Estimating Social Preferences and Kantian Morality in Strategic Interactions

A.  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.⁶ 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; Alger and Weibull 2013) 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 or her material payoff would be if, hypothetically, the opponent were to choose the same strategy. This requires that the interaction be 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: x₁ is the probability that i plays C as a first mover, x₂ the probability that i plays C as a second mover following play C by the opponent, and x₃ 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)(ζ) of each play ζ 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 fig. 2). For example, ${\eta }_{\left(x,y\right)}\left((1,C,C)\right)=x_1{y}_2/2$ and ${\eta }_{\left(x,y\right)}\left((2,D,C)\right)=(1-y_1)x_3/2$.

Turning to the two other game protocols, when the trust game protocol is symmetrically randomized, a behavior strategy is a vector, $x=\left(x_1,x_2\right)\in {\left[0,1\right]}^2$, where x₁ is the probability with which i invests (selects I) and x₂ 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, $x=\left(x_1,x_2\right)\in {\left[0,1\right]}^2$, where x₁ is the probability with which i proposes an equal sharing (selects E), and x₂ 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)(ζ) to denote the probability of each play ζ 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.

B.  Preferences

In our empirical analysis we posit preferences that combine material self-interest, attitudes toward being ahead as well as behind (Fehr and Schmidt 1999), reciprocity (Charness and Rabin 2002), and a Kantian moral concern (Alger and Weibull 2013). Thus, let the expected utility of a subject i playing against a subject j be

$$\begin{array}{}\text{(1)}& \begin{array}{rl}{u}_i\left(x,y\right)=& \left(1-{\kappa }_i\right)\sum _{\zeta }{\eta }_{\left(x,y\right)}\left(\zeta \right){\pi }_i\left(\zeta \right)\\ & -({\alpha }_i+q{\delta }_i)\sum _{\zeta }{\eta }_{\left(x,y\right)}\left(\zeta \right)\max \left\{0,{\pi }_j\left(\zeta \right)-{\pi }_i\left(\zeta \right)\right\}\\ & -({\beta }_i+p{\gamma }_i)\sum _{\zeta }{\eta }_{\left(x,y\right)}\left(\zeta \right)\max \left\{0,{\pi }_i\left(\zeta \right)-{\pi }_j\left(\zeta \right)\right\}\\ & +{\kappa }_i\sum _{\zeta }{\eta }_{\left(x,x\right)}\left(\zeta \right){\pi }_i\left(\zeta \right),\end{array}\end{array}$$

where x and y are i’s and j’s behavior strategies, respectively, π_i(ζ) is i’s material payoff following play ζ, and π_j(ζ) is that of j. The dummy variable q takes the value 1 if j “misbehaved” and 0 otherwise, while the dummy variable p takes the value 1 if j “behaved nicely” and 0 otherwise.⁷ We follow Charness and Rabin (2002) by labeling a first-mover action as misbehavior if it excludes an outcome that has maximal joint monetary payoffs and as nice behavior if it includes an outcome that has maximal joint monetary payoffs.⁸

This utility function has five parameters. Two of them are the familiar measures of inequity aversion. The parameter α_i captures i’s disutility (if ${\alpha }_i>0$) or utility (if ${\alpha }_i<0$) from disadvantageous inequity, that is, from falling short in terms of material payoff in the interaction. Likewise, the parameter β_i captures i’s disutility (if ${\beta }_i>0$) or utility (if ${\beta }_i<0$) from advantageous inequity, that is, from being ahead in terms of material payoff. Then, reciprocity is captured by the parameters δ_i and γ_i. Finally, κ_i captures the Kantian moral concern (à la Homo moralis; Alger and Weibull 2013). It places weight on the expected material payoff that the subject would obtain if, hypothetically, both individuals were to use the subject’s strategy x. Under this hypothesis, the probability that a play ζ would occur is η_(x,x)(ζ). A κ_i-value strictly between zero and 1 represents a partly deontological motivation, an individual who, in addition to the social concern that consists of caring about his or her own material payoff and that to the other individual in the interaction, is also motivated by what is “the right thing to do,” what strategy to use if it were also used by the opponent. To choose a strategy x in order to maximize the last term in (1) is to choose a strategy that maximizes material payoff if used by both subjects (see Alger and Weibull 2013 for a discussion).⁹

The utility function in (1) nests many familiar utility functions in the literature. Clearly, setting all five parameters to zero, ${\alpha }_i={\beta }_i={\kappa }_i={\delta }_i={\gamma }_i=0$, represents pure self-interest and thus amounts to the classical Homo oeconomicus. The Fehr and Schmidt (1999) model of inequity aversion is obtained by setting ${\alpha }_i\ge {\beta }_i>0$ and ${\kappa }_i={\delta }_i={\gamma }_i=0$. One obtains Becker’s (1974) model of pure altruism by setting ${\kappa }_i={\delta }_i={\gamma }_i=0$ and ${\alpha }_i=-{\beta }_i$, for some ${\beta }_i\in \left(0,1/2\right]$.¹⁰ Here β_i is the individual’s “degree of altruism,” the weight placed on the other subject’s material payoff, while the weight $1-{\beta }_i$ is placed on the individual’s own material payoff. Pure Homo moralis preferences are obtained by setting ${\alpha }_i={\beta }_i={\delta }_i={\gamma }_i=0$ and ${\kappa }_i\in \left(0,1\right]$. Here κ_i is the individual’s “degree of Kantian morality,” the weight placed on the material payoff that would be obtained if both subjects in the interaction at hand played x, the strategy used by individual i, while the weight $1-{\kappa }_i$ is placed on the individual’s own material payoff, given the strategy profile (x, y) effectively played. Finally, the utility function in (1) also nests models with reciprocity like in Charness and Rabin (2002) and Bruhin, Fehr, and Schunk (2019) when δ_i, γ_i, α_i, and β_i are non-nil and ${\kappa }_i=0$.

