## Abstract

Psychological games of guilt aversion assume that preferences depend on (beliefs about) beliefs and on the guilt sensitivity of the decision-maker. We present an experiment designed to measure guilt sensitivities at the individual level for various stake sizes. We use the data to estimate a structural choice model that allows for heterogeneity, and permits that guilt sensitivities depend on stake size. We find substantial heterogeneity of guilt sensitivities in our population, with 60% of decision makers displaying stake-dependent guilt sensitivity. For these decision makers, we find that average guilt sensitivities are significantly different from zero for all stakes considered, while significantly decreasing with the level of stakes.

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## Notes

- 1.
In general, psychological game theory takes into account that people’s utilities do not only depend on material payoffs, but also on their beliefs about others’ behavior, as well as beliefs about the beliefs of others. See Geanakoplos et al. (1989) and Battigalli and Dufwenberg (2009) for general frameworks of games with belief-dependent preferences.

- 2.
Self-serving bias on the other hand has not shown to be related to stake sizes. See Babcock and Loewenstein (1997) for a discussion.

- 3.
Holt and Laury (2002) find that the degree of risk aversion increases with stakes. Bellemare et al. (2008) find that the marginal disutility of disadvantageous inequality is decreasing with stakes while the marginal disutility of advantageous inequality is decreasing with stakes for young, highly educated people (see also Yang et al. 2012). Gibson et al. (2013) find that the percentage of truth-tellers decreases with the costs of truthfulness. On the other hand, Carpenter et al. (2005) report that multiplying the stakes by 10 does not have a statistically significant effect on the share a dictator allocates to a passive player. For general surveys, see Camerer and Hogarth (1999) and Gneezy et al. (2011).

- 4.
See Krupka and Weber (2013) for an experimental study that points out the relevance of dictator games for understanding how social norms influence behavior.

- 5.
The method circumvents the problem of possible spurious correlation between beliefs and behavior by letting the dictator make choices conditional on first-order beliefs of the passive player. See Vanberg (2008), Reuben et al. (2009), Ellingsen et al. (2010), and Bellemare et al. (2011) for other methods that circumvent the problem.

- 6.
They model belief-dependent preferences as a combination of guilt sensitivity and intention-based reciprocity, with trustees’ willingness to share being increasing (decreasing) in his second-order belief if guilt sensitivity (intention-based reciprocity) prevails in the questionnaire.

- 7.
Note that unlike in ‘typical’ dictator games, in our experiments, dictators can condition their choice on the first-order belief of the matched passive player.

- 8.
Following Battigalli and Dufwenberg (2007)’s model of simple guilt, we assume that utility functions are linear in the stake level throughout the paper. However, these properties also hold for utility functions that are power functions of the stake level.

- 9.
Trautmann and van de Kuilen (2015) compare the effect of different belief elicitation methods and find that subjects do not behave differently depending on whether beliefs are incentivized or not (see also Armantier and Treich 2013). Therefore, we decided not to incentivize the measurement of first-order beliefs.

- 10.
Supplementary of Appendix includes detailed instructions. Passive players and dictators were respectively referred to as Players A and B during the experiment.

- 11.
This method to elicit switchpoints corresponds to the ‘minimum acceptable offer’ procedure often used to elicit the strategy of responders in ultimatum games (see e.g., Güth et al. 1982; Schotter and Sopher 2007). In another experiment using a mini trust game where players could switch back and forth between

*r*and*l*as much as they liked (reported in Bellemare et al. 2017), we found that the vast majority of players did not switch more than once (see also Attanasi et al. 2013). Furthermore, for those few players that switched more than once or switched in the ‘wrong’ way, no systematic tendency could be discovered in their behavior. - 12.
Due to an error in the computer program, the data from 2 dictators were not usable, leaving us with 140 dictators.

- 13.
Another important reason for not informing the dictators was that we tried to keep instructions as similar as possible to those in other dictator treatments (reported in Bellemare et al. 2017).

