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Spontaneous evolution of social exchange—An experimental study
The Journal of Socio-Economics 37 (2008) 976–997
Spontaneous evolution of social exchange—An experimental study Siegfried K. Berninghaus a,∗ , Werner G¨uth b , Bodo Vogt c b
a University of Karlsruhe, Institute WiOR, Zirkel 2, Rechenzentrum, D-76128 Karlsruhe, Germany Max Planck Institute for the Evolution of Economic Systems, Kahlaische Strasse 10, D-07745 Jena, Germany c IMWF, University of Magdeburg, Universit¨ atsplatz 2, D-39106 Magdeburg, Germany
Accepted 1 December 2006
Abstract Each of several exchange partners owns a specific commodity which she can share with others. Unlike in other social dilemma scenarios like prisoners’ dilemma, public goods games, etc., voluntary cooperation relies on bilateral exchanges whose profitabilities are interdependent. How will mutual sharing evolve? Will it include all group members or will smaller groups be more efficient? Our experimental data shed partly new light on older topics: cooperation is now relation specific, allowing for discrimination; group size effects are explored dynamically rather than in one-shot interaction; and, finally, we have weakened demand effects for voluntary cooperation by realistic efficiency gains. © 2007 Elsevier Inc. All rights reserved. JEL Classification: A13; C72; C92 Keywords: Exchange networks; Bilateral reciprocity; Mutual gift giving; Ostracism
1. Introduction Networks are usually established by introducing “links” between several “nodes”, which can be established either individually or in agreement with others (Bala and Goyal, 2000; Jackson and Wolinsky, 1996). Often a link refers to some possibility of communication or transportation.1 In social exchange, a link denotes neither a way of transportation nor communication but rather an ∗
Corresponding author. Tel.: +44 7216083380; fax: +44 7216083391. E-mail addresses: [email protected] (S.K. Berninghaus), [email protected] (W. G¨uth), [email protected] (B. Vogt). 1 There may be a quality of the link, e.g. the safety of a road linking two parties, what we will neglect, however. 1053-5357/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2006.12.064
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exchange of goods. For divisible exchange commodities a link can be strong with parties sending large amounts, or weak with parties transferring small amounts for exchange. Each individual owns a specific, divisible commodity. Since group members are interested in consuming all the commodities, sharing them (of commodities) is profitable. Links can be established by sending some own commodities to others in the hope that they entertain similar plans. As usual in social dilemma-type situations, it is optimal to keep everything for oneself, although mutual exchange is efficiency enhancing. Exchanging “niceties” is a frequent topic of experimental research, usually focusing on dilemma games like prisoners’ dilemma or public goods games, or sequential gift giving. In our setup, one is not cooperating generally but specifically with several others, which hopefully will shed new light on motives of cooperation in spite of strong free-riding incentives (see Ledyard, 1995 for an earlier survey). One may, for instance, test for symmetric efficiency2 or reciprocity in the form of balanced bilateral exchanges (see e.g. Levati, 2006). In our experiment, the same participants interact for 15 rounds, which is repeated twice after forming new groups. Previous experiences with the same partners should inform participants whom they can trust. We expect stable links to be established after a while and terminated shortly before interaction ends. Groups are randomly rematched between the three rounds with 15 periods each. Participants first play the game twice in groups of three (rounds 1 and 2) and then once in a group of six (round 3). Hence, each participant deals with altogether three different groups, two with only two exchange partners and one with five exchange partners. Symmetric efficiency suggests an obvious behavioral benchmark, namely equal sharing of all commodities by all group members, independent of group size. Nevertheless we expect that, in a larger group, coordination of symmetric exchanges is more problematic since the number of individual transfers is n(n − 1) in a group of size n. By our particular design we hope to answer such research questions as: Will all-inclusive exchange structures (everybody rewards everybody else) evolve, or will only few stable links be observed? Will early attempts of exploitation (keeping too much for oneself) be forgiven later, or will early norm deviations imply isolation (ostracism)? More generally, we explore the formation, maintenance, disruption, and reestablishment of exchange relations. Section 2 relates our study to the literature. In Section 3, the basic model is introduced together with the two benchmarks of general opportunism and symmetric efficiency. The experimental protocol is described in Section 4. We present and analyze the experimental data in Sections 5 and 6. Section 7 concludes. 2. Relation to the literature According to social exchange theory, reciprocal relations develop within structures of mutual dependency due to individually differential access to resources. In the late 1960s, exchange theorists first studied the longitudinal development of exchange relations in the dyad and then focused on more complex exchange networks by asking how social actors’ structural opportunities for exchange with alternative partners affected their power and the use of power. Recently, atten-
2 As in other dilemma situations, common opportunism leads to no cooperation at all, whereas symmetric efficiency requires equal sharing of all commodities.
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tion has shifted to investigating commitment, trust, and norms of distributive justice in exchange relations/networks with a strong experimental tradition.3 The literature, e.g. Molm et al. (2000), distinguishes between negotiated and reciprocal exchange. In negotiated exchange, trade is carried out after successful negotiations, leading to a binding agreement over the terms of trade. Exchanges in this setting are bilateral and mutual. Agreements are voluntary and can thus only be reached if both actors benefit (however unequally); usually no trade occurs if no agreement is reached. Due to mutual commitment power, there is no need for repeated interaction to explain social exchange. Here we focus on reciprocal exchange where the terms of exchange are not negotiated and actors initiate exchange individually (Molm, 1988, 1994, 1997; Michaels and Wiggins, 1976; Molm and Wiggins, 1979). An actor initiates a relation without knowing whether or when another actor is going to reciprocate. At any given exchange opportunity, an actor may give without knowing what she will receive in return. Thus, benefits might flow unilaterally as in other social dilemma situations,4 and an exchange relation takes the form of sequentially dependent acts, with reciprocity and contingency characterizing the dynamics. In this tradition, the major questions are how to initiate and maintain mutually profitable social exchange, and how reciprocal exchange in such repeated interaction will evolve over time. Social exchange theory has extensively examined the patterns of interaction and their longitudinal development. Though Leik (1992) has studied network structures and network power to find out whether actors can add and delete links, Kollock (1994) seems to be the only one to have explored the endogenous evolution of networks “from scratch.” While we try to answer which pattern of stable exchange relations emerges in situations where everyone may interact with everyone else, and how this depends on the size of the society, Kollock (1994) relies on a single transaction per actor per period. A closely related topic is partner selection in repeated exchange.5 Hauk (1999) and Hauk and Nagel (2001) conducted experimental studies exploring the impact of partner selection or outside options in prisoners’ dilemma, but hardly the evolution of interaction structures.6 Exploring group size effects only in one-shot experiments as done, for instance, by Bonacich et al. (1976) or Fox and Guyer (1977) can be very misleading since repeated interaction is crucial in most situations (except for rare spot markets or the like). We therefore explore the dynamic effects of group size variation. To summarize, in most of the related studies, individual choices are more restricted (mostly to induce network externalities).7 Also, previous studies focused on the effect of different (in experiments mostly exogenously given) exchange structures/networks and usually considered rather small (often dyadic) groups. Instead, we vary the group size and explore the endogenous emergence of exchange structures in a setting where repeated interaction, as experimentally implemented by a partner design (the same participants interact repeatedly over 15 periods), is the only institutional prerequisite (for another institutional feature, namely mutual shareholding, see G¨uth et al., 2004). 3
Commitment here means that pairs of social actors choose to exchange repeatedly with the same rather than with varying exchange partners [e.g. due to incomplete trading contracts; see Brown et al., 2004], whereas trust is based on expectations concerning personality traits of the prospective partner (Molm et al., 2000; Kollock, 1994). 4 Kollock (1994) seems to have been the first to have related the negotiated exchange tradition and social dilemma research. 5 See e.g. Hayashi and Yamagishi (1998) for results on computer tournaments. 6 Here subjects’ choices were restricted to one action which is then realized in all maintained relations. 7 We apply a broad notion of ‘network’ as a set of bilateral relations (for a narrower definition, see Yamagishi et al., 1988; Jackson and Wolinsky, 1996; Bala and Goyal, 2000).
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To promote external validity we also limit the gains from cooperation. In our view, many experimental confirmations of voluntary cooperation, e.g. Berg et al. (1995), rely on outrageous efficiency incentives and therefore induce strong demand effects for voluntary cooperation. More moderate and realistic efficiency gains might inspire considerably less voluntary cooperation, if at all. 3. The setup Each of several individuals, belonging to the same group, owns a specific divisible good. Since all individuals like to consume a balanced diet of all goods, there is an incentive to exchange goods as in a barter economy (see Edgeworth, 1881, and more recently Hildenbrand and Kirman, 1988). Actually, symmetric efficiency (formally8 : maximizing an equal monetary payoff for all group members) requires all individuals to consume the same amount of all commodities. Since there are no institutions enabling commitments to trade, exchanging commodities assumes the form of mutual gift giving based on (sequential) reciprocity. By an additive payoff specification (Eq. (A.1) in Appendix A) the efficiency gains are significant but moderately and realistically low. In transferring positive amounts of one’s own commodity to others, one may hope to gain from this but can never be sure to receive something in return. Thus, what can induce reciprocation is only the shadow of the future, i.e., holding back gifts in future rounds. This illustrates that we analyze an almost institution-free setup. All that is required is that the group stays together—an obvious assumption for early hunter and gatherer societies of homo sapiens as well as for many primate species. Experimentally, this is implemented by using a partner design. Sending a gift is costly, i.e., when one individual shares her commodity with another, she must pay the cost of transfer which we assume does not depend on the amount transferred. Whereas (symmetric) efficiency requires all individuals to share all commodities (equally),9 opportunism in the sense of own payoff maximization predicts no sharing at all. Can the pure shadow of the future, i.e., the fact of repeated interaction alone, induce mutual gift giving?10 The experimental scenario provides sufficient incentives for mutual sharing, is easily understood by participants, and excludes network externalities.11 From other social dilemma games like multi-person prisoners’ dilemma or common pool resource or public goods games our setup differs since
8 Unlike in constant pie games, like the dictator or ultimatum game, the payoff sum depends on how it is distributed, i.e., in the game theoretic terminology, one has non-transferrable utilities (all goods that can be freely transferred are commodities). 9 What can be gained by equal sharing depends on several parameters of the formal model (see Appendix A) like the utility parameters, the group size (the number n of individuals), and the fixed cost c of giving a gift. 10 A follow-up study, using the same basic setup, introduces another institutional feature, namely mutual shareholding (G¨uth et al., 2004). 11 To explicitly incorporate network externalities, we could make the cost c of giving a gift to somebody depend on how many gifts the recipient receives in total. Let m(j) = δ(aj (i)) denote the number of players giving positive gifts I=j (aj (i) > 0, implying δ(aj (i)) = 1 and δ(aj (i)) = 0 if aj (i) = 0) to player j = 1, . . . , n. By (i) c = c(m) increasing (in m), we could capture congestion, where the fixed cost of giving a gift increases with the number m of gift givers, modeling a negative network externality or (ii) c = c(m) decreasing (in m) would model a positive network externality. .
