Cooperation of Pairs

Abstract

To examine the stability of prosocial behavior in groups and pairs, we use an indirect approach. We conducted linear public good experiments with two and four subjects repeatedly three times at intervals of one week. All experiments were carried out without providing feedback and used a payment mechanism promoting stable behavior. We study the dynamics of behavior in repeated sessions and find that pairs are much better at establishing and stabilizing cooperation than groups of four. Furthermore, we conducted all experiments in a partner and a stranger design. As is known from the literature, cooperation in a stranger design should be lower than in a partner design. Once again, we are interested in the differences of the strength of this cooperation reducing effect between pairs and groups. Unlike pairs, groups show very low contributions to the public good in the stranger treatment and display a strong tendency to decrease cooperation in the partner treatment. The results in all treatments demonstrate that decreasing cooperation is a stable pattern of behavior in dynamic social dilemma contexts. Finally, we conducted a norm elicitation experiment using a method introduced by Krupka and Weber (2013) and find that in pairs symmetric behavior plays a very important role.

1. Introduction

In the last two decades, experimental and theoretical work on “social preferences” has produced a vast number of interesting new insights. One of the focal points is the question of how to overcome a fundamental cooperation problem paradigmatically described by the prisoner’s dilemma. Modern societies face this cooperation problem in many variations and in very important contexts. Environmental problems, the stability of democratic systems, or more generally, the pursuit of efficiency gains in situations characterized by the fundamental conflict between individually rational, selfish behavior, and collectively rational (efficient) behavior, are examples of social dilemmas. These types of cooperation problems always affect groups of people and the size of these groups ranges from two to seven billion (in cases where the whole of mankind is involved in a global public good problem). We can characterize cooperative behavior as a person’s willingness to sacrifice individual benefit for the sake of increasing the group’s prosperity. Therefore, the opposite of cooperative behavior is the readiness to withhold contributions while at the same time accepting the share of benefits created by the other group members’ sacrifices. Thus, a group’s success in solving a cooperation problem crucially depends on the willingness of its members to forego their own payoffs and to behave unselfishly.

The standard experimental setting in which cooperation is studied is the public good game, in which the public good is created by a particular payoff function displaying the so called voluntary contribution mechanism introduced by [1]. Let z_i denote the initial endowment of group member i, b_i the individual contribution to the provision of the public good and α the return every group member receives if one monetary unit is invested in the production of the public good. The marginal return on the share of z_i that is not invested in the public good is normalized to 1. Then α is identical to the marginal per capita return (MPCR) of investments in the public good. If N is the number of group members, group member i’s payoff π_i is

$${\pi }_i=\left(z_i-b_i\right)+\alpha {{\displaystyle \sum }}_{j=1}^{\mathrm{N}}{b}_j .$$

(1)

A cooperation problem arises if the following holds:

$$\alpha \langle 1;\mathrm{N}\alpha \rangle 1 \mathrm{and}\mathrm{thus}\alpha >\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{N}$}\right.$$

(1a)

An individual investing one monetary unit in the public good receives a return of α. Since α < 1, not investing is, from the individual’s point of view, more profitable because the return of a monetary unit he keeps is equal to 1. However, since α > 1/N, from the group perspective, the efficient solution is to invest the complete endowment in the public good.

Given the payoff Function (1) there is the obvious question whether or not and how the coordination performance of the subjects can depend on the group size N and the MPCR [2] investigate public good experiments with large groups of up to 100 subjects and show that even large groups with very small MPCR are capable of the same cooperative performance as the small groups working in the laboratory with high MPCR.

Large groups of 100 players are an extreme value for N. In this paper we look at the opposite, the smallest group size for which a public good problem can arise, N = 2. We believe that “pairs” are particularly interesting because two-person relationships with recurring cooperation problems play an important and peculiar role in the life of human beings. In most societies, for example, a couple starts a family and the partners remain in a two-person relationship throughout their lives. Two-person interactions are also important in market interactions, where most exchanges are made between two parties, albeit in this context people usually interact with many different partners. A peculiar kind of symmetry is characteristic for pairs: the “rest of the group” is “worth” as much as oneself.

We do not study the behavior of pairs and groups in a single experimental session, but focus on the dynamic of behavior in a series of four experimental sessions conducted with one week between the three repetitions of the starting experiment. We use the first experimental session in this sequence as a calibration device. That means that we compare the dynamics of pair and group behavior relative to the first session. We then conducted an additional treatment with conditions, which are known to have the tendency to reduce the willingness to cooperate. Our hypothesis is that the decrease of cooperation will be stronger in the groups than in the pairs because the cooperation norm is stronger in pairs.

