3.1 The public goods games

Our experiment on endogenous institutions involves choosing between and playing different public goods games. The choice is always between a standard public goods game and a public goods game with an option to exclude members from the group. Participants are divided into groups of $N=5$ members that remain fixed throughout the experiment (partner design).Footnote ⁴ There are four phases which consist of five rounds each, with the game being fixed within a phase. In every round, groups of size $n\le N$ play a public goods game and every player $i\in \left(1,\dots ,n\right)$ receives an endowment $E_p$ of which he or she can contribute to the public good. Player $i^{\prime }s$ contribution is denoted by $g_i$ . The stage game payoff to player $i$ is given by ${\pi }_i=E_p-g_i+a{\sum }_{j=1}^n{g}_j$ and the marginal per capita return (MPCR) is $a=0.4$ .

In every round, players choose simultaneously how much to contribute to the public good. After each round, individual contributions are displayed on the screen in random order, so that it is not possible to track the contribution by other members over time. This ensures that the decision to vote for the exclusion of a player in a given round is not based on the player’s reputation formed in the course of the game, but only on his or her contribution in that round.

To study endogenous institutional choice, we distinguish between three versions of the public goods game which are denoted by $p\in \left(A,B10,B8\right)$ . In game A, players’ endowment is $E_A=10$ . This game does not allow players to exclude other members from the group so that the group size is fixed at $n=N=5$ in all rounds. In game B10, players’ endowment is the same as in game A with $E_{B10}=10$ , whereas in game B8, it is reduced by 20% to $E_{B8}=8$ . Both games, B10 and B8, allow players to exclude members from the group so that $n\le N$ . For this purpose, these games include an additional stage. After having been informed about the individual contributions, players can vote to exclude a member from the group. Next to each contribution, an empty box is shown on the screen which players can tick in order to vote for that player to be excluded. The players are informed about the number of votes they have received but not from whom. Thus, while blind revenge against the group is possible, targeted retaliation after exclusion is not. Each player can cast at most one vote, at no cost, in order to determine who should be excluded. Players cannot vote for themselves but they can decide not to vote at all. Players who receive the votes from more than half of his or her co-players will be excluded from the game for the remaining rounds in that phase. This implies that the group can shrink over time. If the group consists of five members, a player must receive at least three votes in order to be excluded. If the group consists of three or four members, a player must receive at least two votes in order to be excluded. If the group consists of only two members, exclusion is no longer possible. With these voting rules, it is possible but unlikely that two players are excluded from the group at the same time. The only case in which two players could be excluded at the same time is when there are four players and two of them receive exactly two votes. The excluded players receive the endowment, either $E_{B10}$ or $E_{B8}$ , in each round but they are no longer able to contribute to and benefit from the public good. They are able to observe what happens in the public goods game but they are no longer allowed to vote for other players to be excluded.Footnote ⁵ There is no exclusion stage in the last round of a phase. To exclude the ostracized players from the benefits of the public good but not from getting their endowment is a relatively conservative approach. It can be interpreted that the community has the power to exclude individuals from the social benefits but not to take away their source of livelihood.

3.2 Main treatments

At the start of each phase, the full group, consisting of $N=5$ members, chooses the game they want to play, with simple majority deciding. Importantly, the choice is always between the A game and one of the two B games (and never between the two B games). In the treatment called “B10,” players choose between A and B10, while in the treatment called “B8,” players choose between A and B8.Footnote ⁶ The reduced endowment in B8 compared to game A can be interpreted as a collective cost of the exclusion option. We set the fixed cost of the institution to 20% of the endowment, so that it would be challenging but not impossible to compensate for the cost through higher contributions in the B8 game. All members of the group simultaneously vote either for game A or for game B. There are no abstentions. For a game to be selected, at least three out of the five members must vote for that game. Members are informed about which game has been selected, but not about the individual votes. Afterwards, the group plays the chosen game throughout that phase. If the group plays B10 or B8 and a player gets excluded from the group, the exclusion lasts only until the end of the respective phase. At the beginning of the new phase, the excluded player re-enters the group and all players vote again to choose between game A and game B. Figure 1 presents the time line in the experiment.

