Abstract
Antisocial punishment—punishment of pro-social cooperators—has shown to be detrimental for the efficiency of informal punishment mechanisms in public goods games. The motives behind antisocial punishment acts are not yet well understood. This article shows that inequality aversion predicts antisocial punishment in public goods games with punishment. The model by Fehr and Schmidt (Q J Econ 114(3): 817–868, 1999) allows to derive conditions under which antisocial punishment occurs. With data from three studies on public goods games with punishment I evaluate the predictions. A majority of the observed antisocial punishment acts are not compatible with inequality aversion. These results suggest that the desire to equalize payoffs is not a major determinant of antisocial punishment.
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Notes
There are different definitions for the punishment of cooperative subjects in the literature. Herrmann et al. (2008) focus on the bilateral comparison of contributions between punisher and punishee. Falk et al. (2005) investigate motives behind the punishment decision in the prisoners’ dilemma and call the punishment of cooperative subjects ‘spiteful punishment’, indicating that they see the motive to increase payoff differences as a determinant for such punishment acts (see also Masclet et al. 2003). Cinyabuguma et al. (2006) define ‘perverse punishment’ as punishing a subject who contributes more than the group average. They investigate whether second order punishment, i.e., punishing the punishers, eliminates perverse punishment. Nikiforakis (2008) addresses a similar question in a different design. Further data on antisocial punishment are reported by Anderson and Putterman (2006); Ertan et al. (2009), and Gächter and Herrmann (2009, 2011).
In fact, the analysis that follows is applicable to any situation where members of a group of \(n\) players, endowed with some income \(\varvec{w}\), can mutually reduce their incomes by the punishment mechanism described in Eq. (2).
Fehr and Schmidt (1999, p. 841) show that the composition of types and parameters assumed here allows for a fully cooperative equilibrium. To explain punishment we need to assume that one player deviates in the contribution stage.
The inclusion of antisocial punishment does not affect the general structure of the equilibria described in Fehr–Schmidt’s Prop. 5, where all players contribute \(g \in (0,y]\) in the first stage and punishment occurs only off equilibrium.
To keep things simple I assume that all enforcers are homogeneous both in their preferences (equal \(\alpha _E\) and \(\beta _E\)) and in their stage 1 earnings \(w_{Ei}\).
Note that the ERC model by Bolton and Ockenfels (2000) is also capable in predicting antisocial punishment, but does so in a rather trivial way. In this model players care about their share of the total pie (and their own income). If they earn too low a share they can increase their share by punishing any other group member, not just the free rider. Consequently, the ERC model has no predictive power whatsoever about the direction of punishment.
This holds for symmetric equilibria of the punishment subgame, i.e. equilibria where all enforcers punish the other players equally. Because the enforcers do only care about final payoffs they could freely reallocate punishment points among themselves, as long as the total amount of punishment meted out by each punisher and received by each punishee remains constant. In case the group of enforcers is sufficiently large there can be asymmetric equilibria where some enforcers mete out exclusively antisocial punishment.
I use only the data from the sessions with the sequence punishment—no punishment.
This number holds for all situations. It might be more interesting to consider only situations similar to those described in Table 1, where there is a clear free rider. Consider the cases where three subjects contribute strictly more than the group average and one subject contributes strictly less than the group average. In 40.3 % of these cases all three high contributors punish the free rider. Consequently, in 59.7 % of the situations not all three high contributors engage in free-rider punishment, leaving scope for antisocial punishment as explained by inequality aversion.
The number corresponds to the parameter \(p_{ij}\) in equation 2, i.e. to the average reduction of the punishee’s income. In the experiment subjects choose ‘deduction points’, which reduce the punishee’s income by three units and cost the punisher one unit. Deduction points are limited to integers and up to ten points per punishment act. Thus, in the experiment the optimal punishment solution presented in Table 1 is not feasible, because player 2’s income cannot be reduced by more than 30 units.
This comparison controls for the fact that a subject who is among the richest players has more ‘occasions’ to engage in antisocial punishment than a poorer subject, because it is the average number of punishment points assigned in all bilateral comparisons with weakly poorer subjects. Looking at absolute numbers the difference becomes even stronger. From a total of 1188 units of payoff reduction by antisocial punishment 921 (78 %) are caused by the richest subjects in the group.
Like in the first dataset I use only the data from the sessions which started with the one-shot public goods game with punishment.