We study the stability of cooperation in pairs and groups. It is worth noticing, that a direct comparison of pairs and groups is difficult. The reason is that a variation of group size unavoidably also alters other parameters that can potentially influence cooperation. For example, if the group size is varied (and no other parameter of Equation (1)), the effectiveness of contributions to the public good varies too, so that it is not possible to decide what causes a change in the cooperation level. On the other hand, if the MPCR is changed in order to compensate the group size effect on effectiveness, the individual costs of contribution to the public good are altered. Moreover, it is well known, that in small groups a higher MPCR leads to higher cooperation.

It is well known from the literature that in public good experiments, which are repeated within one session, contributions do fall1 [4] introduced a theory, which explains this decay of contributions by focusing on the behavior of conditional cooperators. To repeat session is something different from repeating a game within one session. Two differences are obvious. First, a game repeated in a session is perceived by the players as a supergame and not as independent, recurring events. In the real world, however, only very rarely supergames are played. Much more often, people have to make very similar decisions at different times, which is more like repeating sessions. Secondly, the average opportunity costs in a one session repeated game drop with every replay. In this respect, these are no identical repetitions. This is different for repeated sessions, where the opportunity costs are identical for each session. Due to these differences, it is not easy to conclude from the observations of repeated games in one session what happens if sessions are repeated. For this reason, the experimental design used in this paper has the advantage that it allows not only to observe the different dynamics in pairs and groups, but also to inform about how the repetition of sessions generally affects cooperation behavior. Reference [5] have shown that in modified dictator game experiments the repetition of session leads to a strong erosion of social behavior. The question therefore arises as to whether this can also be observed for cooperative behavior. Our conjecture is that this will be the case and that the decay of cooperation is stronger in groups than in pairs.

Additionally, we conducted each series of experiments in a partner and a stranger treatment. In the partner treatment, group/pair composition did not change over the course of the four waves, whereas in the stranger treatment our main participants were matched with three/one freshly recruited new subject(s) in each wave. It is conceivable that the group bonding and the resulting behavioral norm to contribute to the public good is slightly less strong in the stranger treatments. Thus, employing a partner vs. stranger design enabled us to vary the cooperation norm in pairs and groups. Once again, the comparison with the benchmark treatment (partner, first of four waves) shows us where the easing of the cooperation norm has a stronger effect. Our conjecture is that subjects in pairs feel a stronger obligation to cooperate than subjects in groups even if they are matched with a newly recruited subject in each wave.

Our conjecture is that two-person relationships should be exceptionally powerful in the context of cooperation problems. This should result in higher and more stable cooperation in pairs. Evidence for this conjecture can be taken from oligopoly experiments, which demonstrate that subjects in a duopoly situation are more likely to collude than subjects in markets of more than two firms ([6,7,8,9]). Reference [10] observe a higher degree of cooperation in pairs than in groups of three in the context of n-person prisoner’s dilemma experiments. Reference [11] conducted public good experiments with varying group sizes and reported that the average contribution to the public good is highest for pairs, followed by groups of four, three and eight. Reference [12] show theoretically that the contribution to a public good decreases with the group size.

Our experiments confirm our conjectures. Pairs show more stable cooperation than groups. While the average contribution to the public good is reduced by 42% in groups over the course of four waves, the reduction in pairs is only 21%. To gain a deeper understanding of the underlying norms determining cooperative behavior in pairs and groups, we conducted a norm elicitation experiment using the method introduced by [13]. We elicit both, social approval for a series of contributions over the course of four weeks and social approval for some conditional contributions. Group size serves as a treatment variable. We find that social approval of cooperative moves in general do not differ a lot between pairs and groups but that symmetric behavior plays a very important role in pairs. This finding is in line with the suggestion that the special form of symmetry, which is characteristic for pairs, drives the particular power of the cooperation norm.

Section 2 provides a detailed explanation of our experimental design and exemplifies each aspect with respect to our research question. We report the results of our experiments in Section 3. Section 4 presents the norm elicitation task and in Section 5 we discuss our results.

2. Experimental Design

We employed a setup in this study, which uses the voluntary contribution mechanism represented by Equation (1) (see, e.g., [1] for the stranger treatment and [14] for the partner treatment). Subjects are either interacting in pairs (N = 2) or groups (N = 4) with each subject receiving a monetary endowment of EUR 10. Subjects were assigned randomly to the other subject(s). Subjects then decide on the fraction of the endowment they wish to contribute to a public account. The amount of money that is not contributed to the public account is paid out to the subjects at a rate of 1:1. For each EUR 1 contributed to the public account, each of the 2 (4) group members receives an MPCR of EUR 0.80 (EUR 0.40). The cooperation dilemma arises because the MPCR is smaller than the private return of not contributing (0.8 < 1 and 0.4 < 1), while at the same time N × MPCR is larger than the private return (2 × 0.8 > 1 and 4 × 0.4 > 1). Rational payoff-maximizing subjects will therefore not contribute to the public account, while total return to the group is maximized by contributing the entire endowment. Contributing is thusly interpreted as cooperative, prosocial behavior.