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• it allows a discriminatory treatment of others by each individual, i.e., one can react specifically to each other individual, and • the target specifications are “continuous” with a common bound for their sum. The multi-person character of our setup allows, furthermore, for various forms of gift giving like one-sided exchanges, disjunct sharing circles (subgroups) as well as the grand sharing club of all. 4. Experimental design We ran four sessions, three with 24 participants each and one with 18 participants. Let us refer to the 15 consecutive decisions as periods and to the three parts with 15 successive periods each with the same partners as rounds 1, 2, and 3, respectively. Thus, a participant experienced 45 periods split up into three rounds. For rounds 1 and 2, participants were partitioned into groups of size n = 3 (rounds 1 and 2, in which interaction took place in groups of 3, will be named phase 1), and for round 3 into groups of size n = 6 (referred to as phase 2). Between the first and the second round, participants were rematched into new groups of 3, whereas in round 3 participants were rematched into groups of 6. During the first 30 periods each participant had to decide how much of her own commodity to give to the two other members of her group, while she could share the remainder of her endowment with five other members of her group in the final 15 periods. Participants were students from various disciplines of Jena University who were recruited through notices posted around the campus. The experiment was computerized (Z-tree, Fischbacher, 1999). Each participant was seated at a computer terminal which was visually separated from the other terminals. Participants received written instructions, which were read aloud to make them commonly known. During the experiment no communication was permitted. After the experiment, participants were paid their total payoff earned over the 45 periods in cash. Payoffs were quoted in “Experiment Pfennigs” (in the following referred to as “EP”), where two EP were converted into one euro cent at the end of the experiment. Additionally, each participant received a show-up fee of EUR 2.50. The average total payoff earned was EUR 18.20, the minimum EUR 13.38, and the maximum EUR 20.28. Each participant received an endowment of E = 18 units of her own commodity in each period. Let x denote the fraction that is kept and y and z [u, v and w] what is given away in case of a group of 3 [6].12 No restrictions were imposed on how participants could allocate their endowment between themselves and others in their group. If, for example, in a 3-player group, player A keeps x units of her commodity for herself and passes on y (>0) units to B, z (>0) units to C, while she receives y∗ and z∗ units of B’s and C’s commodity, A’s per-period payoff (in EP) is determined by the additive function √ √ ( x + y∗ + z∗ ) × 10 − 2 with decreasing marginal utility of each commodity. Here we subtract 2 since transportation costs were c = 1 EP per transaction. In our view, the additive utility specification is easily understood by participants although it has to be non-linear to implement decreasing marginal utilities (see Appendix A). We provided payoff tables in the instructions (see Appendix A) and a calculator 12 The condition x + y + z = 18 has to be fulfilled in a 3-player group, whereas x + y + z + u + v + w = 18 must hold in a 6-player group, respectively.
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as part of the experimental software for computational help if needed. A disadvantage of this additive payoff function is that it is rather flat, i.e., even larger deviations from equal shares of all commodities yield substantial positive payoffs. An alternative multiplicative, e.g. Cobb–Douglastype payoff function could have avoided this (see Anderhub et al., 2000, who compare both types of payoff functions for a different no strategic allocation problem), but would have rendered equal sharing by proper subgroups rather unattractive. Since we were interested whether and, if so, which subgroups engage in stable voluntary cooperation, we implemented the additive payoff function above. At the end of each period, each participant was informed about the amounts of the different commodities she had received from the members of her group. In addition, all amounts transferred and received by any participant in the three preceding periods were displayed on the computer screen. In the following, we first present aggregate results by averaging data across groups. In a second step, we distinguish different types of groups, based on group averages over the 15 periods in a round. 5. Group size effects 5.1. Average exchange dynamics Efficiency requires to distribute one’s endowment equally among all group members: participants should give 6 units (3 units) of their commodity (endowment) to each of the two (five) other groups members and keep 6 units (3 units) for themselves. The dominant strategy, however, is to keep one’s initial endowment. We measure the individual level of cooperation by the overall amount of units that player i gives to others, i.e., COi = E − ai (i), where a(i) denotes the amount that i keeps. The efficiency level of individual cooperation equals 12 units in a group size of 3 (COEFF3 = 12) or, respectively, 15 units in a group size of 6 (COEFF6 = 15). To compare across group sizes, we rely on the relative level of cooperation (COOP), which is the actual level of cooperation divided by the efficiency level of cooperation, viz., COOPi = COi /COEFFn . Result 1. There is no significant difference in individuals’ average relative level of cooperation in groups of 3 and groups of 6 members. Except for a pronounced endgame effect, there is also no obvious dynamic trend (see Figs. 1 and 2) . To confirm Result 1, we rely on individual data and compare the relative level of cooperation between rounds 2 and 3, averaged over all periods in rounds 2 and 3. Altogether 52 (of 90) players show a higher level of relative cooperation in round 2 than in round 3. The one-sided binomial test with individual averages as observations rejects the null hypothesis that the relative cooperation level in round 3 is at least as high as the one in round 2 at a 9% significance level. But if we only assume independence of groups due to repeated interaction within groups, this null hypothesis is not rejected. Figs. 3 and 4 show that the evolution of average payoff is highly correlated with the level of cooperation presented in the action space. In Fig. 3, the average payoffs are compared with the opportunistic solution 42.43 (green line) and the symmetric efficiency payoff 71.48 (red line). In Fig. 4, the symmetric payoff line 98.92 (red line) is larger than in Fig. 3, while the line of opportunistic payoff remains unchanged. The degree of correlation between payoff and cooperation level is equal to 0.9522 for the n = 3 treatment and 0.9944 for the n = 6 treatment.
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Fig. 1. Level of cooperation, phase 1.
The average amount per transfer starts in round 1 with an amount slightly above 3 units (see Fig. 5). This amount (the average amount sent) then increases to 3.5 units before, at the end of the round, it sharply drops to an amount between 2 and 2.5 units. This endgame effect is also observed at the end of the second round where the average amount given decreases even more sharply to about 1.5 units. The evolution of average amounts (sent) per transfer in groups of 6 by and large exhibits similar features (see Fig. 6). Starting with an average amount per transfer
Fig. 2. Level of cooperation, phase 2.
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Fig. 3. Average payoff, phase 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)
Fig. 4. Average payoff, phase 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)
Fig. 5. Amounts per transfer, phase 1.
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Fig. 6. Amounts per transfer, phase 2.
of slightly above 2 units, this remains almost constant in the following periods, before it sharply falls (to 0.5 units) in the final periods. Compared to rounds 1 and 2 the amount varies less over time. The findings underline that the theoretical benchmark of no gift giving is not supported by our data, except for the final periods. Result 2. Independently of the group size, players do exchange substantial gifts except for the last periods of a round (endgame effect). Result 2 is not very surprising, being in line with the extensive literature on dilemma and particularly public goods experiments. Players, except for the final periods, often avoid the dominant strategy of the base game when this conflicts with the socially efficient optimum. 5.2. Evolution of individual exchange structures The average number of transfers per capita indicates how inclusive the exchange structure is. A player in a group of 3 (6) can initiate at most 2 (5) transfers. High average transfer rates indicate a tendency to build fairly complete or all-inclusive exchange structures, whereas low average transfer rates suggest rational play or at least more exclusive gift exchanges. Finally, large variances in individual transfers to others suggest bilateral reciprocity as the main behavioral force. We first look at the evolution of average transfers in rounds 1 and 2, i.e., in the 3-player group (see Fig. 7). The trend concerning the average number of transfers parallels the trend observed for the level of cooperation. The initial average number of transfers is 1.6 and remains rather stable until an endgame effect is observed. The same holds for round 2. Here the average transfer rate is slightly closer to the socially efficient level of 2. Round 3 behavior exhibits similar features (see Fig. 8). The average number of transfers oscillates around 4 transfers until it drops sharply at the end of the round.
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Fig. 7. Transfers per player, phase 1.
So far we have only looked at the average tendency to support all-inclusive exchange. In the following, we distinguish different transfer types. In a group of 3 (or, respectively, 6), we focus on the percentage of players exerting 0, 1, or 2 (or, respectively, 0, . . . , 5 in groups of 6) transfers in a given period. A constant phenomenon (see Figs. 9 and 10) is that at the beginning of each round there is a significant (rather great) willingness to establish the maximal and efficient number of links which, however, decreases over time. At the end of each round, the number of transfers drops sharply. In groups of 6, the percentage of intermediate numbers of transfers (2 or 3) is rather low and changes only slightly over time.
Fig. 8. Transfers per player, phase 2.
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Fig. 9. Transfer types, phase 1.
Let us now concentrate on two questions of the evolution of network structures: (i) Is there a tendency of subjects to give equally, that is, to give the same positive amount to all other group members? (ii) Concerning the pairwise exchange relations, is there a tendency to exchange the same amounts on a mutual basis?
Fig. 10. Transfer types, phase 2.
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Fig. 11. Nondiscriminate subjects, phase 1.
We try to answer question (i) by Figs. 11 and 12, showing the fraction of those players who transfer an equal (positive) amount to all other group members. These graphical illustrations of the evolution of group members’ willingness to share equally reveals a strong decline from the start to the end of each round. A generally equal treatment is apparently rather fragile and at best initially strong. As units of analysis, let us use pairwise exchanges rather than players to address question (ii). Here we are interested in learning reciprocal behavior, i.e., to bilaterally exchange equal amounts. More specifically, does pairwise exchange of equal amounts increase or decrease over time?
