Games 2012, 3(4), 138-149; doi:10.3390/g3040138

Letter
Modeling Inequity Aversion in a Dictator Game with Production
Ismael Rodriguez-Lara 1,* and Luis Moreno-Garrido 2
1
ERICES Universidad Valencia, Analisis Economico, Universidad de Valencia, Avda. Naranjos s/n, 46022 Valencia, Spain
2
Universidad de Alicante, San Vicente del Raspeig s/n, Alicante, Spain; Email: luisjoseblas@merlin.fae.ua.es
*
Author to whom correspondence should be addressed; Email: Ismael.Rodriguez@uv.es; Tel.: (+34) 96-3828211; Fax: (+34) 96-3828249.
Received: 21 September 2012; in revised form: 14 October 2012 / Accepted: 16 October 2012 /
Published: 23 October 2012

Abstract

: We expand upon the previous models of inequity aversion of Fehr and Schmidt [1], and Frohlich et al. [2], which assume that dictators get disutility if the final allocation of surplus deviates from the equal split (egalitarian principle) or from the subjects' production (libertarian principle). In our model, dictators may also account for the way in which the surplus was generated. More precisely, our model incorporates the idea of liberal egalitarian ethics into the analysis, making it possible for dictators to divide the surplus according to the accountability principle, which states that subjects should only be rewarded for factors under their control. This fairness ideal does not hold subjects responsible for factors beyond their control in the production of the surplus, an idea that is absent in the models of inequity aversion cited above (JEL Codes: D3, D6, D63).
Keywords:
social preferences; inequity aversion; inequality aversion; egalitarian principle; libertarian principle; accountability principle

1. Introduction

The standard assumption that subjects only care about their own material payoff is frequently used to solve economic models. However, the overwhelming experimental evidence against this assumption (especially in the dictator experiment) indicates that subjects are willing to sacrifice their own material payoff so as to achieve fair allocations (see Camerer [3] and Engel [4]).

In the field of experimental economics, the dictator game with production (e.g., Cappelen et al. [5], Cherry et al. [6] and Konow [7]) has recently emerged as a useful tool to investigate fairness attitudes towards redistribution. In this game, subjects contribute to the production of surplus before a randomly selected dictator chooses a division of this surplus.1 The findings of the literature highlight that dictators are willing to reward other subjects' efforts, even though this behavior contradicts the selfish hypothesis (Frohlich et al. [2], Oxoby and Spraggon [8], Ruffle [9]). It is also found that subjects exhibit heterogeneous preferences with regard to the distribution of the surplus (Cappelen et al. [5], Konow [7], Miller and Ubeda [10], Rodriguez-Lara and Moreno-Garrido [11]), suggesting the existence of different fairness ideals.

Although the importance of the production stage has been crucial to achieve these results, almost no studies of social preferences incorporate the source of the surplus into the theoretical analysis. The models of social preferences that have been put forward to explain dictators’ deviations from narrow self-interest usually focus on the way in which dictators divide the surplus, while leaving aside the way in which the surplus was generated (see Fehr and Schmidt [12] for an overview of the theoretical literature). One exception is Frohlich et al. [2] who generalize the model of inequality aversion of Fehr and Schmidt [1]. In their model, Frohlich et al. [2] consider that dictators' behavior depends on two different costs. On the one hand, dictators' utility decreases if subjects do not receive exactly the same monetary payoff, as in Fehr and Schmidt [1]. On the other hand, there exists a cost for dictators if the final allocation deviates from the subjects' production. By assessing both types of costs, the model of Frohlich et al. [2] allows dictators to choose the egalitarian allocation (which divides the surplus in two identical parts) and the libertarian allocation (which assumes that subjects ought to receive exactly the value of their production).2 Arguably, some factors that determine the subjects' production might be beyond their control, not withstanding that the libertarian allocation holds subjects responsible for all factors determining production.

