18 game protocols, we estimate parameters based on the other 17 game protocols, and use the estimates to predict the choice for the one omitted game protocol. We conduct these analyses both at the individual level and the aggregate level. The model allowing for Kantian morality (α, β, κ) outperforms the other models in terms of predictive accuracy. The (α, β, κ)-model on average predicts 56.0% of choices correctly, somewhat more than the (α, β) and (α, β, δ) models, which give 53.4% and 54.3% average accuracy, respectively. All models allowing for social preferences and/or morality perform much better than when assuming self-interest, which gives 48.1% average accuracy.

The right panel of Figure 8 shows the predictive accuracy of finite mixture models assuming two types. Compared to the individual estimations, the gap between the model allowing for Kantian morality (α, β, κ) and the other models is larger for the two-types models. The two-types model with α, β and κ predicts 54.8% of choices correctly, which is better than the two-types (α, β)-model and reciprocity model (α, β, δ) which give 50.2% and 49.8% accuracy respectively. Note that the predictive accuracy of the twotypes model allowing for Kantian morality (54.8%) is not far from the model allowing for Kantian morality with individual estimates (56.0%). 27 This provides further evidence that the model effectively captures the heterogeneity in preferences.

Concluding discussion

In this paper, we report results from a laboratory experiment designed to evaluate the explanatory power of Kantian morality in standard strategic interactions. To distinguish Kantian morality from other social concerns, we posit a general utility function that nests several much studied preference classes, such as pure self-interest, altruism, spite, and inequity aversion, and of course Kantian morality. We structurally estimate the preference parameters of this utility function, allowing for risk aversion and controlling for the beliefs about opponent's play. We obtain both individual and aggregate estimates, where the latter consists of estimating the parameters for a representative agent, as well as identifying a small number of endogenously determined "preference types".

The individual estimates suggest substantial heterogeneity. This heterogeneity limits the usefulness of a representative agent approach. However, we find that the subjects' behaviors are well captured by models with two or three preference types. The two-types model suggests that 62% of the subjects display a combination of mild inequity aversion with Kantian morality, and the remaining 38% a combination of Kantian morality and strong spite. Within the three-types model, again one type is characterized by a combination of inequity aversion and Kantian morality (representing 30% of the population) and 37% of the population appear to combine Kantian morality with spite. However, now there is another type which displays only Kantian morality, representing 33% of the population. Quite remarkably, all the preference types-both the representative agent and the preference types within the two-types and the three-types model-have an estimated Kantian morality parameter κ k of around 0.2, which given the posited utility function means that the weight attached to the Kantian moral concern is about one quarter of the weight attached to the own material payoff.

Our experimental design was motivated by findings in the theoretical literature that investigates the evolutionary foundations of preferences in strategic interactions (see Alger & Weibull, 2019, for a recent survey). This literature shows that evolution by natural selection favors Kantian morality (see, in particular, [START_REF] Bergstrom | On the evolution of altruistic ethical rules for siblings[END_REF] and [START_REF] Alger | Homo moralis-preference evolution under incomplete information and assortative matching[END_REF]). As it turns out, our results are in fact in line with an even more recent contribution to this theoretical literature. In a model that enables analysis of the long-run impact of population structure on preferences, [START_REF] Alger | Evolution of preferences in structured populations: genes, guns, and culture[END_REF] show that preferences that combine Kantian morality with either altruism or spite are favored by evolution by natural selection. 28Compared with other experimental studies with structural preference estimations, our results agree with those of [START_REF] Bruhin | The many faces of human sociality: Uncovering the distribution and stability of social preferences[END_REF] in that their behavioral data is largely consistent with there being a small number of "preference types". Our findings further agree with [START_REF] Bruhin | The many faces of human sociality: Uncovering the distribution and stability of social preferences[END_REF] in that they do not either find evidence that the purely selfish Homo oeconomicus explains their behavioral data. A more detailed comparison is more involved, since their experimental design differs from ours, and they do not include Kantian morality. Our results further agree broadly with those in the horse race study by [START_REF] Miettinen | Revealed preferences in a sequential prisoners' dilemma: a horse-race between six utility functions[END_REF], although our richer data set allows us to capture the complex combination of subjects' motives that their study cannot address.

