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# The agencies method for coalition formation in experimental games

Contributed by John F. Nash, Jr., September 20, 2012 (sent for review April 22, 2012)

## Abstract

In society, power is often transferred to another person or group. A previous work studied the evolution of cooperation among robot players through a coalition formation game with a non-cooperative procedure of acceptance of an agency of another player. Motivated by this previous work, we conduct a laboratory experiment on finitely repeated three-person coalition formation games. Human players with different strength according to the coalition payoffs can accept a transfer of power to another player, the agent, who then distributes the coalition payoffs. We find that the agencies method for coalition formation is quite successful in promoting efficiency. However, the agent faces a tension between short-term incentives of not equally distributing the coalition payoff and the long-term concern to keep cooperation going. In a given round, the strong player in our experiment often resolves this tension approximately in line with the Shapley value and the nucleolus. Yet aggregated over all rounds, the payoff differences between players are rather small, and the equal division of payoffs predicts about 80% of all groups best. One reason is that the voting procedure appears to induce a balance of power, independent of the individual player's strength: Selfish subjects tend to be voted out of their agency and are further disciplined by reciprocal behaviors.

The evolution of human altruism and cooperation is a puzzle. Unlike other animals, people frequently cooperate even absent of any material or reputational incentive to do so. In this paper we show how a voting procedure to transfer power to another person successfully promotes cooperation by balancing the tension between short-term incentives to defect and long-term incentives to keep cooperation going. Our work is inspired by John Nash (1), who theoretically studied the evolution of cooperation among robot players through acceptance of an agency of another player.

Beyond Nash’s (1) work, there is virtually no work on the agencies method in (experimental) economics as we apply it in our paper. The underlying idea is simple and important: Human subjects can transfer the power to an agency, who determines the final payoff distribution within the group.* Our game reflects that, often, efficiency requires people’s willingness to accept the agency of others, such as political, social, or economic leaders (for voting of an expert, see ref. 7).

In Nash’s (1) work, the robots employed optimal strategies, being the computational result of complex systems of equations. Motivated by Nash’s paper, we study laboratory three-person coalition formation games with a non-cooperative procedure of acceptance of an agency of another player. The base games are finitely repeated for 40 rounds with the same three subjects, allowing cooperation and coordination to evolve. In our games non-cooperative game theory cannot organize behavior because it is basically consistent with any outcome. Thus, even the strategies of fully rational agents cannot be predicted by the theory.^{†} We show, however, that the solution concepts of cooperative game theory together with the equal split solution provide some structure on the emergence of cooperation in our experiment. Yet understanding how cooperation is affected by decisions to transfer power to others requires theories that go beyond these approaches.

More specifically, our model specifies the coalition formation process in extensive form (for more details see *Methods* and Fig. 5). It consists of a coalition formation phase, and second phase in which the final agent distributes the coalition value. A given characteristic function specifies a value for all possible coalitions (Table 1 shows the 10 three-person characteristic function games used in the experiment). In phase 1 each player of a group of three can accept at most one other player as an agent to form a pair. In case nobody accepts, the phase is repeated until a pair is formed or a random break-up rule leads to zero payoffs for all. If one pair is formed, the accepting player becomes inactive and is represented by the accepted player who enters phase 2 together with the remaining player. In case more than one pair is formed in phase 1, a random draw decides which pair is decisive for the next step. In phase 2, each of the two active players has to decide whether to accept the other active player. In case no one accepts, the stage is repeated until a player accepts or a random break-up rule selects the pair of stage 1 as the final coalition. In the latter case the accepted player of stage 1 divides the value of his two-person coalition. In case a player accepts in the second stage, the accepted player divides the three-person coalition value among the three players. If there are two accepted players in the second stage, a random draw selects which of the two players can divide the coalition value.