In this paper, we propose a simple extension of Fehr and Schmidt [1] and Frohlich et al. [2] so as to account for the possibility of dictators dividing the surplus according to the accountability principle. The accountability principle, as first proposed by Konow [13], combines both equity theory (which makes the final allocation proportional to the agents’ inputs) and attribution theory (which considers responsibility or control over inputs). This fairness ideal is related to liberal egalitarian ethics (which assumes that subjects should only be rewarded for factors under their control). There exists overwhelming evidence showing that subjects employ to some extent the accountability principle (see Cappelen et al. [5,14,15], Konow [7], Konow [16], Rodriguez-Lara and Moreno-Garrido [11]). We lack, however, theoretical models that predict this allocation. We posit a utility function in which dictators juggle the tradeoff between subjects' inputs and monetary contributions, so that dictators weigh three different fairness ideals: the egalitarian principle, the libertarian principle and the accountability principle. The latter fairness ideal is disregarded in Fehr and Schmidt [1] and Frohlich et al. [2], but it is relevant whenever subjects' inputs are differently rewarded to determine the size of the surplus, i.e., when there are factors outside the subjects' control (Cappelen et al. [5,14], Konow [7], Rodriguez-Lara and Moreno-Garrido [11]). In this framework, it is plausible to assume that dictators do not only care about the final distribution of payoffs, but also about the way in which entitlements were generated. In particular, our model allows for dictators to evaluate the role of compensation and responsibility, two features that constitute the gist of fairness (see among others Cappelen et al. [17], Fleurbaey and Maniquet [18], Rawls [19] and Roemer [20]).

To the best of our knowledge, there are only two theoretical models that consider the possibility of dictators choosing the accountability principle (Cappelen et al. [5] and Konow [7]). Despite some differences in their functional form, the models assume that dictators endorse a fairness ideal (e.g., the accountability principle) and suffer a cost if they deviate from it. Our approach does also account for this possibility, but generalizes the framework by allowing heterogeneity in fairness preferences to take place within subjects. In that vein, we do not need to consider that dictators are motivated by a single fairness ideal, with dictators being heterogeneous in what they consider as fair. It is also plausible to assume that fairness ideals are tradeoffs against each other, and heterogeneity lies inside each individual (i.e., our model could be used to investigate whether fairness principles are context-dependent, to study the consistency of fairness rules or to analyze the possibility of self-serving bias).3

The rest of the paper is organized as follows. Section 2 briefly describes the models of Fehr and Schmidt [1] and Frohlich et al. [2] and then presents our theoretical model. In Section 3, we discuss our findings and use a couple of examples so as to clarify when our model does yield different predictions than previous models of inequity aversion in the literature. We conclude in Section 4.

2. Results and Discussion

Consider the dictator game in which subjects can be labeled iϵ{d,r}, where d represents the dictator and r represents the recipient. The dictator has to divide a certain surplus Games 03 00138 i001≥ 0 between herself and her counterpart. The size of this surplus depends on both the subjects' monetary contributions, which are denoted by yi≥ 0 for iϵ{d,r}, in particular,

Games 03 00138 i002
where qi≥ 0 represents subject i’s performance in a (previous) production stage, and pi > 0 is the weight assigned to this input, for i ϵ{d,r}. In what follows, we shall think that qi≥ 0 is under the subject i's control (e.g., exerted effort, time of work, money to be invested in a project, etc.). The value of pi > 0 is assumed to be independent of qi ≥ 0 and determines the way in which agent i’s input is transformed into money. We shall think that pi > 0 is outside the subject i's control (e.g., reward level, rate of return, luck, etc.).4

The dictator has to choose a division of the surplus x = (xd, xr) that satisfies xd + xr = Games 03 00138 i001, where xi≥ 0 denotes the monetary payoff that subject i will receive, for i ϵ{d,r}. The model of inequality aversion of Fehr and Schmidt [1] builds on the assumption that the dictator dislikes unequal outcomes (see Nielsen [26]). More specifically, the authors posit the following utility function for the dictator:

Games 03 00138 i003

This function accounts for social preferences because the dictator does not only care about her own monetary payoff. The dictator's utility also depends on the recipient’s payoffs and the relationship between both subjects’ payoffs. In particular, the model of Fehr and Schmidt [1] assumes that it would be costly for the dictator to take more than half of the surplus, but it would also be costly for her to take less. The magnitude of the costs is given by the values of Games 03 00138 i004 and β, which are assumed to satisfy Games 03 00138 i004 > β ≥ 0 and β ≤1. This implies that the dictator cares more for inequality when she has less than half of the surplus than the dictator does when she has more.