Fig. 12. Nondiscriminate subjects, phase 2
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Fig. 13. Equal bilateral exchanges, phase 1.
Fig. 14. Equal bilateral exchanges, phase 2.
Whereas in round 1 even in the last period some (about 6%) reciprocal exchanges (in the sense of equal bilateral gift giving) persist, the corresponding percentage decreases from rounds 1 to 2, and to round 3. Otherwise the percentage of equal reciprocal exchanges lies always between 2% and 23% (see Figs. 13 and 14). Average percentages increase from 11% (in round 1) to 15% (in round 2), and finally to 18% (in round 3), i.e., if two participants cooperate, they learn to exchange symmetrically.13 13 The relatively low percentage of equal reciprocal exchanges in spite of the symmetry of all players is partly due to the rather moderate efficiency gains by voluntary cooperation, as caused by the additive payoff specification (Eq. (A.1)
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Table 1 Logistic regression with random effects (l h = frequency of equal bilateral exchanges)
Constant Round 2 Round 3
Coefficient
S.E.
−1.3534 0.4711 0.5897
0.2950 0.1331 0.1114
t-stat −4.5875 3.5406 5.2962
Prob(T > |t|) 0.0000 0.0004 0.0000
A logistic regression with random effects on the session level uses the frequency of equal bilateral exchanges (related to the possible number of equal bilateral exchanges in the respective group size) as the dependent variable, and independent rounds as well as period dummies as explanatory variables. The period dummies were introduced to control for differences between periods. Table 1 summarizes the main regression results. We can reject the null hypothesis14 of no difference in the frequency of equal bilateral exchanges between rounds 1 and 2 (likelihood ratio test, p = 0.0189), rounds 1 and 3 (likelihood ratio test, p < 0.0001), and rounds 2 and 3 (likelihood ratio test, p = 0.0267). 5.3. Payoff comparisons We measure deviations from efficiency in payoff rather than in action space by the individual (relative) payoff performance (EFF) in the two group sizes (n ∈ {3, 6}).15 General opportunism in the sense of payoff maximizing behavior yields a (relative) payoff performance of 0%, and (symmetrically) efficient behavior a (relative) payoff performance of 100%. Result 3. The average individual payoff performance is higher in groups of 3 than in groups of 6. This result confirms our intuition that (voluntary) cooperation is more likely in small rather than large groups (see Olson, 1971, and game theoretically Selten, 1973, who for a market setting illustrates that 4 are “few” and 6 are “many”). We use individual data to validate Result 3. Comparing the payoff performance of every player averaged over all periods in rounds 2 and 3 shows that 79 (of 90) players achieve a higher payoff in round 2 than in round 3. The null hypothesis that the payoff performance in round 3 equals the performance in round 2 is rejected (two-sided test, 1% significance). If we only assume independence of groups, the null hypothesis is also rejected (5% significance). Thus, there is a statistically reliable difference in individual payoff performance between rounds 2 and 3 as stated in Result 3. Although there is (except for the endgame effect) no trend in the average payoff performance within a given round, there may be a high payoff variance within groups. If subjects need several periods to “learn” to play the socially efficient solution, this variance might decrease over time. We analyzed the variance of payoffs at the group level. According to our statistical analysis, it
in Appendix A). A multiplicative specification (see Anderhub et al., 2000) would have implied more dramatic efficiency incentives, however, at the cost of ruling out opportunism as a reasonable prediction, i.e., by involving a strong demand effect. 14 The hypothesis of larger differences in individual tranfers (as measured by the difference between the maximal and minimal transfer per member) in the larger groups could not be confirmed (see data in the Appendix A revealing also a slightly different evolution of cooperation in different sessions). 15 Formally, (U (a, n) − U (a∗ , n))/(U (a+ , n) − U (a∗ , n)) (see the formal model in the Appendix A for notation). i i i i
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Table 2 Number of groups with larger payoff variance in the respective interval
3-Player groups (round 2) 6-Player groups (round 3)
Periods 3–7
Periods 8–12
14 1
16 15
Table 3 Thresholds used for group categorization
High Medium Low
Level of cooperation (COOP)
Degree of asymmetry (ASYM)
Payoff performance (EFF)
≥75% 50% < COOP < 75% ≤50%
≥75% 25% < ASYM < 75% ≤25%
≥75% 50% < EFF < 75% ≤50%
turns out that there is a significant difference in the temporal evolution of payoff variance in groups of 3 as compared to groups of 6. Only considering round 2 data for groups of 3 leaves us with a total number of 30 groups of 3 and 16 groups of 6. In addition, we dropped the first two and the last three periods and just compared the payoff variance in periods 3–7 to the payoff variance in periods 8–12 (see Table 2 for details). Almost all large groups show an increase in payoff variance over time, whereas for n = 3 the evolution of variance is ambiguous. 6. Group categorization We have categorized group types, based on aggregates over the 15 periods in each round. More asymmetric distributions of exchange activities across the 3 (in groups of 3) or, respectively, 15 (in groups of 6) possible exchange relations renders a group less efficient. Such asymmetry in exchange seems rather plausible if subjects differ in their reliability and – after some experience with different exchange partners – avoid some and strengthen other exchanges.16 Furthermore, we have differentiated groups on the basis of their level of cooperation (COOP), their degree of asymmetry (ASYM), and their overall payoff performance, i.e., their overall efficiency (EFF). For each of these criteria we have distinguished high, medium, and low performance groups such that the ranges provide an overall plausible picture across criteria and apply irrespective of the group size (Table 3.). Our detailed evaluation of the participating groups according to the criteria developed can be found in an earlier working paper (Berninghaus et al., 2004). It is not reported here since it did not reveal a clear-cut group categorization. 7. Conclusion Social exchange has been modelled as mutual gift giving. Each player owns a specific commodity which she can share voluntarily with all or some other players. As a one-shot interaction, 16
The intuition is as follows: Given a certain level of cooperation (COOP), the degree of asymmetry within a group is maximal if all exchanges in the group are between two group members only. On the other hand, the degree of asymmetry equals zero (the group is perfectly symmetric) if an equal (number) amount of units is exchanged between any two group members. Note that this intragroup asymmetry measure compares the intensity of bilateral exchange relations and does not account for the degree of asymmetry within bilateral exchange relations.
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this would be a social dilemma game like multi-person prisoners’ dilemma or public goods games. Repeating the game in constant groups (partner design) may result in stable mutual gift giving, where efficiency considerations suggest that everybody shares with everybody else. Except for the institutional feature of repeated interaction, i.e., the shadow of the future, no social order is presupposed so that the situation is quite anarchic. According to our data, there is a clear tendency to form all-inclusive social exchange structures, suggested by efficiency, at least initially. Many bilateral exchanges, furthermore, are reciprocal, i.e., one gives as much as one receives. The latter result may partly be due to the a priori symmetry of players who all receive the same endowment and whose commodities determine payoffs in symmetric ways.17 The clear-cut termination or endgame effect reveals that players are not intrinsically motivated to give gifts but that mutual gift giving is just a trade (one gives in order to receive something in return). Thus, voluntary social exchange requires stable group structures and is endangered when members leave the group.18 In our experiment, giving imposes two kinds of cost, namely that of not consuming what one gives to others and cost c of establishing a one-way link to each of those to whom one gives. Behaviorally, this should have encouraged higher levels of gift giving whenever giving occurs. We could have varied these costs in order to assess their impact. We have not done so here since we considered it more important to have sufficiently many groups per treatment. An interesting variation of our design would be to allow to refuse a gift by not establishing the corresponding recipient link. There are many other ways to generalize our prototypical design in order to explore how robust our findings are, for example, by incorporating private information or by allowing for substitutes (obtaining a given commodity from more than one trader).19 Compared to other social dilemma games, like prisoners’ dilemma and public goods games, our design allows individuals to differentiate their behavior, depending on whom they encounter. They may be (non)cooperative in certain interactions and react differently in others. In our view, this captures an important aspect of human interaction which has often been neglected. Especially in modern and usually large societies, voluntary cooperation of all members is an illusion and the formation of smaller solidarity groups the only escape. Our design, more specifically the additive payoff specification,20 provides an easily implementable scenario for exploring whether and, if so, which subgroups will manage to establish some stable form of voluntary cooperation, based on bilateral gift exchanges. In reality, such smaller solidarity groups will be based on some form of special relatedness like kinship bounds (see G¨uth et al., 2004) or neighborhoods. Future studies might focus especially on introducing geographic distance, e.g. investigating whether stable solidarity groups are more likely to form on the basis of past mutual sharing, as explained above, than on being close geographically.
17
In case of asymmetric endowments, “symmetry” could assume the form of equally relative or equally absolute exchanges. More generally, endowments may be private information. 18 Due to the strong and robust termination effects, there is no need for controls by performing stranger treatment experiments since the last round of the partner treatment tells us rather reliably what to expect. 19 G¨ uth et al. (2004) have combined and varied both, repeated interaction as in our study as well as mutual shareholding. 20 A multiplicative payoff function, e.g. of the Cobb–Douglas-type, would have rendered voluntary cooperation by proper subgroups rather unattractive.
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Acknowledgements Valuable assistance of Katinka Pantz, MPI Jena, in conducting our experiments is gratefully acknowledged. Appendix A A.1. The formal model Denote by N = {1, . . . , n} the set of individual players. Each player i ∈ N receives the same integer endowment E = kn with k ≥ 1 of her specific commodity. The only decision of a player i ∈ N is the choice of an allocation: a(i) = (a1 (i), . . . , an (i)) with aj (i) = E j
where all components of a(i) are nonnegative integers. All players simultaneously choose such an allocation a(i) of E. Denote by a = (a(1), . . . , a(n)) the vector of individual strategies a(i). The payoff of a player i ∈ N depends on a via ui (a) = α ai (j)p − c δ(aj (i)) with 1 > p > 0 and α > 0 (A.1) j
where δ(aj (i)) =
j=i
1 0
if aj (i) > 0 for aj (i) = 0
Here c (> 0) is the cost of transfer21 more or less of i’s commodity to another player j = i. We assume c to be rather small. Clearly, since i increases her payoff by reducing the number of own links to other players and by increasing the consumption of her own commodity, the only strictly dominant decision is ai∗ (i) = E. Thus, the payoffs in case of general opportunism in the sense of own payoff maximization are ui (a∗ ) = αEp for all players i, where a∗ denotes the strategy vector when all players choose their dominant strategy. The result does not change essentially if the (normal form) game is repeated finitely often. Actually, all that needs to be assumed is a commonly known finite upper bound for the number of repetitions.22 Then there is the last possible round to which our solution applies. But then ai (i) is optimal also in the last but one round, and so forth. Thus, by backward induction23 there is a unique solution prescribing ai∗ (i) = E constantly for all players i = 1, . . . , n.