For a long time the focus of attention in the analysis of coalition formation has been cooperative game theory. One underlying idea of cooperative game theory is that there are no restrictions on how agreements can be reached among players. The coalition formation process, making offers and counteroffers, can thus remain largely unspecified. In our cooperative games it is to the joint benefit of the group to form the grand coalition. Yet, which allocation of the corresponding coalition value is agreeable to all players? We mostly focus on three solution concepts of cooperative game theory: the core, Shapley value, and nucleolus.

Loosely speaking, divisions of the total return are called points of the core if they are stable in the sense that no coalition should have the desire and power to upset the agreement. If, e.g., a coalition of two members is assigned a smaller total payoff in the grand coalition proposal than what the two members coalition can achieve alone, this proposal is inherently instable and thus not in the core. However, one problem is that the core can consist of many points without distinguishing a preferred point, or it may even be empty. The Shapley value (22), on the other hand, assigns to each player a unique payoff (“value”), which may be interpreted as a measure of power of the respective player in the game. One way of arriving at the Shapley value is to suppose that the grand coalition is formed by each player entering into this coalition one by one. As each player enters, he receives a payoff equal to his marginal contribution to the grand coalition payoff. This contribution generally depends on the entering order. The Shapley value is the average payoff to the players if they enter in a random order.^{‡} The nucleolus (23) is neither based on a stability concept like the core nor characterized by principles of power or fairness like the Shapley value. Rather it finds a unique solution of a cooperative game by computing the maximum dissatisfaction with a given allocation across all possible coalitions and then finding the allocation that minimizes the maximum dissatisfaction. So, loosely speaking, the nucleolus serves the most dissatisfied players first. (Dissatisfaction is measured by the maximum total payoff a coalition can reach minus the actual total payoff assigned by the allocation. The nucleolus is always in core, if nonempty.)

We discuss the performance of these solution concepts for our games in the next section (see Table 1 and Fig. 2 for the quantitative solutions for our base games and the theoretical discussion of the solution concepts in *SI Text A*, *SI Text B*, and Table S1).

Built on Nash’s seminal papers in 1950 (24), since the 1980s there have been many attempts to understand coalition formation and the distributions of payoffs as equilibria of non-cooperative games (25⇓–27 and references therein). Starting in the field of industrial organization, extensive game models of oligopolistic competition and the analysis of their subgame perfect equilibria (28, 29) turned out to be a fruitful approach. Harsanyi’s theory of incomplete information (30) opened further opportunities for non-cooperative game models. Somewhat later non-cooperative game modeling spread to many other fields of economic theory and much less attention was paid to cooperative games. There are exceptions. One is the “Gale–Shapley algorithm” (31) that rather recently turned out to be useful in practical market design; Roth (32) surveys the literature. However, the earliest attempt to develop non-cooperative modeling of cooperation and bargaining was by Nash (33) on two-person bargaining. There he not only presented his axiomatic theory but also offered a non-cooperative interpretation. Both Nash’s model (1) and our agencies method ultimately build on this non-cooperative approach.

A description of a strategic situation as a non-cooperative game is much more detailed than for a cooperative game. For experimental purposes an extensive game procedure for coalition bargaining, as we have devised it, has the advantage that the players interact in a formal and anonymous way. Thereby one isolates the strategic situation from social influences like personal sympathies and easily protocols every decision. However, any coalition (including no coalition with zero payoffs for all) can be supported in a pure equilibrium of our base game. The final agent who can be any of the three players takes the entire coalition value. When repeating the base game, as in our experiment, non-cooperative game theory imposes even less structure on behavior and outcomes: In the supergame almost any payoff division can be chosen in equilibrium, supported by a threat to convert to a one-shot base game equilibrium with no acceptances of any player in case of a deviation from the equilibrium path (*SI Text A*). Thus, it is not possible to derive predictions from non-cooperative game theory. [The complexity involved in analyzing supergames was first emphasized by Nash in the context of the theoretically much less demanding repeated prisoner’s dilemma (34) and has been confirmed in the laboratory (35⇓–37).]