As xr = Games 03 00138 i001 - xd , equation (2) can be rewritten as follows:

Games 03 00138 i005
and the value of β determines whether dictators divide the surplus in two identical parts (egalitarian allocation) or keep the entire surplus for herself (selfish allocation). It can be shown that the dictator who maximizes equation (3) chooses the egalitarian allocation Games 03 00138 i006 if Games 03 00138 i007 > 0.5. Otherwise, the dictator behaves selfishly, Games 03 00138 i008).5

The prediction of Fehr and Schmidt [1] goes in line with the observed behavior in many laboratory experiments, where the equal split is a modal outcome (see Camerer [3] and Engel [4] for a review). The model is especially well-suited when the surplus to be divided as "manna from heaven", because it predicts an allocation that ignores the source of the surplus or its size. However, the underlying idea of Fehr and Schmidt [1] contrasts with fairness ideals that plead for a solution in which entitlements to the available surplus are directly determined by subjects' contributions, an idea that goes back to Selten [30] and it presented in Cherry et al. [6], Oxoby and Spraggon [8] or Ruffle [9], among others. The libertarian ethics indeed pursue the idea that subjects ought to receive exactly what they have contributed to surplus (Nozick [31]). The model of Frohlich et al. [2] incorporates this idea by adding two terms to the dictator's utility function:

Games 03 00138 i009

The utility function (4) expands upon the previous one so as to include what the authors call “just deserts”. Frohlich et al. [2] consider that the dictator suffers a cost Games 03 00138 i010 if she does not take her own production, whereas Games 03 00138 i011 represents the cost of taking more than what she produces.6 The new parameters are assumed to satisfy γ ≥ Games 03 00138 i012 >0. The authors note that if Games 03 00138 i004 and β are both zero, then just deserts affect the dictator's choice when γ >1 or ψ>1. If Games 03 00138 i013 $ and β are not zero, two different contexts are at stake: when the dictator accumulates less money than the recipient ( Games 03 00138 i014), and when the dictator accumulates more ( Games 03 00138 i015). In either case, when 2β + ψ <1, the dictator behaves selfishly and keeps the entire surplus, Games 03 00138 i008). If the dictator accumulates less money than the recipient and 2β + ψ > 1, the model predicts that the dictator will either keep half of the surplus or her own production, depending on whether ψ – 1 > 2 Games 03 00138 i004 or ψ – 1 < 2 Games 03 00138 i004. Both the egalitarian allocation, Games 03 00138 i006 and the libertarian allocation Games 03 00138 i016 can also be predicted if the dictator accumulates more money than the recipient.

The model of Frohlich et al. [2] generalizes the model of Fehr and Schmidt [1] and predicts that recipients might receive any payoff xr Games 03 00138 i017. Arguably, the prediction of this model contrasts with liberal egalitarian ethics, which states that subjects should only be rewarded for factors under their control. In that vein, the accountability principle as first proposed by Konow [13] states that the dictator will choose a division that gives to each subject the amount of money that is generated by her input. We denote this amount predicted by the accountability principle as Games 03 00138 i018 for iϵ{d,r}, so that if the dictator ignores those factors beyond the subjects’ control, she will choose the allocation Games 03 00138 i019. We propose to model the dictator's utility function as follows:

Games 03 00138 i020

Our specification assumes that the dictator cares about her own monetary payoff, but juggles the tradeoff between subjects’ inputs and monetary contributions. This implies that the dictator suffers a cost Games 03 00138 i021 if she free-rides on the recipient's effort, but she also suffers a cost Games 03 00138 i022if she takes less money than what corresponds to her input, where ρ ≥ Games 03 00138 i023 > 0 is assumed to be satisfied (i.e., the dictator cares less about the recipient’s disadvantage than about her own).