The data does, however, still not allow to identify sufficient conditions for antisocial punishment. To do so would require control over the subjects’ beliefs about all other punishment decisions.
The analysis presented here uses data from a dynamic game to test a static prediction. This could be problematic because of strategic incentives in early rounds of the game. To check whether the results are robust with regard to this concern I ran the analysis for the last period only. The last punishment subgame played presents a true one-shot game. The results remain qualitatively unchanged.
See also Houser and Xiao (2010).
Such a relative payoff maximizer could be characterized by having a utility function as shown in Eq. (3) with \(\beta <0\). For example, a player with \(\beta =-\alpha \) would always want to punish other players (irrespective of whether they are poorer or richer) if \(cn<1+c\) and \(\alpha >\frac{c( n-1) }{1+c-cn}\). For the parameters used in Fehr and Gächter (2002) and Herrmann et al. (2008) the first condition holds with equality and requirements for \(\alpha \) go to infinity.
Due to the linearity in payoff differences in the Fehr–Schmidt utility function \(E\) is indifferent between shifting punishment points from one richer player to another richer player. Thus, if a player \(E\) is ready to punish one richer player by, say \(\epsilon \), then she is also ready to punish two richer players by \(\frac{\epsilon }{2}\).
Here I assume that all enforcers use the same punishment strategy so that no inequality towards other enforcers arises. In case of \(n^{\prime }>1\) Eq. 5 describes a joint optimization for all enforcers.
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Acknowledgments
For helpful comments and suggestions I thank two anonymous referees, Simon Gächter, Louis Putterman, Jonathan Schulz and the participants of the Thurgau Experimental Economics Meeting and the meeting of the Economic Science Association.
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Appendix
Appendix
1.1 Prerequisites
There are two types of players: a homogenous subgroup of \(n^{\prime }\!<\!n\) enforcers \(E1, E2\), ..., \(En^{\prime }\) with \(\alpha ,\beta >0\) and \(n-n^{\prime }\) other players with \(\alpha ,\beta =0\). All enforcers have identical provisional income, i.e. \(w_{E1}=w_{E2}=\ldots =w_{En^{\prime }}\). In case of \(n^{\prime }>1\) there is usually an infinite number of equilibria in which all enforcers spend an equal amount on punishment. In the following I will derive the symmetric equilibria most efficient for the enforcers. The homogeneity assumption facilitates the derivation of equilibria because all enforcers either punish or do not punish. In equilibrium all enforcers punish equally and no inequality among them arises.
What are the conditions under which Fehr–Schmidt players might engage in antisocial punishment? Enforcers face a distribution of provisional payoffs \(\varvec{w}\) and seek to maximize their utility by choosing a punishment vector \({\varvec{p}}_{\varvec{E}\varvec{k}}\) for all \(k=1, \ldots , n^{\prime }\). In the group there are \(n-n^{\prime }\) other players. These players shall be ordered according their initial income \(w_i\), such that the incomes of the players are \(w_1 \geqslant w_2 \geqslant \ldots \geqslant w_r \geqslant w_{Ek} > w_{r+1} \geqslant \ldots \geqslant w_{n-n^{\prime }}\), i.e. Player 1 is the richest player, \(r\) players are weakly richer than the enforcers, and \(n-n^{\prime }-r\) players are strictly poorer. The variable \(r\) is an indicator for the enforcers’ position in the income hierarchy. For \(r=0\) the enforcers are the strictly richest players and there are \(n-n^{\prime }\) poorer players. In case of \(r=n-n^{\prime }\) the enforcers are (among) the poorest players.
Proof of C1
Will the enforcers ever punish players with a lower income? Due to \(c<1\) this increases inequality towards the punished player. Furthermore, it reduces the enforcers’ incomes by \(c\) and increases the inequality towards all players who are or become richer than the enforcers. The only benefit the enforcers can get from punishing a poorer player is that the reduction of their income reduces inequality towards the other players with lower incomes by \(c\). The case most favorable for punishing a poorer player is thus the situation where a single enforcer is the richest player, i.e. \(r=0\). In such a situation it would be utility enhancing to punish another player if benefits outweigh costs, i.e.