A detailed account of how our experimental design allows us to disentangle the motivations is provided in section II.D. However, the qualitative distinction between reciprocity and a Kantian moral concern is already clear from (1): while the reciprocity parameters δ_i and γ_i are akin to modifications of the parameters α_i and β_i that capture the subject’s attitude toward the other actual subject’s payoff, the Kantian moral concern instead makes the subject evaluate what material outcome he himself would obtain if his strategy were universalized.

C.  Experimental Procedures

In total, 136 subjects (69 men, 67 women) participated in the experiment. We conducted eight 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),¹¹ for different monetary payoff assignments T, R, P, and S, listed in table 1.¹²

All payoffs are denoted in “points,” where 1 point is equivalent to 17 eurocents. At the beginning of each session, the order of the 18 game protocols was fully randomized, meaning that participants could for example play a UG protocol first, then a TG protocol, followed by an SPD, and then another TG. For each game protocol, subjects indicated first what they would do at each decision node and second what they believed others would do at each decision node.¹³ In all game protocols, we used neutral labels. Two of the 18 game protocols were randomly selected for payment. To minimize the possibility to hedge, 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 Eckel and Grossman (2002). Self-reported demographic data were 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 A7. Sessions took around 1 hour and subjects earned between €10.50 and €26.90 with an average of €18.80. Key features of the experimental design and main analyses were preregistered.¹⁴

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 (x₁) and trustworthiness (x₂). In the UGs, lower offers (P) are accepted less frequently (x₂). Moreover, on average actions (x) and beliefs (y) are highly correlated (see also fig. A.1 in app. A2). Table A.1, in appendix A2, presents all decisions in the risk elicitation task. Based on their lottery choice, most subjects (83%) are classified as being risk averse.

D.  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, as shown in detail in appendix A1. 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. Here we explain why.

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 by using the strategy method). Put differently, an important consequence of Kantian morality is that a subject’s preferences regarding 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 distributional concerns such as altruism, inequity aversion, or spite (whether or not augmented by reciprocity). We illustrate this with the trust game protocol shown in figure 3, for two different values of R: 50 and 70. Consider a subject who, for both values of R, believes that the opponent will keep (K) as a second mover. If driven by purely distributional concerns (or reciprocity), such a subject should choose the same strategy for both values of R, since G is off the equilibrium path. By contrast, if the subject has a Kantian moral concern, and would himself choose G as a second mover, the value of R does matter in his evaluation of all the behavior strategies. In particular, if the subject selects N for $R=50$, a sufficiently high κ can make the subject switch to I for $R=70$.

More generally and more formally, consider a (symmetrically randomized) trust game protocol (see fig. 1B) with $2R>T+S$. Suppose that an individual i believes that the opponent will play K (“keep”) as second mover and I (“invest”) as a first mover. The conditions for i to choose I as first mover and G (“give back”) as second mover, respectively, are then¹⁵

$$\begin{array}{}\text{(2)}& (1-{\kappa }_i)(S-P)-{\alpha }_i(T-S)+{\kappa }_{i}2(R-P)\ge 0,\end{array}$$

$$\begin{array}{}\text{(3)}& (1-{\kappa }_i)(R-T)+({\beta }_i+p{\gamma }_i)(T-S)+{\kappa }_i(2R-S-T)\ge 0.\end{array}$$

The first condition, which pertains to the choice as first mover, shows formally the observation made above: the value of R, which given the subject’s posited belief is off the equilibrium path, matters if and only if ${\kappa }_i>0$. Indeed, Kantian morality makes this individual evaluate the improvement in the material payoff he would obtain from selecting I instead of N, under the hypothetical scenario in which the opponent would also pick G as second mover: collecting the terms multiplying κ_i, this improvement equals $R-P/2-S/2$ (the probability $1/2$ has been omitted in [2]). Turning now to the choice as second mover, a positive κ_i makes the individual evaluate the increase in expected material payoff (the expectation being taken over the two player roles) he would obtain if he as well as the opponent (hypothetically) were to choose G rather than K as second mover, given that he himself picks I as first mover: collecting the terms multiplying κ_i, this equals $(1/2)(R-S)$ (the probability $1/2$ has been omitted in [3]).

Two important implications appear from conditions (2) and (3). First, payoffs off the equilibrium path may matter: for example, condition (2) shows that a change in the payoff R (which is off the equilibrium path if the individual at hand moves first and his beliefs about his opponent are correct) can make the individual switch from N to I. Second, condition (3) reveals that in a model in which the Kantian moral concern is omitted, an individual must be averse to being ahead (${\beta }_i>0$) for him to choose G. By contrast, an individual with a positive degree of morality ${\kappa }_i>0$ may choose G even if ${\beta }_i=0$. In fact, if κ_i is large enough, he can even be spiteful (${\beta }_i<0$) and still choose G.