The individual payoff function for subject $i$ in EUR is given by:

$$\mathrm{N}=2: {\pi }_i=\left(10-x_i\right)+0.8{{\displaystyle \sum }}_{j=1}^2{x}_j$$

(2)

$$\mathrm{N}=4:{\pi }_i=\left(10-x_i\right)+0.4{{\displaystyle \sum }}_{j=1}^4{x}_j$$

(3)

where $x_i$ denotes the amount of money contributed to the public account. Equations (2) and (3) do not only differ with respect to group size, but also with respect to the MPCR. The MPCRs are chosen in a way that the extent of the efficiency gain from contributing to the public account is identical in both treatments (2 × 0.8 = 4 × 0.4). However, the private costs of contributing (1 − MPCR) are higher for N = 4 than for N = 2. On the other hand, leveling the private costs of contribution between treatments would lead to a significant difference with respect to the efficiency gain of contribution. Because of the interaction of group size, the MPCR and the effectiveness of contribution and the costs of contribution, it is not possible to create two or more treatments in a public good experiment that solely differ with respect to group size. Therefore, we concentrate on the stability of cooperation within a particular group size. For small groups lowering the MPCR leads to a sharp decrease in contribution [15]). To test our conjecture even though the MPCR’s differ, we focus on the dynamics of behavior and not on the absolute level of contributions. To examine the behavioral dynamics over time, our experiments were repeated three times at intervals of one week between each wave. Our conjecture is that the stronger social norms developed in pairs lead to a stabilization of cooperative behavior. Thus, if contribution declines the decay should appear stronger in groups than in pairs.

The key feature of our design is a series of four identical one-shot experiments. Subjects live through the entire experience of taking part in an experiment in each wave, which includes coming to the laboratory, reading instructions and making choices within the very same decision context each time. The benefit of such a setup is that subjects are more likely to perceive all the experiments as completely independent events as compared to a setup where the same decision is made repeatedly within a single session. The downside is a loss of control, because we cannot know whether or not behavior is influenced by events happening outside the laboratory between waves. Under the premise that potentially relevant effects happen randomly, we therefore concentrated on treatment effects when interpreting our results.

All experiments were conducted at the MaXLab (Otto-von-Guericke-University of Magdeburg). We used ORSEE [16] and hroot [17] for the recruitment of subjects. In the invitation, subjects were asked to commit to the total duration of the series of experiments if they wished to participate and were told that failure to show up for any wave would result in all earnings from the experiment being forfeited. In the stranger treatment, each subject was matched with a stranger who took part in the experiment in only one wave. Subjects were not told that they were to face the same decision situation in each wave.

In order to rule out systematic effects on behavior due to subjects discussing the experiments with each other, every subject was assigned an individual meeting point inside the faculty building, picked up by an experimenter and escorted to the laboratory. Subjects were then led into individual soundproof and opaque booths. After the end of each wave, all subjects left the laboratory on their own. This procedure ruled out the possibility that two subjects learned about each other’s participation in the same experiment.

To avoid confusion on the subjects’ behalf with regard to the actual public good game, every participant had to complete a set of control questions before the start of each experiment (see Appendix B1 and Appendix B2). The experiments started only after each subject had answered all control questions correctly.

Four treatments were conducted. In the partner treatments which were conducted for both N = 2 and N = 4, the composition of pairs and groups never changed over the course of the experiment. This was made known to all subjects through written instructions (see Appendix A1, Appendix A2, Appendix A3 and Appendix A4). However, subjects never learned the identity of the other participants, which ensured total anonymity. In the stranger treatments, which were also conducted for both N = 2 and N = 4, the main participants who took part in all four waves were matched with freshly recruited new subjects in each wave. These subjects could only take part once and this was also made known to the main participants through written instructions. In contrast to other well-known experiments from the literature, we did not employ a partner vs. stranger design based on random re-matching after each round of play, but instead recruited completely new subjects for each new wave. Total anonymity was ensured for everybody involved in the stranger treatments as well.

Table 1. Treatment overview.

		Group Composition
		Partner Treatments	Stranger Treatments
Group size	N = 2	Independent observations: 22 ⌀ Earnings per wave: EUR 12.71	Independent observations: 24 ⌀ Earnings per wave: EUR 13.13
	N = 4	Independent observations: 25 ⌀ Earnings per wave: EUR 11.75	Independent observations: 21 ⌀ Earnings per wave: EUR 13.09

lists all treatments including the number of participants2 in all treatments and average earnings per wave.

In each wave, a full set of conditional preferences is elicited from the subjects by using Selten’s strategy elicitation method [18]. In the partner treatments, we adopt the elicitation mechanism introduced for public good experiments by [14]. The mechanism consists of two tasks. In the direct response task, subjects indicate their unconditional decision with respect to the amount of money contributed to the public account. In the second task, subjects indicate their preferred choices conditional upon the other subject’s contribution (N = 2) or the other group members’ average contribution (N = 4) accordingly. This second task requires eleven choices to be made, one for each possible level of (average) contribution chosen by the other subject (other group members) in the direct response task. In each treatment (N = 2 and N = 4), one subject is randomly determined for whom the actual contribution level is taken from the conditional response task, while for the other subject (all other group members), the choice made in the direct response task is relevant to the payoff.