As for all laboratory experiments, establishing external validity would be highly desirable [START_REF] Levitt | What do laboratory experiments measuring social preferences reveal about the real world[END_REF]. It would further be interesting to examine whether results similar to ours also obtain in a representative sample, along the lines of the studies by [START_REF] Bellemare | Measuring inequity aversion in a heterogeneous population using experimental decisions and subjective probabilities[END_REF] and [START_REF] Cettolin | Return on trust is lower for immigrants[END_REF]. Also, while our experiment was conducted on a WEIRD population [START_REF] Henrich | Most people are not weird[END_REF], evolutionary theory suggests that the qualitative nature of preferences guiding behavior in strategic interactions should be similar across the world, although certain differences between populations may be expected to influence the relative importance of self-interest, social concerns, and Kantian morality. In particular, since evolutionary theory suggests that migration patterns and the involvement in inter-group conflict are expected to impact preferences guiding behavior in strategic interactions [START_REF] Alger | Evolution of preferences in structured populations: genes, guns, and culture[END_REF][START_REF] Choi | The coevolution of parochial altruism and war[END_REF], this theory delivers testable predictions that may help explain cross-cultural differences [START_REF] Falk | Global evidence on economic preferences[END_REF] and also perhaps differences between men and women [START_REF] Croson | Gender differences in preferences[END_REF]. Note: All estimates of α i , β i and κ i larger than 2 in absolute value are grouped in bins ("<" and ">") at the extremes of the horizontal axis. Notes: Standard errors in parentheses. For all types, we impose log-utility (r k = 1). Estimation results from models with 1, 2 and 3 types can be found in Table 3. Based on our 'core sample' of 109 subjects. Notes: Relative frequencies (in %) of chosen strategies based on the 1, 2, and three-types models reported in Table 3. Subjects are assigned a type based on the type posterior probability τ i,k (that subject i belongs to type k, see eq. ( 14)).

FigureFigure 2 :

 2 Figure 1: Game protocols

  Each dot represents one subject. Dotted lines indicate linear predictions (intercept+slope). Specifically, we estimate β i = -0.05-0.44α i , κ i = 0.19+0.28α i and κ i = 0.22 -0.11β i . Figure based on the 109 subjects for whom the α i , β i and κ i estimates have absolute value below 2. in line with spiteful/competitive preferences, i.e., an individual dislikes being behind but likes being ahead of the other. The middle panel of Figure 4 reveals a strong and positive correlation between α i and κ i estimates (Spearman's ρ = 0.423, p < 0.001, n = 109). This means that many individuals combine a distaste for disadvantageous inequity, or, as Bruhin et al. (

Figure 5

 5 Figure 5 shows the distributions of the estimated posterior probability τ i,k (of individual i belonging to type k) for the two-type and three-type models. In all cases, most estimated τ i,k are very close to zero or 1, which implies that most individuals are quite precisely assigned to a type. For the two-types model, virtually all estimated τ i,k are close to zero or one. For the three-types model, some individuals are imprecisely classified to either Type 1 or Type 2. Bruhin et al. (2019) use the 'normalized entropy criterion' (NEC, Celeux & Soromenho,

Figure 5 :

 5 Figure 5: Posterior probabilities of type classifications

  than 2 in absolute value. Dotted lines indicate 45 degree lines. Figure based on our 'core sample' of 109 subjects.

  than 2 in absolute value. Dotted lines indicate 45 degree lines. Figure based on our 'core sample' of 109 subjects. a lower ICL score than the risk-neutral model. For the three-types model, for example, the ICL score under the CRRA assumption is quite a bit lower than under risk neutrality (4343.6 versus 4515.6), showing that the CRRA model considerably improves the fit over the risk-neutrality model.

Figure 8

 8 Figure 8 illustrates the results, by comparing the predictive accuracy of the model with α, β and κ, to self-interest (α = β = κ = 0), a model without Kantian morality (α, β) and a model allowing for negative reciprocity (α, β, δ). The left panel of Figure 8

Figure 8 :

 8 Figure 8: Accuracy of out-of-sample predictions

Figure

Table 1 :

 1 Game protocols: monetary payoffs, actions and beliefs Here x 1 , x 2 and x 3 denote action frequencies. In the SPDs, x 1 is the frequency by which the first mover plays C, x 2 the frequency by which the second mover plays C after C, and x 3 the frequency by which she plays C after D. In the