We find that the agencies method for coalition formation is an effective mechanism to promote efficient cooperation and balanced payoffs. In particular, we observe that even though the players’ strengths differ, long-run payoffs aggregated over all rounds tend to converge to the equal division. This is consistent with Nash’s (1) simulations with robots in a similar context, as well as with parts of the behavioral economics literature, indicating a general attraction for payoff equality in bargaining and cooperation games, especially when payoff comparisons between players are possible.^{§}

However, equality is not generally the leading principle in each round separately. Here, many agencies seem to succumb to short-term incentives and allocate a significantly larger portion of the payoffs to themselves. Yet, they generally resist taking the whole surplus, as would have been predicted by non-cooperative game theory for the one-shot version of the game. Rather, in the short run, many agents appear to be guided by strength comparisons as captured by the Shapley value and the nucleolus. So, although there is inequality at many snapshots, some of which is organized by cooperative game theory, the repeated and symmetric voting procedure makes sure that, ultimately, everybody is taken care of equally—even to the extent that it mitigates strong differences in the subjects’ strengths.

## Experimental Results

Each of the 10 games in Table 1 was played by 10 different groups of three players. The same group interacted for 40 rounds, and all players maintained their strengths (associated with player roles *A*, *B*, and *C*) throughout. This procedure gives us 10 independent observations per game. We start our analyses with results on voting behavior, before studying payoff consequences. We generally do not find significant differences in behavior and outcomes across games. Thus, most of the time, we pool the data. If there are differences, we report them.

### Coalition Building.

Overall, cooperation is very successful. Only 1% of all 4,000 rounds end in no agreement, 7.5% in two-person coalitions, and 91.5% in the grand coalition.^{¶} The high level of efficiency appears to be only to a small extent the result of the low exogenously given probability of conflict (10%) in case of an impasse, because the random mechanism (see Fig. 5, steps 4 and 10) was rarely used: An impasse occurred only in 5% of all possible cases in stage 1, of which in 75% voting was repeated only once or twice. Stage 2 saw more deadlocks, but still the random mechanism was used only in 14% of the rounds and at most twice in 8%. Although neither cooperative nor non-cooperative game theory predicts how a grand coalition can emerge, one might speculate that the key to successful cooperation is a commonly accepted, stable agency.^{║} However, although player *A* has some advantage,** in no group is the same player always the final agent; in 14/100 groups, the same player (*A* in 8 of these groups) is the final agent 80% or more of the time. In 50 groups the same player (*A* in 27 of these groups) becomes the final agent between 50% and 80% of the time.

We conclude that the voting procedure does not strongly and consistently discriminate according to the players’ strengths. Rather, the symmetric, procedural aspect of coalition formation in our game gives all players a chance in creating and leading the grand coalition, whether weak or strong.

### Payoff Distribution.

#### Equal splits.

The equal split in the grand coalition has considerable attraction. We observe it in 54% of the 4,000 rounds. (As a contrast, in 3% we observe that a player takes all.) Fig. 1 shows the distribution of the 100 groups according to the number of times with exact equal splits of the 40 rounds. (Figs. S1 and S2 give a more detailed picture of both the evolution and the distribution of payoff distributions for each group.) In 29 of the 100 groups, the final agents chose the equal split between 36 and 40 times and thus are homogenous groups with respect to payoff distributions.^{††} For the remaining 71 groups Fig. 1 reveals large differences in equal-splits usage. The next paragraphs investigate the guiding behavioral principles in these remaining groups.

#### Payoff distributions in nonequal-split groups.

Whereas non-cooperative game theory is consistent with basically any payoff distribution in our games, cooperative game theory predictions are not. Our results demonstrate that some solution concepts from cooperative game theory can structure the data of the 71 nonhomogeneous groups in some (qualitative) ways. We concentrate on three concepts, the Shapley value, the nucleolus, and the core. We also consider the equal split.

The average payoff distributions across all rounds are typically closest to the equal split (72%, 51 groups), whereas 25% (18 groups) distribute near the Shapley value and 3% (2 groups) near the nucleolus (Fig. S3).^{‡‡} The core distribution has predictive power only in game 10 with all theoretical solutions, the equal split, and all data points lying in the core. No actual payoff distribution of the other games lies, however, in the core. None of the payoff splits are close to the corners of the triangles, the selfish splits.

