The interplay of self‐interest and equity in coalition formation

In this paper we investigated the interplay of self‐interest and equity concerns in coalition formation by manipulating the number of units in which the coalition payoff is made available, and by manipulating the way people are allowed to interact. Results of three experiments showed that when the coalition payoff was such that members of each possible coalition could obtain an equitable payoff share, the outcome tended to be coalitions that also maximized the payoff of its members. However, when the payoff was such that people had to make trade‐offs between maximizing their payoff share and obtaining an equitable payoff share, it took longer to form a coalition and it was harder to maintain a coalition. Moreover, depending on the way people were allowed to interact, the final outcome was a coalition that maximized the payoff of its members or a coalition that provided them with an equitable payoff share. Copyright © 2004 John Wiley & Sons, Ltd.     

Coalition Preference as a Function of Expected Values in a Tetradic Weighted-Majority Game

Eighty male subjects participated in a tetradic weighted-majority game in which a coalition was required to control the division of the payoff. The amount of the payoff and/of the probability of receiving the payoff were varied. The data strongly supported the hypothesis that subjects would prefer the coalition with the largest expected value (size of payoff x probability of success = expected value) rather than the coalition with the largest payoff or the coalition with the highest probability of success. However, contrary to expectations, the 'division of rewards was found to be a function of the relative assigned resources of the coalition members, despite the lack of a logical connection between the assigned resources and the rules governing obtaining the payoff.     

Coalition formation in characteristic function games: Competitive tests of three theories

Two experiments tested the bargaining, minimum resource, and minimum power theories of coalition formation in situations involving different payoffs for some of the winning coalitions (characteristic function games). In the first experiment, a triadic resource distribution was employed and payoffs for the coalitions were specified in such a way that each of the three theories predicted the formation of a different coalition. The coalition predicted by minimum power theory formed the most frequently, and the mean divisions of the payoffs among coalition members were also closest to the predictions of minimum power theory. However, the most frequent coalition was not only the one predicted by minimum power theory, it was also the one having the largest payoff per member. There-fore, a second experiment was conducted, which employed a tetradic resource distribution and specified the coalition payoffs in such a way that (a) each of the theories predicted the formation of a different coalition and (b) none of the predicted coalitions was the one with the largest payoff per member. Although the mean payoff divisions in the coalitions in this experiment were closest to those predicted by bargaining theory, the coalition that formed most frequently was not one of those predicted by any of the theories. Rather, it was the one having the largest payoff per member. None of the three theories is able to account adequately for the results of both experiments. The difficulties that the theories have in dealing with coalition formation in situations in which there are different payoffs for winning are discussed.     

Get angry, get out: The interpersonal effects of anger communication in multiparty negotiation

Research on multiparty negotiation has investigated how parties form coalitions to secure payoffs but has not addressed how emotions may affect such coalition decisions. Extending research on bilateral negotiations which has generally argued that it is beneficial to communicate anger, we argue that it constitutes a considerable risk when there are more than two people present at the negotiation table. Using a computer-mediated coalition game we show that communicating anger is a risky strategy in multiparty bargaining. The main findings of three studies were that participants: (1) form negative impressions of players who communicate anger and therefore (2) exclude such players from coalitions and from obtaining a payoff share, but (3) make considerable concessions on those rare occasions that they choose to form a coalition with an angry player, or (4) when they had to form a coalition with an angry player. We discuss the implications of these results for theorizing on emotions, negotiations, and coalition formation.     

The excluded player in coalition formation

Coalition research generally assumes that people strive to maximize their share of the coalition payoff and that they exclude others from joining a coalition if these others are not needed to obtain the coalition payoff. In two experiments, the authors show that this view is too narrow and that willingness to include such others is dependent on the extent to which people feel that exclusion affects the payoff of the excluded player. This finding was moderated by social value orientations. Proselfs were not affected by the consequences for the excluded players. Prosocials were less willing to exclude others the more harmful were the consequences of exclusion. Results are related to research on social exclusion, the do-no-harm principle, and social value orientations.     

The effect of social motives,communication and group size on behaviour in an N-person multi-stage mixed-motive game

Two experiments investigated the impact of social motives or individuals' preferences for specific self—other outcome distributions, on behaviour in an n-person game. Subjects' social motives (altruistic, cooperative, individualistic, competitive) were assessed prior to the decision-making in either 7-person games (Experiment 1) or 20-person games (Experiment 2). A modification of the n-person game format normally employed is introduced in this research to permit the choices made by players on a given trial to modify the payoff matrix available to self and others on subsequent trials. The game format, a simulated social dilemma, was presented in terms of a conservation of resources problem. In Experiment 1 communication opportunities were manipulated. As predicted, there were consistent differences between the four classes of social motivation in the amount of the resources taken for self competitive subjects took the most, individualistic subjects took less than the competitive ones but more than the average, while cooperative and altruistic subjects took the smallest amount of the resources for themselves. Moreover, competitive subjects expected the others to take fewer resources than they intended to themselves, and altruistic subjects expected the others to take more resources than they intended to themselves. These findings are only partly consistent with existing theories concerning the relationship between behaviour and expected behaviour of others. In addition, when communication was allowed, significantly fewer resources were taken for self Contrary to the predictions based on previous research findings, subjects in the 20-person groups did not take more resources for self than subjects in the 7-person groups.     