In the stranger treatments, there is no need for the direct response task. Therefore, the main participants only submit their set of conditional preferences with respect to the direct responses made by the freshly recruited subjects they are matched with in that particular wave. In any treatment, data collection is conducted with pencil and paper (see Appendix A1, Appendix A2, Appendix A3 and Appendix A4).

In none of the treatments do the subjects learn the outcome of the experiment immediately after each wave. All information is only revealed after the end of the final wave. This procedure has many important advantages in regard to our research strategy. First of all, it rules out effects due to reputation building. Subjects also cannot learn the relevant moral norm through observation of other subjects’ choices. Furthermore, subjects cannot update their beliefs based on the other subjects’ behavior. This is a necessary pre-condition for exercising imperfect conditional preferences, which have shown to be a relevant factor in the context of public good experiments by [4]. In combination with employing the strategy elicitation method, withholding feedback until after the end of the last experiment also rules out several other effects that are discussed in the literature with respect to cooperation levels and group size. Punishment, for example, is not possible and the “bad apple” hypothesis (see, e.g., [19]) is rendered irrelevant, since bad apples do not become salient and preferences are stated conditionally anyway.

Withholding feedback requires withholding payment after each wave as well, since the amount of money paid out would otherwise serve as indirect feedback. For this reason, subjects are paid only after the end of the final wave. In doing so, we employed the following payment mechanism: subjects do not receive the sum of earnings from all waves but instead receive the earnings from a randomly determined wave, which is multiplied by four. This mechanism rules out portfolio choices and increases the validity of our findings because the mechanism provides an incentive for stable behavior to risk-averse subjects3. A side benefit of this payoff mechanism is the credible threat of all earnings being forfeited in the case of failure to show up for one of the experiments. This results in a very low number of no-shows, which in turn rules out possible selection biases.

To summarize, by employing the strategy method, withholding feedback and using a payment mechanism promoting stable behavior, we created an experimental design that strongly favored stability of cooperation over the four waves. Nevertheless, we expect contributions to decrease and our hypothesis is that the moral norm for prosocial behavior is stronger in pairs than in groups of four. We therefore expect more stable behavior in pairs than in groups.

3. Results

Figure 1, Figure 2, Figure 3 and Figure 4 show the extent of conditional cooperation over the course of the four waves in all four treatments of our experiment. For our analysis we focus on the average of contributions that where made conditional upon the other subject’s contribution. Figure 1, Figure 2, Figure 3 and Figure 4 summarizes the average conditional cooperation in all 16 waves of our experiment. Figure 5 shows the average contributions in all treatments.

The results are very conclusive. Focusing on the behavioral dynamics, we observed a monotonous but statistically insignificant and economically negligible decline in prosocial behavior over time in the N = 2 partner treatment (see Row (f) in

Table 2. Statistical significance in the N = 2 partner treatment (Wilcoxon signed-rank test).

	Treatment	⌀ = 0	⌀ = 1	⌀ = 2	⌀ = 3	⌀ = 4	⌀ = 5	⌀ = 6	⌀ = 7	⌀ = 8	⌀ = 9	⌀ = 10
(a)	W1 vs. W2	0.084 ↓	0.894	0.985	0.741	0.753	0.985	0.769	0.741	0.752	0.504	0.900
(b)	W2 vs. W3	1.000	0.099 ↓	0.259	0.171	0.095 ↓	0.052 ↓	0.015 ↓	0.046 ↓	0.026 ↓	0.123	0.900
(c)	W3 vs. W4	0.317	0.306	0.961	0.961	0.755	0.478	0.851	0.859	0.828	0.927	0.602
(d)	W1 vs. W3	0.084 ↓	0.091 ↓	0.805	0.898	0.714	0.476	0.426	0.663	0.448	0.917	0.721
(e)	W2 vs. W4	0.317	0.083 ↓	0.284	0.167	0.348	0.251	0.048 ↓	0.099 ↓	0.108	0.227	0.330
(f)	W1 vs. W4	0.528	0.579	0.791	0.844	0.475	0.638	0.402	0.454	0.329	0.541	0.491

^↓ Statistically significant decline of prosocial behavior.

for statistical significance with ⌀ = x indicating an average contribution of the other subjects of x) as the average contribution dropped from EUR 3.60 in the first wave to EUR 3.00 in the fourth wave. In stark contrast, cooperative behavior in the N = 4 partner treatment declined substantially by 39 percent from the first wave to the last wave (from EUR 3.62 to EUR 2.21). The decline over time is statistically significant (Row (f) in

Table 3. Statistical significance in the N = 4 partner treatment (Wilcoxon signed-rank test).