Rearranging leads to \(\beta - \frac{\beta }{c(n-1)} > 1\). This inequality can only be satisfied for \(\beta >1\), which is ruled out by the parameter restrictions of the Fehr–Schmidt model. Thus, irrespective of the position in the income distribution the enforcers will never punish a player with lower income than themselves. Note that this does not exclude that punishment of players with \(w_i<w_E\) eventually takes place, when the enforcers become poorer than other players due to their punishment of free riders. However, if the enforcers are among the richest players in the first place, then this situation cannot occur and, consequently, antisocial punishment can be ruled out. \(\square \)
Proof of C2
What is the structure of the optimal punishment strategy? The examples in the main text already demonstrated that it can be utility enhancing for the enforcers to punish free riders. The crucial question is not whether, but how many richer players the enforcers are ready to punish.Footnote 20 Punishing \(r\) weakly richer player has costs and benefits: (i) punishment has direct costs of \(rc\), (ii) due to \(c<1\) this decreases the disadvantageous inequality towards the punished player and (iii) it reduces the enforcers’ payoff advantage towards the other \(n-r-n^{\prime }\) players with lower income by \(c\). Taken together, punishing all \(r\) weakly richer players pays if Footnote 21
This expression allows to identify the effect of changes in the preference parameters: punishing richer players is more likely to be profitable if the enforcers’ inequality aversion becomes stronger (\(\alpha \) and \(\beta \)). Equalizing benefits and cost in Eq. (5) and solving for \(r\) gives the maximum integer number of other players that will be punished
It is easy to show that \(\tilde{r}\) increases in the number of enforcers \(n^{\prime }\). The expression is decreasing in \(n\), which is due to the fact that in larger groups the inequality towards the \(r\) richer players has less weight in an enforcer’s utility function. Furthermore, \(\tilde{r}\) is decreasing in \(c\), i.e. more expensive punishment reduces the number of other players the enforcers are willing to punish.
What is the optimal amount of punishment? As demonstrated in Table 2 we have to check whether punishment is constrained by the income of the next poorer player or not. For unconstrained punishment all enforcers and weakly richer players have the same final payoff, i.e. we have to solve the following system of equations:
where the final income of an enforcer \(k\) is on the left hand side and the right hand side shows the income of a weakly richer player \(i\). To simplify matters I assume that punishment of a player \(i\) is split equally among the enforcers. This allows to replace \(p_{Ekj}\) by \(p_{Ej}\) and the sum on the right hand side by \(n^{\prime }p_{Ei}\). Thus, dependent on \(r\), optimal punishment points are
Total expenditures of an enforcer for punishment in the unconstrained case are, therefore,
Clearly the amount of punishment necessary to bring down the \(r\) weakly richer players decreases in \(n^{\prime }\). When is punishment unconstrained? If \(w_E - \pi _E^r \geqslant w_{r+1}\) then there is enough ‘room’ to punish all richer players without undershooting the income of the next poorer player.
Otherwise the enforcers are in the constrained case. Here the enforcers’ incomes will touch \(w_{r+1}\) before they could equate all incomes of the \(r\) richer players with their own incomes. If this happens the number of weakly richer players increases by one and the optimal punishment for \(r+1\) comes into action. The enforcers will include further players into the group of punishees until either (i) there is no strictly richer player anymore or (ii) the group of weakly richer players exceeds \(\tilde{r}\). To conclude, punishment expenditures depend on the distribution of the preliminary incomes \(\varvec{w}\) which is characterized by \(r\), the number of weakly richer players. Punishment expenditures are
In all but the first and last case the enforcers reduce the incomes of richer players such that all \(r\) players earn the same income as the enforcers. In doing so the group of weakly richer players might increase up to a maximum of \(\tilde{r}\). Depending on the provisional income of player \(\tilde{r}+1\) the enforcers equalize the incomes of all \(\tilde{r}\) players or mete out punishment such that their income is equal to \(w_{{\tilde{r}}+1}\). In the latter case the free riders will keep some of their monetary payoff advantage relative to the enforcers (as demonstrated by the case of the weakly inequality averse player in Table 2).
To conclude, if none of the players are richer than the enforcers then no one will be punished. If some of the players are richer and others are poorer than the enforcers, then the poorer players are punished if and only if the enforcers become poorer than some of these players due to the punishment of free riders. Consequently, antisocial punishment can only occur in combination with free-rider punishment. \(\square \)
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Thöni, C. Inequality aversion and antisocial punishment. Theory Decis 76, 529–545 (2014). https://doi.org/10.1007/s11238-013-9382-3
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DOI: https://doi.org/10.1007/s11238-013-9382-3