The utility function (5) allows dictators to take into account the way in which inputs are transformed into money so as to "compensate" for those factors outside the subjects’ control. Our prediction is that dictators might (i) behave selfishly Games 03 00138 i024 (ii) choose the egalitarian allocation and divide the surplus in two identical parts Games 03 00138 i006, (iii) choose the libertarian allocation that is based on subjects' production Games 03 00138 i025 or (iv) behave according to the accountability principle that is based solely on factors under the subjects’ control Games 03 00138 i019.7 We summarize our results in Table 1. As we argue, the dictator's choice depends on the relationship between factors within and beyond the subjects’ control, which determines the relationship between Games 03 00138 i026, Games 03 00138 i027 and Games 03 00138 i028, as well as the values of the parameters.

We derive these results in the supplementary material. Our model predicts that the dictator will behave selfishly, if 2β + ψ + ω < 1. If it is not the case, the dictator's decision depends on the subjects’ inputs (qi≥ 0) as well as on the weight that is assigned to these inputs (pi > 0). Consider that the dictator is at a relative advantage with regard to the accumulation of money ( Games 03 00138 i029) so that the dictator's production in terms of money Games 03 00138 i027 would be greater than her production in terms of inputs Games 03 00138 i030 The values of qi ≥ 0 and pi > 0 determine in this framework whether these contributions ( Games 03 00138 i027 and Games 03 00138 i028) are above half of the surplus ( Games 03 00138 i031) or not. Assume that Games 03 00138 i032 is satisfied. It is clear that the dictator gets the highest monetary payoff by choosing the egalitarian allocation Games 03 00138 i033 The dictator will choose this division of the surplus as long as the cost of deviating from the accountability principle (that yields the smallest payoff) is small enough. More precisely, Games 03 00138 i034 will be chosen if ω < 1 + 2 Games 03 00138 i013– ψ. If the cost is very high and the condition ω > 1 + 2 Games 03 00138 i004 + γ holds, the dictator will choose to divide the surplus according to the accountability principle Games 03 00138 i019 that yields the dictator the smallest monetary payoff. If Games 03 00138 i035, then the libertarian allocation will be chosen because it would be very costly to divide the surplus according to the egalitarian allocation, but the cost is not sufficiently high to enforce the use of the accountability principle. We can follow this reasoning so as to explain the prediction of our model in Table 1. We observe that there always exists a tradeoff between a higher monetary payoff and the cost of deviating from the fairness ideal that yields the smallest payoff.

Table 1. Predictions of our model.

Click here to display table

Table 1. Predictions of our model.
Games 03 00138 i036

Note. In all the cases above, it is assumed that Games 03 00138 i037. Otherwise, the dictator behaves selfish, Games 03 00138 i038. We do not consider the possibility of Games 03 00138 i039, because (by definition) Games 03 00138 i040 in that case our predictions would be identical to the ones in Frohlich et al. (2004).

3. Applications of our model

Overall, our model in Section 2 relies on liberal egalitarian ethics, which states that subjects should be only rewarded for factors under their control. The model is especially useful for those cases in which the dictator is at a relative advantage with regard to the accumulation of money (pd > pr), because in this case our model predicts a larger set of transfers from the part of the dictator than previous models of inequity aversion. In this section, we present a couple of examples so as to illustrate this feature. We also mention at the end of this section some experimental papers that produce behavior that is consistent with our model, and then discuss the relevance of our approach.

To start with, let us consider a numerical example. Imagine that subjects solve a questionnaire during the production stage. In particular, assume that qd = 10 and qr = 15 are the number of correct answers, which are assumed to be controlled by the subjects (i.e., the number of correct answers depends on the exerted effort in the questionnaire). Consider that these questions are rewarded at pd = 1.5 and pr = 1, respectively, where the reward level is exogenously determined and is independent of performance.8 In that case, the surplus to be distributed is Games 03 00138 i001= 30 and the subjects’ contributions to the surplus is given by yd = yr =15. Since both subjects contribute the same (i.e., yd = yr = Games 03 00138 i031), the models of Fehr and Schmidt [1] and Frohlich et al. [2] predict that the recipient will receive at most half of the surplus. It is worth noting, however, that the dictator might consider this allocation as unfair because the recipient has more questions correct at a lower reward level. Liberal egalitarian ethics states that the recipient ought to receive the part of the surplus that is due to her performance, which would compensate recipients for the lower reward. In our example, Games 03 00138 i041 18. This amount represents 60-percent of the surplus. Importantly, our model is able to predict this allocation, although this behavior cannot be predicted by equations (3) or (4). In fact, the nonlinear versions of our model in the supplementary material can predict any dictator's giving xr in the interval [15,18], which are above the equal split allocation.