Fig. 1 Voting rounds and phases

A few things about our design are worth noting. First, players can abstain from the exclusion vote but not from the vote on the institution. There are several reasons for this. The nature of our research question, which is endogenous institutional choice and its consequences, requires an active game choice by the participants. Allowing for abstention from the institutional vote would have introduced behavioral issues out of our control. For example, playing game B would not necessarily imply that the majority has voted in favor of B. Another reason is to avoid practical inconvenience. Assume that all five players abstain from voting or there is a tie. A random device would have been needed then to determine which game is played, since one of the two games must be played. In this situation, the institutional choice would not have been endogenous. In the case of the exclusion vote these factors are less of an issue. The option to abstain is necessary here for situations in which all group members make equally high (or low) contributions. Second, given the MPCR of $a=0.4$ , contributing to the public good is inefficient once the group has shrunk to just two members. In this case, the collective benefit of contributing one unit to the public good is smaller than the cost ( $0.8<1$ ). This could have been avoided by a higher MPCR. If, for example, the MPCR was increased to 0.6, contributing to the public good would be efficient even with two players only. However, in the initial group of five players, the full cooperative payoff would then be three times as large as the Nash payoff and thus create strong incentives to cooperate even without the exclusion institution. Alternatively, we could have restricted the voting rule in the B games by capping the number of excludable players at two but this would have facilitated the institutional choice between game A and game B. In our design, if players choose the B game their challenge is to maintain both a high cooperation level and a large enough group. Third, our groups start the experiment by choosing between the games with no prior experience. Therefore, all learning is endogenous as it depends on how groups choose and play over the course of the experiment. Experience has been shown to be critical for institutional choice, so a natural extension of our study would be to have subjects gain some experience in one or both games before they choose between them (Markussen et al. Reference Markussen, Putterman and Tyran2014; Barrett and Dannenberg Reference Barrett and Dannenberg2017).

3.3 Exogenous control treatments

With endogenous choice of the institution, where groups select themselves into the different games, it is not clear if the institution is successful because it attracts the most cooperative groups or because the institution changes the incentives to cooperate, regardless of whether the groups are particularly cooperative or not. In order to distinguish between the effect of self-selection and the effect of the institution, we conducted two additional treatments, B10-exo and B8-exo, in which groups played games A and B over the same number of rounds but, unlike the groups in the endogenous treatments, these groups could not vote on the two games but had to play the game that was announced by the computer.Footnote ⁷ For each group in the endogenous treatments, we had one group in the exogenous treatments that played the exact same sequence of A games and B games (perfect matching groups). This means that, in each phase, the distribution of groups between the two games in the exogenous treatments is identical to the distribution in the corresponding endogenous treatment. To keep the difference to the endogenous treatments to a minimum, players in the exogenous treatments were not informed about the sequence in advance but learned which game they would play only at the beginning of each phase. Apart from the missing voting stage and the way the games were chosen, everything in the exogenous treatments was identical to the endogenous treatments. The exogenous treatments also allow us to compare the results with the previous literature.

3.4 Implementation

The experimental sessions were held in a computer lab at the University of Magdeburg, Germany, using undergraduate students recruited from the general student population. In total, 460 students participated in the experiment with each one taking part in one treatment only (between-subject design). For our main treatments, we conducted eight sessions in June 2016 and assigned them randomly to B10 and B8.Footnote ⁸ For the exogenous control treatments, we conducted ten sessions in September and November 2018 at the same computer lab and assigned them randomly to B10-exo and B8-exo.Footnote ⁹ For each of the four treatments, we had 23 groups that consisted of five players each. In each session, subjects were seated at linked computers (game software z-Ttree; Fischbacher Reference Fischbacher2007) and randomly divided into five-person groups. Subjects did not know the identities of their co-players, but they did know that the membership of their group remained unchanged throughout the session. The experimental instructions were handed out to the students and also read aloud to ensure common knowledge. They carefully explained both games, A and B, and included several numerical examples. Before subjects began playing the games, they had to answer a number of control questions. The control questions tested subjects’ understanding of the games to ensure that they were aware of the available strategies and the implications of making different choices. The experiment began only when all participants had answered the control questions correctly. Questions during this process were answered privately. During the game, earnings were displayed in tokens. It was public knowledge that payments would be calculated by summing up the number of tokens earned over all rounds and by applying an exchange rate of €.05 per token. At the end of the experiment, subjects were paid their earnings privately in cash.