21
Since transportation costs are independent of the amount of commodities transferred, they could be interpreted as the costs of establishing a link to another player (connection costs). 22 The assumption always holds in experiments despite the claim to have implemented an infinite horizon as made by Weg et al. (1990). 23 Backward induction can also be described here as repeated elimination of dominated strategies. Since the number of elimination steps (requiring various levels of “all knowing that all know . . . that all are rational”) is finite, this is weaker than commonly known rationality (requiring infinitely many steps of “all knowing that all know . . . that all are rational”).
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Fig. A.1. Absolute (p, c) payoff increase.
As an alternative benchmark, let us explore symmetric efficiency. Since i’s payoff is strictly increasing and strictly concave in her own consumption ai (i), the candidate is a+ = (a+ (1), . . . , a+ (n)),
with aj+ (i) = k for i, j = 1, . . . n
where a+ denotes the commodity vector which results when all players i realize the symmetric exchange pattern aj+ (i) = k, for all j = 1, . . . , n. The payoff implications of this efficiency benchmark a+ are ui (a+ ) = αnkp − (n − 1)c
for all i = 1, . . . , n
In the following, let us denote the efficiency change from a∗ to a+ by D(·). The function D(·) depends on the parameters n, c, and p as D(n, c, p) := αnkp − (n − 1)c − α(nk)p = αkp (n − np ) − (n − 1)c Since we want to confront participants with a social dilemma environment, we focus on sufficiently small levels of c to render D(n, c, p) positive. In Figs. A.1 and A.2, we have illustrated for k = 2, α = 1 as well as n = 3, n = 6, and c = 0.1 how the efficiency increase D(n, c, p) from a∗ to a+ depends on p ∈ (0, 1). Fig. A.1 shows, how the absolute efficiency increase and Fig. A.2 shows, how the relative efficiency increase: RD(n, c, p) :=
D(n, c, p) αnkp − (n − 1)c
depend on p, is shown in Figs. A.1 and A.2, respectively. A.2. Translated instructions You will participate in an experiment which will last for approximately 1 h.24 For your participation you will be paid in cash at the end of the experiment. Your payoff depends partly on 24 We provided subjects with complete instructions for the 3-player group and gave them additional instructions before the start of round 3, spelling out the modifications. Since these are straightforward, we do not include them here.
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Fig. A.2. Relative (p, c) payoff increase.
your own and partly on the decisions of the other participants. The entire experiment is computerized. We provided subjects with complete instructions for the 3-player group and gave them additional instructions before the start of round 3, spelling out the modifications. Since these are straightforward, we do not include them here. We will not provide any information concerning the identity of the other participants you were interacting with in any of the periods. Neither will any other participant find out from us which role you played in the experiment and how much you earned. We will now describe the experimental rules. When all participants have read the instructions, one of the experimenters will read them out aloud once. A.2.1. Groups and assignment of roles The experiment consists of three rounds, each comprising 15 periods. In the first two rounds, participants are assigned to groups of three, and in the last round to groups of six persons. It will not be possible for you to identify those participants that have been assigned to your group among the 24 that are present. Within one group, participants will be denoted as persons A–C (and as persons A–F in round 3, respectively). Your role will be displayed on your screen. Participants are assigned randomly to the different groups. Groups will stay together for the duration of a round (15 periods). Between rounds, participants are randomly assigned to new groups. A.2.2. Your decision At the beginning of each period, each participant is endowed with 18 units of a commodity which no other member of her group owns. More precisely, person A is the only one endowed with commodity A, person B is the only one endowed with commodity B, and person C is the only one endowed with commodity C (the same applies to persons D–F in the final round). In each period, each person can decide whether or not she wants to give any part of her commodity to one or several of the other members in her group, i.e., she can decide to give between 0 and 18 units of her commodity. You make your decision without knowing what you receive from the other group members, while the same holds for them.
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To summarize, you have to take the following decisions in each period: • I keep x units of my commodity for myself. • I give y units of my commodity to one group member. • I give z units of my commodity to the other group member. It is up to you how much you keep or give away. You do not have to give anything to any other person and can keep everything for yourself. It is also possible that you give different amounts of your commodity to different persons in your group. But clearly, only your own endowment is at your disposal. For x, y, and z the following condition has to be satisfied x + y + z = 18 Whenever you give part of your commodity to another member in your group, you incur transportation costs. If you give part of your commodity only to one person, transportation costs equal 1 Experiment Pfennig. If you give part of your commodity to two other group members, transportation costs equal 2 Experiment Pfennigs (if – in a group of 6 – you give part of your commodity to three other group members, costs equal 3 Experiment Pfennigs; if you give part of your commodity to four group members, costs equal 4 Experiment Pfennigs; and correspondingly costs equal 5 Experiment Pfennigs if you give part of your commodity to all other group members in a group of 6). Your payment depends on how much of each commodity type you own at the end of a period. Your payoff per period is the sum of the payoffs derived from your stock of commodity A, commodity B, and commodity C at the end of a period. Let x denote the amount you kept of your own commodity, y∗ and/or z∗ denote the respective amounts you received from the two other group members. The payoff of this period denoted by U is computed according to the following formulas: • If you did not give anything to any other person: √ √ U = ( x + y∗ + z∗ ) × 10Pfg • If you gave something to one other person: √ √ U = ( x + y∗ + z∗ ) × 10Pfg − 1 • If you gave something to two other persons: √ √ U = ( x + y∗ + z∗ ) × 10Pfg − 2
Your payment depends on the amount you kept of your own commodity as well as on the amounts you received from others’ commodities. To help you, the following table provides a translation of different amounts of the different commodities into monetary payoffs (in Pfennigs):
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Amount of commodity (x, y∗ , z∗ )
Payment (in Pfg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0 10 14.1 17.3 20 22.4 24.5 26.5 28.3 30 31.6 33.2 34.6 36.1 37.4 38.7 40 41.2 42.4
In addition, we provide you with a payoff calculator. You can start this program by pressing the keys “Alt” and “Tab” on your keyboard. The payoff calculator can be used by you to enter how many units of your commodity you want to keep, to how many persons you want to give, and which amounts of the other commodities you expect to receive. Accordingly, the calculator will provide you with the corresponding per-period payoff. At the end of each period, every group member is informed about how many commodity units she received from the other group members and how this translates into payoff. On the decision screen beneath the boxes in which you enter how many units to keep or how many to give to which other group members, a reminder is displayed of the amounts you received from each of the other group members for the respective three previous periods. At the end of the 15 periods, every group member is informed about her total payments over all periods. A.2.3. Payment Your total payment consists of the sum of all period payments over the 45 periods, with 2 Experiment Pfennigs corresponding to 1 euro cent. In addition, you will receive a fixed amount of EUR 2.50 for participating in this experiment. In case you have any immediate questions, please raise your hand. Should any questions emerge during the experiment, you can ask one of the experimenters at any time. Please do not ask aloud. References Anderhub, V., G¨uth, W., M¨uller, W., Strobel, M., 2000. An experimental analysis of intertemporal allocation behavior. Experimental Economics 3, 137–152. Bala, V., Goyal, S., 2000. A noncooperative model of network formation. Econometrica 68, 1181–1229. Berg, J., Dickhaut, J., McCabe, K., 1995. Trust, reciprocity, and social history. Games and Economic Behavior 10, 122–142. Berninghaus, S.K., G¨uth, W., Pantz, K., Vogt, B., 2004. Evolution of Spontaneous Social Exchange—An Experimental Study. Discussion Papers on Strategic Interaction 17-2004. Max Planck Institute for Research into Economic Systems.