To further illustrate that our model extends upon the previous ones, we consider Figure 1. Along the horizontal axis, we plot the proportion of the surplus that is due to the recipient’s production (yr/ Games 03 00138 i001). We use the vertical axis to represent the proportion of the surplus that the dictator gives away to the recipient (xr/ Games 03 00138 i042. As a consequence, the 45-degree line represents the appropriate theoretical prediction in Frohlich et al. [2], because observations on this line indicate that recipients are being transferred exactly the proportion of the surplus that they have contributed, i.e., the 45-degree line represents the libertarian allocation xr = yr. In Figure 1, we also plot the horizontal line (xr/ Games 03 00138 i001 = 0.5), which represents the egalitarian allocation xr = Games 03 00138 i0432 (Fehr and Schmid [1]). The dotted curve depicts the accountability principle xr=ar. Therefore, allocations on this curve indicate that recipients are being transferred exactly the proportion of the surplus that is due to their effort.9 The difference between pd and pr establishes the concavity of the dotted curve xr = ar and determines those allocations that cannot be predicted by Fehr and Schmidt [1] or Frohlich et al. [2]. In particular, the nonlinear version of these models predicts no allocations above both “just dessert” and the horizontal line xr = Games 03 00138 i031. Graphically, this implies that the models predict any giving in the shadowed area, except the striped area (i.e., the models predict xr ≤ max{ Games 03 00138 i031,yr}). However, our model takes into account the accountability principle so that our prediction includes the striped area (i.e., our model predict xr ≤ max{ Games 03 00138 i031,yr,ar}).

Games 03 00138 g001 200
Figure 1. Graphical representation of our predictions if pd > pr.

Click here to enlarge figure

Figure 1. Graphical representation of our predictions if pd > pr.
Games 03 00138 g001 1024

All the allocations on the striped area give some weight to the accountability principle, such that these allocations cannot be explained with the models of inequity aversion of Fehr and Schmidt [1] and Frohlich et al. [2], in which the idea of liberal egalitarian ethics is disregarded (i.e., Games 03 00138 i045 Games 03 00138 i023 = 0 is assumed).

We find that our model generalizes the previous ones and can be used to explain dictators’ behavior, especially when the production stage involves factors within and beyond the subject's control. In that context, the accountability principle is likely to lead dictators' behavior, especially when dictators act as a third party in the distributional problem (e.g., Cappelen et al. [17] and Konow [7]). The results in Cappelen et al. [5], Konow [16] and Rodriguez-Lara and Moreno-Garrido [11] do also produce behavior that is also consistent with the idea of subjects who endorse the accountability principle. In particular, Cappelen et al. [5] consider a dictator game with production and estimate that 38 percent of dictators are closest to satisfying the accountability principle, 43 percent to being egalitarians and 18 percent to being libertarians when dividing the surplus. If we use the reported data in Cappelen et al. [5] and focus our analysis on the cases in which dictators are at relative advantage with regard to the accumulation of money (pd > pr), we will find that 15-percent of the dictators behave according to the accountability principle. In Figure 1, these observations would lie on the striped area; therefore these allocation choices cannot be predicted by the previous models of inequity aversion. Similar results can be derived from the reported results in Rodriguez-Lara and Moreno-Garrido [11] or Ubeda [24]. For instance when pd > pr, roughly 17-percent of the data in Rodriguez-Lara and Moreno-Garrido [11] is incompatible with Fehr and Schmid [1] and Frohlich et al. [2], because it includes dictators who transfer xr ≥ max{ Games 03 00138 i031,yr}.10

