Games and Economic Behavior 66 (2009) 202–220 www.elsevier.com/locate/geb

Cooperation in viscous populations—Experimental evidence Veronika Grimm a,b , Friederike Mengel c,d,∗ a University of Erlangen-Nuremberg, Lehrstuhl für Volkswirtschaftslehre, insb. Wirtschaftstheorie,

Lange Gasse 20, D-90403 Nürnberg, Germany b University of Cologne, Köln, Germany c University of Alicante, Departamento de Fundamentos del Análisis Económico, Campus San Vicente del Raspeig, 03071 Alicante, Spain d University of Maastricht, Netherlands Received 29 June 2007 Available online 15 June 2008

Abstract We experimentally investigate the effect of population viscosity (an increased probability to interact with others of one’s type or group) on cooperation in a standard prisoner’s dilemma environment. Subjects can repeatedly choose between two groups that differ in the defector gain in the associated prisoner’s dilemma. Choosing into the group with the smaller defector-gain can signal one’s willingness to cooperate. We find that viscosity produces an endogenous sorting of cooperators and defectors and persistently high rates of cooperation. Higher viscosity leads to a sharp increase in overall cooperation rates and in addition positively affects the subjects’ preferences for cooperation. © 2008 Elsevier Inc. All rights reserved. JEL classification: C70; C73; C90 Keywords: Experiments; Cooperation; Group selection; Norms; Population viscosity

1. Introduction Population viscosity refers to a tendency of agents in a population to interact with increased probability with other individuals of their own type. The concept is often used in evolutionary biology to explain the emergence of altruistic behavior in both animals and humans.1 The intuition is simply that if altruists interact with (sufficiently) increased probability among themselves they benefit more often from the cooperative behavior of others and thus enjoy higher fitness than selfish types. While there is a lot of theoretical work on viscosity, to our knowledge viscosity has not been tested experimentally.2 * Corresponding author.

E-mail addresses: [email protected] (V. Grimm), [email protected] (F. Mengel). 1 See Mitteldorf and Wilson (2000), Boyd and Richerson (2005), Richerson et al. (2003), Myerson et al. (1991), Mengel (2007) or Wilson and

Sober (1994), among others. Early references are Hamilton (1964) or Price (1970). 2 An exception is Grimm and Mengel (2007). There are also several experimental works on group selection (e.g. Page et al., 2005 or Bohnet and Kübler, 2005) but none of them investigate population viscosity. 0899-8256/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.geb.2008.05.005

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In this experiment we investigate the effect of population viscosity on cooperation and the strength of preferences for cooperation in a standard prisoner’s dilemma environment. More precisely we address the following questions. • Can population viscosity induce an endogenous sorting of cooperators and defectors? • Can persistently high rates of cooperation be sustained in (sufficiently) viscous populations? • Does viscosity have an effect on preferences for cooperation? To address these questions we chose an experimental design that we consider both simple and natural. Subjects in our experiment play 100 rounds of a prisoner’s dilemma game. They can repeatedly choose between two groups. In one of the groups (group A) the defector payoff is lower, making cooperation less costly relative to defection. Agents might choose that group to signal their willingness to cooperate. The degree of population viscosity, i.e. the extent to which the probability of interacting with others of one’s own group is increased, is varied across treatments. Our experimental evidence allows us to answer all three questions raised above affirmatively. In particular, we find that participants choose group A if and only if the degree of viscosity is high enough. Also, while most of the agents in the “cooperative” group cooperate whenever the degree of viscosity is high, agents in the other group almost never cooperate. The proportion of agents who choose into the “cooperative” group and the proportion of agents who cooperate both rise sharply and monotonically with the degree of population viscosity. With high viscosity 35–60% of all subjects choose group A and cooperate most of the time. However it is not clear a priori whether the sorting we observe is indeed a sorting of types with more “cooperative” and less “cooperative” preferences. Moreover, if cooperators were motivated by purely altruistic motives, viscosity would not be necessary in our experiment to sustain cooperation. Our results (in particular the breakdown of cooperation under low viscosity) indicate that subjects, rather than being altruists, are conditional cooperators, i.e. are willing to cooperate only if they expect many others to do so. To further investigate these conjectures we use a random utility model and estimate the subjects’ preferences for cooperation as a function of the expected probability of cooperation of others. We find that most subjects are indeed conditional cooperators. Furthermore we find that those subjects that predominantly choose group A have much stronger preferences for cooperation than those that predominantly choose group B.3 This leads us to conclude that an endogenous sorting according to the strength of preferences for cooperation does take place. We contrast cooperation rates in group A with a control treatment where all subjects play the game with the low defector payoff and find that endogenous sorting produces clearly higher cooperation rates than a situation where all subjects play the group A-game. We explain this result with self-exclusion of non-cooperative types that increases the willingness of conditional cooperators to cooperate in group A. Finally, we are able to identify feedback effects from the matching structure to the agents’ preferences which are due to changes in their beliefs about others. These results can have important implications for the design of institutions. If a designer has the possibility to design an institution that makes defection less profitable in bilateral interactions, our results show that she might be well advised to make participation in this mechanism voluntary. As an example, think of workers in a firm that face bilateral helping effort games. If a manager can implement a punishment mechanism for those who do not provide helping effort, she should make participation in this mechanism voluntary if the degree to which participants and non-participants interact is sufficiently low. The results can also help to understand interactions in social dilemma situations, where viscosity between groups plays an important role. As an example, think of the interaction between friendship groups, where in some of them defection is less worthwhile, e.g. because of social disapproval. Let us finally relate our study to the experimental literature. Only very recently experimental economics has started to focus on the relation between interaction structures and cooperation. Coricelli et al. (2004), Engelmann and Grimm (2006), or Page et al. (2005) are examples of studies in which agents can endogenously choose interaction partners.4 In Bohnet and Kübler (2005) subjects can choose between groups that are perfectly separated. In this study, however, cooperation rates remain low. Two studies investigate group selection in the presence of punishment institutions. Guererk et al. (2006) and Grimm and Mengel (2007) show that subjects learn to choose into a group where a punishment 3 The pattern of cooperator types we find is roughly consistent with results by Fischbacher et al. (2001), Fischbacher and Gächter (2006) or Brandts and Schram (2001). 4 See also Ones and Puttermann (2007).

