Dictator choice and causal attribution of recipient endowment

Abstract

In a laboratory experiment, two dictators give serially to a common recipient. In all treatment conditions, the second dictator knows the outcome in the first game. We vary the nature of the dictator in the first game across different treatments. We ask if the resulting variation in the attribution of intent to the prior causal dictator affects giving in the second game. We find that causal attribution has no effect on average giving, but may impact marginal giving. In particular, giving in the second game is negatively correlated with that in the first only when the first dictator has self-interest. We further find that giving is not affected by knowledge of recipient endowment and falls over the sequence of games.

Appendices

Appendix A: Theoretical background

1.1 A1: Deriving the hypotheses

Focusing on the second game, suppose we interpret any endowment e present with a recipient as arising from the amount received in a prior dictator game. Theoretical models of dictator giving posit utility functions which are defined effectively over ex post wealth of the recipient, or beliefs over it. Since these theories are familiar, we relegate a detailed discussion (including the derivation of relationships we refer to in this section) to Appendix A3, and consider here for example the reference point-based dictator utility function below, based on Bolton and Ockenfels (2000), where y is the amount the dictator keeps for herself out of an endowment normalized to 1, $\sigma$ is the norm or social reference point for giving (the reference point of ex post-equality is typically assumed to be the social norm, with $\sigma =\frac{1+e}{2}$), u is an increasing and concave function, and v is a concave function reaching a maximum at y = $\sigma$

$$U\left(y;\sigma \right)=u\left(y\right)+v(y-\sigma )=u\left(y\right)+v\left(y-\frac{1+e}{2}\right)$$

As can be seen, the dictator’s utility is independent of the source of the recipient’s endowment. Hence, attributional concerns cannot affect the amount given by C, which we denote as x^*. In our experiment, since the distribution of allocations in the first game are the same across Human, Human3U, Human3R, and CompU by design, these models predict for the first question that average second game choices are identical across these four conditions. Similarly, for the second question, these predict that as long as there are sufficiently many C subjects choosing to give positive amounts and sufficiently many B subjects with positive endowment, there should be a negative relationship between e and x^* in any of Human, Human3U, Human3R, and CompU. Also similarly, for the comparison of average giving across the first and second games for conditions Human, Human3U, Human3R, and CompU (part of the third question), these predict that as long as there are sufficiently many C subjects choosing to give positive amounts and sufficiently many B subjects with positive endowment, average giving should be higher by A subjects compared to C subjects.

For the rest of the answer to the third question, and for the answer to the fourth question, we need to specify the belief of a C subject in HumanN, who does not know what endowment her matched B subject has, but knows it may be positive. Suppose she conjectures that the endowment is $\hat{e}$ ≥ 0. For the comparison across average giving in the first and second games for HumanN (third question), the models then predict that as long as sufficiently many C subjects conjecture $\hat{e}$ > 0, giving should be less in the second game. Hence, for the third question as a whole, allocation by the second dictator is lower than that by the first, because the former anticipates that the recipient’s endowment may be positive.

The answer to the fourth question depends on the comparison between the distribution of $\hat{e}$ among C subjects in HumanN and the actual distribution of giving by A subjects in Human. If these are the same, then all the models predict that information on the amount received by the matched recipient should not impact average dictator giving in the second game, leading to indistinguishable second game choices on average across HumanN and Human. For the first and fourth questions together, the prediction is therefore that the distribution of allocations by second dictators when they know the allocations by the first dictators equals the distribution when they do not know yet correctly anticipate the aggregate distribution. Overall, the prediction for any of the four questions is invariant across these models.

1.2 A2: A model allowing for attributional effects

On the other hand, if attributional concerns can affect choice in the dictator game, as Blount (1995) and others found for ultimatum and other games, then average second game choices may differ across Human, Human3U, Human3R and CompU (first question), and the relationship between e and x^* (second question) may not be negative in any of these four conditions, and may be different across them. Here, we sketch a simple extension of the reference point-based model above, based on Cappelen et al. (2007), López-Pérez (2008) and Kessler and Leider (2012), which permits attributional concerns to affect choice. Specifically, we allow the reference point for C to depend not only on what B received from her earlier interaction, but also on how it was obtained, i.e., the nature or intention of the prior decision-maker.