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Bonacich, P.F., Shure, G.H., Kahan, J.P., Meeker, R.J., 1976. Cooperation and group size in the N-person prisoners’ dilemma. Journal of Conflict Resolution 20, 687–706. Brown, M., Falk, A., Fehr, E., 2004. Relational contracts and the nature of market interactions. Econometrica 72, 747–780. Edgeworth, F.Y., 1881. Mathematical Psychics. P. Kegan, London. Fischbacher, U., 1999. z-Tree – zurich toolbox for readymade economic experiments – ex-perimenter’s manual. Working Paper, 21. Institute for Empirical Research in Economics at the University of Zurich. Fox, J., Guyer, M., 1977. Public choice and cooperation in n-person Prisoners’ dilemma. Journal of Conflict Resolution 22, 469–481. G¨uth, W., Levati, M.V., Von Wangenheim, G., 2004. Relatives Versus Neighbors—An Experiment Studying Spontaneous Social Exchange. Discussion Papers on Strategic Interaction 33-2004. Max Planck Institute for Research into Economic Systems. Hauk, E., 1999. Multiple Prisoner’s Dilemma Games With(out) an Outside Option: An Experimental Study. Mimeo. Hauk, E., Nagel, R., 2001. Choice of partners in multiple two-person prisoner’s dilemma games: an experimental study. Journal of Conflict Resolution 45, 770–793. Hayashi, N., Yamagishi, T., 1998. Selective play: choosing partners in an uncertain world. Personality and Social Psychology Review. Hildenbrand, W., Kirman, A., 1988. Equilibrium Analysis. North-Holland, New York. Jackson, M.O., Wolinsky, A., 1996. A strategic model of social and economic networks. Journal of Economic Theory 71, 44–74. Kollock, P., 1994. The emergence of exchange structures: an experimental study of uncertainty, commitment, and trust. American Journal of Sociology 100, 313–345. Ledyard, J., 1995. Public goods: a survey of experimental research. In: Kagel, J., Roth, A. (Eds.), Handbook of Experimental Economics. Princeton University Press, Princeton, pp. 111–195. Leik, R.K., 1992. New directions for network exchange theory: strategic manipulation of network linkages. Social Networks 14, 309–323. Levati, M.V., 2006. Explaining private provision of public goods by conditional cooperation: an indirect evolutionary approach. Metroeconomica 57, 68–92. Michaels, J.W., Wiggins, J.A., 1976. Effects of mutual dependency and dependency asymmetry on social exchange. Sociometry 39, 368–376. Molm, L.D., 1988. The structure and use of power: a comparison of reward and punishment power. Social Psychology Quarterly 51, 108–122. Molm, L.D., 1994. Dependence and risk: transforming the structure of social exchange. Social Psychology Quarterly 57, 163–176. Molm, L.D., 1997. Risk and power use: constraints on the use of coercion in exchange. American Sociological Review 62, 113–133. Molm, L.D., Takahashi, N., Peterson, G., 2000. Risk and trust in social exchange: an experimental test of a classical proposition. American Journal of Sociology 105, 1396–1427. Molm, L.D., Wiggins, J.A., 1979. A behavioral analysis of the dynamics of social exchange in the dyad. Social Forces 57, 1157–1179. Olson, M., 1971. The logic of collective action. Harvard University Press. Selten, R., 1973. A simple model of imperfect competition, where 4 are few and 6 are many. International Journal of Game Theory 2, 141–201. Weg, E., Rapoport, A., Felsenthal, D.S., 1990. Two-person bargaining behavior in fixed discounting factor games with infinite horizon. Games and Economic Behavior 2, 76–95. Yamagishi, T., Gillmore, M.R., Cook, K.S., 1988. Network connections and the distribution of power in exchange networks. American Journal of Sociology 93, 833–851.
Spontaneous evolution of social exchange—An experimental study Siegfried K. Berninghaus a,∗ , Werner G¨uth b , Bodo Vogt c b
a University of Karlsruhe, Institute WiOR, Zirkel 2, Rechenzentrum, D-76128 Karlsruhe, Germany Max Planck Institute for the Evolution of Economic Systems, Kahlaische Strasse 10, D-07745 Jena, Germany c IMWF, University of Magdeburg, Universit¨ atsplatz 2, D-39106 Magdeburg, Germany
Accepted 1 December 2006
Abstract Each of several exchange partners owns a specific commodity which she can share with others. Unlike in other social dilemma scenarios like prisoners’ dilemma, public goods games, etc., voluntary cooperation relies on bilateral exchanges whose profitabilities are interdependent. How will mutual sharing evolve? Will it include all group members or will smaller groups be more efficient? Our experimental data shed partly new light on older topics: cooperation is now relation specific, allowing for discrimination; group size effects are explored dynamically rather than in one-shot interaction; and, finally, we have weakened demand effects for voluntary cooperation by realistic efficiency gains. © 2007 Elsevier Inc. All rights reserved. JEL Classification: A13; C72; C92 Keywords: Exchange networks; Bilateral reciprocity; Mutual gift giving; Ostracism
1. Introduction Networks are usually established by introducing “links” between several “nodes”, which can be established either individually or in agreement with others (Bala and Goyal, 2000; Jackson and Wolinsky, 1996). Often a link refers to some possibility of communication or transportation.1 In social exchange, a link denotes neither a way of transportation nor communication but rather an ∗
Corresponding author. Tel.: +44 7216083380; fax: +44 7216083391. E-mail addresses: [email protected] (S.K. Berninghaus), [email protected] (W. G¨uth), [email protected] (B. Vogt). 1 There may be a quality of the link, e.g. the safety of a road linking two parties, what we will neglect, however. 1053-5357/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2006.12.064
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exchange of goods. For divisible exchange commodities a link can be strong with parties sending large amounts, or weak with parties transferring small amounts for exchange. Each individual owns a specific, divisible commodity. Since group members are interested in consuming all the commodities, sharing them (of commodities) is profitable. Links can be established by sending some own commodities to others in the hope that they entertain similar plans. As usual in social dilemma-type situations, it is optimal to keep everything for oneself, although mutual exchange is efficiency enhancing. Exchanging “niceties” is a frequent topic of experimental research, usually focusing on dilemma games like prisoners’ dilemma or public goods games, or sequential gift giving. In our setup, one is not cooperating generally but specifically with several others, which hopefully will shed new light on motives of cooperation in spite of strong free-riding incentives (see Ledyard, 1995 for an earlier survey). One may, for instance, test for symmetric efficiency2 or reciprocity in the form of balanced bilateral exchanges (see e.g. Levati, 2006). In our experiment, the same participants interact for 15 rounds, which is repeated twice after forming new groups. Previous experiences with the same partners should inform participants whom they can trust. We expect stable links to be established after a while and terminated shortly before interaction ends. Groups are randomly rematched between the three rounds with 15 periods each. Participants first play the game twice in groups of three (rounds 1 and 2) and then once in a group of six (round 3). Hence, each participant deals with altogether three different groups, two with only two exchange partners and one with five exchange partners. Symmetric efficiency suggests an obvious behavioral benchmark, namely equal sharing of all commodities by all group members, independent of group size. Nevertheless we expect that, in a larger group, coordination of symmetric exchanges is more problematic since the number of individual transfers is n(n − 1) in a group of size n. By our particular design we hope to answer such research questions as: Will all-inclusive exchange structures (everybody rewards everybody else) evolve, or will only few stable links be observed? Will early attempts of exploitation (keeping too much for oneself) be forgiven later, or will early norm deviations imply isolation (ostracism)? More generally, we explore the formation, maintenance, disruption, and reestablishment of exchange relations. Section 2 relates our study to the literature. In Section 3, the basic model is introduced together with the two benchmarks of general opportunism and symmetric efficiency. The experimental protocol is described in Section 4. We present and analyze the experimental data in Sections 5 and 6. Section 7 concludes. 2. Relation to the literature According to social exchange theory, reciprocal relations develop within structures of mutual dependency due to individually differential access to resources. In the late 1960s, exchange theorists first studied the longitudinal development of exchange relations in the dyad and then focused on more complex exchange networks by asking how social actors’ structural opportunities for exchange with alternative partners affected their power and the use of power. Recently, atten-
2 As in other dilemma situations, common opportunism leads to no cooperation at all, whereas symmetric efficiency requires equal sharing of all commodities.
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tion has shifted to investigating commitment, trust, and norms of distributive justice in exchange relations/networks with a strong experimental tradition.3 The literature, e.g. Molm et al. (2000), distinguishes between negotiated and reciprocal exchange. In negotiated exchange, trade is carried out after successful negotiations, leading to a binding agreement over the terms of trade. Exchanges in this setting are bilateral and mutual. Agreements are voluntary and can thus only be reached if both actors benefit (however unequally); usually no trade occurs if no agreement is reached. Due to mutual commitment power, there is no need for repeated interaction to explain social exchange. Here we focus on reciprocal exchange where the terms of exchange are not negotiated and actors initiate exchange individually (Molm, 1988, 1994, 1997; Michaels and Wiggins, 1976; Molm and Wiggins, 1979). An actor initiates a relation without knowing whether or when another actor is going to reciprocate. At any given exchange opportunity, an actor may give without knowing what she will receive in return. Thus, benefits might flow unilaterally as in other social dilemma situations,4 and an exchange relation takes the form of sequentially dependent acts, with reciprocity and contingency characterizing the dynamics. In this tradition, the major questions are how to initiate and maintain mutually profitable social exchange, and how reciprocal exchange in such repeated interaction will evolve over time. Social exchange theory has extensively examined the patterns of interaction and their longitudinal development. Though Leik (1992) has studied network structures and network power to find out whether actors can add and delete links, Kollock (1994) seems to be the only one to have explored the endogenous evolution of networks “from scratch.” While we try to answer which pattern of stable exchange relations emerges in situations where everyone may interact with everyone else, and how this depends on the size of the society, Kollock (1994) relies on a single transaction per actor per period. A closely related topic is partner selection in repeated exchange.5 Hauk (1999) and Hauk and Nagel (2001) conducted experimental studies exploring the impact of partner selection or outside options in prisoners’ dilemma, but hardly the evolution of interaction structures.6 Exploring group size effects only in one-shot experiments as done, for instance, by Bonacich et al. (1976) or Fox and Guyer (1977) can be very misleading since repeated interaction is crucial in most situations (except for rare spot markets or the like). We therefore explore the dynamic effects of group size variation. To summarize, in most of the related studies, individual choices are more restricted (mostly to induce network externalities).7 Also, previous studies focused on the effect of different (in experiments mostly exogenously given) exchange structures/networks and usually considered rather small (often dyadic) groups. Instead, we vary the group size and explore the endogenous emergence of exchange structures in a setting where repeated interaction, as experimentally implemented by a partner design (the same participants interact repeatedly over 15 periods), is the only institutional prerequisite (for another institutional feature, namely mutual shareholding, see G¨uth et al., 2004). 3
Commitment here means that pairs of social actors choose to exchange repeatedly with the same rather than with varying exchange partners [e.g. due to incomplete trading contracts; see Brown et al., 2004], whereas trust is based on expectations concerning personality traits of the prospective partner (Molm et al., 2000; Kollock, 1994). 4 Kollock (1994) seems to have been the first to have related the negotiated exchange tradition and social dilemma research. 5 See e.g. Hayashi and Yamagishi (1998) for results on computer tournaments. 6 Here subjects’ choices were restricted to one action which is then realized in all maintained relations. 7 We apply a broad notion of ‘network’ as a set of bilateral relations (for a narrower definition, see Yamagishi et al., 1988; Jackson and Wolinsky, 1996; Bala and Goyal, 2000).