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mechanism is in place.5 All these studies (except for Grimm and Mengel, 2007) deal only with the case of perfect separation of groups. The paper is organized as follows. Section 2 describes the experimental design. The results from the experiment are presented and discussed in Section 3. In Section 4 we derive a random utility model in order to estimate the preferences that guide the subjects’ behavior and report the results. Section 5 concludes. The Appendix contains regression tables and a translation of the experimental instructions. 2. The experiment In our experiment 224 participants (with no, or very little, prior exposure to game theory) anonymously interacted in a social dilemma situation for 100 rounds.6 We ran 6 treatments in total. Treatments T0, T1 and T2 are our baseline treatments. Treatments TP, TA and TB were control treatments. In treatments T0, T1 and T2 each round consisted of three stages. In the first stage subjects chose their group. In the second stage they had to give an estimate of the cooperativeness of their next round match. And at the last stage they played a prisoner’s dilemma game. Treatments T0, T1 and T2 differed in the matching technology. In all three treatments matching took place in a viscous population, meaning that individuals faced an increased probability to interact with others of their own group. The degree of viscosity is measured by the parameter x ∈ [0, 1]. The case x = 1 corresponds to (unbiased) random matching. The case x = 0 implies that the population is completely separated. Hence agents interact with probability 1 with agents of the same group and never with agents from another group. Suppose nA (nB ) is the number of agents in group A (B) and define n = nA + nB to be the total number of agents. In a viscous society with parameter x nB A −1 )]x = n−1 x and the the probability for an agent from group A to interact with a member of group B is [1 − ( nn−1 nA −1 nB probability to interact with a member of group A is (1 − (1 − [ n−1 )]x) = 1 − n−1 x. The matching probabilities are summarized in Table 1. In the experiment we chose the values x ∈ {0, 13 , 23 } for our three treatments T0, T1 and T2. There were 32 students in the laboratory in each session, but one population consisted of 8 subjects only. The participants were not informed about the size of a population. The members of a population (8 students) were initially randomly assigned to groups A and B in equal proportions. At the first stage of each round, two of the eight subjects could decide to either join the other group, or to stay in their own group. Each subject could make this decision every fourth round. We did not allow all subjects to switch simultaneously in each round, as we wanted to create a more stable environment for learning. At the second stage of each round subjects were asked for their expectation on the cooperation probability of their match.7 At the third stage subjects played the prisoner’s dilemma game with payoffs as given by Table 2 with an interaction partner who was assigned randomly according to the matching technology. As can be seen from Table 2, the two payoff-matrices differ only in the defector’s payoff. Independently of their group membership, subjects face a prisoner’s dilemma game. However, in group A the defector’s payoff is lower (i.e. cooperation is less costly as compared to defection). Throughout the paper we will use the term “group” to refer to the set of subjects who have chosen a particular set of payoffs in a particular round. Prior to playing the game, subjects were informed about (a) the percentage of subjects in groups A and B, and (b) their individual probability to meet a group A and group B member, respectively. When choosing an action in the bilateral game at stage three, agents had incomplete information about the group membership (i.e. the type) of their match.8 They had to estimate the type of their match from the information we gave to them. Table 1 Matching probabilities A A B

nB 1 − n−1 x nA x n−1

B

nB n−1 x nA 1 − n−1 x

5 See also Goette et al. (2006) for a field study on these issues. 6 We excluded economics and business students from the experiment. 7 We did not provide material incentives for correct answers because we wanted to avoid subjects trying to trade off earnings from correct guessing and from the game’s payoffs. 8 In T0, of course, agents were certain to interact with a member of their own group.

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Table 2 Payoffs in the prisoner dilemma games Group A other C D me

C

800

100

D

850

150

Group B me

205

other C

D

C

800

100

D

1100

400

Since in our experiment the population was necessarily finite, one-to-one matching was not feasible. Consequently our subjects did not play against another participant, but against what we called an “interaction partner” in the experiment. This “interaction partner” was determined as follows. First we realized a random draw with the probabilities given in Table 1 to decide whether a subject’s “interaction partner” was from group A or B. Then the “interaction partner” of any given subject played the actions “cooperate” or “defect” with probabilities that corresponded to the proportions with which all other subjects in the respective group (except, possibly, the subject herself) chose these actions (in that round). In the unlikely event that only one subject remained in a group (either A or B) and the first random draw determined that she had to play against a member of her own group, the subject’s “interaction partner” was preprogrammed to play C or D with equal probabilities.9 After each of the 100 rounds, subjects were informed of whether their “interaction partner” belonged to group A or B, of her action, and their own monetary payoffs. Finally the three additional treatments TA, TB, and TP were run in order to check the robustness of our results and to get additional insights. Treatment TP was essentially a robustness check. It coincides with T0 except for the fact that in addition to the information given in T0, at the end of each round subjects were informed about the average payoff obtained in each group (A and B). In treatments TA and TB there was no group choice option. In TA (TB) subjects played for 100 rounds with the payoffs corresponding to group A (B). In those treatments, each round consisted of only two stages. At stage 1 they had to give an estimate of the cooperativeness of their next round match. And at stage 2 they played the respective prisoner’s dilemma game. Treatments T0, T1 and T2 were conducted in four sessions in October, 2006 (1.5 sessions for T0 and T1 and one session for T2). Treatments TP, TA and TB were run in three sessions (one each) in January 2008. All experimental sessions were computerized.10 Written instructions were distributed at the beginning of the experiment.11 Each session took approximately 90–120 minutes (including reading the instructions, answering a post-experimental questionnaire and receiving payments). Subjects participating in the experiment received 2.50 Euros just to show up. On average, subjects earned a little more than 15 Euros (all included). 3. Results 3.1. Group choice Fig. 1 illustrates the effect population viscosity has on group choice. While under perfect separation of groups (treatment T0) a high proportion (59.2%) of subjects joins group A, the proportion of subjects who choose into group A is 36.8% on average in T1. These proportions do not decrease in the later periods. In treatment T2, on the contrary, group A shrinks and finally disappears (the average proportion of subjects in group A is 14.7%). Pairwise comparison of the three treatments shows that all differences in group choice are highly significant (Mann–Whitney test, p < 0.0001).12 Summing up, we observe an endogenous sorting of the subjects in the two groups under high viscosity (T1 and T2), whereas under low viscosity (T2) almost all subjects pool in the same group (group B). 9 The subjects were informed that the interaction partner would use a preprogrammed strategy in this case. 10 The experiment was programmed and conducted with the software z-Tree (Fischbacher, 2007). Subjects were recruited using the Online

Recruitment System by Greiner (2004). 11 The instructions for T1 (x = 1 ), translated from German into English, can be found in the Appendix. Instructions for the remaining treatments 3 are available upon request. 12 The Null-hypothesis in the Mann–Whitney test is that the distributions (of the variable in question) are identical in the two treatments compared. Whenever we say that two treatments are significantly different, we mean that the test rejects this hypothesis at the p-value indicated.