	No. T	R	P	S	x 1	x 2	x 3	y 1	y 2	y 3
	Sequential Prisoner's Dilemmas
	1	90 45 15 10 0.18 0.15 0.10 0.33 0.20 0.13
	2	90 55 20 10 0.24 0.20 0.06 0.30 0.21 0.07
	3	80 65 25 20 0.35 0.29 0.13 0.32 0.30 0.16
	4	90 65 25 10 0.29 0.31 0.03 0.31 0.25 0.08
	5	80 75 30 20 0.43 0.50 0.04 0.40 0.41 0.11
	6	90 75 30 10 0.30 0.40 0.01 0.33 0.33 0.08
	All SPDs				0.30 0.31 0.06 0.33 0.28 0.11
	Trust Games
	7	80 50 30 20 0.44 0.27	.	0.41 0.23	.
	8	90 50 30 10 0.18 0.18	.	0.33 0.19	.
	9	80 60 30 20 0.56 0.35	.	0.47 0.30	.
	10	90 60 30 10 0.35 0.25	.	0.37 0.24	.
	11	80 70 30 20 0.62 0.51	.	0.54 0.42	.
	12	90 70 30 10 0.46 0.40	.	0.42 0.31	.
	All TGs				0.44 0.33	.	0.42 0.28	.
	Ultimatum Games
	13	60 50 40 10 0.49 0.96	.	0.48 0.91	.
	14	65 50 35 10 0.52 0.96	.	0.49 0.88	.
	15	70 50 30 10 0.46 0.96	.	0.47 0.87	.
	16	75 50 25 10 0.43 0.90	.	0.47 0.83	.
	17	80 50 20 10 0.60 0.88	.	0.51 0.79	.
	18	85 50 15 10 0.60 0.81	.	0.55 0.72	.
	All UGs				0.51 0.91	.	0.50 0.83	.

Notes:

TGs, x 1 is the frequency by which the first mover plays I, and x 2 the frequency by which the second mover plays G after I. For the UGs, x 1 is the frequency by which the first mover plays E, and x 2 the frequency by which the second mover plays A after U . Likewise, y 1 , y 2 and y 3 are the mean values of the stated beliefs about x 1 , x 2 and x 3 . Table based on all 136 subjects.

  Note: Figure based on the 109 subjects for whom the α i , β i and κ i estimates have

	Figure 3: Distributions of individual parameter estimates
	Frequency
	-2	-1.6	-1.2	-.8	-.4	0 estimated α i	.4	.8	1.2	1.6	2
	Frequency
	-2	-1.6	-1.2	-.8	-.4	0 estimated β i	.4	.8	1.2	1.6	2
	Frequency
	-2	-1.6	-1.2	-.8	-.4	0 estimated κ i	.4	.8	1.2	1.6	2
	absolute value below 2. The (blue) lines indicate fitted Gumbel distributions
	(see Appendix A2 for details). Figure A.2 shows a similar figure based on all
	136 subjects.

Table 2 :

 2 Individual parameter estimatesTable based on the 109 subjects for whom the α i , β i and κ i estimates have absolute value below 2. Table A.3 shows a similar table based on all 136 subjects.

	Parameter	Median	Mean	S.D.	Min	Max
	α i	0.14	0.19	0.38	-0.89	1.75
	β i	-0.06	-0.14	0.51	-1.97	1.37
	κ i	0.18	0.24	0.22	-0.10	1.10

Notes:

Table 3 :

 3 Estimates at the aggregate level

		1 type	2 types		3 types
		Rep. agent	Type 1	Type 2	Type 1	Type 2	Type 3
	α k	0.14	0.06	0.27	-0.01	0.12	0.27
		(0.02)	(0.03)	(0.03)	(0.05)	(0.08)	(0.03)
	β k	0.00	0.09	-0.31	-0.07	0.24	-0.31
		(0.03)	(0.04)	(0.06)	(0.07)	(0.06)	(0.08)
	κ k	0.21	0.23	0.18	0.21	0.23	0.18
		(0.01)	(0.02)	(0.02)	(0.03)	(0.06)	(0.02)
	λ k	0.25	0.28	0.16	0.32	0.22	0.15
		(0.02)	(0.02)	(0.01)	(0.05)	(0.03)	(0.02)
	φ k	1.00	0.62	0.38	0.33	0.30	0.37
		(-)	(0.05)	(0.05)	(0.07)	(0.08)	(0.05)
	ln L	-2336.9	-2154.4		-2131.7
	EN (τ )	0.00		4.25		17.80
	ICL	4692.6	4355.3		4346.9
	NEC	-	0.023		0.087

Notes: Bootstrapped standard errors in parentheses. Table based on our 'core sample' of 109 subjects. In these estimations, we impose r k = 1 (i.e. logarithmic utility) for all types. Table A.4 in Appendix A3 shows estimates based on the full sample. Table A.5 in Appendix A3 shows the estimates of a 4-type model.

