Carrots and sticks: new evidence in public goods games with heterogeneous groups

Abstract

Standard public goods games often assign group members the same marginal per capita returns to public goods production, but in reality group members facing differential individual returns often must collaborate to produce a public good. This paper uses a laboratory experiment to investigate the comparative efficacy of punishment and reward in heterogeneous groups. Punishment and reward are implemented by allowing every member to incentivize other members at a cost. Contrary to the common belief that punishment is more effective than reward, I find that reward increases group contributions and efficiency, but punishment does not. Reward increases cooperation because all members are happy to reward cooperators. Punishment is ineffective because high-benefit members assign antisocial punishment toward others, whereas low-benefit members refrain from punishing other low-benefit members.

Main Appendix

1.1 Theoretical predictions

To predict contribution behavior of members, two theories are adopted here. Homo oeconomicus model assumes rational and purely selfish agents. The other one is a model of with inequality aversion from Fehr and Schmidt (1999). Both models can tractably derive explicit predictions for a one-shot public goods game¹³. Importantly, this experiment is by no means a rigorous test of the theories. Rather, theories are adopted here to facilitate constructions of theoretical predictions.

Homo oeconomicus predictions. A player predicted by this theory cares only about his own payoffs (see Online Appendix B.2). For any $\frac{1}{n}<a<1$, the equilibrium solution is zero. Further, any individual’s contribution is independent of other members’ contributions. With incentive opportunities, theoretical predictions do not change. Any individual player will contribute zero to the public good project. Because no selfish player would actually enforce the incentive scheme at own cost.

For all MPCRs smaller than 1, the Nash equilibrium for a rational person is to contribute 0 in the first stage, and never punish or reward in the second stage. However, this prediction is only based on the assumption of a first-order public good (Foster et al. 2017). An individual may commit to contribute more, to punish, or to reward, despite of others’ contribution decisions, when he or she is concerned about a second-order public good (e.g. maintain a social norm, form an institution etc.). Existing literature has repeatedly found violations of the equilibrium prediction: when incentives are available, participants will utilize the incentives to promote greater cooperation, or to elicit own preferences (Herrmann et al. 2008). Thus, I expect to see deviations from Nash equilibrium prediction.

Predictions with Fehr and Schmidt (1999) utilities. Assuming there exists subjects who dislike inequitable outcomes, both when they are better off and when they are worse off in monetary payoffs compared to another player. Additionally, subjects dislike material disadvantage more than material advantage. The utility function is as below.

$$\begin{aligned} U_i(\pi )=\pi _i-\alpha _i\frac{1}{n-1}\sum _{j\ne i}\max \{\pi _j-\pi _i,0\}-\beta _i\frac{1}{n-1}\sum _{j\ne i}\max \{\pi _i-\pi _j,0\} \end{aligned}$$

where n denotes number of subjects in a group, $\alpha$ denotes level of disadvantageous inequality aversion, and $\beta$ denotes level of advantageous inequality aversion ($\alpha >\beta$).

This paper modifies the theoretical model of Fehr and Schmidt (1999) to account for having a high-benefit member in three steps (see Online Appendix B.3). First, taking into account Sutter et al. (2010)’s extension of the original inequality aversion model, I reconstruct the conditions for standard public goods game. Second, following the same structure, I calculate changes in utilities in public goods games with a high-benefit member. Third, theoretical predictions are derived for public goods game with heterogeneous MPCRs. The predictions for a one-shot public goods game are listed below.

• Prediction 1: Groups with a high-benefit member will generate fewer zero contributions compared to groups without such a member.¹⁴

• Prediction 2: There is a continuum of positive contribution bundles forming Nash Equilibria. If a high-MPCR member has $\beta _i>1/15$, then he always contributes full endowment, while all low-MPCR players contribute a positive amount c. Otherwise, the high-MPCR member will contribute the same amount as everyone else does.

• Prediction 3: Reward is more credible than punishment in heterogeneous groups, because the binding conditions for punishment are always more stringent than those for reward.

• Prediction 3.1: Reward opportunities more effectively increase group cooperation than punishment opportunities in heterogeneous groups.

Table 6 Treatment effects: tobit regression results

1.2 Additional Tables

Table 7 Punishment/reward depending on contributions and first-stage earnings

Table 8 Punishment/reward depending on contributions and first-stage earnings

Table 9 Punishment/reward depending on contributions and first-stage earnings