4.1 Standard preferences model

In the standard preferences model, zero contribution by all players is the unique Nash equilibrium (NE) of the stage game. This equilibrium is Pareto dominated by the outcome in which all players contribute their entire endowment as long as the group has more than two members. By backward induction it obtains that the unique subgame perfect Nash equilibrium (SPNE) of the repeated game is zero contribution by all players in each round, regardless of the game played. Thus, players are indifferent between game A and game B10, but prefer A to B8 as the former gives a higher endowment and so a higher payoff. Hence, the standard preferences model predicts that game B8 is never played when the choice is between A and B8. When the choice is between A and B10, each game will be played half the time. If B10 is chosen, then any configuration of votes and group sizes can be part of an equilibrium because exclusion in our setting is costless and players are thus indifferent between excluding and not excluding a group member (see Appendix A.1 of ESM).

4.2 Inequality aversion model

In the inequality aversion model by Fehr and Schmidt (Reference Fehr and Schmidt1999), players derive utility from the material earnings resulting from the public good, and they derive disutility if their earnings are higher than those of other group members (advantageous inequality aversion) or if their earnings are lower than those of other group members (disadvantageous inequality aversion). Specifically, the inequality averse utility function is:

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

where ${\pi }_i$ is player $i^{\prime }s$ material payoff from the public good, ${\alpha }_i$ measures the aversion to disadvantageous inequality and $0\le {\beta }_i<1$ captures the aversion to advantageous inequality. Moreover, ${\alpha }_i\ge {\beta }_i$ such that players are more averse to disadvantageous than to advantageous inequality.

With these preferences, any weakly positive contribution level $g_i=g\in [0,E_p],$ for all i, can be supported as an equilibrium of the stage game if all group members are sufficiently averse to advantageous inequality, i.e. ${\beta }_i\ge 1-a=0.6$ . We call these players conditional cooperators, following the original paper. This equilibrium exists in both games and it makes no use of the exclusion option in game B. However, it requires coordination on a certain contribution level for which full contributions seems to be a natural focal point as it is Pareto dominant. By backward induction it obtains that full contributions in each round is a SPNE, regardless of the game played. Since the exclusion option in game B is not used, the choice between the games is governed by the contribution level on which players coordinate in each game. If there is coordination on the same contribution level across the games, then groups of inequality averse players are indifferent between playing A and playing B10, but they strictly prefer A to B8 (see Appendix A.2 of ESM).

With one selfish player in the group, with ${\beta }_i<1-a=0.6$ , the unique equilibrium of the stage game is zero contribution by all players, since this strategy is dominant for the selfish player.Footnote ¹⁰ Given this, it is also the best response of the remaining conditionally cooperative players, i.e. those players for which ${\beta }_j\ge 0.6$ .Footnote ¹¹ However, in game B, the conditional cooperators can use the exclusion institution against the selfish player. It can be shown that, although in the first round all players contribute zero due to the presence of the selfish player, the conditional cooperators exclude her after this round and cooperation is restored for the remaining rounds of play. Because exclusion is not possible in game A, the only SPNE of game A is zero contribution by all players in every round. Given these equilibrium outcomes, the selfish player either strictly prefers game A over both B8 and B10 (if ${\alpha }_i>0$ ) or she prefers A over B8 and is indifferent between A and B10 (if ${\alpha }_i=0$ ).Footnote ¹² The conditional cooperators strictly prefer B10 to A and they prefer B8 to A if they coordinate on a high enough contribution level after excluding the selfish player. With our experimental parameters, these players should contribute more than 5 tokens for game B8 to be preferred (see Appendix A.2 of ESM).Footnote ¹³

4.3 Reciprocity model

The reciprocity model of Rabin (Reference Rabin1993) assumes that people derive utility from reciprocation of kindness and unkindness, in addition to the monetary gains. We base our analysis on the multi-player extension of this model by Nyborg (Reference Nyborg2017) and define the reciprocal utility as