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Fig. 1. The proportion of subjects in group A (per treatment).

Result 1 (Group Choice). The more viscous the population, the higher is the proportion of subjects that is in group A. In T0 and T1 a constant proportion of around 60% and 35%, respectively, is in group A. In treatment T2 the proportion of subjects in group A decreases until it is finally zero. Taken by itself Result 1 only shows that some sorting took place. The more interesting question is whether the observed sorting does enhance cooperative behavior. In the following we investigate whether cooperative types use the group choice to signal their willingness to cooperate and if so, if high cooperation rates can be sustained. 3.2. Cooperation Analyzing cooperation rates separately for the two different groups (A and B) reveals that in all treatments the majority of subjects in group A cooperates, while almost no group B member does. As Fig. 2 illustrates, the proportion of cooperating subjects in group A is constantly around 60% in treatments T0 and T1. In treatment T2 the cooperation rate in group A fluctuates a lot, which is mainly due to the low number of subjects in group A. Note that Figs. 1 and 2 also show that—unlike in most other experimental studies of cooperation— cooperation does not break down at the end of the experiment, i.e. there is no so-called “endgame effect.” This

Fig. 2. Proportions of cooperators in group A (per treatment).

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Fig. 3. Proportions of cooperators in group B (per treatment). Table 3 Average cooperation rates Treatment Group A B T0 T1 T2 TA (only A) TB (only B)

62.4 67.1 59.5 50.9 –

9.9 3.2 6.7 – 9.8

Overall 41.0 26.7 11.9

illustrates (a) that viscosity produces persistent cooperation and (b) that the observed behavior does not seem to be driven by the use of repeated game strategies by players with limited foresight. Fig. 3 shows that in group B initial cooperation quickly breaks down and that from round 20 on, almost no one cooperates in that group. Table 3 gives an overview over average cooperation rates, separately for both groups and for the whole population. The table illustrates that higher viscosity leads to a sharp increase in overall cooperation rates. Pairwise comparison of overall cooperation rates in treatments T0, T1 and T2 shows that all differences are highly significant. (Mann–Whitney test, p < 0.0001). Viscosity seems an effective means to achieve persistently high cooperation rates. Result 2. Subjects in group A cooperate significantly more than subjects in group B. Consequently the overall cooperation rate is highest in treatment T0 and lowest in treatment T2. Our results suggest that viscosity can sustain an endogenous sorting of more cooperative and more selfish types into the two groups, A and B. In order to further investigate the impact of endogenous sorting on cooperation rates we conducted two additional treatments (TA and TB), where subjects did not have the possibility to switch groups but instead played one of the two games for 100 rounds. In treatment TA all subjects faced the payoff matrix designed for group A (which we call “game A” in the following), in TB all subjects played the game corresponding to group B (“game B“). The two lower rows of Table 3 report the cooperation rates in those two treatments. Obviously the incentives provided in game A are such that cooperation can be sustained at a high level (50.9%), whereas the incentives in group B are such that cooperation breaks down (9.8%). More interestingly, though, the rate of cooperation in group A is clearly higher in the treatments with endogenous group choice (62.4% in T0 and 67.1% in T1) than in the treatment where game A is played by all subjects (50.9% in TA).13 The difference in cooperation rates in group A between treatment T0 and treatment TA is highly significant (Mann–Whitney test p < 0.0001). Com13 By choosing the payoffs in the two prisoner’s dilemma appropriately this difference can of course be made larger or smaller. Interesting is only the fact that there is a difference. Ahn et al. (2003), among others, discuss the motivational implications of different payoff configurations.

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parison of cooperation rates in group B between T0 and TB yields no significant difference, however (Mann–Whitney test p = 0.9157). These observations can be rationalized by assuming that subjects have heterogeneous preferences for cooperation and that there are many conditional cooperators, who cooperate only if a sufficient fraction of other subjects also cooperate.14 Cooperation breaks down in game B, where the share of cooperation is too low to trigger cooperative behavior from “conditionally cooperative” types. In game A, however, under endogenous group choice “less cooperative” subjects choose into group B, which increases average cooperation in group A and in addition triggers a higher propensity to cooperate from cooperative types in this group. Consequently, cooperation rates in game A are higher under endogenous group choice. These results have important implications for the design of institutions. On the one hand incentives have to be such that cooperation has a chance at all (this is the case for group A payoffs, but not for group B payoffs). On the other hand cooperation rates are higher if participation in an institution is voluntary, as agents with preferences that are very averse to the institution will opt out of it (choose group B). In fact—as we will see in the next section—even average payoffs (across all subjects) are clearly higher in T0 as opposed to TA, stressing the effectiveness of endogenous sorting. 3.3. Payoffs The group and action choice behavior we observed had clear consequences on profits. Note that overall rates of cooperation in the population were higher, the higher the degree of population viscosity was (compare also Table 3). Consequently, payoffs were highest in treatment T0, lowest (and close to the payoffs from mutual defection) for T2, and in between for the remaining treatment T1. Table 4 reports the profits obtained in the different treatments. Result 3. (i) Average profits in the population are highest in treatment T0, followed by T1 and T2. (ii) The profit of a group A member is higher than the profit of a group B member in treatments T0 and T1 and the opposite is true in T2. The payoff differences in group A are highly significant between all treatments T0, T1 and T2 (Mann–Whitney test, p < 0.0001). There are no significant payoff differences in group B between treatments T0 and T1 (Mann–Whitney test, p = 0.6732). Payoffs in T0 and T2 as well as in T1 and T2, though, are significantly different. (Mann–Whitney test, p = 0.0516 (0.0055)). The intuition simply is that, since in T2 overall cooperation converges quickly to zero, there are no possibilities for exploitation. The differences, though, are small (and for T0 only weakly significant) as possibilities for exploitation are small (or even non-existent) in T0 and T1 because of high viscosity. More striking is the comparison of payoffs in T0 and TA. It can be seen clearly that average payoffs across all subjects are higher in T0 (518) than in TA (481). (Mann–Whitney test p = 0.0001). The possibility of subjects with weak (or no) preferences for cooperation to opt out of group A (and switch to group B and defect) increases efficiency in the whole population. As already mentioned at the end of the previous section, this has most interesting implications for the design of institutions. In particular, it is not necessarily desirable to enforce an institution in society. Voluntary participation can lead to better outcomes, as it is the case in our experiment. Note that, in addition to the fact that Table 4 Profits Treatment T0 T1 T2 TA TB

