Inequality aversion and antisocial punishment

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.

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.

$$\begin{aligned} \frac{\beta }{n-1}(n-2)c > c + \frac{\beta }{n-1}(1-c). \end{aligned}$$

(4)

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.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 ²¹

$$\begin{aligned} \underbrace{\frac{}{}rc}_{\mathrm{i}} < \underbrace{\frac{\alpha }{n-1}r(n^{\prime }-rc)}_{\mathrm{ii}} + \underbrace{\frac{\beta }{n-1}(n-r-n^{\prime })rc}_{\mathrm{iii}}. \end{aligned}$$

(5)

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

$$\begin{aligned} {\tilde{r}}=\left\lfloor \frac{\alpha n^{\prime }}{c\left( \alpha +\beta \right) }+\frac{\beta n-\beta n^{\prime }-n+1}{\left( \alpha +\beta \right) }\right\rfloor . \end{aligned}$$

(6)

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:

$$\begin{aligned} w_{Ek} - c \sum _{j=1}^{r} p_{Ekj} = w_i - \sum _{k=1}^{n^{\prime }} p_{Eki} \quad \quad \forall \; i=1,\ldots , r \;\quad and \quad k=1,\dots ,n^{\prime }, \end{aligned}$$

(7)

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

$$\begin{aligned} p_{Ei}=\frac{w_{i}}{n^{\prime }}+\frac{c\sum _{j=1}^{r}w_{j}-n^{\prime }w_{E}}{n^{\prime }(n^{\prime }-rc)}. \end{aligned}$$

(8)

Total expenditures of an enforcer for punishment in the unconstrained case are, therefore,

$$\begin{aligned} \pi _E^r=c \sum _{i=1}^r p_{Ei} = \frac{c\sum _{i=1}^{r}w_{i}-rcw_{E}}{n^{\prime }-rc}. \end{aligned}$$

(9)

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

$$\begin{aligned} \pi _E(\varvec{w})= {\left\{ \begin{array}{ll} 0 &{} \text{ if } r=0 \text{ or } r>{\tilde{r}} \\ \pi _E^r &{} \text{ else } \text{ if } \pi _E^r \leqslant w_E-w_{r+1} \\ \pi _E^{r+1} &{} \text{ else } \text{ if } \pi _E^{r+1} \leqslant w_E-w_{r+2} \\ \vdots \\ \pi _E^{\tilde{r}} &{} \text{ else } \text{ if } \pi _E^{\tilde{r}} \leqslant w_E-w_{\tilde{r}+1} \text{ or } r=\tilde{r} \\ w_E-w_{{\tilde{r}}+1} &{} \text{ else } \end{array}\right. } \end{aligned}$$

(10)

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. $\square$