We provide a detailed analysis of the first-order conditions for the three game protocols in appendix A1.2. Furthermore, as an illustration of how Kantian morality may lead to different behavioral predictions than the social concerns included in the posited utility function (1), we compare the predicted behavior for five different preference types for all the payoffs used in the experiment in table 2. The types are pure self-interest (${\alpha }_i={\beta }_i={\gamma }_i={\delta }_i={\kappa }_i=0$), behindness aversion (${\alpha }_i=0.4$, ${\beta }_i={\gamma }_i={\delta }_i={\kappa }_i=0$), altruism (${\alpha }_i=\mathrm{-0.2}$, ${\beta }_i=0.5$, ${\gamma }_i={\delta }_i={\kappa }_i=0$), a combination of altruism and reciprocity (${\alpha }_i=\mathrm{-0.2}$, ${\beta }_i=0.5$, ${\gamma }_i={\delta }_i=0.1$, ${\kappa }_i=0$), or Kantian morality (${\alpha }_i={\beta }_i={\gamma }_i={\delta }_i=0$, ${\kappa }_i=0.2$). The behindness-averse type and the type combining altruism and reciprocity qualitatively resemble the behindness-averse and strongly altruistic type estimated by Bruhin, Fehr, and Schunk (2019).

All types display different behavior. In the sequential prisoner’s dilemma protocols, both self-interest and behindness aversion lead to unconditional defection as a second mover, but self-interested types will more frequently (opportunistically) cooperate as a first mover. An altruist will frequently unconditionally cooperate as a second mover, unless defection after cooperation leads to higher joint payoffs (SPD 1), or when punishment becomes sufficiently attractive (SPD 6). When enriching altruism with reciprocity, conditional cooperation emerges. Likewise, an individual motivated by Kantian morality (Homo moralis) will typically conditionally cooperate, unless the benefits to joint cooperation become too small (SPDs 1 and 2). In particular, note that if $S+T>2R$ (as in SPD 1), the convex combination of self-interest and Kantian morality entails a behavior not seen in any of the other types. By contrast to self-interest and behindness aversion, the Kantian moral concern entails a second-mover behavior that maximizes the expected material payoff from an ex ante perspective (i.e., cooperating following defection and vice versa). However, by contrast to the altruistic type in table 2, which also selects this second-mover behavior, the type that combines self-interest and Kantian morality defects as a first mover: given that such an individual would cooperate as a second mover following defection, both the self-interest part and the Kantian part of the utility function indeed entail a wish to defect.

The behavior of those motivated by Kantian morality differs even more strongly from those exhibiting a combination of altruism and reciprocity in the trust game and ultimatum game protocols. In the trust game protocols, (strong) altruists will always invest (I) as first mover and “give back” (G) as a second mover, while individuals motivated by Kantian morality will play “keep” (K) when R is relatively low.¹⁶ In the ultimatum game, those motivated by Kantian morality will make unequal offers (U) and accept any offer (A), while those motivated by altruism and negative reciprocity will propose equal splits (E).

A.  Individual Preferences

For each subject i, we estimate the individual’s social and moral preference parameters α_i, β_i, δ_i, γ_i, and κ_i as specified in (1), using a standard additive error specification. We refer to these preference parameters using the vector ${\theta }_i=\left({\alpha }_i,{\beta }_i,{\delta }_i,{\gamma }_i,{\kappa }_i\right)$. 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 ${\widehat{y}}_i$ is i’s expectation about his opponent’s behavior, is a random variable of the additive form

$${\tilde{u}}_i(x_i,{\widehat{y}}_i,{\theta }_i)=u_i(x_i,{\widehat{y}}_i,{\theta }_i)+{\epsilon }_{i{x}_i},$$

where $u_i(x_i,{\widehat{y}}_i,{\theta }_i)$ is the expected utility of using strategy x_i given beliefs ${\widehat{y}}_i$ following from the utility function in (1), and ${\epsilon }_{i{x}_i}$ is a random variable representing idiosyncratic tastes not picked up by the hypothesized utility (for the estimates we divide $u_i(x_i,{\widehat{y}}_i,{\theta }_i)$ by 1/2, the role randomization factor). Such a random utility specification sometimes induces choices of actions that do not maximize the deterministic component $u_i(x_i,{\widehat{y}}_i,{\theta }_i)$. Assuming that the noise terms ${\epsilon }_{i{x}_i}$ 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 ${\widehat{y}}_i$ about the opponent’s play, is given by the familiar logit formula (McFadden 1974)

$$\begin{array}{}\text{(4)}& p_i\left(x_i,{\widehat{y}}_i,{\theta }_i,{\lambda }_i\right)=\frac{\exp \{[u_i(x_i,{\widehat{y}}_i,{\theta }_i)]/{\lambda }_i\}}{{\sum }_{x^{\prime }\in X_g}\exp \{[u_i(x^{\prime },{\widehat{y}}_i,{\theta }_i)]/{\lambda }_i\}},\end{array}$$

where ${\lambda }_i>0$ 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 $g\in G$, 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 $u_i(x_i,{\widehat{y}}_i,{\theta }_i)$. We use maximum likelihood to estimate the preference parameter vector ${\theta }_i=\left({\alpha }_i,{\beta }_i,{\delta }_i,{\kappa }_i\right)$ and the “noise” parameter λ_i for each individual i.¹⁷ Then, the probability density function can be written as