Group A B 560 494 350 481 –

457 460 452 – 438

Overall 518 472 436

14 In Section 4 we estimate the subjects preferences for cooperation and find that indeed most subjects are “conditional cooperators.”

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society on average benefits more in the case of voluntary participation in the institution, cooperative agents receive a much higher share of the additional benefits. Note also that—even if there is some interaction between groups (x = 13 )—agents in group A are still slightly better off than in TA, although here the difference is not significant (Mann–Whitney test p = 0.1596). Table 4 also shows that group choice is not an equilibrium (given action choice). In particular agents in treatment T0 have strong incentives to switch from group B to group A (irrespective of whether they correlate group choice with action choice). This is interesting and raises the question of why subjects do not switch from group B to group A in this treatment. In order to gain more insight into this issue we analyze the subjects’ beliefs about cooperation rates in both groups (A and B) in the next subsection. 3.4. Beliefs In the following we analyze the subjects’ beliefs (which we elicited in the second phase of each round) in order to investigate whether the participants have a good sense of how likely it is to meet a cooperator (depending on the group they are in). This serves both as evidence for how well they understood the instructions and for how much they learn from their experience during the experiment. Table 5 compares the subjects’ average beliefs (over all rounds) with the true probabilities of meeting a cooperator, separately for group A and group B. In all cases average beliefs are very accurate, except for group B in treatment T0.15 To investigate why this might be the case we now inspect the beliefs of subjects who do not switch groups very often. In particular we focus on subjects who spent more than two thirds of the time (i.e. at least 67 rounds) in only one of the two groups in treatment T0. We find that in T0 24 subjects spent most of the time in group A, 9 subjects in group B and 15 subjects switched between the two groups very often. Table 6 reports the beliefs of those subjects who do not switch groups very often. Table 6 illustrates that both types—those that spend most of the time in group A and those that spend most of the time in group B—have a good feeling for cooperation rates in group A. Interestingly, cooperation rates in group B are substantially overestimated by both types, but subjects who spend most of the time in group B are particularly bad at estimating these rates. One could argue that too high estimates among these subjects are due to random responses emerging from frustration over low cooperation rates. This explanation is not convincing, though, for two reasons. First, if subjects had correct beliefs (and answer wrongly because of frustration) they should switch to group A. Second, the beliefs become on average more accurate over time, even for these subjects. The frustration hypothesis would predict, though, that beliefs get worse over time. Table 5 Average beliefs and average cooperation rates Treatment Group A B T0 T1 T2

belief cooperation belief cooperation belief cooperation

59.8 62.5 60.0 56.8 43.8 37.9

33.6 9.8 13.8 10.4 14.4 14.5

Table 6 Beliefs of subjects who spent most rounds in group A (B)

group A group B

> 23 of all rounds in A

> 23 of all rounds in B

58.4 24.5

58.8 39.6

15 Once cooperation breaks down almost completely in group B (i.e. from round 20 on) beliefs become more accurate but stay substantially above true cooperation rates (i.e. 24.5% estimated vs. 4.5% true cooperation in the last 25 rounds). In all other treatments beliefs converge quickly and are very close to true cooperation rates throughout (almost) the whole experiment. The time series are available upon request.

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Table 7 Complete group separation with (T0) and without (TP) payoff-information A B T0 TP T0 TP group size cooperation rate beliefs payoffs

59.2 62.4 59.8 559.8

67.4 64.9 61.0 569.3

40.8 9.9 33.6 456.7

32.6 10.0 31.4 475.2

Calculating the (subjective) expected profits from defection in group A and group B reveals that—given their wrong beliefs—it is rational for the subjects who spend most of the time in group B to actually stay in B and defect. Given their beliefs from Table 5, expected material payoffs from defecting in group B are 677 versus 562 from defecting in group A.16 On the other hand, performing the same calculation for those subjects who spend most of their time in group A yields the result that they should switch groups (given their (wrong) beliefs). As we will show in the next section it is their strong subjective preferences for cooperation that prevent them from doing so. These results obviously raise the question whether subjects stayed in group B only because they were not aware of the fact that they could earn more in group A. To test this hypothesis we ran an additional treatment TP with completely separated groups (x = 0) where, at the end of each round, subjects were informed of the average payoff of group A- and group B-members. Table 7 compares the results (group choice, cooperation rates, payoffs and beliefs) of this additional treatment TP with the baseline treatment T0. As the table clearly illustrates, additional information on payoffs does not affect any of the results qualitatively. While there are somewhat more subjects in group A if there is more payoff-information, neither cooperation rates, nor beliefs, nor payoffs significantly differ between the two treatments.17 This is quite interesting, as it shows that the results are very robust to different levels of information. The results also illustrate that it is not due to missing information about potential payoffs that subjects do not switch to group A to exploit cooperators. As a matter of fact, subjects who spend most of the time in group B tend to cooperate quite a bit (52% in T0 and 54% in TP) of the time that they spend in group A.18 This is a general observation. Almost all subjects (those that spend most of the time in group A, those that switch a lot and those that spend most of the time in group B) strongly correlate their action and group choice, cooperating in group A and defecting in group B. In the next section we try to shed more light on all the issues raised above, by estimating the participants’ subjective preferences for cooperation. We will find that the subjects’ elicited beliefs and estimated heterogeneous preferences for cooperation can explain both the observed group choices and cooperation rates. 4. Feedback effects between interaction structure and preferences for cooperation Our results have illustrated that viscosity plays an important role in sustaining cooperation in the experiment. Note that if cooperators were motivated by unconditional altruism, sorting (or viscosity) would not be necessary to sustain cooperation in our setting. If, on the other hand, subjects are conditional cooperators (cooperating only if they believe that their match is likely to do so as well), viscosity can become necessary to sustain cooperation.19 To test the conjecture of conditional cooperation is one objective of this section. Also, as we have seen in Section 3, the possibility to switch between groups leads to (a) a sorting of agents into group A and group B and (b) raises cooperation rates in group A compared to a treatment where all agents face the payoff matrix from group A (and there 16 In T1 beliefs are “more correct” and the reverse is true, even though here the payoff-difference is considerably smaller, promising 523 in group A (for a belief of 55% cooperation in group A) and 490 in group B (for a belief of 12% cooperation in group B). 17 The only variable that might be marginally different is cooperation in group A (Mann–Whitney test, p = 0.0662). 18 Also the number of subjects who spend more than 66 rounds in group A (B) is not substantially different between the two treatments, even though a few more subjects spend most of the time in group A. In T0 50% (22%) of the subjects spend more than 66 rounds in group A (B) as opposed to 62% (19%) in TP. 19 Other studies have shown that subjects often behave as conditional cooperators. See Fischbacher et al. (2001) or Fischbacher and Gächter (2006). See Mengel (2007, in press) for a theoretical analysis of the relation between viscosity and the emergence of conditionally cooperative behavior.