On a more qualitative level, we find that in 37% of all groups the payoffs are ordered as suggested by the cooperative solutions concepts, with player *A* earning the highest payoff and *C* the lowest (a random order would produce 17% of this particular order). In 24% of all groups *B* receives the highest payoff and in 7%, *C*. Further, in 32% *A* receives the highest payoff, but the order is reversed between *B* and *C* or one of these gets the same as *A*. Thus there is a significant difference between the payoff ordering between the three players, using the Friedman test (*P* < 0.001), with each group as an independent observation.

So far we have compared aggregate payoff distributions over all rounds with cooperative concepts and the equal split. However, it is also interesting to compare the theoretical concepts with the average proposal of the final agent. In all rounds the final agent either splits the coalition payoff in his favor or uses an equal split. Thus, the proposals of players *B* and *C* are inconsistent with the strengths of the players. Therefore it makes sense to compare only the proposal of the strong player, *A*, with the theoretical concepts. Fig. 2 shows the average payoff distributions proposed by *A* in each group and game with the theoretical solutions and equal split. Unlike aggregate comparisons over all rounds, the equal split best explains *A*’s average payoff distribution 34% of the time, whereas the Shapley value reaches 40% and the nucleolus, 23%. The corner solution, with *A* earning (close to) 120, is the best descriptor 4% of the time.

Summing up, overall payoff differences mirror differences in the players’ strengths and are thus qualitatively captured by cooperative concepts. In particular, focusing on the divisions by the strong player, outcomes are better organized by the Shapley value than by equality and selfishness. [Kahan and Rapoport (41) and references therein summarize many of the experiments that competitively test several solution concepts; interestingly, in this literature, the Shapley value is generally not supported.] Nucleolus also organizes a significant share of observations well. The core does not have predictive value. However, over all rounds the equal split outperforms other principles of behavior.

#### Reciprocity.

How can average payoff vectors chosen by *A* be successfully organized by the Shapley value and, partly, the nucleolus, whereas equality is the dominant principle across all rounds and players? Before starting the experiment, our hypotheses were guided by the simulation in Nash (1) that cooperation can emerge only when no demand is selfish or when a selfish demand is matched with forgiving play. As mentioned above, 29 groups of the 100 groups showed equal splits across most rounds. Yet, in the remaining 71 groups our voting procedure produces just the opposite pattern: The more aggressive the demand of one player is, the more aggressive are those of the others. This kind of reciprocity is possible, because bargaining strength is offset by the voting procedure as shown below.

Fig. 3 illustrates the reciprocal behavioral pattern among all three players. The strong positive relationship indicates that the gifts of the three players in a group, made across rounds, are positively correlated. The Spearman rank correlation coefficient of mutual payoff gifts across all rounds is 0.50 (*P <* 0.0001) for *A* and *B*; 0.75 (*P* < 0.0001) for *A* and *C*, and 0.57 (*P* < 0.0001) for *B* and *C*. We also observe that these correlations are high from the very beginning and do not increase over time. An analogous analysis regarding each player’s demands for himself when being the final agent yields very similar patterns. The Spearman rank correlation coefficient across all rounds is 0.24 (*P* < 0.05) for *A* and *B*, 0.41 (*P* < 0.001) for *A* and *C*, and 0.30 (*P* < 0.02) for *B* and *C*. This coefficient increases over time for *A* and *C* demands.

The reciprocal relationship between gifts and demands as revealed by the correlations shows that payoff mitigation is made possible through a “fair” voting mechanism that disciplines too selfish demands. In particular, Fig. 4 illustrates the negative relationship between average payoff demand and the number of times being the final agent [the Spearman rank correlation coefficient is −0.24 (*P* < 0.05) for *A*, −0.26 (*P* < 0.05) for *B*, and −0.36 (*P* < 0.005) for *C*].