Table A .

 A 8 in Appendix A3). As a result, the estimated social preference and Kantian morality parameters change very little.21

Table 4 :

 4 Estimates at the aggregate level (assuming risk neutrality)

		1 type	2 types		3 types
		Rep. agent	Type 1	Type 2	Type 1	Type 2	Type 3
	α k	0.17	0.13	0.19	0.18	0.01	0.19
		(0.02)	(0.02)	(0.02)	(0.03)	(0.04)	(0.02)
	β k	0.25	0.36	0.00	0.26	0.50	0.00
		(0.03)	(0.05)	(0.04)	(0.06)	(0.07)	(0.03)
	κ k	0.10	0.11	0.11	0.11	0.14	0.10
		(0.01)	(0.02)	(0.01)	(0.02)	(0.05)	(0.01)
	λ k	7.62	8.98	4.01	9.29	6.92	3.79
		(0.60)	(0.95)	(0.51)	(1.17)	(0.79)	(0.36)
	φ k	1.00	0.64	0.36	0.48	0.18	0.34
		(-)	(0.07)	(0.07)	(0.06)	(0.06)	(0.05)
	ln L	-2426.8	-2247.6		-2217.9
	EN (τ )	0.00	5.31			14.20
	ICL	4872.5	4542.7		4515.6
	NEC	-	0.030		0.068

Notes: Bootstrapped standard errors in parentheses. Table based on our 'core sample' of 109 subjects.

Table 5 :

 5 Estimates at the aggregate level (assuming rational expectations)

		1 type	2 types		3 types
		Rep. agent	Type 1	Type 2	Type 1	Type 2	Type 3
	α k	0.30	0.08	0.67	0.04	0.13	0.68
		(0.05)	(0.03)	(0.07)	(0.05)	(0.10)	(0.09)
	β k	-0.28	0.04	-0.52	-0.03	0.19	-0.51
		(0.05)	(0.06)	(0.09)	(0.07)	(0.14)	(0.11)
	κ k	0.36	0.31	0.31	0.18	0.49	0.32
		(0.03)	(0.04)	(0.03)	(0.02)	(0.10)	(0.05)
	λ k	0.31	0.28	0.23	0.28	0.24	0.21
		(0.02)	(0.02)	(0.02)	(0.02)	(0.04)	(0.03)
	φ k	1.00	0.54	0.46	0.42	0.19	0.39
		(-)	(0.05)	(0.05)	(0.05)	(0.04)	(0.05)
	ln L	-2462.0	-2157.1		-2097.6
	EN (τ )	0.00		2.58		7.52
	ICL	4942.7	4539.0		4270.1
	NEC	-	0.008		0.025

Notes: Bootstrapped standard errors in parentheses. Table based on our 'core sample' of 109 subjects. For all types, we assume logarithmic utility (r k = 1) and rational expectations.

Table 6

 6 shows the results. The left panel shows which model provides the best fit according to BIC. For 37 subjects (33.9%) pure self-interest (α i = β i = κ i = 0) has the lowest BIC score. For the remaining 72 subjects, some combinations of social preferences and/or moral concerns improve the model's fit. For 23 subjects, (21.1%) pure Homo moralis preferences (α i = β i = 0, κ i 0) provides the best individual fit. For another 11 subjects, models with κ i in combination with α i and/or β i have the lowest BIC scores. In sum, for 34 subjects (31.2%), the model with the lowest BIC score includes κ i . In comparison, α i and β i are included in the model with the lowest BIC score for 23 subjects (21.1%) and 35

Table 6 :

 6 Best individual fit Entries indicate the number of subjects for whom the specific model provides the lowest BIC or AIC score respectively. Table based on our 'core sample' of 109 subjects.

			BIC	AIC
	Parameters	Frequency	Percentage	Frequency	Percentage
	α i , β i , κ i	2	1.8	6	5.5
	α i , β i	7	6.4	6	5.5
	α i , κ i	5	4.6	8	7.3
	β i , κ i	4	3.7	10	9.2
	α i	9	8.3	11	10.1
	β i	22	20.2	18	16.5
	κ i	23	21.1	24	22.0
	-	37	33.9	26	23.9

Notes:

Table A .