	Treatment	⌀ = 0	⌀ = 1	⌀ = 2	⌀ = 3	⌀ = 4	⌀ = 5	⌀ = 6	⌀ = 7	⌀ = 8	⌀ = 9	⌀ = 10
(a)	W1 vs. W2	0.157	0.026 ↓	0.089 ↓	0.028 ↓	0.052 ↓	0.001 ↓	0.010 ↓	0.015↓	0.007 ↓	0.036 ↓	0.830
(b)	W2 vs. W3	1.000	0.564	0.207	0.432	0.233	0.141	0.204	0.070↓	0.596	0.368	0.078 ↓
(c)	W3 vs. W4	0.977	0.548	0.328	0.641	0.632	0.957	0.844	0.712	0.655	0.986	0.564
(d)	W1 vs. W3	0.157	0.008 ↓	0.016 ↓	0.009 ↓	0.003 ↓	0.000 ↓	0.004 ↓	0.001 ↓	0.003 ↓	0.005 ↓	0.144
(e)	W2 vs. W4	0.977	0.966	0.655	0.681	0.426	0.418	0.169	0.066 ↓	0.238	0.257	0.219
(f)	W1 vs. W4	0.580	0.069 ↓	0.075 ↓	0.015 ↓	0.017 ↓	0.003 ↓	0.002 ↓	0.002 ↓	0.003 ↓	0.016 ↓	0.139

^↓ Statistically significant decline of prosocial behavior.

) and is most pronounced from the first to the second wave (EUR 3.62 to EUR 2.71, Row (a) in Table 3). The later decline in the third and fourth waves is also insignificant (Rows (b,c) in Table 3). The results show that pairs achieve much more stable cooperation than groups in the partner treatment condition, indicating a stronger moral norm demanding stable cooperation in pairs than in groups.

This conjecture is backed up by the behavioral dynamics in the stranger treatments. In both the pairs and the groups, contribution to the public account declined strongly and significantly from the first wave to the second wave (Row (a) in

Table 4. Statistical significance in the N = 2 stranger treatment (Wilcoxon signed-rank test).

	Treatment	⌀ = 0	⌀ = 1	⌀ = 2	⌀ = 3	⌀ = 4	⌀ = 5	⌀ = 6	⌀ = 7	⌀ = 8	⌀ = 9	⌀ = 10
(a)	W1 vs. W2	0.083 ↓	0.044 ↓	0.056 ↓	0.055 ↓	0.024 ↓	0.013 ↓	0.059 ↓	0.040 ↓	0.111	0.083 ↓	0.401
(b)	W2 vs. W3	1.000	0.564	0.965	0.655	0.150	0.275	0.060 ↑	0.072 ↑	0.091 ↑	0.145	0.307
(c)	W3 vs. W4	1.000	0.564	1.000	0.564	0.096 ↓	0.099 ↓	0.032 ↓	0.031 ↓	0.096 ↓	0.026 ↓	0.046 ↓
(d)	W1 vs. W3	0.083 ↓	0.084 ↓	0.065 ↓	0.039 ↓	0.091 ↓	0.114	0.575	0.432	0.656	0.471	0.728
(e)	W2 vs. W4	1.000	1.000	0.581	0.581	0.581	0.680	0.563	0.779	0.290	0.410	0.598
(f)	W1 vs. W4	0.083 ↓	0.027 ↓	0.037 ↓	0.034 ↓	0.014 ↓	0.057 ↓	0.209	0.160	0.459	0.311	0.643

^↓ Statistically significant decline of prosocial behavior, ^↑ Statistically significant increase of prosocial behavior.

and Row (a) in

Table 5. Statistical significance in the N = 4 stranger treatment (Wilcoxon signed-rank test).

	Treatment	⌀ = 0	⌀ = 1	⌀ = 2	⌀ = 3	⌀ = 4	⌀ = 5	⌀ = 6	⌀ = 7	⌀ = 8	⌀ = 9	⌀ = 10
(a)	W1 vs. W2	0.604	0.026 ↓	0.026 ↓	0.959	0.009 ↓	0.123	0.155	0.133	0.070 ↓	0.181	0.491
(b)	W2 vs. W3	0.157	1.000	0.317	0.046 ↓	0.046 ↑	0.895	0.589	0.416	0.547	0.834	0.631
(c)	W3 vs. W4	1.000	1.000	0.564	0.545	0.622	0.672	0.786	0.277	0.323	0.414	1.000
(d)	W1 vs. W3	0.3173	0.026 ↓	0.026 ↓	0.046↓	0.368	0.450	0.303	0.103	0.087 ↓	0.408	0.200
(e)	W2 vs. W4	0.157	1.000	0.545	0.106	0.166	0.251	0.296	0.747	0.240	0.468	0.446
(f)	W1 vs. W4	0.317	0.026 ↓	0.026 ↓	0.213	0.600	0.508	0.453	0.587	0.719	0.139	0.050 ↓

^↓ Statistically significant decline of prosocial behavior, ^↑ Statistically significant increase of prosocial behavior.