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To promote external validity we also limit the gains from cooperation. In our view, many experimental confirmations of voluntary cooperation, e.g. Berg et al. (1995), rely on outrageous efficiency incentives and therefore induce strong demand effects for voluntary cooperation. More moderate and realistic efficiency gains might inspire considerably less voluntary cooperation, if at all. 3. The setup Each of several individuals, belonging to the same group, owns a specific divisible good. Since all individuals like to consume a balanced diet of all goods, there is an incentive to exchange goods as in a barter economy (see Edgeworth, 1881, and more recently Hildenbrand and Kirman, 1988). Actually, symmetric efficiency (formally8 : maximizing an equal monetary payoff for all group members) requires all individuals to consume the same amount of all commodities. Since there are no institutions enabling commitments to trade, exchanging commodities assumes the form of mutual gift giving based on (sequential) reciprocity. By an additive payoff specification (Eq. (A.1) in Appendix A) the efficiency gains are significant but moderately and realistically low. In transferring positive amounts of one’s own commodity to others, one may hope to gain from this but can never be sure to receive something in return. Thus, what can induce reciprocation is only the shadow of the future, i.e., holding back gifts in future rounds. This illustrates that we analyze an almost institution-free setup. All that is required is that the group stays together—an obvious assumption for early hunter and gatherer societies of homo sapiens as well as for many primate species. Experimentally, this is implemented by using a partner design. Sending a gift is costly, i.e., when one individual shares her commodity with another, she must pay the cost of transfer which we assume does not depend on the amount transferred. Whereas (symmetric) efficiency requires all individuals to share all commodities (equally),9 opportunism in the sense of own payoff maximization predicts no sharing at all. Can the pure shadow of the future, i.e., the fact of repeated interaction alone, induce mutual gift giving?10 The experimental scenario provides sufficient incentives for mutual sharing, is easily understood by participants, and excludes network externalities.11 From other social dilemma games like multi-person prisoners’ dilemma or common pool resource or public goods games our setup differs since
8 Unlike in constant pie games, like the dictator or ultimatum game, the payoff sum depends on how it is distributed, i.e., in the game theoretic terminology, one has non-transferrable utilities (all goods that can be freely transferred are commodities). 9 What can be gained by equal sharing depends on several parameters of the formal model (see Appendix A) like the utility parameters, the group size (the number n of individuals), and the fixed cost c of giving a gift. 10 A follow-up study, using the same basic setup, introduces another institutional feature, namely mutual shareholding (G¨uth et al., 2004). 11 To explicitly incorporate network externalities, we could make the cost c of giving a gift to somebody depend on how many gifts the recipient receives in total. Let m(j) = δ(aj (i)) denote the number of players giving positive gifts I=j (aj (i) > 0, implying δ(aj (i)) = 1 and δ(aj (i)) = 0 if aj (i) = 0) to player j = 1, . . . , n. By (i) c = c(m) increasing (in m), we could capture congestion, where the fixed cost of giving a gift increases with the number m of gift givers, modeling a negative network externality or (ii) c = c(m) decreasing (in m) would model a positive network externality. .
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• it allows a discriminatory treatment of others by each individual, i.e., one can react specifically to each other individual, and • the target specifications are “continuous” with a common bound for their sum. The multi-person character of our setup allows, furthermore, for various forms of gift giving like one-sided exchanges, disjunct sharing circles (subgroups) as well as the grand sharing club of all. 4. Experimental design We ran four sessions, three with 24 participants each and one with 18 participants. Let us refer to the 15 consecutive decisions as periods and to the three parts with 15 successive periods each with the same partners as rounds 1, 2, and 3, respectively. Thus, a participant experienced 45 periods split up into three rounds. For rounds 1 and 2, participants were partitioned into groups of size n = 3 (rounds 1 and 2, in which interaction took place in groups of 3, will be named phase 1), and for round 3 into groups of size n = 6 (referred to as phase 2). Between the first and the second round, participants were rematched into new groups of 3, whereas in round 3 participants were rematched into groups of 6. During the first 30 periods each participant had to decide how much of her own commodity to give to the two other members of her group, while she could share the remainder of her endowment with five other members of her group in the final 15 periods. Participants were students from various disciplines of Jena University who were recruited through notices posted around the campus. The experiment was computerized (Z-tree, Fischbacher, 1999). Each participant was seated at a computer terminal which was visually separated from the other terminals. Participants received written instructions, which were read aloud to make them commonly known. During the experiment no communication was permitted. After the experiment, participants were paid their total payoff earned over the 45 periods in cash. Payoffs were quoted in “Experiment Pfennigs” (in the following referred to as “EP”), where two EP were converted into one euro cent at the end of the experiment. Additionally, each participant received a show-up fee of EUR 2.50. The average total payoff earned was EUR 18.20, the minimum EUR 13.38, and the maximum EUR 20.28. Each participant received an endowment of E = 18 units of her own commodity in each period. Let x denote the fraction that is kept and y and z [u, v and w] what is given away in case of a group of 3 [6].12 No restrictions were imposed on how participants could allocate their endowment between themselves and others in their group. If, for example, in a 3-player group, player A keeps x units of her commodity for herself and passes on y (>0) units to B, z (>0) units to C, while she receives y∗ and z∗ units of B’s and C’s commodity, A’s per-period payoff (in EP) is determined by the additive function √ √ ( x + y∗ + z∗ ) × 10 − 2 with decreasing marginal utility of each commodity. Here we subtract 2 since transportation costs were c = 1 EP per transaction. In our view, the additive utility specification is easily understood by participants although it has to be non-linear to implement decreasing marginal utilities (see Appendix A). We provided payoff tables in the instructions (see Appendix A) and a calculator 12 The condition x + y + z = 18 has to be fulfilled in a 3-player group, whereas x + y + z + u + v + w = 18 must hold in a 6-player group, respectively.
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as part of the experimental software for computational help if needed. A disadvantage of this additive payoff function is that it is rather flat, i.e., even larger deviations from equal shares of all commodities yield substantial positive payoffs. An alternative multiplicative, e.g. Cobb–Douglastype payoff function could have avoided this (see Anderhub et al., 2000, who compare both types of payoff functions for a different no strategic allocation problem), but would have rendered equal sharing by proper subgroups rather unattractive. Since we were interested whether and, if so, which subgroups engage in stable voluntary cooperation, we implemented the additive payoff function above. At the end of each period, each participant was informed about the amounts of the different commodities she had received from the members of her group. In addition, all amounts transferred and received by any participant in the three preceding periods were displayed on the computer screen. In the following, we first present aggregate results by averaging data across groups. In a second step, we distinguish different types of groups, based on group averages over the 15 periods in a round. 5. Group size effects 5.1. Average exchange dynamics Efficiency requires to distribute one’s endowment equally among all group members: participants should give 6 units (3 units) of their commodity (endowment) to each of the two (five) other groups members and keep 6 units (3 units) for themselves. The dominant strategy, however, is to keep one’s initial endowment. We measure the individual level of cooperation by the overall amount of units that player i gives to others, i.e., COi = E − ai (i), where a(i) denotes the amount that i keeps. The efficiency level of individual cooperation equals 12 units in a group size of 3 (COEFF3 = 12) or, respectively, 15 units in a group size of 6 (COEFF6 = 15). To compare across group sizes, we rely on the relative level of cooperation (COOP), which is the actual level of cooperation divided by the efficiency level of cooperation, viz., COOPi = COi /COEFFn . Result 1. There is no significant difference in individuals’ average relative level of cooperation in groups of 3 and groups of 6 members. Except for a pronounced endgame effect, there is also no obvious dynamic trend (see Figs. 1 and 2) . To confirm Result 1, we rely on individual data and compare the relative level of cooperation between rounds 2 and 3, averaged over all periods in rounds 2 and 3. Altogether 52 (of 90) players show a higher level of relative cooperation in round 2 than in round 3. The one-sided binomial test with individual averages as observations rejects the null hypothesis that the relative cooperation level in round 3 is at least as high as the one in round 2 at a 9% significance level. But if we only assume independence of groups due to repeated interaction within groups, this null hypothesis is not rejected. Figs. 3 and 4 show that the evolution of average payoff is highly correlated with the level of cooperation presented in the action space. In Fig. 3, the average payoffs are compared with the opportunistic solution 42.43 (green line) and the symmetric efficiency payoff 71.48 (red line). In Fig. 4, the symmetric payoff line 98.92 (red line) is larger than in Fig. 3, while the line of opportunistic payoff remains unchanged. The degree of correlation between payoff and cooperation level is equal to 0.9522 for the n = 3 treatment and 0.9944 for the n = 6 treatment.
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Fig. 1. Level of cooperation, phase 1.
The average amount per transfer starts in round 1 with an amount slightly above 3 units (see Fig. 5). This amount (the average amount sent) then increases to 3.5 units before, at the end of the round, it sharply drops to an amount between 2 and 2.5 units. This endgame effect is also observed at the end of the second round where the average amount given decreases even more sharply to about 1.5 units. The evolution of average amounts (sent) per transfer in groups of 6 by and large exhibits similar features (see Fig. 6). Starting with an average amount per transfer
Fig. 2. Level of cooperation, phase 2.
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Fig. 3. Average payoff, phase 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)
Fig. 4. Average payoff, phase 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)
Fig. 5. Amounts per transfer, phase 1.
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Fig. 6. Amounts per transfer, phase 2.
of slightly above 2 units, this remains almost constant in the following periods, before it sharply falls (to 0.5 units) in the final periods. Compared to rounds 1 and 2 the amount varies less over time. The findings underline that the theoretical benchmark of no gift giving is not supported by our data, except for the final periods. Result 2. Independently of the group size, players do exchange substantial gifts except for the last periods of a round (endgame effect). Result 2 is not very surprising, being in line with the extensive literature on dilemma and particularly public goods experiments. Players, except for the final periods, often avoid the dominant strategy of the base game when this conflicts with the socially efficient optimum. 5.2. Evolution of individual exchange structures The average number of transfers per capita indicates how inclusive the exchange structure is. A player in a group of 3 (6) can initiate at most 2 (5) transfers. High average transfer rates indicate a tendency to build fairly complete or all-inclusive exchange structures, whereas low average transfer rates suggest rational play or at least more exclusive gift exchanges. Finally, large variances in individual transfers to others suggest bilateral reciprocity as the main behavioral force. We first look at the evolution of average transfers in rounds 1 and 2, i.e., in the 3-player group (see Fig. 7). The trend concerning the average number of transfers parallels the trend observed for the level of cooperation. The initial average number of transfers is 1.6 and remains rather stable until an endgame effect is observed. The same holds for round 2. Here the average transfer rate is slightly closer to the socially efficient level of 2. Round 3 behavior exhibits similar features (see Fig. 8). The average number of transfers oscillates around 4 transfers until it drops sharply at the end of the round.