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is no group B). This suggests that agents with weaker preferences for cooperation choose group B and that the sorting we observe is a sorting of types with “more cooperative” and “less cooperative” preferences. To test this conjecture is another objective of this section. Finally we can also gain additional insights into why some subjects do not switch from group B to group A in spite of the fact that this would be profitable for them. 4.1. The model To estimate the subjective preferences for cooperation we use a random utility model. Consider the following payoff matrix (payoffs for row player, denoted player i). In Table 8 the term wi measures player i’s subjective preference for cooperation, δBi (t) = 1 if i is in group B at time t and zero otherwise, and εiC , εiD are error terms. We define εi = εiD − εiC . We denote by cˆit the probability with which an agent i believes that her match cooperates at time t (we elicit this probability in our experiment). Say cit = 1 if agent i cooperates at time t and cit = 0 otherwise. Then the expected utility of an agent can be written as   (1) U (cit ) = 800cˆit + 100(1 − cˆit ) + εiCt +(1 − cit ) 50 + 250δB (t) − wi + εit .       exp. profit from coop.

add. perceived profit from defection

The choice rule of an expected utility maximizing agent then predicts that whenever the additional perceived profit of defection as compared to cooperation is negative, i.e. whenever   cit∗ = wi − 50 + 250δB (t) + εit is positive, a subject cooperates whereas if it is negative, the subject defects. The choice rule is,  ∗ cit = 1 if cit  0, 0 otherwise. In the same way—for any given wi (and thus cooperation decision)—agent i should choose the group (A or B), that maximizes her expected utility. If we now assume that the εit are i.i.d. extreme value distributed we have a logit choice model. We want to allow the subjective preferences of the agent to depend on the beliefs c. ˆ In the estimation we consider the following quadratic specification for the preference trait w(·), ˆ = ni + mi cˆ + li (c) ˆ 2. wi (c) The case where w(c) ˆ is a constant and does not depend on cˆ and the case where w(c) ˆ is linear in cˆ will be treated as special cases of the quadratic model where the coefficients of higher order terms are zero. We estimate a logit choice model where the independent variables are the expected monetary payoff of defection as compared to cooperation ˆ 2 . The logit model for the ( = 50 + 250δB ) as well as the agent’s preference parameter wi = ni + mi cˆ + li (c) quadratic case can be written Pr(cit = 1) =

eμ(ni +mi cˆit +li (cˆit ) 1+e

2 −50−250δ i (t)) B

i (t)) μ(ni +mi cˆit +li (cˆit )2 −50−250δB

(2)

.

Regressing Pr(cit = 1) on cˆit , (cˆit )2 and δBi (t), we have that (given our utility model) the coefficient for the constant term (of the regression) is an estimate of μ(mi − 50), the coefficient on δBi (t) is an estimate of −250μ, the coefficient on cˆit is an estimate of μni and the coefficient on (cˆit )2 is an estimate of μli . Comparing these coefficients allows us to eliminate μ and thus to distinguish between the effect of noise and that of the preference parameter.20 Table 8 Random utilities other me (player i)

C

D

C

800 + εiC

100 + εiC

D

i −w +ε 850 + 250δB i iD

i −w +ε 150 + 250δB i iD

20 Note that δ i (t) is also a choice variable of the agents. In addition we thus ran simultaneous regressions of δ i (t) and c and tested whether the it B B error terms of the two regressions are correlated. We cannot reject the hypothesis of non-correlation for a test of size 5% in any of the treatments.

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4.2. Results We find that on average the subjective preferences for cooperation depend as follows on cˆ in our three treatments21 : w0 (c) ˆ = −103.77 + 838.01cˆ − 545.21(c) ˆ2

in T0,

ˆ = −111.3 + 812.25cˆ − 543.71(c) ˆ2 w1 (c)

in T1,

ˆ = −119.72 + 1104.0cˆ − 995.91(c) ˆ2 w2 (c)

in T2.

The result is illustrated in Fig. 4. The lower of the two similar lines represents T1, whereas the most “U-shaped” one T2. For very small values the preference function w(c) ˆ is actually negative. This could indicate the presence of motives like “spite” or “anger.” But then the function is rapidly increasing in cˆ in all treatments indicating that conditional cooperation is a strong behavioral motive. All three functions are concave in c, ˆ i.e. subjective preferences to cooperate seem to rise faster with cˆ for smaller levels of cooperation. This is well in line with other experimental findings, like Fischbacher et al. (2001) or Fischbacher and Gächter (2006) who observe conditionally cooperative behavior. If subjects were altruists (as opposed to conditional cooperators), their functions w(c) ˆ should be approximately constant, as their subjective preference for cooperation should be independent of the rate of cooperation they expect in the population. We also checked whether motives like (either positive or negative) reciprocity can explain the observed behavior.22 Note that—unlike in other experiments—reciprocity could only be directed towards the group and not towards an individual in our setting, as subjects do not know whom they are matched with in each round. In particular we tested how subjects react to the experience of being exploited (i.e. cooperating when their opponent defects) and find that this experience does not (at least not immediately) lead to a change in action choice. We conclude that (negative) reciprocity does not seem to be a major driving force of the subjects’ decisions. Fig. 4 also shows that our utility model seems to capture well the feedback effects viscosity has on preferences for cooperation. In addition to the indirect effect (via beliefs) there also seems to be a small direct effect. This effect is illustrated by the fact that under full separation (x = 0) the subjective preference for cooperation seems slightly stronger than for x = 1/3. In treatment T2 the function w(c) ˆ is inversely U-shaped. Note though, that there are almost

Fig. 4. Estimated w(·) and material incentives (all subjects).