$$\begin{array}{}\text{(5)}& f({\mathbf{x}}_i,{\widehat{\mathbf{y}}}_i,{\theta }_i,{\lambda }_i)=\prod _{g\in G}\prod _{x\in X_g}{p}_i{\left(x,{\widehat{y}}_i,{\theta }_i,{\lambda }_i\right)}^{I(i,g,x)},\end{array}$$

where x_i is the vector of the observed pure strategies of individual i, ${\widehat{\mathbf{y}}}_i$ is the vector of stated beliefs of individual i about the opponent’s strategy in all the game protocols, and I(i, g, x) is an indicator function that equals 1 if i plays strategy x in game protocol g and 0 otherwise.

B.  Aggregate Estimations

We estimate preference parameters for both 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, Fehr, and Schunk (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 $k=\{1,\dots ,K\}$, we estimate the parameter vector ${\theta }_k=\left({\alpha }_k,{\beta }_k,{\delta }_k,{\kappa }_k\right)$ and the noise parameter λ_k. The log likelihood is then given by

$$\begin{array}{}\text{(6)}& \ln L=\sum _{i=1}^N\ln [\sum _{k=1}^K{\varphi }_k\cdot f({\mathbf{x}}_i,{\widehat{\mathbf{y}}}_i,{\theta }_k,{\lambda }_k)],\end{array}$$

where ϕ_k is the population share of type k in the population. To maximize the log likelihood in (6), we use an expectation-maximization (EM) algorithm (see, e.g., McLachlan, Lee, and Rathnayake 2019).¹⁸ As part of the EM algorithm, we estimate the posterior probabilities τ_i,k that individual i belongs to type k by

$$\begin{array}{}\text{(7)}& {\tau }_{i,k}=\frac{{\varphi }_k\cdot f({\mathbf{x}}_i,{\widehat{\mathbf{y}}}_i,{\theta }_k,{\lambda }_k)}{{\sum }_{m=1}^K{\varphi }_m\cdot f({\mathbf{x}}_i,{\widehat{\mathbf{y}}}_i,{\theta }_m,{\lambda }_m)}.\end{array}$$

A.  Individual Preferences

Figure 4 shows the marginal distributions of the estimated individual preference parameters α_i, β_i, and κ_i for our core sample of 112 subjects.²⁰ For all three parameters, we observe considerable heterogeneity. Most estimates of α_i, β_i, and κ_i are positive, and signed-rank tests confirm that the parameter distributions are located to the right of zero ($p<.001$ for either α_i, β_i, and κ_i estimates).

Table 3, which shows summary statistics for the parameter estimates, provides further support for the pattern observed in figure 4. Median and mean estimates are positive for α_i, β_i, and κ_i. Moreover, the relatively large standard deviations indicate that there is considerable heterogeneity in social preferences and Kantian morality.

Figure 5 illustrates the pairwise correlations between the three preference parameter estimates. The left panel of figure 5 shows that the estimates for α_i and β_i are negatively correlated (Spearman’s $\rho =\mathrm{-.235}$, $p=.013$, $n=112$). For many individuals we observe a combination of ${\alpha }_i>0$ and ${\beta }_i>0$, in line with inequality-averse preferences. However, we also observe a number of individuals for whom ${\alpha }_i>0$ and ${\beta }_i<0$, in line with spiteful or competitive preferences. The middle panel of figure 5 reveals a strong and positive correlation between α_i and κ_i estimates (Spearman’s $\rho =.427$, $p<.001$, $n=112$). This means that many individuals combine behindness aversion with Kantian morality. For the estimates of β_i and κ_i we find a negative correlation (Spearman’s $\rho =\mathrm{-.217}$, $p=.022$, $n=112$). 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 A3 provides more details.

B.  Aggregate Estimations

We now turn to estimation of preferences at the aggregate level (see sec. III.B for details). To distinguish these estimates from the individual ones, we use an index k to designate the type. Table 4 presents the estimates of the finite mixture models for one, two, and three types.

3.  Comparing the One-, Two-, and Three-Type 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 such as 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, McLachlan, Lee, and Rathnayake (2019) recommend using the “integrated completed likelihood” (or “integrated classification,” ICL; Biernacki, Celeux, and Govaert 2000). This criterion is approximated by

$$\begin{array}{}\text{(8)}& \text{ICL}=\mathrm{-2}\ln L+d\ln N+\text{EN}(\mathit{\tau }),\end{array}$$

where the log-likelihood function lnL is defined as in (6), d is the number of estimated parameters, and N is the number of individuals in our sample. The last term in (8) is the entropy

$$\begin{array}{}\text{(9)}& \text{EN}(\mathit{\tau })=-\sum _{k=1}^K\sum _{i=1}^N{\tau }_{i,k}\ln {\tau }_{i,k},\end{array}$$

where τ_i,k is the estimated posterior probability of individual i belonging to type k, as defined in (7). This implies that the more strongly individuals are assigned to types (i.e., all τ_i,k’s close to zero or 1), 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.

Figure 6 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’s are very close to zero or 1, which implies that most individuals are quite precisely assigned to a type. For the two-type model, virtually all estimated τ_i,k’s are close to zero or 1. For the three-type model, a few individuals are imprecisely classified.