One important feature of our model is that dictators are allowed to weigh three different fairness ideals instead of only one. Graphically, this implies that any allocation on the shadowed area can be derived after considering that dictators weigh the lines xr = 0, xr = Games 03 00138 i001/ 2, xr = ar and xr = yr. This feature of our model generalizes the idea that each dictator is motivated by a single fairness view (i.e., by a single “line”), with dictators being heterogeneous in what they consider as fair, and the marginal utility loss of deviating from this fairness ideal increasing in the size of the deviation. Our way of modeling social preference is also valid under the assumptions that fairness ideals are tradeoffs against each other, and heterogeneity lies inside each subject rather, than taking place across subjects. It remains to be discussed whether we should expect subjects to hold different fairness ideals at each time. Using a questionnaire, the seminal paper of Yaari [25] provides evidence that subjects choose different solutions for the same distribution problems, depending on the prevalence of tastes or needs in the story underlying each question (see also Messick and Sentis [23]). Recent experimental findings are also in line with the idea of within-subject heterogeneity (e.g., Croson and Konow [22], Konow [16], Ubeda [24], Blanco et al. [33] and Miller and Ubeda [10]). In their paper, Blanco et al. (2011) test the consistency of the Fehr and Schmidt's model and find that it fails to predict behavior at the individual level, as subjects exhibit different fairness attitudes across games (i.e., the estimation of and Games 03 00138 i046 and β for a given subject changes depending on the game). The work of Miller and Ubeda [10] focuses on a repeated dictator game with production and highlight that women are likely to switch between the three different fairness ideals presented in the current paper.

4. Conclusion

We have presented a theoretical model of social preferences that expands upon Fehr and Schmidt [1] and Frohlich et al. [2]. The underlying idea of our model is that dictators who are involved in a dictator game with production care about three different features. First, dictators get disutility if the final allocation deviates from the equal division. Second, dictators consider that subjects should receive exactly what they have produced. Finally, dictators account for the way in which entitlements are generated and try to compensate for factors beyond subjects' control.

Our contribution to the literature is to provide a model that predicts the accountability principle in Konow [13], which states that subjects should only be rewarded for factors under their control. This behavior is in accordance with liberal egalitarian ethics and cannot be predicted by the models of inequity aversion of Fehr and Schmidt [1] and Frohlich et al. [2], although it is observed in many laboratory experiments.

One novelty of our approach with respect to Cappelen et al. [5], Cappelen et al. [15] or Konow [7] is that we do not need to assume that, "individuals have a fairness ideal that is independent of the distributional function in which they find themselves [Cappelen et al. 2007, page 824].” Our model is suitable for this case of between-subjects heterogeneity, but it does also accommodate for the possibility of heterogeneity of fairness ideals lying inside each individual.

Acknowledgments

We would like to thank two anonymous referees for their comments and suggestion, which helped to shape the exposition of the paper. We are also grateful to Elisabet Rutstrom, Glenn W. Harrison, Giovanni Ponti, Paloma Ubeda, Juan D. Moreno-Ternero, Hubert Janos Kiss, Alfonso Rosa-Garcia, David Gill, Vicente Calabuig, Gonzalo Olcina and Jaromir Kovarik for valuable comments. Financial support from the Instituto Valenciano de Investigaciones Economicas (IVIE) and the Spanish Ministry of Education and Science under the projects SEJ2007-62656 and ECO2011-29230 is gratefully acknowledged.