 A 2: Lottery choices Lottery choices in the[START_REF] Eckel | Sex differences and statistical stereotyping in attitudes toward financial risk[END_REF] risk elicitation task.'Outcomes' are the payoffs denoted in "points", see Appendix A4 for the instructions. The final column lists the implied r i parameters for each lottery choice. Note that after the first session, we slightly adjusted the outcomes to better estimate r i . Table based on all 136 subjects.

		Outcomes
	Lottery	A	B	Frequency	Percentage	r i
	Sessions 2-8
	1	18	18	50	43.9%	1.61
	2	22	15	24	21.1%	1.00
	3	26	12	18	15.8%	0.39
	4	30	9	3	2.6%	0.25
	5	34	6	8	7.0%	0.08
	6	37	2	11	9.7%	-0.09
	Session 1
	1	18	18	5	22.7%	4.71
	2	22	16	3	13.6%	2.95
	3	26	14	6	27.3%	1.19
	4	30	12	4	18.2%	0.77
	5	34	10	2	9.1%	0.32
	6	40	4	2	9.1%	-0.13

Notes:

Table A .

 A 3: Individual parameter estimates (all subjects)Table based on estimates from all 136 subjects.

	Figure A.2: Distributions of individual parameter estimates (all subjects)
	Frequency
		<	-2	-1.6 -1.2	-.8	-.4 estimated α i 0	.4	.8	1.2	1.6	2	>
	Frequency
		<	-2	-1.6 -1.2	-.8	-.4 estimated β i 0	.4	.8	1.2	1.6	2	>
	Frequency
		<	-2	-1.6 -1.2	-.8	-.4 estimated κ i 0	.4	.8	1.2	1.6	2	>
	Parameter	Median	Mean		S.D.		Min			Max
	α i		0.17	599.72	5938.54		-0.89	68 186.74
	β i	-0.11	50.30	697.75		-496.78		8105.22
	κ i		0.20	189.62	2200.83		-0.29	25 666.71

Notes:

Table A .

 A 4: Estimates at the aggregate level (all subjects) Standard errors in parentheses. Table based on all 136 subjects. For all types, we assume logarithmic utility (r k = 1).

		1 type	2 types		3 types
		Rep. agent	Type 1	Type 2	Type 1	Type 2	Type 3
	α k	0.15	0.08	0.26	0.11	0.03	0.24
		(0.02)	(0.03)	(0.04)	(0.05)	(0.09)	(0.05)
	β k	0.00	0.11	-0.39	0.02	0.24	-0.45
		(0.02)	(0.03)	(0.11)	(0.05)	(0.07)	(0.11)
	κ k	0.21	0.24	0.17	0.19	0.35	0.16
		(0.01)	(0.02)	(0.02)	(0.02)	(0.10)	(0.03)
	λ k	0.26	0.29	0.17	0.31	0.22	0.16
		(0.01)	(0.02)	(0.02)	(0.02)	(0.03)	(0.02)
	φ k	1.00	0.60	0.40	0.48	0.17	0.36
		(-)	(0.05)	(0.05)	(0.06)	(0.05)	(0.04)
	ln L	-2898.0	-2638.3		-2587.6
	EN (τ )	0.00		6.15		14.87
	ICL	5815.7	5326.9		5258.8
	NEC	-	0.024		0.048

Notes:

Table A

 A

		.5: The 4-types model
		Type 1	Type 2	Type 3	Type 4
	α k	-0.02	0.47	0.27	0.13
		(0.07)	(0.12)	(0.04)	(0.11)
	β k	0.17	-1.10	-0.37	0.04
		(0.08)	(0.19)	(0.11)	(0.11)
	κ k	0.29	0.52	0.18	0.18
		(0.06)	(0.07)	(0.02)	(0.05)
	λ k	0.23	1.07	0.15	0.24
		(0.03)	(0.18)	(0.02)	(0.05)
	φ k	0.23	0.08	0.32	0.37
		(0.05)	(0.10)	(0.06)	(0.11)
	ln L		-2130.5
	EN (τ )		23.95
	ICL		4374.1
	NEC		0.116