Consider the first game in conditions HumanN and Human, where the recipient B has no endowment or history of prior interaction. This corresponds to the version of the dictator game commonly studied in the literature. We assume the ex post-equality norm continues to apply in this benchmark case unaltered. Letting ${\sigma }_{1}$ and ${\sigma }_{2}$, respectively, denote reference points in games 1 and 2, we have ${\sigma }_{1}=\frac{1}{2}$.

Now, consider dictator C in the second game in condition HumanN, who knows B has interacted with an earlier dictator. Although he does not know how much B received in that interaction, he may consider the possibility that she received a positive amount. It seems reasonable to posit that C’s desire to divide his own endowment equally may then be impaired relative to a dictator in game 1, since he takes into account that B may already be in possession of some wealth.⁸ We suppose therefore for this case that the norm of how much to keep is adjusted relative to ${\sigma }_{1}$: ${\sigma }_{2}^{HumanN}=\frac{1}{2}+{\varphi }^{HumanN}$, where ${\varphi }^{HumanN}$ is the adjustment factor. Let ê be his conjecture regarding how much A gave to B and $\stackrel{\sim }{{e}}$ be his belief or position, which could be socially determined or influenced, with respect to how much B should have received from A. We assume $\varphi^{HumanN} \left( {\hat{e},\tilde{e}} \right) = \alpha^{HumanN} \left( {\hat{e} - \tilde{e}} \right) + \beta^{HumanN} \left( {\tilde{e}} \right)$, $\alpha^{HumanN} \left( 0 \right) = 0$, $\alpha^{{HumanN^{\prime}}} > 0$, $\beta^{HumanN} \left( 0 \right) = 0$, and $\beta^{{HumanN^{\prime}}} > 0$. The factor thus is the sum of an absolute component, dependent on position alone, and a relative component, an adjustment depending on whether A’s conjectured giving exceeds or falls short of that position. For the determination of $\hat{e}$, we assume that the dictator considers himself representative. We take this to imply in this context a belief that B has received what C would have given her had he been in the role of A, so that $\hat{e} ={x}_{1}^{*}=1-{y}_{1}^{*}$, where ${y}_{1}^{*}$ and ${x}_{1}^{*}$ are, respectively, the amounts individual C would have optimally kept for himself and given to the recipient in the benchmark environment. For the determination of $\tilde{e }$, we appeal again to representativeness and take it to imply in this context that C holds that A should have given what C would have given had he been in the role of A, so that $\tilde{e }={x}_{1}^{*}$. We therefore have ${\sigma }_{2}^{HumanN}=\frac{1}{2}+{\beta }^{HumanN}\left({x}_{1}^{*}\right)$. The benchmark norm is thus restored if C believes A gave nothing. As long as solutions are interior, giving by C should be lower compared to giving by A under these assumptions.

We now consider the second game in conditions Human, Human3U, Human3R, and CompU, and assume an adjustment factor similar to the one in IB, with the difference that C knows e, the amount B received from the earlier interaction. Let therefore ${\sigma }_{2}^{j}=\frac{1}{2}+{\varphi }^{j}$, and ${\varphi }^{j}\left(e,\tilde{e }\right)={\alpha }^{j}\left(e-\tilde{e }\right)+{\beta }^{j}\left(\tilde{e }\right)$, ${\alpha }^{j}\left(0\right)=0$,${\alpha }^{j{^{\prime}}}>0$, ${\beta }^{j}\left(0\right)=0$ and ${\beta }^{j{^{\prime}}}>0$,$j\in \{Human, Human3U, Human3R,CompU\}$.⁹ Appealing to representativeness again yields $\tilde{e }={x}_{1}^{*}$, so that ${\varphi }^{j}={\alpha }^{j}\left(e-{x}_{1}^{*}\right)+{\beta }^{j}\left({x}_{1}^{*}\right)$ and ${\sigma }_{2}^{j}=\frac{1}{2}+{\alpha }^{j}\left(e-{x}_{1}^{*}\right)+{\beta }^{j}\left({x}_{1}^{*}\right)$.