Bruhin, Fehr, and Schunk (2019) use the “normalized entropy criterion” (NEC; Celeux and Soromenho 1996), which is defined as

$$\begin{array}{}\text{(10)}& \text{NEC}=\frac{\text{EN}(\mathit{\tau })}{\ln L(K)-\ln L(1)},\end{array}$$

where lnL(1) is the log likelihood of the representative agent model and lnL(K) the log likelihood 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 4 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-type model and the ICL selects the three-type model. Table A.5, in appendix A2, shows estimates and goodness-of-fit metrics for a four-type model. The four-type model performs worse on NEC but better on ICL than the two- and three-type models in table 4. Note that marginal improvement in the ICL score is largest when going from the representative agent to the two-type model. In sum, assuming two types instead of a representative agent brings us a long way in capturing the heterogeneity in the population.

C.  Robustness

Here we examine the robustness of the results reported above by allowing for risk aversion, rational expectations, and game-specific noise parameters λ. We only discuss the main findings; more details are provided in appendix A4.

A.  Aggregate Estimations

Table 6 shows the finite mixture estimates for the model allowing for distributional preferences (α_k, β_k), Kantian morality (κ_k), and reciprocity (δ_k, γ_k). Including the reciprocity parameters δ_k and γ_k has limited effect on the parameter estimates compared to the preregistered models in table 4. In particular, the estimated κ_k is nearly identical for all the types. Also, for most types the parameter estimates of α_k and β_k are nearly identical, and the estimates of δ_k and γ_k are not significantly different from zero. The only major change appears for type 2 in the two-type model and type 3 in the three-type model, for which the estimates of β_k are now positive and much larger than those in table 4, and both the δ_k and γ_k estimates are significantly different from zero.²⁶ Interestingly, these δ_k and γ_k estimates are all negative.²⁷ Thus, subjects classified as belonging to these types appear to exhibit less behindness aversion when the opponent has “behaved unkindly” as a first mover, and less aheadness aversion when the opponent has “behaved kindly” as a first mover. Although counterintuitive, these findings are in line with those of Charness and Rabin (2002) (see their tables VI and VII and the associated discussion). By contrast, Bruhin, Fehr, and Schunk (2019) find conjectured signs for the reciprocity parameter estimates that are significantly different from zero (see their tables 1 and 2).

To study the added value of Kantian morality and reciprocity, we compare one-, two-, and three-type models allowing for combinations of distributional preferences (α, β), Kantian morality (κ), and reciprocity (δ, γ). Tables 4 and 6 showed the results for models allowing for (α, β, κ) and (α, β, κ, δ, γ), respectively. In tables A.10 and A.11, in appendix A2, we further report the results for models allowing for only distributional preferences (α, β) and distributional preferences in combination with reciprocity (α, β, δ, γ), respectively. Figure 7 shows the ICL scores for these models. Three things become clear from this figure. First, as lower ICL scores indicate a more preferred model, all multiple-type models strongly outperform the one-type (representative agent) models. Second, adding either Kantian morality (κ) or reciprocity (δ, γ) to the pure distributional model (α, β) reduces the ICL scores substantially, showing that both Kantian morality (κ) and reciprocity (δ, γ) improve the fit of the model. Third, Kantian morality (κ) and reciprocity (δ, γ) are not substitutes: adding both Kantian morality (κ) and reciprocity (δ, γ) to the pure distributional model (α, β) yields a decrease in the ICL score that is approximately twice as large as adding only Kantian morality or only reciprocity.²⁸

B.  Individual Estimations

Figure 8 shows the individual parameter estimates when allowing for distributional preferences (α_i, β_i), Kantian morality (κ_i), and reciprocity (δ_i, γ_i). As in the preregistered model without reciprocity, most individual estimates of α_i, β_i, and κ_i are positive. For the reciprocity parameters δ_i and γ_i, we observe considerable heterogeneity, and both negative and positive estimates. table A.12, in appendix A2, shows summary statistics.

To study the added value of the different preference parameters, we consider all models that are nested in (1) and apply standard information criteria.²⁹ We use both the Bayesian information criterion (BIC) and the Akaike information criterion (AIC), each of which is based on the log likelihoods and adds a penalty for each parameter. The lower the score, the better the fit. More precisely, the criteria are

$$\begin{array}{}\text{(11)}& \text{BIC}=\mathrm{-2}\ln (L)+d\ln (18)\end{array}$$

and

$$\begin{array}{}\text{(12)}& \text{AIC}=\mathrm{-2}\ln (L)+2d,\end{array}$$

where ln(18) in (11) comes from the 18 observations per subject. Since $\ln 18\approx 2.89>2$, BIC gives a heavier penalty per parameter than AIC.