). However, it should also be noted that the decline was more pronounced in the N = 4 stranger treatment (48%) than in the N = 2 condition (30%). Neither the third nor fourth wave differed significantly from the second wave in the N = 2 or the N = 4 stranger treatments (Rows (b,e) in Table 4 and Rows (b,e) Table 5 respectively). This indicates that a lower willingness to cooperate also becomes a factor in pairs when there is a new partner in the next wave to cooperate with. Strictly speaking, however, pairs maintained a significantly higher level of cooperation than groups in which cooperation was extremely weak through waves two to four.

As mentioned before, comparisons across group size treatments and between stranger and partner treatments should be interpreted with caution. Focusing on our benchmark, which was the first wave of the partner experiments, we found that the extent of prosocial behavior was almost identical in both partner treatments (EUR 3.60 contributed to the public account is the average level for N = 2, EUR 3.62 is the average for N = 4, see Row (a) in

Table 6. Statistical significance for pairwise comparisons of first wave behavior (Mann-Whitney U tests).

	Treatment	⌀ = 0	⌀ = 1	⌀ = 2	⌀ = 3	⌀ = 4	⌀ = 5	⌀ = 6	⌀ = 7	⌀ = 8	⌀ = 9	⌀ = 10
(a)	N = 2 partner vs. N = 4 partner	0.869	0.789	0.406	0.149	0.389	0.333	0.679	0.862	0.957	0.905	0.212
(b)	N = 2 stranger vs. N = 4 stranger	0.196	0.313	0.298	0.009	0.046	0.016	0.040	0.027	0.108	0.118	0.031
(c)	N = 2 partner vs. N = 2 stranger	0.686	0.560	0.536	0.332	0.536	0.775	0.599	0.780	0.425	0.328	0.455
(d)	N = 4 partner vs. N = 4 stranger	0.404	0.430	0.162	0.001	0.010	0.001	0.001	0.008	0.007	0.021	0.109

). While cooperation levels in the first wave did not differ between the two N = 2 treatments (Row (c) in Table 6), we found that cooperation was significantly weaker in the stranger condition of the two N = 4 treatments (Row (d) in Table 6). When subjects in groups knew that they would be matched with freshly recruited new subjects in the next waves, the extent of prosocial behavior in the first wave was much smaller than in the partner treatment (EUR 3.62 in the partner treatment and EUR 2.04 in the stranger treatment).

It is worthwhile noting that the low extent of cooperative behavior in the first wave of the N = 4 stranger treatment cannot be explained by the lack of opportunity for reputation building because attempts to trigger high cooperation through one’s own substantial sacrifices are also ruled out in the N = 4 partner treatment by withholding feedback. It can be concluded that a lesser extent of bonding lowers the willingness to cooperate in the N = 4 stranger treatment. Contrarily, no such partner vs. stranger effect can be found in the N = 2 treatment, which indicates a strong bond in pairs regardless of group composition.

In the stranger treatment, we asked subjects state their unconditional contribution. This was necessary to find determine the payoff-relevant conditional contribution. Figure 6 shows the average unconditional contributions for pairs and groups in the partner treatment. Three results are worth noticing. First, unconditional contributions are between 4.28 and 7.86 and therefore arguable high. This is significantly more than the average conditional contribution. This finding is in line with [14], who also find a higher unconditional contribution compared to the conditional contribution. Second, unconditional contributions are significantly higher in pairs compared to groups. Third, while unconditional contributions decrease in both treatments over time, they are more stable in pairs. The total decrease from Wave 1 to Wave 4 is 18.82% in pairs and 29.14% in groups.

Thus far, we can summarize our results:

• Despite having designed the experiment in a way that promotes stable behavior, we generally find declining prosocial behavior in all treatments.
• There is a strong norm demanding subjects to engage in cooperative behavior in the first wave. Under partner conditions, this norm is similar in groups and pairs. Under stranger conditions, this norm is stronger in pairs.
• The norm mentioned in 2 apparently allows pairs a much more stable cooperation over time. This result is backed by both the partner and the stranger treatment in terms of conditional cooperation and for unconditional cooperation in the stranger treatment.

We shall note another observation backing up our findings up to this point. Examining the behavioral dynamics in Figure 1, Figure 2, Figure 3 and Figure 4, we find that in the N = 4 treatments, contributions in six cases (waves two to four in both the stranger and the partner treatment) do not increase any further when the average contribution from the other group members is ≥EUR 6. Conditional responses to an average cooperation level ≥EUR 6 are constant at approximately EUR 3 in the partner treatment and approximately EUR 2 in the stranger condition. Contrarily, contribution in the pair treatments is also positively correlated with a high average contribution from the partner in each of the eight waves in the N = 2 treatments (

Table 7. Correlation of average conditional contribution and contribution from other group member(s) when contribution from other group member(s) ≥6.