The formulation above allows for parsimonious description. Knowledge of a second dictator’s decision, her belief with regard to how much a first dictator should have given, i.e., how much she would have given had she been the first dictator, and her functional forms $u$, $v$, $\alpha$, and $\beta$, permit the use of the decision as second dictator as a proxy for decision as the first.

In terms of this model, findings presented in Sects. 4.1 and 4.2, and Appendix B1 indicate that for $j,j{^{\prime}}\in \{Human3U,Human3R,CompU\}$, ${\alpha }^{j{^{\prime}}}={\alpha }^{j}=0$ and ${\beta }^{j}={\beta }^{IB}$, whereas ${\alpha }^{SI{^{\prime}}}>0$.¹⁰ An interesting finding (Sect. 4.1 and Appendix B2) is that average second game giving in condition Human is the same as that in any other condition. It is implied if the adjustment ($\alpha$) in giving by C in Human is on average zero and ${\beta }^{Human}={\beta }^{HumanN}$. The extended model thus allows a unified explanation for all our findings under the maintained assumptions.

1.3 A3: Theories of giving by dictators

Several theoretical frameworks have been advanced to explain why dictators may give positive amounts. We briefly describe examples of some prominent such theories below. Specifically, we discuss altruism theory, models where preferences are influenced by reference points regarding posterior wealth distributions, and hybrid models combining distributional preferences with reactivity to attributions of intent.

As a note, models based on preferences for aggregate efficiency or maximin preferences (e.g., Charness & Rabin, 2002) have also been fruitfully applied to explain data from one-shot distribution experiments. Korenok et al (2012) report poor performance of such models in an experimental setting close to ours; we hence do not focus on these. Similarly, models based on guilt aversion (Charness & Dufwenberg, 2006) can explain positive giving in the dictator game. Under the usual interpretation of guilt aversion, however, the recipient's belief is over how much he will get from the dictator, resulting in no direct link between the dictator's giving and the recipient's prior receipt, a key consideration of this paper. We thus do not focus on such models either.

In the impure altruism theory of Andreoni (1989, 1990), a dictator choosing how much of a surplus to share with the recipient obtains utility from (a) the amount she keeps for herself, as in classical models, (b) the well-being or ex-post payoff of the recipient (pure altruism, Becker, 1974), and (c) the amount she gives to the recipient (warm glow, the source of impurity in altruism). As Konow (2010) points out, the warm glow element, originally introduced to address empirical findings related to public good games, has little marginal importance in the dictator game. We represent the overall utility function of an altruistic dictator using the classical and purely altruistic parts, with the warm glow element eliminated. It is easy to show in the current environment that qualitative results are invariant to the presence of this element (see Konow, 2010, Sect. 3.1), leading to essential equivalence between models of pure and impure altruism. This is not true in all environments, though: for an example, see Chowdhury and Jeon (2014).

Suppose a dictator is deciding how much to give to a recipient out of a surplus, normalized to 1. Let x be amount she gives to the recipient, and y be the amount she keeps for herself, with x + y = 1. We assume that i) the dictator has no endowment other than the surplus to be divided, and ii) the recipient has an endowment $e\in \left[\mathrm{0,1}\right]$, which cannot be shared. The overall utility function U_A (assuming separability for simplicity) is as follows¹¹:

$$U_{A} \left( {y;e} \right) = u\left( y \right) + f\left( {e + x} \right) \ldots$$

(1)

u(.) represents material utility, while f(.) represents altruistic utility in (1).¹² Both functions are assumed to be differentiable, monotone increasing and strictly concave: u′ > 0, u″ < 0, f′ > 0, f″ < 0.