Table 7 shows the results. Columns 1 and 2 show which model provides the best fit according to BIC. For 21 subjects (18.8%) pure self-interest (${\alpha }_i={\beta }_i={\kappa }_i={\delta }_i={\gamma }_i=0$) has the lowest BIC score. This contrasts with the aggregate estimates (table 6), where no purely self-interested type emerges. This difference may be the result of the relatively small number of observations for each individual estimation, giving less power to reject self-interest at the individual level. For the remaining 91 subjects, some combinations of social preferences and/or moral concerns improve the model’s fit. In sum, for 21 subjects (18.8%), the model with the lowest BIC score includes κ_i. In comparison, α_i, β_i, δ_i, and γ_i are included in the model with the lowest BIC score for 28 subjects (25.0%), 77 subjects (68.8%), 16 subjects (14.3%), and 45 subjects (40.2%), respectively. In particular β_i (aheadness aversion) and γ_i (positive reciprocity) play a big role and improve the fit for 69% and 40% of subjects, respectively, while the added value of each of the other three preference parameters is roughly in the same ballpark. Columns 3 and 4 of table 7 show 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 25 subjects (or 22.3%): again, a smaller number of subjects than for β_i (82 subjects, or 73.2%) and γ_i (53 subjects, or 47.3%), but a number of subjects closer to α_i (40 subjects, or 35.7%) and also δ_i (24 subjects, or 21.4%).

C.  Out-of-Sample Predictions

So far, we have 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 at both the individual level and the aggregate level.

Figure 9 illustrates the results, by comparing the predictive accuracy of models allowing for distributional preferences (α, β), distributional preferences in combination with either reciprocity (α, β, γ, δ) or Kantian morality (α, β, κ), or with both (α, β, κ, γ, δ).³⁰ Figure 9A compares the predictive accuracy based on individual estimates. All models clearly outperform random choice (which would lead to 20.8% accurate predictions in expectation). All models allowing for distributional preferences perform much better than when assuming self-interest, but the differences in predictive accuracy between these models are small. On average, the (α, β, κ)-model on average predicts 55.5% of choices correctly, somewhat more than the (α, β)-model, which predicts 55.2% of choices correctly, and less than the (α, β, δ, γ)- and (α, β, κ, δ, γ)-models, which give 59.1% and 58.8% average accuracy, respectively. All models allowing for distributional preferences, reciprocity, and/or Kantian morality perform much better than when assuming self-interest, which gives 48.8% average accuracy.

The other panels of figure 9 show the predictive accuracy of finite mixture models. Figure 9D shows that assuming a representative agent (one type) leads to much lower predictive accuracy. On average, the models assuming a representative agent achieve between 43.9% and 48.9% accuracy, much below the accuracy of the models allowing for individual estimates. The three-type and two-type models, however, perform much better. As for the individual estimates, the two-type and three-type models with distributional preferences, reciprocity, and/or Kantian morality outperform self-interest, but the differences between these models are modest. For the two-type models that go beyond self-interest, average accuracy is between 53.4% and 54.5%, while for the three-type models the range is 55.9% to 57.6%. This provides further evidence that the two-type model effectively captures the heterogeneity in preferences.

In sum, allowing for distributional preferences substantially improves the predictive accuracy over self-interest. However, the added value of Kantian morality and reciprocity over distributional preferences is limited in the out-of-sample predictions. This contrasts with the improved within-sample fit when allowing for Kantian morality and reciprocity that we observed in figure 7.³¹

1.  Dictator Game Protocols

To see why dictator game protocols would not allow us to distinguish between social preferences and Kantian morality à la Homo moralis, consider such a 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 $m>1$.³³ Suppose that both players face an equal probability of being the donor, and denote by $x\in [0,w]$ and $y\in [0,w]$ their respective strategies (how much to give in the donor role). Consider first a pure altruist i, with ${\beta }_i=-{\alpha }_i\ge {\kappa }_i={\delta }_i={\gamma }_i=0$, and thus a utility function of the form (the factor $1/2$ represents nature’s draw of roles)

$$\begin{array}{}\text{(13)}& u_i\left(x,y\right)=\frac{1}{2}\left[(1-{\beta }_i)(w-x+my)+{\beta }_i(mx+w-y)\right].\end{array}$$

If instead i is a pure Homo moralis, with ${\kappa }_i\ge {\alpha }_i={\beta }_i={\delta }_i={\gamma }_i=0$, then his or her expected utility is

$$\begin{array}{}\text{(14)}& u_i\left(x,y\right)=\frac{1}{2}\left[(1-{\kappa }_i)(w-x+my)+{\kappa }_i(mx+w-x)\right].\end{array}$$

Comparison of the second terms in these utility functions reveals that while an altruist cares about the other individual’s monetary payoff $(mx+w-y)/2$ (which depends on the other’s strategy y), an individual driven by Kantian morality instead cares about the monetary payoff $(mx+w-x)/2$, which would result if both players were to use i’s strategy x. Nonetheless, as shown by the derivatives with respect to the individual’s own strategy x, the trade-off for altruists and Kantian moralists is the same here:

$$\begin{array}{}\text{(15)}& \frac{d{u}_i(x,y)}{dx}=\frac{1}{2}\left[{\beta }_{i}m-(1-{\beta }_i)\right]\end{array}$$

and

$$\begin{array}{}\text{(16)}& \frac{d{u}_i(x,y)}{dx}=\frac{1}{2}\left({\kappa }_{i}m-1\right).\end{array}$$

Whether an altruist or a Kantian moralist, the individual gives either the whole endowment or nothing at all: indeed, dividing the right-hand side of (15) by $1-{\beta }_i$, and letting ${\sigma }_i\equiv {\beta }_i/(1-{\beta }_i)$, we see that the altruist gives everything if σ_i exceeds $1/m$ while the Kantian moralist gives everything if κ_i exceeds $1/m$.³⁴ Therefore, we would be unable to separate altruism from a Kantian concern using dictator games.³⁵

2.  The Ultimatum, Trust, and Sequential Prisoner’s Dilemma Game Protocols

Here we write the full expected utility expressions of a subject i with a utility function as in (1) in each of the three game protocols. The objective is to show the qualitative difference between Kantian morality on the one hand (as captured by κ_i) and social preferences on the other hand (as captured by α_i, β_i, δ_i, and γ_i).