Table A

 A

			.6: Strategies by type
	1 type	2 types		3 types
	Rep. agent	Type 1	Type 2	Type 1	Type 2	Type 3
	Sequential Prisoner's Dilemmas
	C, C, C	2%	3%	0%	2%	4%	0%
	C, C, D	21%	32%	5%	21%	43%	5%
	C, D, C	0%	1%	0%	1%	0%	0%
	C, D, D	9%	12%	4%	19%	4%	4%
	D, C, C	2%	3%	0%	3%	2%	0%
	D, C, D	7%	9%	3%	6%	13%	3%
	D, D, C	2%	3%	2%	5%	1%	2%
	D, D, D	57%	38%	87%	43%	24%	87%
	Trust Games
	I, G	29%	43%	5%	31%	56%	5%
	I, K	17%	23%	7%	38%	6%	7%
	N , G	5%	5%	3%	2%	9%	3%
	N , K	50%	28%	85%	30%	28%	84%
	Ultimatum Games
	E, A	44%	57%	23%	52%	61%	22%
	E, F	8%	12%	3%	9%	15%	3%
	U , A	48%	32%	74%	39%	24%	74%
	U , A	0%	0%	0%	0%	0%	0%

Table A .

 A 8: Estimates at the aggregate level, incl. CRRA parameter r k Bootstrapped standard errors in parentheses. Table based on our 'core sample' of 109 subjects.Table A.9: Estimates at the aggregate level (without morality) Standard errors in parentheses. Based on our 'core sample' of 109 subjects. For all types, we assume logarithmic utility (r k = 1).

		1 type		2 types		3 types
		Rep. agent	Type 1		Type 2	Type 1	Type 2	Type 3
	α k	0.15 1 type	0.07	0.28 2 types	0.06	0.11 3 types	0.28
		(0.02) Rep. agent	(0.02) Type 1		(0.07) Type 2	(0.05) Type 1	(0.05) Type 2	(0.07) Type 3
	β k α k	0.03 0.03	0.13 -0.12		-0.34 0.09	0.11 -0.16	0.19 -0.08	-0.37 0.09
		(0.03) (0.02)	(0.03) (0.03)		(0.17) (0.02)	(0.12) (0.05)	(0.07) (0.07)	(0.14) (0.02)
	κ k β k	0.19 0.15	0.21 0.27		0.19 -0.22	0.18 0.10	0.26 0.45	0.20 -0.21
		(0.01) (0.03)	(0.02) (0.04)		(0.03) (0.07)	(0.03) (0.05)	(0.03) (0.05)	(0.03) (0.08)
	λ k λ k	0.37 0.26	0.44 0.28			0.12 0.15	2.74 0.30	0.10 0.23	0.11 0.15
		(0.11) (0.02)	(0.20) (0.02)		(0.06) (0.01)	(0.44) (0.05)	(0.07) (0.03)	(0.07) (0.01)
	r k φ k	0.88 1.00	0.86 0.64			1.07 0.36	0.33 0.36	1.26 0.29	1.14 0.36
		(0.09) (-)	(0.12) (0.05)		(0.29) (0.05)	(0.27) (0.07)	(0.21) (0.06)	(0.25) (0.05)
	φ k ln L	1.00 -2425.2	0.63	0.37 -2244.4	0.30	0.31 -2217.1	0.39
	EN (τ )	(-) 0.00	(0.05)	4.96	(0.05)	(0.06)	(0.06) 17.55	(0.05)
	ln L ICL EN (τ ) NEC	-2335.1 4864.4 0.00 -		-2152.5 4526.6 4.17 0.027		4503.3 -2122.4 14.80 0.084
	ICL	4693.6		4360.8		4339.3
	NEC	-		0.023		0.070

Notes:

Notes:

Table A .

 A 10: Estimates at the aggregate level (reciprocity) Standard errors in parentheses. Based on our 'core sample' of 109 subjects. For all types, we assume logarithmic utility (r k = 1).

		1 type	2 types		3 types
		Rep. agent	Type 1	Type 2	Type 1	Type 2	Type 3
	α k	-0.08	-0.19	0.12	-0.13	0.12	-0.24
		(0.02)	(0.03)	(0.04)	(0.08)	(0.04)	(0.04)
	β k	0.15	0.26	-0.23	0.45	-0.21	0.10
		(0.03)	(0.03)	(0.06)	(0.05)	(0.07)	(0.05)
	δ k	0.16	0.28	-0.07	0.22	-0.07	0.32
		(0.03)	(0.04)	(0.05)	(0.14)	(0.05)	(0.08)
	λ k	0.25	0.26	0.15	0.22	0.15	0.28
		(0.02)	(0.02)	(0.01)	(0.03)	(0.02)	(0.05)
	φ k	1.00	0.64	0.36	0.29	0.36	0.35
		(-)	(0.05)	(0.05)	(0.07)	(0.05)	(0.07)
	ln L	-2409.6	-2214.0		-2186.2
	EN (τ )	0.00		3.84		16.24
	ICL	4838.0	4474.1		4454.3
	NEC	-	0.020		0.073

Notes:

By a "game protocol", we mean a game tree and associated monetary payoffs.