Summing up, whereas the strong player’s behavior is better organized by the Shapley value and, partly, by the nucleolus, reciprocity explains the strong prominence of the equal split in the aggregate. The three players mimic each other, so that both gifts and demands are highly correlated between players.

## Summary and Conclusion

The agencies method by Nash (1) is very effective in promoting human cooperation and fair outcomes: Full efficiency is almost always reached in our laboratory coalition formation game, and the divisions of payoffs across rounds are much less extreme than one might expect from a non-cooperative analysis of the base game. The tension between short-term incentives of not sharing the coalition value with others and the long-term concern to keep cooperation going is, by the strong player, often solved approximately in line with the Shapley value and the nucleolus. Also, the players’ average payoff differences reflect the different strengths of players as measured by these concepts. However, over all rounds the payoff differences are rather small, and the equal division is the concept best describing 80% of all average payoff vectors. One reason is that the symmetry of the voting procedure induces a balance of power: Selfish agents tend to be voted out of their agency and are disciplined by reciprocal behavior. In fact, all players have a good chance to become the final agent. As a result, even if the short-run round payoffs are dispersed, long-run average payoffs tend to converge.

We use the non-cooperative approach to clearly define and control the coalition formation process. Yet non-cooperative theory does not structure the behavior as the base game solutions are inconsistent regarding final payoffs and voting behavior. This complements earlier research in one-shot characteristic function games, where a great number of different extensive game procedures have been employed [see, in particular, the work on demand commitment models (41⇓⇓–44)]. For instance, the non-cooperative theoretical analysis of these procedures suggests that the results depend strongly on procedural details. In fact, however, human behavior depends less on such details than predicted. Humans often seem to analyze the situation more in the flavor of cooperative game theory, ignoring the strategic consequences of the specific procedures used (38). Similarly, in earlier work on repeated asymmetric cooperation games, behavior could not be explained by optimizing behavior but rather by fairness criteria and cooperative goals (37).

The cooperative solution concepts, on the other hand, can help us organize the payoff division data, but they do not capture the effect of the underlying institutions and procedures. Whereas the strength of the players captures some of the average payoff differences when the strong player is in charge, voting and long-run distribution behavior was essentially independent of the characteristic function. Here, the repeated voting procedure, which gives all an equal weight when transferring power to an agency, leads to rather equal total payoffs. This mitigating effect of the voting procedure is not captured by theory. (The distribution of power across subjects in our experiments—as is generally the case in experimental economics—was random, which may also contribute to the attractiveness of the equal split.)

We conclude that other approaches to modeling human cooperation and coalition formation are needed, models that take people’s cognitive and motivational limits in dealing with institutions and other players seriously. In this connection, an interesting related experimental study is the “three-person cooperative game with no side payments” by Kalisch, Milnor, Nash, and Nering (4). This study is one of the first experimental economics studies of negotiation and characteristic function games. In one treatment (section IV of their paper), two players could vote for another player; yet a player attracting two votes could not choose the distribution but was automatically awarded 40 monetary units, whereas the other two lost 20 each (otherwise, all payoffs were zero). They observed, like we do, that in the long run players typically equalized payoffs. Sometimes this was accomplished by randomization and sometimes by sequential reciprocity (“if you vote for me, I’ll vote for you”).

In the same paper the authors suggested to investigate these two mitigating mechanisms in an asymmetric setting as a robustness check for their findings. Although our experiment differs in some other ways too, we implement asymmetric characteristic functions—and observe the same two basic mechanisms at work in the following sense. Randomizing can be interpreted as a fair procedure, because it equalizes expected payoffs in the (base) game, where a deterministic equal outcome is not feasible (45, 46). Voting in our experiment can similarly be qualified as a fair procedure, balancing negotiation power in an otherwise asymmetric situation, because “one man, one vote” levels the playing field for everybody independent of a player’s strength.