21 We used a random effects panel logit model (selecting between random and fixed effects using the Hausman test). We had 48 groups (individuals) in treatments T0 and T1 and 32 in treatment T2. From each individual we had 100 observations. Standard errors were estimated robustly. The regression tables are given in the Appendix. 22 See, among others, Fehr and Gächter (2000).

V. Grimm, F. Mengel / Games and Economic Behavior 66 (2009) 202–220

213

no observations for high beliefs in this treatment. We conjecture that with more observations in this range (and higher rates of cooperation) the function in treatment x = 23 would be of a similar shape than that of the other treatments.23 We then estimated the function w(c) ˆ separately for those subjects who spend more than 23 of all rounds in group A 24 and group B, respectively. We did this only for treatments T0 and T1, as in T2 almost all subjects spend most of the time in group B. Fig. 5 illustrates the preference for cooperation of the different subsets of participants as well as the material incentives (straight horizontal lines). The dashed lines correspond to subjects who spend most of the time in group B and the continuous lines to those that spend most of the time in group A. Black lines correspond to treatment T0 and gray lines to treatment T1.25 In treatment T0 subjects who are mostly in group A have much stronger subjective preferences to cooperate than subjects who spend most of the time in group B. They are willing to cooperate in group A if they believe that roughly 20% of all others will also do so. For those subjects who spend most of the time in group B this share needs to be around 50%. This illustrates that it is indeed the case that subjects with a stronger preference for cooperation endogenously sort into group A. The weak preferences for cooperation of group B members together with their wrong beliefs (see Section 3.4) can explain why those subjects do not choose group A more often.26 Given their elicited beliefs of 39.6% and the estimated preferences, it can be seen that they would be better off choosing defection in group A (as opposed to defection in group B) if and only if they believed that the cooperation rate in group A exceeded 81.6%. On the other hand they are willing to cooperate in group A if and only if their belief exceeds 48.7%. On occasion these subjects experimented with switching into group A. It seems that experiencing defection in group A shocked their beliefs below the 50% threshold, inducing them to switch back into group B. Subjects who spend most of the time in group A on the other hand stay there even for lower beliefs, because their intrinsic incentives to cooperate are higher. Given their estimated preferences for cooperation and their (wrong) beliefs about cooperation rates in group B, their psychological payoff loss of switching to group B and defecting would be w(0.25)  90, which is not compensated by the higher material payoffs defecting in group B promises (given again their (wrong) beliefs).

Fig. 5. Preferences of agents who spend most of the time in group A (B).

23 There are very few data points for high values of cˆ in treatment T2. (In fact, only 11 guesses exceed 90% in this treatment.) 24 The regression tables are given in the Appendix. 25 For subjects who spend most of their time in group A in treatment T1 the coefficient on the quadratic term is not significant. The function w A (c) 1 ˆ is therefore estimated without this term. Comparing the model with and without the quadratic term using different information criteria showed that the model without this term can explain the data better. 26 Indeed in T0, out of the subjects who spend more than two thirds of the rounds in group A (B), 42% (67%) spend more than 90% of the rounds in group A (B). On the other hand 0% (22%) of all subjects spend more than 95% of the time in group A (B). In T1 these numbers are a lot lower, namely 18% (0%) for group A and 56% (25%) for group B.

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In treatment T1 the willingness to cooperate of subjects mostly in groups A and B, respectively, is much more different for small beliefs and then this difference becomes smaller as the belief rises. The curves even intersect at a belief of roughly 65%. The interpretation is clear. If x = 13 , the agents who are most of the time in group A have very strong preferences for cooperation. The “altruistic” component (ni ) in their function is more prominent as compared to the “conditional” component (mi , li ). Note also that these agents (on average) do not seem to have a “spite motive.” Agents who strongly condition their behavior on that of the others seem to be mostly in group B in this treatment, which implies that the function of agents mostly in group B is steeper in T1 than in T0. We also ran individual regressions for each subject in each treatment. This allowed us to classify the subjects into six types that displayed different kinds of preferences for cooperation. The results are given in Table 9. A subject is attributed the model for w(c) ˆ that fits best (constant, linear or quadratic). To be attributed any model at all, the coefficients of the regression have to be significant at 10% at least.27 Subjects who cooperate (defect/follow a strategy of “C in A and D in B”) at least 91 out of 100 times are classified into always C (“D/ C in A, D in B”) instead of being attributed the model that fits best. Those who cannot be classified in any of the categories are classified into “none.” Non-surprisingly the proportion of flat defectors (“Always D”) increases as population viscosity decreases. In the next table we show which proportion of all agents of a given category chose group A (B) more than 23 of the time in treatments T0 and T1. To interpret the table it is helpful to remember first that in T0 50% of all subjects spend most of the time in group A and 18% of subjects are mostly in B, whereas in T1 the corresponding numbers are 23% in A and 52% in B. Table 10 illustrates that among all cooperator types, those subjects with altruistic preferences (“Constant model” and “Always C”) are always in group A, whereas flat defectors are mainly in group B in both treatments.28 Conditional cooperators, however, are mainly in group A in T0 but mainly in group B in T1. The different size of group A in T0 and T1 is, thus, due to the fact that in T1 conditional cooperators are mostly in group B. The fact that in T1 subjects with altruistic preferences account for a large part of the population in group A also explains the very flat curve w(·) among group A members on average. Non-surprisingly there are no “altruists” (“Always C”) among those who spend most of the time in group B.