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Fig. 7. Transfers per player, phase 1.
So far we have only looked at the average tendency to support all-inclusive exchange. In the following, we distinguish different transfer types. In a group of 3 (or, respectively, 6), we focus on the percentage of players exerting 0, 1, or 2 (or, respectively, 0, . . . , 5 in groups of 6) transfers in a given period. A constant phenomenon (see Figs. 9 and 10) is that at the beginning of each round there is a significant (rather great) willingness to establish the maximal and efficient number of links which, however, decreases over time. At the end of each round, the number of transfers drops sharply. In groups of 6, the percentage of intermediate numbers of transfers (2 or 3) is rather low and changes only slightly over time.
Fig. 8. Transfers per player, phase 2.
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Fig. 9. Transfer types, phase 1.
Let us now concentrate on two questions of the evolution of network structures: (i) Is there a tendency of subjects to give equally, that is, to give the same positive amount to all other group members? (ii) Concerning the pairwise exchange relations, is there a tendency to exchange the same amounts on a mutual basis?
Fig. 10. Transfer types, phase 2.
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Fig. 11. Nondiscriminate subjects, phase 1.
We try to answer question (i) by Figs. 11 and 12, showing the fraction of those players who transfer an equal (positive) amount to all other group members. These graphical illustrations of the evolution of group members’ willingness to share equally reveals a strong decline from the start to the end of each round. A generally equal treatment is apparently rather fragile and at best initially strong. As units of analysis, let us use pairwise exchanges rather than players to address question (ii). Here we are interested in learning reciprocal behavior, i.e., to bilaterally exchange equal amounts. More specifically, does pairwise exchange of equal amounts increase or decrease over time?
Fig. 12. Nondiscriminate subjects, phase 2
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Fig. 13. Equal bilateral exchanges, phase 1.
Fig. 14. Equal bilateral exchanges, phase 2.
Whereas in round 1 even in the last period some (about 6%) reciprocal exchanges (in the sense of equal bilateral gift giving) persist, the corresponding percentage decreases from rounds 1 to 2, and to round 3. Otherwise the percentage of equal reciprocal exchanges lies always between 2% and 23% (see Figs. 13 and 14). Average percentages increase from 11% (in round 1) to 15% (in round 2), and finally to 18% (in round 3), i.e., if two participants cooperate, they learn to exchange symmetrically.13 13 The relatively low percentage of equal reciprocal exchanges in spite of the symmetry of all players is partly due to the rather moderate efficiency gains by voluntary cooperation, as caused by the additive payoff specification (Eq. (A.1)
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Table 1 Logistic regression with random effects (l h = frequency of equal bilateral exchanges)
Constant Round 2 Round 3
Coefficient
S.E.
−1.3534 0.4711 0.5897
0.2950 0.1331 0.1114
t-stat −4.5875 3.5406 5.2962
Prob(T > |t|) 0.0000 0.0004 0.0000
A logistic regression with random effects on the session level uses the frequency of equal bilateral exchanges (related to the possible number of equal bilateral exchanges in the respective group size) as the dependent variable, and independent rounds as well as period dummies as explanatory variables. The period dummies were introduced to control for differences between periods. Table 1 summarizes the main regression results. We can reject the null hypothesis14 of no difference in the frequency of equal bilateral exchanges between rounds 1 and 2 (likelihood ratio test, p = 0.0189), rounds 1 and 3 (likelihood ratio test, p < 0.0001), and rounds 2 and 3 (likelihood ratio test, p = 0.0267). 5.3. Payoff comparisons We measure deviations from efficiency in payoff rather than in action space by the individual (relative) payoff performance (EFF) in the two group sizes (n ∈ {3, 6}).15 General opportunism in the sense of payoff maximizing behavior yields a (relative) payoff performance of 0%, and (symmetrically) efficient behavior a (relative) payoff performance of 100%. Result 3. The average individual payoff performance is higher in groups of 3 than in groups of 6. This result confirms our intuition that (voluntary) cooperation is more likely in small rather than large groups (see Olson, 1971, and game theoretically Selten, 1973, who for a market setting illustrates that 4 are “few” and 6 are “many”). We use individual data to validate Result 3. Comparing the payoff performance of every player averaged over all periods in rounds 2 and 3 shows that 79 (of 90) players achieve a higher payoff in round 2 than in round 3. The null hypothesis that the payoff performance in round 3 equals the performance in round 2 is rejected (two-sided test, 1% significance). If we only assume independence of groups, the null hypothesis is also rejected (5% significance). Thus, there is a statistically reliable difference in individual payoff performance between rounds 2 and 3 as stated in Result 3. Although there is (except for the endgame effect) no trend in the average payoff performance within a given round, there may be a high payoff variance within groups. If subjects need several periods to “learn” to play the socially efficient solution, this variance might decrease over time. We analyzed the variance of payoffs at the group level. According to our statistical analysis, it
in Appendix A). A multiplicative specification (see Anderhub et al., 2000) would have implied more dramatic efficiency incentives, however, at the cost of ruling out opportunism as a reasonable prediction, i.e., by involving a strong demand effect. 14 The hypothesis of larger differences in individual tranfers (as measured by the difference between the maximal and minimal transfer per member) in the larger groups could not be confirmed (see data in the Appendix A revealing also a slightly different evolution of cooperation in different sessions). 15 Formally, (U (a, n) − U (a∗ , n))/(U (a+ , n) − U (a∗ , n)) (see the formal model in the Appendix A for notation). i i i i
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Table 2 Number of groups with larger payoff variance in the respective interval
3-Player groups (round 2) 6-Player groups (round 3)
Periods 3–7
Periods 8–12
14 1
16 15
Table 3 Thresholds used for group categorization
High Medium Low
Level of cooperation (COOP)
Degree of asymmetry (ASYM)
Payoff performance (EFF)
≥75% 50% < COOP < 75% ≤50%
≥75% 25% < ASYM < 75% ≤25%
≥75% 50% < EFF < 75% ≤50%
turns out that there is a significant difference in the temporal evolution of payoff variance in groups of 3 as compared to groups of 6. Only considering round 2 data for groups of 3 leaves us with a total number of 30 groups of 3 and 16 groups of 6. In addition, we dropped the first two and the last three periods and just compared the payoff variance in periods 3–7 to the payoff variance in periods 8–12 (see Table 2 for details). Almost all large groups show an increase in payoff variance over time, whereas for n = 3 the evolution of variance is ambiguous. 6. Group categorization We have categorized group types, based on aggregates over the 15 periods in each round. More asymmetric distributions of exchange activities across the 3 (in groups of 3) or, respectively, 15 (in groups of 6) possible exchange relations renders a group less efficient. Such asymmetry in exchange seems rather plausible if subjects differ in their reliability and – after some experience with different exchange partners – avoid some and strengthen other exchanges.16 Furthermore, we have differentiated groups on the basis of their level of cooperation (COOP), their degree of asymmetry (ASYM), and their overall payoff performance, i.e., their overall efficiency (EFF). For each of these criteria we have distinguished high, medium, and low performance groups such that the ranges provide an overall plausible picture across criteria and apply irrespective of the group size (Table 3.). Our detailed evaluation of the participating groups according to the criteria developed can be found in an earlier working paper (Berninghaus et al., 2004). It is not reported here since it did not reveal a clear-cut group categorization. 7. Conclusion Social exchange has been modelled as mutual gift giving. Each player owns a specific commodity which she can share voluntarily with all or some other players. As a one-shot interaction, 16
The intuition is as follows: Given a certain level of cooperation (COOP), the degree of asymmetry within a group is maximal if all exchanges in the group are between two group members only. On the other hand, the degree of asymmetry equals zero (the group is perfectly symmetric) if an equal (number) amount of units is exchanged between any two group members. Note that this intragroup asymmetry measure compares the intensity of bilateral exchange relations and does not account for the degree of asymmetry within bilateral exchange relations.
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this would be a social dilemma game like multi-person prisoners’ dilemma or public goods games. Repeating the game in constant groups (partner design) may result in stable mutual gift giving, where efficiency considerations suggest that everybody shares with everybody else. Except for the institutional feature of repeated interaction, i.e., the shadow of the future, no social order is presupposed so that the situation is quite anarchic. According to our data, there is a clear tendency to form all-inclusive social exchange structures, suggested by efficiency, at least initially. Many bilateral exchanges, furthermore, are reciprocal, i.e., one gives as much as one receives. The latter result may partly be due to the a priori symmetry of players who all receive the same endowment and whose commodities determine payoffs in symmetric ways.17 The clear-cut termination or endgame effect reveals that players are not intrinsically motivated to give gifts but that mutual gift giving is just a trade (one gives in order to receive something in return). Thus, voluntary social exchange requires stable group structures and is endangered when members leave the group.18 In our experiment, giving imposes two kinds of cost, namely that of not consuming what one gives to others and cost c of establishing a one-way link to each of those to whom one gives. Behaviorally, this should have encouraged higher levels of gift giving whenever giving occurs. We could have varied these costs in order to assess their impact. We have not done so here since we considered it more important to have sufficiently many groups per treatment. An interesting variation of our design would be to allow to refuse a gift by not establishing the corresponding recipient link. There are many other ways to generalize our prototypical design in order to explore how robust our findings are, for example, by incorporating private information or by allowing for substitutes (obtaining a given commodity from more than one trader).19 Compared to other social dilemma games, like prisoners’ dilemma and public goods games, our design allows individuals to differentiate their behavior, depending on whom they encounter. They may be (non)cooperative in certain interactions and react differently in others. In our view, this captures an important aspect of human interaction which has often been neglected. Especially in modern and usually large societies, voluntary cooperation of all members is an illusion and the formation of smaller solidarity groups the only escape. Our design, more specifically the additive payoff specification,20 provides an easily implementable scenario for exploring whether and, if so, which subgroups will manage to establish some stable form of voluntary cooperation, based on bilateral gift exchanges. In reality, such smaller solidarity groups will be based on some form of special relatedness like kinship bounds (see G¨uth et al., 2004) or neighborhoods. Future studies might focus especially on introducing geographic distance, e.g. investigating whether stable solidarity groups are more likely to form on the basis of past mutual sharing, as explained above, than on being close geographically.