Given the repeated structure both in our study and in ref. 4, reciprocity comes in as an additional, dynamic balancing mechanism, further reinforcing the convergence of power and payoffs in the groups. This suggests that the interaction of “fair institutions,” such as voting, randomization, and reciprocity might be a key ingredient of the evolution of cooperation. However, it is captured neither by cooperative nor by non-cooperative theory, and it has been rarely studied outside these two papers. (An exception is ref. 47, which assumes a role of “fair chance” in coalition games and on this basis applies a probabilistic choice model for light guessing behavior to coalition choice problems.) We hope that our findings and our framework inspire more research in this field.

## Methods

### Subjects.

We invited 300 subjects, mostly economics students, into the Cologne Laboratory for economic research. For each game (Table 1) we ran one session with 10 independent groups of 3 subjects. Each group interacted via computer terminals for 40 rounds without knowing the identity of other subjects, using the coalition formation procedure explained below. Each subject could participate only in one supergame, maintaining the same position [strong (*A*), medium (*B*), or weak (*C*) player], which was known to the subjects. At the end each subject was paid individually according to the points obtained throughout the 40 rounds.

### Experimental Design and Task.

In each round of each game, each group of three bargained in two steps to elect an agent (or representative). If no member wished to be represented by another group member, all members received a payoff of zero for this round. If only one member wished to be represented, the representative could divide the corresponding coalition payoff among himself and the represented member; the third member receives a payoff of zero for this round. If two members were represented by the third member, the representative could divide the grand coalition payoff among the three members of the group.

More specifically, in step 1 each member could select at most one other agent to form a pair. If no member wished to be represented by any other (no pair was formed), the round ended with probability 10% with zero payoffs for all three; otherwise (with probability 90%) the first step was repeated. If there is more than one pair of a member and his representative, one of the pairs is chosen with equal probability. In this case, or when there is only one pair, step 1 ends and the represented group member remains passive for the rest of this round.

In step 2, each of the two active members had the option to choose the other as his representative. If no active member chose the other as his representative, the second step was repeated with probability 90%; otherwise (with probability 10%) the round ended and the representative chosen in step 1 could divide the corresponding coalition payoff among himself and the represented member. In this case, the third member received a payoff of zero for this round. If both opted for the other as their representative, the actual representative was chosen with equal probability. In this case, or when only one representative was voted for in step 2, this final agent divided 120 experimental currency units (ECU) at his discretion among all three members of the group. After this, the first step of the next round began.

The flowchart in Fig. 5 illustrates more formally our base game as implemented in the experiment. See *SI Text C* for the instructions to subjects and screenshots.

## Acknowledgments

We thank Felix Lamouroux for research assistance and help with the collection of data and Aniol Llorente, Riccardo Pedersini, and Jonathan Hersh for technical help and two invited reviewers for excellent comments. A.O. acknowledges financial support from the German Science Foundation (Deutsche Forschungsgemeinschaft) through the Leibniz Program and through the research group “Design & Behavior.” R.N. thanks the Spanish Ministry of Education (SEJ2005-08391 and ECO2008-01768), Barcelona Graduate School of Economics, and the Generalitat de Catalunya for support. By accepting to participate in our experiments the subjects, who sign up voluntarily, agree to the norms and rules of an experiment within our laboratory.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: xkjfnj{at}princeton.edu.

Author contributions: J.F.N., R.N., A.O., and R.S. designed research; R.N. and A.O. performed research; R.N., A.O., and R.S. analyzed data; J.F.N., R.N., A.O., and R.S. contributed to the theory and the supporting material; and J.F.N., R.N., A.O., and R.S. wrote the paper.

The authors declare no conflict of interest.

Data deposition: The data reported in this paper are available in Dataset S1.