Beginning with the ultimatum game protocol, as in figure 1C, i obtains the following expected utility from using behavior strategy $x=\left(x_1,x_2\right)$ when he believes that the opponent will use behavior strategy $\widehat{y}=\left({\widehat{y}}_1,{\widehat{y}}_2\right)$ (the randomization factor $1/2$ has been omitted):

$$\begin{array}{}\text{(17)}& \begin{array}{rl}{u}_i\left(x,\widehat{y}\right)=& (1-{\kappa }_i)[x_{1}R+\left(1-x_1\right){\widehat{y}}_{2}T+\left(1-x_1\right)\left(1-{\widehat{y}}_2\right)S\\ & +{\widehat{y}}_{1}R+\left(1-{\widehat{y}}_1\right)x_{2}P+\left(1-{\widehat{y}}_1\right)\left(1-x_2\right)S]\\ & -\left[({\alpha }_i+{\delta }_i)\left(1-{\widehat{y}}_1\right)x_2+{\beta }_i\left(1-x_1\right){\widehat{y}}_2\right]\left(T-P\right)\\ & +{\kappa }_i[x_{1}R+\left(1-x_1\right)x_{2}T+\left(1-x_1\right)\left(1-x_2\right)S\\ & +x_{1}R+\left(1-x_1\right)x_{2}P+\left(1-x_1\right)\left(1-x_2\right)S].\end{array}\end{array}$$

The partial derivatives with respect to x₁ and x₂ are thus

$$\begin{array}{}\text{(18)}& \begin{array}{rl}\frac{\partial u_i\left(x,\widehat{y}\right)}{\partial x_1}=& (1-{\kappa }_i)[R-{\widehat{y}}_{2}T-\left(1-{\widehat{y}}_2\right)S]+{\beta }_i{\widehat{y}}_2\left(T-P\right)\\ & +{\kappa }_i\left[2\left(R-S\right)-x_2\left(T+P-2S\right)\right],\end{array}\end{array}$$

$$\begin{array}{}\text{(19)}& \frac{\partial u_i\left(x,\widehat{y}\right)}{\partial x_2}=(1-{\kappa }_i)\left(1-{\widehat{y}}_1\right)\left(P-S\right)-({\alpha }_i+{\delta }_i)\left(1-{\widehat{y}}_1\right)\left(T-P\right)+{\kappa }_i\left(1-x_1\right)\left(T+P-2S\right).\end{array}$$

Note that in this game protocol behindness aversion matters only following the unfair offer, in which case it is augmented by the negative reciprocity parameter δ_i. Hence, in the following discussion we will refer to the term ${\alpha }_i+{\delta }_i$ simply as behindness aversion. To see the two key effects of Kantian morality mentioned in the main text, we compare an individual who is inequity averse but does not have a Kantian concern ($({\alpha }_i+{\delta }_i){\beta }_i>0={\kappa }_i$) to one who has a Kantian concern but is not inequity averse (${\kappa }_i>0={\alpha }_i+{\delta }_i={\beta }_i$). First, when considering the effect of his choice as a first mover, x₁, the inequity-averse individual pays no attention to his choice as a second mover, while the Kantian moralist does (i.e., x₂ shows up in the derivative in [18] if and only if ${\kappa }_i\ne 0$). Likewise, when considering the effect of his choice as a second mover, x₂, the inequity-averse individual pays no attention to his choice as a first mover, while the Kantian moralist does (i.e., x₁ appears in [19] if and only if ${\kappa }_i\ne 0$). Second, the expressions (18) and (19) 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 in which $1={\kappa }_i>{\alpha }_i+{\delta }_i={\beta }_i=0$, the Kantian moralist chooses the strategy that would maximize the expected material payoff should both players choose it, irrespective of what she or he believes the opponent will play.

In the trust game protocol (fig. 1B), a behavior strategy is a vector $x=\left(x_1,x_2\right)\in X={\left[0,1\right]}^2$, where x₁ is the probability with which the player trusts the receiver, and x₂ the probability with which he honors trust (if the sender trusts him).³⁶ Then the expected utility (as defined in [1]) from playing $x=\left(x_1,x_2\right)$ against $y=\left(y_1,y_2\right)$ is (omitting the factor $1/2$)

$$\begin{array}{}\text{(20)}& \begin{array}{rl}{u}_i\left(x,y\right)=& (1-{\kappa }_i)\{x_1\left[y_{2}R+\left(1-y_2\right)S\right]+\left(1-x_1\right)P\}\\ & +(1-{\kappa }_i)\{y_1\left[x_{2}R+\left(1-x_2\right)T\right]+\left(1-y_1\right)P\}\\ & +{\kappa }_i\left\{x_1\left[x_{2}R+\left(1-x_2\right)S\right]+\left(1-x_1\right)P\right\}\\ & +{\kappa }_i\left\{x_1\left[x_{2}R+\left(1-x_2\right)T\right]+\left(1-x_1\right)P\right\}\\ & -\left[{\alpha }_i{x}_1\left(1-y_2\right)+({\beta }_i+{\gamma }_i)y_1\left(1-x_2\right)\right]\left(T-S\right).\end{array}\end{array}$$