Treatment	Wave 1	Wave 2	Wave 3	Wave 4
N = 2 Stranger	0.909 **	0.912 **	0.961 ***	0.966 ***
N = 2 Partner	0.959 ***	0.985 ***	0.930 **	0.898 **
N = 4 Stranger	0.938 **	−0.467	−0.237	−0.116
N = 4 Partner	0.815 *	0.776	0.286	0.676

*** p < 0.01; ** p < 0.05; * p < 0.1.

).

This allows the conclusion that members of a group of four are much more willing to engage in uncooperative behavior than partners in pairs do. Choosing low levels of contribution in cases of high contribution by others underlines a strong readiness to benefit from the other group members’ sacrifices without contributing to the cause itself. This particular behavior is prevalent in those waves where we suspected a weak cooperation norm to be particularly relevant. We will come back to this point after presenting the norm elicitation experiment.

4. Norm Elicitation

The experimental results show very clearly that pairs are capable of more stable cooperation than groups. At the same time, however, they also show that the identical repetition of the experiments leads to a significant decline in cooperation—especially in groups. The question of the causes arises for both phenomena. Reference [5] observe that the gifts in modified dictator game experiments also decrease when the experiment is repeated. They offer two explanations which are not mutually exclusive. First, an experimenter demand effect, which weakens when the experiment is repeated, and second, moral self-licensing. This means that people take the fact that they have behaved socially as an opportunity to think more about themselves at the next opportunity.

Reference [20] show that moral self-licensing can be rationalized if one assumes that there is a substitutional relationship between social recognition and monetary payout and if the repetition of social actions leads to an increase in social recognition for these actions. The norm elicitation experiment, which will be described now, primarily has the task of checking whether the second condition for rational moral self-licensing is fulfilled. It also serves to investigate whether the behavior in groups and in pairs is fundamentally assessed differently or whether it is assessed in the same moral way. One possible difference could be that one does depend more on another in pairs than in groups. Therefore, the act to contribute the same amount as the other player is more important in pairs. This could possible reflect in a norm where the same social approval is attached to “matching the other persons contribution” in pairs and to give slightly less in groups.

4.1. Description of the Experiment

We use an incentivized coordination game to elicit behavioral norms. Reference [13] introduced this method. Subjects state their belief with respect to what the majority of the other participants felt about the social appropriateness of certain behaviors. The true norm serves as a focal point for this coordination game. Subjects whose stated belief matches that of the majority are financially rewarded.

Since we are interested in the behavioral norm for contributing to a public good conditional on the group size in the one-shot context and the repeated context, we invited two groups (A and B) of 50 subjects to take part in the elicitation experiments (between-subject design). All participants were recruited using hroot [17]. The subjects were separately seated at the same time in a large lecture hall with more than 500 seats so that all data could be collected in a single session.

Each treatment consisted of three parts. Written instructions (see Appendix C1, Appendix C2, Appendix C3 and Appendix C4) were handed as soon as all subjects finished a prior part. Subjects received the identical description of the public good game as we described above. The treatment difference was the group size (N = 2 or N = 4) of the public good game. The instructions were numbered with an ID, which served as a means to run the norm elicitation process anonymously. The subjects picked up the instructions by themselves in such a way that the ID was not observable for the experimenter.

In the first task, subjects in treatment N = 2 were instructed to evaluate the social appropriateness of four different contributions (B = 2; B = 4; B = 6; B = 8) conditional on the contributions of another player (A = 2; A = 4; A = 6; A = 8). In treatment N = 4 the contribution was conditional on the average contributions of three other players. Subjects were able to choose from four evaluations: ‘very desirable’, ‘somewhat desirable’, ‘somewhat undesirable’ and ‘very undesirable’.

After all the subjects had stated their belief about the assessments of the majority on a sheet of paper, one out of the twelve evaluations was randomly drawn to be payoff relevant and the results for this particular evaluation were calculated on the spot. Of those subjects who marked the assessment, the majority had chosen to receive a payoff of EUR 10, whereas all other subjects received a show up fee of EUR 5 only.

In the second part of the elicitation experiment, the subjects were told that the same public good (N = 2 or N = 4) experiment was played four times with a timespan of one week between each repetition. All subjects were confronted with four sequences (6-6-4-0; 4-4-4-4; 8-8-2-0 and 8-6-2-0) of contributions of one particular player. Subjects were asked to evaluate the social appropriateness for each contribution within a sequence. Part two was incentivized as in part A, but with an additional payoff of 2 euros.

The third part was identical for both treatments. Subjects were asked if the contributions (same as in part A) deserved “more”, “less” or “equally” social recognition if the group size was two or four.

4.2. Results

Following Krupka and Weber, all evaluations are assigned with a value: ‘very desirable’ = 1, ‘somewhat desirable’ = 0.33, ‘somewhat undesirable’ = −0.33 and ‘very undesirable’ = −1.