In the process of selecting the number of game protocols and the monetary payoffs, we conducted simulations to verify if we could retrieve the original parameters.

See https://aspredicted.org/blind.php?x=4u5nu8.

In the maximum likelihood estimations, we use 7 different starting values for each parameter, so 7 4 = 2, 401 starting values per individual i.

The finding that a sizeable share of the subjects (here 38%) are both spiteful (α k > 0 and β k < 0) and moral (κ k > 0) agrees with a recent theoretical result that preference evolution in some settings leads to a combination of self-interest, spite and Kantian morality (see[START_REF] Alger | Evolution of preferences in structured populations: genes, guns, and culture[END_REF].

In panel A of TableA.7 (see Appendix A3), we show a transition matrix for the two-types and three-

in the three-types model. All subjects who were classified as Type 1 in the two-types model are now distributed across the new Types 1 and 2.20 This is in line with the findings by[START_REF] Bruhin | The many faces of human sociality: Uncovering the distribution and stability of social preferences[END_REF].

Table A.7 (panels F and G) shows that the assignment of subjects to types is similar under subjective and rational expectations.

When allowing for Kantian morality (α, β, κ), a model with a representative agent (1 type) performs worse (51.4% accuracy) than the two-types model, and a model with three-types performs only slightly better (55.4%) than the two-types model.

This result does not contradict that of[START_REF] Alger | Homo moralis-preference evolution under incomplete information and assortative matching[END_REF], which is shown by[START_REF] Alger | Evolution of preferences in structured populations: genes, guns, and culture[END_REF]

Since each player has only one decision node, the distinction between mixed and behavioral strategies is immaterial.

sibility (Toulouse) for helpful suggestions and comments. I.A. acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 789111 -ERC EvolvingEconomics) and IAST funding from the French National Research Agency (ANR) under grant ANR-17-EURE-0010 (Investissements d'Avenir program).

Appendix A1 Distinguishing Kantian morality from social preferences

The Ultimatum Game protocol having been analyzed in detail in the main text (Subsection 2.3), we here analyze the other two game protocols. Throughout we assume risk neutrality; this is only for notational simplicity, the only difference being that the monetary payoffs would be replaced by the associated monetary utilities.

In the Trust Game protocol (Figure 1b), a behavior strategy is a vector x = (x 1 , x 2 ) ∈ X = [0, 1] 2 , where x 1 is the probability with which the player trusts the receiver, and x 2 the probability with which he honors trust (if the sender trusts him). 29 Then the expected utility (as defined in (1)) from playing x = (x 1 , x 2 ) against y = (y 1 , y 2 ) is (omitting the factor 1/2):

Hence, for a subject who believes that the opponent plays ŷ:

and ∂u i (x, ŷ)

The social preference parameters α i and β i represent consequentialistic motives: they give weight to the monetary payoff consequences given what the subject believes about the opponent's actual play. By contrast, the Kantian morality parameter κ i captures a deontological motive, such as "duty" or "to do the right thing", which ((following [START_REF] Alger | Homo moralis-preference evolution under incomplete information and assortative matching[END_REF] we take to be to evaluate one's strategy in the light of what would happen if, hypothetically, the opponent would also use the same strategy.

Turning now to the Sequential Prisoners' Dilemma game protocol (as in Figure 1a), denote by x 1 the probability of playing C when moving first, x 2 the probability of playing C when moving second after play of C by the opponent, and and x 3 the probability of playing C when moving second after play of D by the opponent. Hence, the vector x = (x 1 , x 2 , x 3 ) ∈ [0, 1] 3 is the player's behavior strategy in the symmetrically randomized sequential prisoners' dilemma. Then the expected utility (as defined in (1)) from playing

x = (x 1 , x 2 , x 3 ) against y = (y 1 , y 2 , y 3 ) is (again omitting the factor 1/2):

Hence, for a subject who believes that the opponent would play ŷ one obtains:

and

Again, these equations show that an individual with a Kantian moral concern (κ i > 0) is not only influenced by his belief about the opponent's strategy, but also by what he would himself do at every decision node of the game tree.