17
In case of asymmetric endowments, “symmetry” could assume the form of equally relative or equally absolute exchanges. More generally, endowments may be private information. 18 Due to the strong and robust termination effects, there is no need for controls by performing stranger treatment experiments since the last round of the partner treatment tells us rather reliably what to expect. 19 G¨ uth et al. (2004) have combined and varied both, repeated interaction as in our study as well as mutual shareholding. 20 A multiplicative payoff function, e.g. of the Cobb–Douglas-type, would have rendered voluntary cooperation by proper subgroups rather unattractive.
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Acknowledgements Valuable assistance of Katinka Pantz, MPI Jena, in conducting our experiments is gratefully acknowledged. Appendix A A.1. The formal model Denote by N = {1, . . . , n} the set of individual players. Each player i ∈ N receives the same integer endowment E = kn with k ≥ 1 of her specific commodity. The only decision of a player i ∈ N is the choice of an allocation: a(i) = (a1 (i), . . . , an (i)) with aj (i) = E j
where all components of a(i) are nonnegative integers. All players simultaneously choose such an allocation a(i) of E. Denote by a = (a(1), . . . , a(n)) the vector of individual strategies a(i). The payoff of a player i ∈ N depends on a via ui (a) = α ai (j)p − c δ(aj (i)) with 1 > p > 0 and α > 0 (A.1) j
where δ(aj (i)) =
j=i
1 0
if aj (i) > 0 for aj (i) = 0
Here c (> 0) is the cost of transfer21 more or less of i’s commodity to another player j = i. We assume c to be rather small. Clearly, since i increases her payoff by reducing the number of own links to other players and by increasing the consumption of her own commodity, the only strictly dominant decision is ai∗ (i) = E. Thus, the payoffs in case of general opportunism in the sense of own payoff maximization are ui (a∗ ) = αEp for all players i, where a∗ denotes the strategy vector when all players choose their dominant strategy. The result does not change essentially if the (normal form) game is repeated finitely often. Actually, all that needs to be assumed is a commonly known finite upper bound for the number of repetitions.22 Then there is the last possible round to which our solution applies. But then ai (i) is optimal also in the last but one round, and so forth. Thus, by backward induction23 there is a unique solution prescribing ai∗ (i) = E constantly for all players i = 1, . . . , n.
21
Since transportation costs are independent of the amount of commodities transferred, they could be interpreted as the costs of establishing a link to another player (connection costs). 22 The assumption always holds in experiments despite the claim to have implemented an infinite horizon as made by Weg et al. (1990). 23 Backward induction can also be described here as repeated elimination of dominated strategies. Since the number of elimination steps (requiring various levels of “all knowing that all know . . . that all are rational”) is finite, this is weaker than commonly known rationality (requiring infinitely many steps of “all knowing that all know . . . that all are rational”).
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Fig. A.1. Absolute (p, c) payoff increase.
As an alternative benchmark, let us explore symmetric efficiency. Since i’s payoff is strictly increasing and strictly concave in her own consumption ai (i), the candidate is a+ = (a+ (1), . . . , a+ (n)),
with aj+ (i) = k for i, j = 1, . . . n
where a+ denotes the commodity vector which results when all players i realize the symmetric exchange pattern aj+ (i) = k, for all j = 1, . . . , n. The payoff implications of this efficiency benchmark a+ are ui (a+ ) = αnkp − (n − 1)c
for all i = 1, . . . , n
In the following, let us denote the efficiency change from a∗ to a+ by D(·). The function D(·) depends on the parameters n, c, and p as D(n, c, p) := αnkp − (n − 1)c − α(nk)p = αkp (n − np ) − (n − 1)c Since we want to confront participants with a social dilemma environment, we focus on sufficiently small levels of c to render D(n, c, p) positive. In Figs. A.1 and A.2, we have illustrated for k = 2, α = 1 as well as n = 3, n = 6, and c = 0.1 how the efficiency increase D(n, c, p) from a∗ to a+ depends on p ∈ (0, 1). Fig. A.1 shows, how the absolute efficiency increase and Fig. A.2 shows, how the relative efficiency increase: RD(n, c, p) :=
D(n, c, p) αnkp − (n − 1)c
depend on p, is shown in Figs. A.1 and A.2, respectively. A.2. Translated instructions You will participate in an experiment which will last for approximately 1 h.24 For your participation you will be paid in cash at the end of the experiment. Your payoff depends partly on 24 We provided subjects with complete instructions for the 3-player group and gave them additional instructions before the start of round 3, spelling out the modifications. Since these are straightforward, we do not include them here.
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Fig. A.2. Relative (p, c) payoff increase.
your own and partly on the decisions of the other participants. The entire experiment is computerized. We provided subjects with complete instructions for the 3-player group and gave them additional instructions before the start of round 3, spelling out the modifications. Since these are straightforward, we do not include them here. We will not provide any information concerning the identity of the other participants you were interacting with in any of the periods. Neither will any other participant find out from us which role you played in the experiment and how much you earned. We will now describe the experimental rules. When all participants have read the instructions, one of the experimenters will read them out aloud once. A.2.1. Groups and assignment of roles The experiment consists of three rounds, each comprising 15 periods. In the first two rounds, participants are assigned to groups of three, and in the last round to groups of six persons. It will not be possible for you to identify those participants that have been assigned to your group among the 24 that are present. Within one group, participants will be denoted as persons A–C (and as persons A–F in round 3, respectively). Your role will be displayed on your screen. Participants are assigned randomly to the different groups. Groups will stay together for the duration of a round (15 periods). Between rounds, participants are randomly assigned to new groups. A.2.2. Your decision At the beginning of each period, each participant is endowed with 18 units of a commodity which no other member of her group owns. More precisely, person A is the only one endowed with commodity A, person B is the only one endowed with commodity B, and person C is the only one endowed with commodity C (the same applies to persons D–F in the final round). In each period, each person can decide whether or not she wants to give any part of her commodity to one or several of the other members in her group, i.e., she can decide to give between 0 and 18 units of her commodity. You make your decision without knowing what you receive from the other group members, while the same holds for them.
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To summarize, you have to take the following decisions in each period: • I keep x units of my commodity for myself. • I give y units of my commodity to one group member. • I give z units of my commodity to the other group member. It is up to you how much you keep or give away. You do not have to give anything to any other person and can keep everything for yourself. It is also possible that you give different amounts of your commodity to different persons in your group. But clearly, only your own endowment is at your disposal. For x, y, and z the following condition has to be satisfied x + y + z = 18 Whenever you give part of your commodity to another member in your group, you incur transportation costs. If you give part of your commodity only to one person, transportation costs equal 1 Experiment Pfennig. If you give part of your commodity to two other group members, transportation costs equal 2 Experiment Pfennigs (if – in a group of 6 – you give part of your commodity to three other group members, costs equal 3 Experiment Pfennigs; if you give part of your commodity to four group members, costs equal 4 Experiment Pfennigs; and correspondingly costs equal 5 Experiment Pfennigs if you give part of your commodity to all other group members in a group of 6). Your payment depends on how much of each commodity type you own at the end of a period. Your payoff per period is the sum of the payoffs derived from your stock of commodity A, commodity B, and commodity C at the end of a period. Let x denote the amount you kept of your own commodity, y∗ and/or z∗ denote the respective amounts you received from the two other group members. The payoff of this period denoted by U is computed according to the following formulas: • If you did not give anything to any other person: √ √ U = ( x + y∗ + z∗ ) × 10Pfg • If you gave something to one other person: √ √ U = ( x + y∗ + z∗ ) × 10Pfg − 1 • If you gave something to two other persons: √ √ U = ( x + y∗ + z∗ ) × 10Pfg − 2
Your payment depends on the amount you kept of your own commodity as well as on the amounts you received from others’ commodities. To help you, the following table provides a translation of different amounts of the different commodities into monetary payoffs (in Pfennigs):
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Amount of commodity (x, y∗ , z∗ )
Payment (in Pfg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0 10 14.1 17.3 20 22.4 24.5 26.5 28.3 30 31.6 33.2 34.6 36.1 37.4 38.7 40 41.2 42.4
In addition, we provide you with a payoff calculator. You can start this program by pressing the keys “Alt” and “Tab” on your keyboard. The payoff calculator can be used by you to enter how many units of your commodity you want to keep, to how many persons you want to give, and which amounts of the other commodities you expect to receive. Accordingly, the calculator will provide you with the corresponding per-period payoff. At the end of each period, every group member is informed about how many commodity units she received from the other group members and how this translates into payoff. On the decision screen beneath the boxes in which you enter how many units to keep or how many to give to which other group members, a reminder is displayed of the amounts you received from each of the other group members for the respective three previous periods. At the end of the 15 periods, every group member is informed about her total payments over all periods. A.2.3. Payment Your total payment consists of the sum of all period payments over the 45 periods, with 2 Experiment Pfennigs corresponding to 1 euro cent. In addition, you will receive a fixed amount of EUR 2.50 for participating in this experiment. In case you have any immediate questions, please raise your hand. Should any questions emerge during the experiment, you can ask one of the experimenters at any time. Please do not ask aloud. References Anderhub, V., G¨uth, W., M¨uller, W., Strobel, M., 2000. An experimental analysis of intertemporal allocation behavior. Experimental Economics 3, 137–152. Bala, V., Goyal, S., 2000. A noncooperative model of network formation. Econometrica 68, 1181–1229. Berg, J., Dickhaut, J., McCabe, K., 1995. Trust, reciprocity, and social history. Games and Economic Behavior 10, 122–142. Berninghaus, S.K., G¨uth, W., Pantz, K., Vogt, B., 2004. Evolution of Spontaneous Social Exchange—An Experimental Study. Discussion Papers on Strategic Interaction 17-2004. Max Planck Institute for Research into Economic Systems.
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