The dictator maximizes (1) subject to $y\in \left[\mathrm{0,1}\right]$ and x + y = 1. The solution for x is unique and the first-order condition is $u{^{\prime}}\left(1-x\right)=f{^{\prime}}\left(e+x\right)$. For interior solution with x^* > 0, Konow (2010, Proposition 1) shows that $\frac{\mathrm{d}{x}^{*}}{\mathrm{d}e}=\frac{-{f}^{{^{\prime}}{^{\prime}}}}{{u}^{{^{\prime}}{^{\prime}}}+{f}^{{^{\prime}}{^{\prime}}}}<0$.

In models based on reference points, a dictator’s utility from giving an amount is dependent on a reference point for giving or posterior inequality. Here, we present a typical model, with a reference point defined purely in terms of payoff-linked variables such as giving or posterior wealth positions. An example would be the inequality aversion model of Bolton and Ockenfels (2000) or the conditional altruism model of Konow (2010), the latter with any warm glow element suppressed (an extension with a non-consequentialist reference point is presented in Appendix A1). Two factors affect the utility a dictator gets from giving a part of the surplus to her matched recipient: (a) the amount she keeps for herself (y), as in altruism theory, and (b) the difference between the amount she keeps for herself and the amount to be kept ($\sigma$), as specified by a norm or social reference point for a dictator in her situation. The reference point of ex post-equality is typically assumed to be the social norm: $\sigma =\frac{1+e}{2}$. Using the same notation and assumptions as for altruism theory, we represent the overall utility function (again assuming separability) of a dictator as follows:

$$U_{I} \left( {y;\sigma } \right) = u\left( y \right) + v\left( {y - \sigma } \right) \to U_{I} \left( {y;e} \right) = u\left( y \right) + v\left( {y - \frac{1 + e}{2}} \right) \ldots$$

(2)

e is recipient endowment, and x + y = 1 (v is assumed to be defined over the difference between y and $\sigma$ for simplicity). u(.) has the same interpretation and properties as in (1). For v, we assume that ${{v}^{{\prime}}\left(y\right)}_{y=\sigma }=0$ and ${v}^{{{\prime\prime}}}\left(y\right)<0$. The dictator maximizes (2) subject to $y\in \left[\mathrm{0,1}\right]$. The solution for y is unique, y^* ≥ $\frac{1+e}{2}$, and the first-order condition is ${u}^{{\prime}}\left(y\right)+{v}^{{\prime}}\left(y-\frac{1+e}{2}\right)=0$. For interior solution with y^* < 1, it follows that $\frac{\mathrm{d}{y}^{*}}{\mathrm{d}e}=\frac{{v}^{{^{\prime}}{^{\prime}}}}{2\left({u}^{{^{\prime}}{^{\prime}}}+{v}^{{^{\prime}}{^{\prime}}}\right)}$. Since y^* > $\frac{1+e}{2}=\sigma$, the properties assumed for u and v imply $\frac{\mathrm{d}{y}^{*}}{\mathrm{d}e}>0$, and so $\frac{\mathrm{d}{x}^{*}}{\mathrm{d}e}<0$.

Findings from various distribution experiments have led to models (e.g., Falk & Fischbacher, 2006; Trautmann, 2009) which combine distributional preferences with a role for perceived intentions or procedural considerations (Dufwenberg & Kirchsteiger, 2004; Rabin, 1993). We focus on Falk and Fischbacher (2006) as they provide an explicit analysis of the dictator game. In their model, developed to analyze sequential move games, a player’s utility depends on (a) her own ex post payoff, as in other theories, (b) her perception of the ‘kindness’ of the other player, as measured by expected ex post payoff difference, and (c) her ‘reciprocal response’, as measured by how her action affects the ex post-expected payoff of the other player.¹³ They impose some linearity assumptions for their applications. For the dictator game, the utility function for a dictator reduces to (the notation is as above).¹⁴

$$U_{F} \left( {y; \, \tilde{x},e} \right) = y + \frac{1}{b}\left( {1 - 2\tilde{x} - e} \right)\left( {x - \tilde{x}} \right) \ldots$$