  • 1 Hereafter we focus our exposition on the dictator game with production, which has been an important device for studying social preferences and distributional justice. We note that the existence of the production stage is important to disentangle the effect of fairness concerns and property rights in the final outcome. As noted by Cherry et al. [6], if there is not a production stage, then dictators might give money away, because they are fair-minded, or simply because the surplus to be divided is "manna from heaven", and dictators do not feel any right to keep the entire surplus for themselves.
  • 2 Frohlich et al. [2] use the concept of “just desert” to identify the division of the surplus that corresponds to the subjects’ production. We borrow the terminology in Cappelen et al. [5,14] and refer to this division as the libertarian allocation. Hereafter, we follow Cappelen et al. [5,14] in labeling the rest of the fairness ideals presented in the paper.
  • 3 The approach in Cappelen et al. [5,14] relies on the assumption of between-subject heterogeneity; i.e., subjects differ with respect to their consideration of fairness ideals (see also Engel [4] or Visser [21] for evidence of between-subject heterogeneity). However, there exists experimental evidence suggesting the existence of within-subject heterogeneity; i.e., the same subject might choose different allocations (or fairness ideals) depending on the context (see, among others, in Croson and Konow [22], Konow [16], Messick and Sentis [23], Miller and Ubeda [10], Ubeda [24] or Yaari [25]).
  • 4 We acknowledge that it might be hard to disentangle which variables are under the subjects’ control and which variables are outside their control in some situations. We find, however, that the classification of factors within and beyond individuals’ control is beyond the scope of this paper. For further discussion on this topic, the interested reader can see Fleurbaey and Maniquet [18]. Konow [27] is also an excellent overview of various theories of justice that deals with this feature.
  • 5 For the special case in which β = 0.5, the dictator is indifferent between any share xdϵ Games 03 00138 i047]. As Fehr and Schmidt [1] argue, the nonlinear versions of their model lead also to predictions x= (xd, xr) that satisfy xdϵ Games 03 00138 i048] and xr = Games 03 00138 i001- xd≥ 0. The models of Bolton and Ockenfelds [28] and Charness and Rabin [29] do also integrate equality as an argument. In particular, Bolton and Ockenfelds [28] show that the results of Fehr and Schmidt [1] are robust to non-linearity as well as incomplete information.
  • 6 We note that we have rewritten the original equation in Frohlich et al. [2] so as to follow our reasoning in equation (3). In particular, because yd + yr = xd + xr, the latter term in equation (4) can also be thought of as the cost of not giving to the recipient her production.
  • 7 Nonlinear versions of our model would predict interior results that lead to compromises between these fairness ideals. The interested reader can find a brief discussion about the linearity assumption in the supplementary material. We note that Bolton and Ockenfelds [28] is an extension of Fehr and Schmidt [1] to the non-linear case. The model of Bolton et al. [32] incorporates an idea that is related to the accountability principle in the context of procedural fairness; i.e., the authors use different games and investigate the extent to which subjects care about fair procedures.
  • 8 The experimental design in Rodriguez-Lara and Moreno-Garrido [11] relies on this production stage. In Konow [7], subjects are differently rewarded for a real-effort task that involves stuffing letters into envelops. In Ubeda [24], the task consists of solving puzzles, whereas in Cappelen et al. [5], subjects have to choose how much money to invest in an investment game in which the rate of return is exogenously determined.
  • 9 Recall that we focus on the case in which the dictator is rewarded at a higher rate; therefore the recipient's monetary contribution to the surplus will be below her contribution in terms of inputs (ar > yr). Graphically, this implies that the dotted curve (the accountability principle) is above the 45-degree line (the libertarian principle). Both principles coincide when (yr / Games 03 00138 i001) = 0 and (yr / Games 03 00138 i001) = 1. In the first case, all the available surplus is due to the dictators' inputs (qr = 0). The contrary is true if (yr / Games 03 00138 i001) = 1.We also note that we represent (yr / Games 03 00138 i001) on the horizontal line and assume that pd = 1.5 and pr = 1. Thus, the egalitarian principle and the libertarian one coincide when (yr / Games 03 00138 i001) = 0.5 (i.e., when yr = Games 03 00138 i049), whereas the egalitarian principle and the accountability one coincide if ar / Games 03 00138 i001 = 0.5 (i.e., when ar = Games 03 00138 i049). In order to satisfy this latter condition, both subjects should have exactly the same number of correct answers (qd = qr), which implies that yr / Games 03 00138 i050 = 0.4.
  • 10 We note that our model cannot be tested using the reported data in other experiments such as Bolton and Ockenfelds [28], Charness and Rabin [29] or Fehr and Schmidt [1], which do not consider a production stage.

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