$u_i={\pi }_i+{\beta }_i{R}_i,$

where ${\pi }_i$ is the material payoff from the public good, ${\beta }_i$ is the weight attributed to reciprocation, and R _i is the reciprocation term. We use the same measure of kindness as in Nyborg (Reference Nyborg2017) and define the reciprocation term as:

$R_i=\frac{1}{n-1}\left(\sum _{j\ne i}{\tilde{f}}_{\mathrm{ji}}+\sum _{j\ne i}{f}_{\mathrm{ij}}{\tilde{f}}_{\mathrm{ji}}\right),$

where $f_{\mathrm{ij}}$ is the kindness of player $i$ towards player $j$ and ${\tilde{f}}_{\mathrm{ji}}$ is $i^{\prime }\text{s}$ belief about the kindness of $j$ towards $i.$ If all players have a sufficiently high concern for reciprocation, i.e. ${\beta }_i=\beta >2E_p\left(1-a\right),\forall i=1,\dots ,n,$ then the stage game has two pure-strategy NE, one in which all players contribute zero and one in which all players contribute their full endowment. For the SPNE, in which either of the stage-game equilibria is repeated every round, the symmetry of the equilibrium leaves the exclusion institution in game B unused (or ineffective if used in the zero-contribution equilibrium). Hence, groups of highly reciprocal players are indifferent between A and B10, but prefer A to B8 due to the higher endowment (see Appendix A.3 of ESM). If players are not sufficiently reciprocal, i.e. $0<\beta \le 2E_p\left(1-a\right),$ then zero contribution by all players is the only equilibrium.

When there is one non-reciprocal player with ${\beta }_k=0,$ but ${\beta }_i=\beta >0,i\ne k$ the stage game again has two pure-strategy NE. The non-reciprocal player contributes zero, regardless of what the reciprocal players do. Apart from the equilibrium in which all players contribute zero, there is a pure-strategy equilibrium in which the reciprocal players contribute their full endowment, but only if they are highly reciprocal, i.e. $\beta >2E_p(1-a)\frac{n-1}{n-3}$ . These two types of equilibria exist both in game A and in game B. Therefore, the repetition of either of the two pure-strategy NE is a SPNE of game A. However, in game B, the SPNE that involves the full-contribution equilibrium by the highly reciprocal players in all rounds includes the exclusion of the non-reciprocal player after the first round. If the reciprocal players are only moderately reciprocal, i.e. $2E_p(1-a)<\beta <2E_p(1-a)\frac{n-1}{n-3}$ , then in game B there is yet a third SPNE in which all players contribute zero in the first round, the reciprocal players exclude the non-reciprocal player after this round and contribute their full endowments thereafter.

For the choice between the games we assume that the reciprocal players play consistently across the two games either the zero- or the full-contribution equilibrium, when they exist in both games. If the zero-contribution equilibrium is played, then players are indifferent between B10 and A, but strictly prefer A to B8. If the full-contribution equilibrium is played, then both game B10 and game B8 are preferred by the reciprocal players. Even if the reciprocal players are only moderately reciprocal they prefer B10 and B8 to game A and use the exclusion institution. The non-reciprocal player strictly prefers game A over B10 and B8, since game A allows her to benefit from the public good while defecting in all rounds (see Appendix A.3 of ESM).

4.4 Differences between standard and social preferences

In summary, in the standard preferences model, the exclusion institution does not change the zero-contribution equilibrium, as the threat of exclusion is not sufficient to sustain cooperation. When social preferences are assumed, the composition of the group and the ability of the social players (with strong preferences for equality or reciprocity) to coordinate towards the Pareto-superior equilibrium are crucial for the game choice. Groups consisting solely of social individuals can sustain cooperation in both games. If at all, they choose the exclusion institution only if it is costless and they do not use it in equilibrium. If there is a selfish player in the group who does not care much about equality or reciprocity and if the social players coordinate successfully, they implement the exclusion institution, exclude the selfish player from the group, and cooperate thereafter. With some restrictions, this is also true when the exclusion institution is costly. While it is not our intention to test the two theories of social preferences with this experiment, as has been done for example by Blanco et al. (Reference Blanco, Engelmann and Normann2011), we use them to provide possible explanations for why players may vote for and use the exclusion institution.