(3)

x, y, and e are as defined before, so x + y = 1, and $b$ is a positive parameter. $\tilde{x }$ is the dictator’s belief regarding the recipient’s belief about the amount the dictator will give. Their solution concept (reciprocity equilibrium; see Falk & Fischbacher, 2006, Sect. 3.4) derives $\tilde{x }$ from the first-order condition when the right-hand side of (3) is differentiated with respect to x: $\tilde{x }=\frac{1-b-e}{2}$. Replacing in (3), it follows (see Proposition 8 in Falk & Fischbacher, 2006) that the solution for x is unique and given by ${x}^{*}=\mathrm{max}\left[0,\frac{1}{2}\left(1-b-e\right)\right]$. For ${x}^{*}>0$, this implies that $\frac{\mathrm{d}{x}^{*}}{\mathrm{d}e}=\frac{-1}{2}<0$.

Appendix B: Results with respect to hypotheses 3 and 4

2.1 B1: Comparing average giving across first and second dictators

Here, we explore the third question, and compare average giving across A and C subjects.¹⁵ Giving should be higher by A subjects to be in accordance with Hypothesis 3. Table 3 presents mean amount given by A and C subjects in various conditions. Average giving in the first game is the same across conditions Human, Human3U, Human3R, and CompU as observations generated in Human were implemented in Human3U, Human3R, and CompU. The overall mean for game 1 giving is 37.65 (using 90 unique observations), while it is 30.80 for game 2 giving (with 225 unique observations). These are different at the 1% level using either a t test or an MW test (two-sided p values: t test = 0.0024, MW = 0.0012).

Table 3 Mean amounts given by A and C subjects

We also compare A and C dictator giving independently by condition and find that the overall effect of A giving being greater than C giving is driven by Human3R (t test p value = 0.0471) with the difference statistically insignificant at the 5% level for other conditions. As further robustness checks, we run two partially pooled OLS regressions, one with data from only HumanN and Human (which have human agents with intent as dictators in both games), and the other using data only from Human, Human3U, Human3R, and CompU (which have identical distributions of giving for game 1, and in which game 2 dictators know recipient endowment). Finally, we run a regression with data pooled over all five conditions. The dependent variable in these regressions is dictator giving. Regressors consist of condition dummies and their interactions with the game 2 dummy. In the pooled regression of Human and HumanN, we find no significant regressors. In the latter two regressions, we find that only the interaction of the period 2 dummy with condition Human3R to be significant (p value = 0.043). This is unsurprising as we found the difference between game 1 and game 2 giving to be significant at the 5% level only for condition Human3R. These regressions are not reported for the sake of brevity.¹⁶ We thus conclude that second dictators give less than first dictators, with this effect disappearing once we put in condition specific controls and interactions. Overall, we find qualified support for Hypothesis 3.

2.2 B2: The effect of providing information on amount received earlier on average giving

HumanN and Human are identical except that C subjects in Human are told the exact amount their matched B subjects received from the earlier game. For the fourth question, comparing giving by C subjects across these two conditions thus yields the impact of information regarding the amount received.

Tests comparing giving by C subjects across HumanN and Human yield insignificant difference (two-sided p values: t test = 0.523, MW = 0.638). In addition to these two-group tests, we also compare C giving from HumanN and Human using Wald test contrasts that report an F statistic. We find that ${\overline{c}}_{Human}$ and ${\overline{c}}_{HUmanN}$ are not statistically different (Wald F test p value = 0.4284).

Specific information on amount received earlier thus does not leave an impact on giving on average. This suggests, in accordance with Hypothesis 4, that the actual distribution of giving by A subjects in Human is comparable to the distribution of giving by A subjects as expected by C subjects in HumanN, i.e., C dictators, on average, correctly anticipate the decisions made by the A dictators. Together with the finding from Appendix B1 above, this indicates that the effect of sequential dictatorship is independent of whether or not a later dictator knows the exact prior receipts of her matched recipient.