Table 9 Subject classification in the three treatments Treatment T0 T1 T2 Constant model Linear model Quadratic model C in A, D in B Always C Always D None

13% 38% 19% 6% 4% 10% 10%

6% 36% 10% 13% 2% 27% 6%

3% 16% 13% 3% – 53% 12%

Table 10 Group choice of the different cooperator-types Treatment T0 (A) T0 (B) T1 (A) T1 (B) Constant model Linear model Quadratic model C in A, D in B Always C Always D None

50% 67% 55% 33% 100% 20% 20%

– 11% 11% 33% – 80% 20%

100% 6% 20% 20% 100% 7% 100%

– 53% 60% 40% – 85% –

27 Mostly the significance level is 1%. 28 Even among those who are most of the time in group A there is also one “flat defector” in both treatments, T0 and T1. Some subjects (albeit

few), thus, act according to the “correct” material incentives and exploit the cooperators in group A.

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215

5. Conclusion In this paper we have experimentally investigated the impact of population viscosity on cooperation in social dilemma situations. Participants in our experiment could repeatedly choose between two groups, where in one of them the defector payoff was reduced. We found that under high population viscosity many subjects chose into the group with the lower defector payoff, possibly to signal their willingness to cooperate. In all treatments a significant proportion of subjects actually cooperated in that group, while almost no subject cooperated in the other group. The proportion of participants that choose into the “cooperative” group rises with the degree of population viscosity. Population viscosity thus seems to enable an endogenous sorting of cooperative and non-cooperative agents and to sustain persistent cooperation. We also find evidence for a positive relation between cooperative preferences and population viscosity. In a nutshell, population viscosity seems a powerful and important mechanism, not only for sustaining cooperation given the preferences of individuals, but (via a change in beliefs) it also positively affects their preferences for cooperation. To further understand the way population viscosity acts on economic incentives and subjective preferences gives rich potential for further research, both theoretically and experimentally. Acknowledgments We thank Ana Babus, Lola Collado, Dirk Engelmann, Urs Fischbacher, Sven Fischer, Marco van der Leij, Karl Schlag, Jakub Steiner, Christian Traxler, Axel Ockenfels, three anonymous referees as well as seminar participants at Amsterdam, Bonn, Budapest (EEA), Cologne, Cornell, Goslar, Granada (SAE 2007), Magdeburg and Siena (LABSI 2007) for helpful comments and suggestions, and Rene Cyranek, Meike Fischer, and Felix Lamoroux for most valuable research assistance. Financial support by the Deutsche Forschungsgemeinschaft, the Instituto Valenciano de Investigaciones Económicas (IVIE) and the Spanish Ministery of Education and Science (Grant SEJ 2004-02172) is gratefully acknowledged. Appendix A. Instructions treatment x =

1 3

Welcome and thanks for participating at this experiment. Please read these instructions carefully. They are identical for all the participants with whom you will interact during this experiment. If you have any questions please raise your hand. One of the experimenters will come to you and answer your questions. From now on communication with other participants is forbidden. If you do not conform to these rules we are sorry to have to exclude you from the experiment. Please do also switch off your mobile phone at this moment. For your participation you will receive 2,50 Euro. During the experiment you can earn more. How much depends on your behavior and the behavior of the other participants. During the experiment we will use ECU (Experimental Currency Units) and at the end we will pay you in Euros according to the exchange rate 1 Euro = 2500 ECU. All your decisions will be treated confidentially. A.1. The experiment At the beginning of the experiment we will split you and the other participants equally into two groups—group A and group B. In each round of the experiment you play a game against a “representative member” either from group A or group B that we will call in the following your interaction partner. Each round has three phases: • Phase 1: Each round some participants can decide whether to change groups or not. You can make this decision for the first time between round 1 and 4 and from then on every 4 rounds. • Phase 2: You are asked to give an estimate about your interaction partner’s likely behavior. • Phase 3: You play the game that we will describe in the next section. The experiment consists of 100 rounds.

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A.2. The game and the payments Independently from which group (A or B) you are in, you play during the first 4 rounds and in the second phase of every following round the following game with a randomly selected interaction partner: In each round you and your interaction partner can choose between two alternatives, C and D. How much you earn in each round depends on what you and your interaction partner have chosen and in which group you are. Each member of group A receives the following payments: Group A you choose

your match chooses C D C

800 ECU

100 ECU

D

850 ECU

150 ECU

The table reads as follows: • • • •

If both choose D, each gets 150 ECU (down right). If you choose D and your interaction partner C, you get 850 ECU (down left). If you choose C and your interaction partner D, you get 100 ECU (up right). If both choose C, each gets 800 ECU (up left). Each member of group B receives the following payments: Group B you choose

your match chooses C D C

800 ECU

100 ECU

D

1000 ECU

490 ECU

The table reads as follows: • • • •

If both choose D, each gets 400 ECU (down right). If you choose D and your interaction partner C, you get 1100 ECU (down left). If you choose C and your interaction partner D, you get 100 ECU (up right). If both choose C, each gets 800 ECU (up left).

A.3. Who do I play with and how does this depend on my group? In each round your interaction partner is determined randomly. The probability to interact with someone of your own group differs from that of interacting with someone from the other group. The following is true: • The more members a group has the more likely it is to meet a member of that group. • Relatively it is more likely to meet someone from your own group. The tables below give you an overview of the probabilities to interact with a member of group A or B respectively, depending on whether you yourself are in group A or B. If you are in group A the relevant table is Table A.1. If you are in group B the relevant table is Table A.2. The tables are for your orientation. It can happen that the actual proportion of participants in group A is not listed in the table. Each time we will thus calculate the corresponding probabilities for you and inform you about them before the start of Phase 2.