Figure 7 and Figure 8 show the average social recognition of four consecutive decisions (across waves). They hardly distinguish between groups and pairs. A contribution of 0 is rated very negatively in pairs and groups. Even if high contributions were previously made, this does not change the social disregard for low cooperation. It is striking that a constant contribution of 4 leads to a slight decrease in recognition. If 6 is contributed in the first two waves, a contribution of 4 in the third wave is rated significantly worse than with constant output. Otherwise, the amount of social recognition depends solely on the amount of the contribution.

Figure 7 and Figure 8 clearly show that repeated cooperation does not lead to an increase in social recognition. This speaks against the interpretation that the decline in cooperation is due to a moral self-licensing effect. Conversely, this suggests that the second hypothesis of [5]), that a weakening experimenter demand effect is responsible for the decline, is becoming more likely.

Figure 9, Figure 10, Figure 11 and Figure 12 show the average social assessments of contributions in groups and pairs conditional on the contributions of the other player or players. Contribution A denotes the given contribution of the “others”. A very clear pattern can be seen in all four illustrations. The highest social recognition is achieved in pairs if the contributions are symmetrical. This result is significant (MW-Test: p = 0.0544 for A = 2; p = 0.0999 for A = 4; p = 0.000 for A = 6 and A = 8). Symmetrical behavior is obviously of central importance in pairs. This explains why the cooperation does not collapse in pairs even with high contributions of the other player (as in the groups). Even if the other player contributes beyond 6, the partner is still involved because this is the only way to maintain symmetry in the pair. For the groups, the contributions of the others seem to be less relevant. For example, social recognition at A = 4 and A = 6 is almost identically distributed. If the other group members cooperate very little (A = 2), contributions are socially recognized. However, recognition decreases significantly (p = 0.004) if one’s own contribution goes far beyond the group average. If the rest of the group is very cooperative (A = 8), however, a contribution of at least 6 is required in order to receive social recognition. When asked directly (experimental part C) to compare the same contribution in a 2-player group and in a 4 player group, we find no significant difference.

5. Discussion

Two main conclusions can be taken from this study. First, the extent of prosocial behavior found in non-repeated experiments should be seen as the upper bound of cooperative behavior. The identical repetition of experimental sessions leads also in a public good experiment to a considerable decrease in social behavior. This is remarkable in that this observation has so far only been made for the dictator game experiment, in which there is no strategic interaction between the subjects. Obviously, repeating sessions also reduces social behavior in games with strategic interaction. Therefore, in real world situations where people are repeatedly put in similar cooperation dilemma contexts, we should expect to observe less cooperation as in non-repeated experiments.

Second, the mechanics of cooperation in pairs differ significantly from those in groups. The peculiar characteristics of two-person interactions highlighted in the introduction apparently promote stable cooperative behavior. Our results are in line with findings recently reported by [11] who also found that cooperation in their public good experiments was strongest for N = 2, followed by N = 4, and N = 3.

In our analysis we focus on the conditional contributions, because unconditional contributions are harder to interpret because they depend on unobservable elements as beliefs and expectations. Nevertheless, our results suggest that these conclusions can be drawn for both, unconditional and conditional decisions. Both show a high pro social behavior in the first decision. However, the unconditional contribution is more than twice as high as the average conditional contribution. One reason for this could be, as [21] lined out, that leaders who make an unconditional contribution, expect conditional cooperators as followers. The same observation was made by [22]. Therefore, they can maximize their payoff by contributing a relatively high amount. Our results show that conditional cooperation is especially stable in pairs when compared to groups. If the leader in a sequential framework expects this relative stability of pairs, then her unconditional contribution will be relatively stable as well. This is out of self-interest because of the before mentioned reason. In the consequence, we find a more stable unconditional contribution pattern in pairs compared to groups.

From our norm elicitation experiment, we can learn that symmetric behavior is of particular importance in pairs. This finding corresponds to the fact that pairs are insofar a symmetric “group” as the weight of each member is the same as the weight of the rest of the group. The second important observation made in the norm elicitation experiment is that the repetition of a cooperative contribution does not lead to increased social recognition for it. However, this would be a condition for moral self-licensing being a rational reaction in repeated experiments and therefore decreasing social behavior.

We can only speculate on the underlying foundations of pairs being able to achieve and maintain such high levels of cooperation. One might reason, for example, that people who constantly experience and therefore learn the mutual benefits of anonymous two-person trade interactions typical for market-oriented economies internalize a behavioral norm demanding cooperation in other contexts as well. Irrespective of whether or not such an explanation is indeed true, our findings illustrate that partners in pairs achieve cooperation quite successfully, which in turn supports the hypothesis that institutional arrangements based on two-person interactions allow for particularly stable cooperation. We can therefore expect that market interactions can be functional even in the absence of regulative interventions designed to enforce cooperation.