- 14.
In a similar vein, it is possible to show predictions under non-linear inequality aversion. Suppose that the utility of choosing the selfish option

*r*is equal to \(54 s - \gamma \cdot m(32s)\) where the function*m*() captures the disutility from advantageous inequality aversion. It can easily be seen that dictators with quadratic inequality aversion \(m(a)=a^2\) choose*l*for all*s*if \(\gamma \ge 0.0039\) and*r*otherwise. Moreover, dictators with square-root inequality aversion \(m(a)=\sqrt{a}\) choose*l*for \(s=1,2,4\) if \(\gamma \ge 0.943\). They choose*r*for \(s=1\) and*l*for \(s=2,4\) if \(\gamma \ge 0.666\), and choose*r*for 1, 2 and*l*for \(s=4\) if \(\gamma \ge 0.472\). - 15.
Our identification regions are based on the model of simple guilt aversion by Battigalli and Dufwenberg (2007). They should be considered as conservative lower bounds in case players are motivated by simple guilt aversion as well as distributional concerns. Assuming that dictators are also motivated by Fehr and Schmidt (1999)’s model of inequality aversion implies that they should choose

*l*if \(\beta \ge \frac{2{-}15\gamma }{13\theta }\). The higher the dictator’s sensitivity to advantageous inequality, the lower the threshold. For \(\gamma\) bigger or equal to 2/15 (0.13333), the dictator should choose*l*independent of \(\beta\). Fehr and Schmidt (1999) report an average sensitivity to advantageous inequality of 0.315 (in Table 3 on page 844). Using this coefficient implies that all dictators in our experiment that are motivated both by guilt aversion and inequality aversion should have chosen the nice option*l*independent of \(\beta\). - 16.
We experimented with a specification which modeled the distribution of \(\left( \theta ^1,\theta ^2,\theta ^4\right)\) as a discrete distribution: \(\left( \theta ^1_k,\theta ^2_k,\theta ^4_k\right)\) with probability \(\omega _k\) for \(k = 1,2,\ldots ,7\). This specification led to a lower log-likelihood function (when evaluated at the solution) than our preferred specification despite the higher number of model parameters of the discrete specification (the discrete and our preferred specifications model the joint distribution of \(\left( \theta ^1,\theta ^2,\theta ^4\right)\) using respectively 27 and 11 parameters). Numerically unstable solutions were obtained when we further increased the number of mass points of the discrete distribution.

- 17.
- 18.
The predicted conditional posterior probability of player

*i*is obtained by evaluating \(\left( \omega _{1i} \frac{1}{R}\sum_{r=1}^{R}\left[ \Pi_{\forall s} \Pi_{j=1}^{11}l_{ij}\left( y_{ij}^{s};\theta _{i}^{s,r},\lambda \right) \right] \right) /\widetilde{L}_i\) at the estimated values of the model parameters, where \(\widetilde{L}_i\) is given in (11). - 19.
This is because the threshold for the second-order belief at which guilt-sensitive individuals switch from

*right*to*left*(condition (2)) decreases as stakes increase. Intuitively, increasing stakes has the same effect as increasing \(\rho\) when the utility function is \(u(x)=1-e^{-\rho x}\). - 20.
Calculations available upon request.

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## Acknowledgements

The authors thank audiences of seminars at the universities of Amsterdam, Copenhagen, Frankfurt, Gothenborg, Heidelberg, Innsbruck, Mannheim, Munich, Pennsylvania, and Tilburg, at the Max Planck Institute of Economics in Jena, at the Nordic Conference of Behavioral and Experimental Economics in Stockholm, and at IMEBE in Madrid for helpful comments. Furthermore, we would like to thank the editor and two anonymous referees for very helpful comments. Suetens acknowledges financial support from the Netherlands Organization for Scientific Research (NWO) through the VIDI program (Project No. 452-11-012).

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Bellemare, C., Sebald, A. & Suetens, S. Heterogeneous guilt sensitivities and incentive effects.
*Exp Econ* **21, **316–336 (2018). https://doi.org/10.1007/s10683-017-9543-2

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### Keywords

- Guilt sensitivity
- Psychological game theory
- Heterogeneity
- Stakes
- Dictator game

### JEL Classification

- A13
- C91