V. Grimm, F. Mengel / Games and Economic Behavior 66 (2009) 202–220

Table A.1 You are in group A percentage of participants in group A: percentage of participants in group B: in which percent of all cases do A B I meet someone from group

25 75 71 29

Table A.2 You are in group B percentage of participants in group A: percentage of participants in group B: A in which percent of all cases do I meet someone from group B

0 100 0 100

50 50 81 19

25 75 10 90

75 25 90 10

50 50 19 81

217

100 0 100 0

75 25 29 71

A.4. Your interaction partner Your interaction partner in each round is not another participant of the experiment, but a “representative member” of either group A or group B. At each round it is determined randomly (according to the probabilities given above), whether your interaction partner at this round is from group A or group B. If your interaction partner is from group A, she chooses the actions C and D with probabilities that correspond to the proportions with which the members of group A (excluding you) have chosen C and D. If your interaction partner is from group B, she chooses the actions C and D with probabilities that correspond to the proportions with which the members of group B (excluding you) have chosen C and D. If you are the only member of your group, the behavior of your interaction partner will be simulated by the computer (but only in this case). In all other cases the behavior of your interaction partner depends exclusively on the behavior of the other members of the respective group. These rules obviously are the same for all other participants of the experiment. Example. You are in group A and your interaction partner is also from group A. • if among the other members in group A 70% chose action C and 30% chose action D, your interaction partner will choose with probability 70% action C and with probability 30% action D. • if all other members of group A have chosen action C, your interaction partner will choose action C with probability 100%. A.5. Information you receive Survey of the three phases and the information you are given • Phase 1: Some participants can change their group. • Phase 2: (a) We inform you about ∗ your current group, ∗ which proportion of participants is in group A and B respectively, ∗ with which probability you meet a participant of group A or B. (b) You give an estimate about the behavior of your match. Specifically we will ask you the following question: How likely do you think it is (in percent) that your match in the next round will choose action C? • Phase 3: You play the game described above with a randomly chosen match. After the third phase you are informed about which action you and your match have chosen and about your payment.

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Appendix B. Regression results B.1. Regression results cooperation, all treatments

Table B.1 Random effects logit regression cooperation (128 groups, 12800 observations) ci

Coef.

Std. Err.

p > |z|

[95% conf. interval]

constant cˆi i δB Treatment 1 (T1) Treatment 2 (T2) i T1 ∗ δB

−.5211202 .0291948 −2.84822 −.7628143 .2530817 −.8251537

.1092775 .0013919 .1158171 .1492469 .3071179 .1797644

.000 .000 .000 .000 .410 .000

−.7353002, .0264667, −3.075217, −1.055333, −.3488583, −1.177485,

−.3069402 .0319228 −2.621223 −.4702957 .8550217 −.4728219

.3893097 1.548935 .421721

.2087595

.062

−.0198515,

.7984708

i T2 ∗ δB σu ρ

B.2. Estimated “average” preference for cooperation Remark. From the regression in the x = 0 treatment we excluded 2 out of 48 students because their behavior did not produce enough variation.

Table B.2 Random effects logit regression, cooperation (46 groups, 4600 observations) Treatment T0 ci

Coef.

Std. Err.

z

p > |z|

[95% conf. interval]

constant cˆi (cˆi )2 i δB σu ρ

−1.516847 .0826623 −.0005378 −2.466043 1.598325 .4371011

.2404864 .0081416 .0000715 .1262566

−6.31 10.15 −7.52 −19.53

.000 .000 .000 .000

−1.988192, .066705, −.0006779, −2.713502,

−1.045503 .0986197 −.0003976 −2.218585

Table B.3 Random effects logit regression, cooperation (48 groups, 4800 observations) Treatment T1 ci

Coef.

Std. Err.

z

p > |z|

[95% conf. interval]

constant cˆi (cˆi )2 i δB σu ρ

−2.098794 .1056852 −.0007066 −3.252846 1.587211 .4336705

.2365592 .0108448 .0001133 .1574204

−8.87 9.75 −6.24 −20.66

.000 .000 .000 .000

−2.562442, .0844298, −.0009286, −3.561384,

−1.635146 .1269405 −.0004845 −2.944308

V. Grimm, F. Mengel / Games and Economic Behavior 66 (2009) 202–220

Table B.4 Random effects logit regression, cooperation (32 groups, 3200 observations) Treatment T2 ci

Coef.

Std. Err.

z

p > |z|

[95% conf. interval]

constant cˆi (cˆi )2 i δB σu ρ

−1.453488 .0945457 −.0008529 −2.141006 1.166163 .2924717

.2358278 .0087753 .0001005 .171408

−6.16 10.77 −8.48 −12.49

.000 .000 .000 .000

−2.562442, .0844298, −.0009286, −3.561384,

−1.635146 .1269405 −.0004845 −2.944308

Table B.5 Random effects logit regression, cooperation (24 groups, 2400 observations) Treatment T0—mainly group A ci

Coef.

Std. Err.

z

p > |z|

[95% conf. interval]

constant cˆi (cˆi )2 i δB σu ρ

−1.417036 .0737343 −.0003181 −3.21969 1.800229 .496245

.261111 .0107859 .0001111 .2065388

−5.43 6.84 −2.86 −15.59

.000 .000 .004 .000

−1.928805, .0525942, −.0005359, −3.624498,

−0.9052681 .0948743 −.0001002 −2.814881

Table B.6 Random effects logit regression, cooperation (9 groups, 900 observations) Treatment T0—mainly group B ci

Coef.

Std. Err.

z

p > |z|

[95% conf. interval]

constant cˆi (cˆi )2 i δB σu ρ

−2.906708 .0943457 −.0007122 −2.063492 1.912685 .5265172

.5988917 .0177721 .0001461 .3018012

−4.85 5.31 −4.87 −6.84

.000 .000 .000 .000

−4.080514, .0559513, −.0009986, −2.655011,

−1.732902 .1291784 −.0004258 −1.471972

Table B.7 Random effects logit regression, cooperation (11 groups, 1100 observations) Treatment T1—mainly group A p > |z|

ci

Coef.

Std. Err.

constant cˆi (cˆi )2 i δB σu ρ

.1054158 .0100486 n.s. −2.808757 1.527525 .4149475

.0543221 .0036544

8.85 7.21

.000 .000

.3228432

−6.56

.000

z

[95% conf. interval] .0511943, .0059703, −3.457851,

.1797412 .01296684 −2.224542

Table B.8 Random effects logit regression, cooperation (23 groups, 2300 observations) Treatment T1—mainly group B ci

Coef.

Std. Err.

z

p > |z|

[95% conf. interval]

constant cˆi (cˆi )2 i δB σu ρ

−3.822882 .1377454 −.0010114 −2.16048 1.131976 .2803114

.4789548 .0167564 .0001704 .2465972

−7.98 8.22 −5.94 −8.76

.000 .000 .000 .000

−4.761617, .1049034, −.0013453, −2.643801,

−2.884148 .1705874 −.0006774 −1.677158

219

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