Appendix A2 Copula estimation

We use copula methods to describe the joint parameter distributions for the individual estimates of α i , β i and κ i . For this, let X α , X β and X κ be random variables, possibly statistically dependent, with marginal CDFs F α , F β and F κ . By Sklar's Theorem, their joint CDF can be written in the form

We follow a two-step approach [START_REF] Joe | The estimation method of inference functions for margins for multivariate models[END_REF][START_REF] Cherubini | Copula methods in finance[END_REF]. First, we fit the marginal distributions. For this, we assume that each preference parameter follows a Gumbel distribution, with CDF

where a ∈ R is usually called the location, and b > 0 the scale. The associated PDF is

The empirical distributions of α i and κ i have a relatively long right tail (see 3, which fits well with the Gumbel distribution. The empirical distribution of β i has a relatively long left tail, therefore, we fit the reverse distribution, i.e. we fit the distribution of -β i .

In the second step, we estimate the copula. We assume a Gumbel copula, which has the form:

for some ω ≥ 1, where ω = 1 represents statistical independence.

In both steps we use maximum likelihood to estimate parameters. Table A.1 shows the estimated parameters, and Figure ?? plots the estimated marginal distributions together with the empirical distributions. For the joint distribution, we estimate ω = 1.30. To put this into perspective, this estimate implies a Kendall's tau of τ = 1 -1 1.30 = 0.23. This compares well to the bivariate correlations (see Section 4.1). Expressed in Kendall's tau, the correlation between α i and -β i is τ = 0.21, for α i and κ i we obtain τ = 0.29 and for -β i and κ i we obtain τ = 0.11. Notes: Each panel shows transition matrices between types in different finite mixture models. Subjects are assigned a type based on the posterior probability τ i,k (that subject i belongs to type k, see eq. ( 14)).

Appendix A3 Additional tables and figures

Appendix A4 Experimental instructions Welcome

Welcome to this experiment. All subjects receive the same instructions. Please read them carefully.

Do not communicate with any of the other subjects during the entire experiment. If you have any questions, raise your hand and wait until one of us comes to you to answer your question in private.

During the experiment you will receive points. These points are worth money. How many points (and hence how much money) you get depends on your own decisions, the decisions of others, and chance. At the end of the experiment the points that you got will be converted to euros and the amount will be paid to you privately, in cash.

Every point is equivalent to 0.17 euro.

Your decisions are anonymous. They will not be linked to your name in any way.

Other subjects can never trace your decisions back to you.

Today's experiment consists of two parts. At the beginning of each part, you will receive new instructions. Your decisions made in one part will never affect outcomes in another part, so you can treat both parts as independent.

Decision situations I

In this part, you will participate in 18 different decision situations. For each decision situation, you will be randomly paired with someone else in the lab. Therefore, in each decision situation you will (most likely) be paired with a different subject than in the previous situation. You will never learn with whom you are paired.

The 18 decision situations will all be different, but they all involve two persons, and in all the decision situations one person is assigned to Role A (person A) while the other is assigned to Role B (person B). There are then two kinds of situations, as depicted in At the beginning of each decision situation, you and all others in the lab will be informed of the values.

Example

The figure below gives an example of a decision situation. This decision situation is randomly selected. Remember that each of the 18 decision situations will be different.

In this example:

• Both your decisions and your guesses will determine how many euros you get at the end of the experiment. Specifically, at the end of today's experiment, two of the 18 decision situations will be randomly selected for payment: for one of these situations you get points from the decisions, while for the other situation you get points from your guesses. The same two decision situations will be selected for everyone in the lab. Your decisions For one decision situation you and the others in the lab get points from the decisions.

For this situation, either you or the person you are paired with is assigned to Role A, while the other is assigned to Role B, with equal probability for each case. The number of points you and this other person get is then determined by your decision in the role to which you were assigned and the decision of the other person in the role to which (s)he was assigned.

Note that it is equally likely that your choices in role A or role B count.

Decision screens

Below you can see and try the decision screens. First, you will see the screen where you will be asked for a decision in a decision situation. If you make a decision, you will be taken to the screen where you will be asked for a guess about what others will do.

In the examples below, all decision situations are chosen randomly. You can try the decision screens as often as you want.

Show example

Quiz questions I