Appendix C: Instructions

The subjects were given roles Red, Green, and Blue, corresponding, respectively, to A, B, and C.

1. General instructions—given at the beginning of any session to all subjects.

Hello and welcome. Today, you will participate in a study on decision-making. Depending on chance, some of you will have to make a decision in today’s session based on which you will get a payment in rupees to take home after the session. You will receive Rs 50 for showing up and completing the session. Depending on your decision or those of others, you may receive another payment which will be added to this show-up fee of Rs. 50.

Your decisions are strictly confidential and your identity will never be disclosed to anyone. You will be identified by a unique ID which cannot be traced to you, so there is no chance that others will know how you decided. In making decisions, please think carefully and perform the action of your choice, not what you think others may want to see.

Today, you will be participating in either Red, Green or Blue roles. A box containing some colored slips of paper will be brought to you shortly. Please choose a slip from the box. The color of the paper you draw will determine your role (red, green, or blue). Please follow instructions thereafter.

Thank you for your interest in this project, and please wait for further instructions.

2. Instructions to Green subjects (B) given in all conditions (Red subjects (A) given in conditions Human3U, Human3R and CompU).

You have been paired with two other participants, one red (green) and one blue. Their identities will not be revealed to you now or later. Furthermore, these other participants will also never learn your identity. You will merely be conveyed their decisions at the end of the session.

You do not have to make a decision in today’s session. Please wait quietly, do not communicate with anyone and await further instructions.

3. Instructions to Red subjects (A) given in conditions HumanN and Human.

You have been paired with a green participant. His/her identity, however, will not be revealed to you now or later. Furthermore, the other person will also never learn your identity. The individual will merely be conveyed your decision at the end of the session.

Now the game. Suppose you have been allocated Rs. 100. Your task is to divide Rs. 100 between yourself and this other person (integer division only please). In the designated space below, please put down the amount out of this you wish to share with this other individual. Whatever you do not share is yours to take home. Thus, if you give nothing, you get 100, and if you give everything, you get 0. The entire range in between and including 0 and 100 is open to you.

I give to the other person _________________

Before submitting this sheet, please check that the number above is (a) an integer, (b) not negative, and (c) not greater than 100.

4. Instructions to Blue subjects (C).

Instructions given to Blue subjects were the same as those given to Red subjects in condition HumanN and Human, except that a third paragraph was added after the second (beginning with ‘Now the game’). This paragraph differed across the five conditions. These paragraphs are reproduced below.

I. Condition HumanN—Additionally, in another identical earlier interaction, this same person was paired with a red participant, i.e., the red participant had an opportunity to give out of 100 to this green participant, who is now paired with you. The red participant was not aware at the time of this earlier interaction that the green participant would later be paired again.

II. Condition Human—Additionally, in another identical earlier interaction, this same person was paired with a red participant and received ______ out of 100, i.e., the red participant had an opportunity to give out of 100 to this green participant, who is now paired with you, and gave the specified amount. The red participant was not aware at the time of this earlier interaction that the green participant would later be paired again.

III. Condition Human3U—Additionally, just before this session, this same person was paired with a red participant. A neutral third party, who is a doctoral student and present among us, then divided 100 among the two, selecting from among a set of possible divisions. The green participant received ____. This neutral third party was not aware at the time of this earlier interaction that the green participant would later be paired again, and did not receive any compensation for performing the division.

IV. Condition Human3R—Additionally, just before this session, this same person was paired with a red participant. A neutral third party, who is a doctoral student and present among us, then divided 100 among the two, using a random process to select from among a set of possible divisions. The green participant received ____. This neutral third party was not aware at the time of this earlier interaction that the green participant would later be paired again, and did not receive any compensation for performing the division.

V. Condition CompU—Additionally, just before this session, this same person was paired with a red participant. A computer then divided 100 among the two, selecting from among a set of possible divisions. The green participant received ____.