↵*Unlike in our case, where there is no precommitment to a particular policy prior to voting, most models in the theoretical voting literature assume that candidates are fully committed to their campaign policy proposals. Thus, by assumption, when a candidate is elected, he implements the policies that he promised to his constituency during the campaign. There are a few papers that drop the assumption of full commitment and analyze the strategic policy choice of candidates after they are elected. In one-shot elections the only possible outcome is the implementation of the most preferred policy of the winning candidate (2). In repeated elections, the value of reputation allows candidates to make credible policy proposals in equilibrium (3). In the experimental literature an important exception is ref. 4; see also

*Summary and Conclusion*. A related literature on voting experiments can be found in ref. 5 with experiments on voting for fully committed candidates, the voting paradox, and also some experiments on the issue of voting over redistribution; e.g., a proposal by one or more players, by the experimenter, or through the rules of the game is either accepted or dismissed (6).↵

^{†}See ref. 8 for the history of experimental testing of game-theoretic hypotheses and different approaches to model human behavior (9⇓⇓⇓⇓–14). Unlike in the context of cooperation and bargaining base games with multiple equilibria, the evolution of behavior in simple coordination games appears to be relatively well understood since Schelling’s (15) seminal work on coordination problems (refs. 16⇓–18 and references therein). There is also a considerable literature on how reputation-building institutions may affect cooperation (refs. 19⇓–21 and references therein).↵

^{‡}The values can also be characterized by the Shapley axioms of fairness: (*a*) the sum of all players’ values is the grand coalition value, (*b*) players contributing equally to any coalition have the same value, (*c*) players neither harming nor helping any coalition have a zero value, and (*d*) the value of two games played at the same time is equal to the sum of values played at different times. Note that the Shapley value is not necessarily in the core, even if the core is nonempty.↵

^{§}See also, e.g., the role of equity considerations in coalition bargaining (38) and a survey of recent evidence in various non-cooperative games (39). We caution that, because unlike Nash we implement a finite multistage game, Nash’s simulation results are only suggestive for our data analysis, which is why we complement our analysis with hypotheses from cooperative and non-cooperative game theory.↵

^{¶}There is no significant difference between the frequency of two-person coalitions in games 1–4 (146 of the 301 rounds with two-person coalitions of all games), with the two-person coalition implying full efficiency (*v*(*A*,*B*) = 120), and the frequency in games 5–10 (151 of 301), when full efficiency is reached only in the grand coalition (using the Mann–Whitney*U*test, using each individual group as an independent observation,*P*> 0.12). In 32% of all 100 groups there are never two-person coalitions, and in 18% it happens only once per group.↵

^{║}Recall, however, that non-cooperative game theory predicts that, in every pure equilibrium of the base game where one player is voted for, this player will be voted for by either one or two players but the voted player will never vote for another player. In our supergame, we find that players*A*do not vote in 40% of the cases when they are voted for by either or both of the other players, whereas players*B*and*C*do not vote in 33% and 30% of the cases, respectively, with no significant differences between the players (sign tests based on group level with*P*= 0.38, 0.31, and 0.18 for the three comparisons).↵**Over all groups, player

*A*is most often the final agent in 42% of all rounds [16.7 rounds (SD = 10.4)], player*B*in 32% [12.8 rounds (SD = 9.1)], and player*C*in 25% [10.1 rounds (SD = 8.6)], and in 1% no coalition is formed. Rank ordering*A*,*B*, and*C*players in a group by number of being representative, there is a significant difference between the three players (Friedman’s test, two-way analysis on ranks,*P*< 0.01).↵

^{††}Twenty-three of these 29 groups choose the equal split from the very beginning, whereas only 6 groups manage to converge to the equal-split norm when the first three rounds are not equal-split proposals (see ref. 40 for how norms may emerge in competitive environments). However, even when the equal split is the start-off norm, 20 groups fail to maintain it throughout.↵

^{‡‡}Our measure of success for a particular prediction is the mean squared error (MSE) between the payoff vector of the coalition formation solution and the actual average payoff vector of a group. The theoretical concept having the smallest MSE to the actual average data best describes a particular group on average.This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1216361109/-/DCSupplemental.

Freely available online through the PNAS open access option.

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- Siegel S,
- Siegel AE,
- Andrews JM

- Kalisch GK,
- Milnor JW,
- Nash JF,
- Nering ED

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