Note that in this game protocol it is the positive reciprocity parameter γ_i that appears: it augments aheadness aversion following a “nice” first move by the opponent. Hence, for a subject who believes that the opponent plays $\widehat{y}$,

$$\begin{array}{}\text{(21)}& \frac{\partial u_i\left(x,\widehat{y}\right)}{\partial x_1}=(1-{\kappa }_i)[S-P+{\widehat{y}}_2\left(R-S\right)]+{\kappa }_i\left[x_2\left(2R-S-T\right)+S+T-2P\right]-{\alpha }_i\left(1-{\widehat{y}}_2\right)\left(T-S\right)\end{array}$$

and

$$\begin{array}{}\text{(22)}& \frac{\partial u_i\left(x,\widehat{y}\right)}{\partial x_2}=(1-{\kappa }_i){\widehat{y}}_1\left(R-T\right)+{\kappa }_i{x}_1\left(2R-S-T\right)+({\beta }_i+{\gamma }_i){\widehat{y}}_1\left(T-S\right).\end{array}$$

Again, the individual’s own play as second mover, x₂, appears in the derivative for play as first mover, x₁, if and only if ${\kappa }_i\ne 0$ (see in [21]). Likewise, the individual’s own play as first mover, x₁, appears in the derivative for play as second mover, x₂, if and only if ${\kappa }_i\ne 0$ (see in [22]).

We turn finally to the sequential prisoner’s dilemma game protocol (as in fig. 1A). As noted in the main text, “defect” by the first mover is classified as misbehavior if and only if $2R>S+T$, while “cooperate” by the first mover is classified as nice behavior if and only if $2R>S+T$. To account for this in the expression below, let q and p be dummy variables that take the value 1 if $2R>S+T$ and 0 otherwise. The expected utility (as defined in [1]) from playing $x=\left(x_1,x_2,x_3\right)$ against $y=\left(y_1,y_2,y_3\right)$ is then (again omitting the factor $1/2$)

$$\begin{array}{}\text{(23)}& \begin{array}{rl}{u}_i\left(x,y\right)=& (1-{\kappa }_i)[x_1{y}_{2}R+x_1\left(1-y_2\right)S+\left(1-x_1\right)y_{3}T+\left(1-x_1\right)\left(1-y_3\right)P]\\ & +(1-{\kappa }_i)[y_1{x}_{2}R+y_1\left(1-x_2\right)T+\left(1-y_1\right)x_{3}S+\left(1-y_1\right)\left(1-x_3\right)P]\\ & +{\kappa }_i\left[x_1{x}_{2}R+x_1\left(1-x_2\right)S+\left(1-x_1\right)x_{3}T+\left(1-x_1\right)\left(1-x_3\right)P\right]\\ & +{\kappa }_i\left[x_1{x}_{2}R+x_1\left(1-x_2\right)T+\left(1-x_1\right)x_{3}S+\left(1-x_1\right)\left(1-x_3\right)P\right]\\ & -{\alpha }_i{x}_1\left(1-y_2\right)\left(T-S\right)-({\alpha }_i+q{\delta }_i)\left(1-y_1\right)x_3\left(T-S\right)\\ & -{\beta }_i\left(1-x_1\right)y_3\left(T-S\right)-({\beta }_i+p{\gamma }_i)y_1\left(1-x_2\right)\left(T-S\right).\end{array}\end{array}$$

Hence, for a subject who believes that the opponent would play $\widehat{y}$ one obtains

$$\begin{array}{}\text{(24)}& \begin{array}{rl}\frac{\partial u_i\left(x,\widehat{y}\right)}{\partial x_1}=& (1-{\kappa }_i)[S-P+{\widehat{y}}_2\left(R-S\right)-{\widehat{y}}_3\left(T-P\right)]\\ & +{\kappa }_i\left[x_2\left(2R-S-T\right)+\left(1-x_3\right)\left(S+T-2P\right)\right]\\ & +{\beta }_i{\widehat{y}}_3\left(T-S\right)-{\alpha }_i\left(1-{\widehat{y}}_2\right)\left(T-S\right),\end{array}\end{array}$$

$$\begin{array}{}\text{(25)}& \frac{\partial u_i\left(x,\widehat{y}\right)}{\partial x_2}=(1-{\kappa }_i){\widehat{y}}_1\left(R-T\right)+{\kappa }_i{x}_1\left(2R-S-T\right)+({\beta }_i+p{\gamma }_i){\widehat{y}}_1\left(T-S\right),\end{array}$$

and

$$\begin{array}{}\text{(26)}& \frac{\partial u_i\left(x,\widehat{y}\right)}{\partial x_3}=(1-{\kappa }_i)\left(1-{\widehat{y}}_1\right)\left(S-P\right)+{\kappa }_i\left(1-x_1\right)\left(T+S-2P\right)-({\alpha }_i+q{\delta }_i)\left(1-{\widehat{y}}_1\right)\left(T-S\right).\end{array}$$

Again, these equations show that an individual with a Kantian moral concern (${\kappa }_i>0$) 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.