Coordination in Market Entry Games with Symmetric Players

We report the results of two experiments designed to study tacit coordination in a class of market entry games with linear payoff functions, binary decisions, and zero entry costs, in which each of n = 20 players must decide on each trial whether or not to enter a market whose capacity is public knowledge. The results show that although the subjects differ considerably from one another in their decision policies, tacit coordination emerges quickly on the aggregate level and is accounted for most successfully by the Nash equilibrium solution for noncooperative n-person games.     

Strength-is-Weakness: The (ir)relevant relation between resources and payoffs in coalition formation

A key observation in coalition formation is that bargainers who control many resources are often excluded from coalitions by bargainers who control few resources, the Strength-is-Weakness effect. We argue that this effect is contingent on whether resources provide a legitimate claim to be included in a coalition. Across three incentivized coalition experiments (n = 2745; 915 triads), three participants (player A had four resources, player B had three resources, player C had two resources) negotiated about a payoff of 90 monetary units. Depending on condition, these resources were obtained randomly, earned, or earned and proportionally linked to the payoff. Results showed player As were less included when resources were obtained randomly and more often included in coalitions when resources were earned and/or proportionally linked to the payoff. This provides evidence that the Strength-is-Weakness is contingent on the legitimacy of the resources.     

The equal division kernel: An equity approach to coalition formation and payoff distribution in N-person games

Most existing models of coalition formation and payoff distribution in groups rest upon normative considerations and are ambiguous in their predictions insofar as they do not determine which of several coalitions will most probably result. The paper sketches the basic features of a model derived from social psychological exchange-and equity-theory which predicts coalitions and payoff distributions for a variety of situations. The evaluation of the model by the results of several experiments indicates that it provides a reasonable starting point for further theoretical developments that are based empirical studies.     

When three is not always two against one: Coalitions in experimental three-person cooperative games

Four models of coalition formation are tested in a computer-controlled experiment in which three players negotiated to form coalitions in order to gain rewards for themselves. Formation of the grand coalition of all three players as well as any of the three possible 2-person coalitions is permitted. The results indicate that the grand coalition is the most likely to form, followed by the coalition between the two strongest players. The individually rational bargaining set model of coalition formation is supported over an egalitarian model, the Shapley value, and the coalitionally rational bargaining set model. Various measures of the progress of negotiations are introduced, examined, and discussed in an effort to provide an increased data base for the establishment of a behavioral theory of bargaining and coalition formation.     

Resources and alternatives in coalition formation: The effects on payoff,self‐serving behaviour,and bargaining length

A consistent finding in coalition research is that the payoff of coalition members is related to (a) the resources they contribute to the coalition, and to (b) the number of alternative coalitions they can form. These two factors are, however, often intertwined. A greater number of resources tends to go hand in hand with a greater number of alternatives, leaving unanswered how both factors affect coalition behaviour. This paper attempts to clarify the interplay of resources and alternatives by disentangling the two in a newly developed coalition paradigm. Results indicated that participants base their payoff allocation on both resources and alternatives, but suggest that self‐serving behaviour is more related to alternatives. Furthermore, resources and alternatives had a distinct effect on the bargaining process. It was shown that differences in alternatives led to longer bargaining. Copyright © 2004 John Wiley & Sons, Ltd.     

The competitive bargaining set for cooperative n-person games

The continuum of solutions constituting the bargaining set for n-person games in characteristic function form (n ≥ 4) reflects various degrees of determinateness to form a coalition, different requirements of stability, or different levels of competition among the players. At one end of this continuum is the kernel solution. A new solution, called the competitive bargaining set, constitutes the other extreme. It incorporates the concept of a multiobjection, which is based on the idea that threats may be perceived and considered simultaneously even though their simultaneous implementation is impossible.For the simple homogeneous majority games [n ? 2, 1, 1, …, 1] the competitive bargaining set (1) yields unique outcomes for each coalition structure, (2) coincides with von Neumann and Morgenstern's main simple solution for certain coalition structures, and (3) converges to Shapley value as n → ∞ when the grand coalition forms.Mathematical and experimental implications of the new solution are discussed briefly.     

The Negative Effect of Probability Assessments on Decision Quality

It is commonly assumed that uncertain information can be reduced to numerical probabilities without biasing preferences. It is also implicitly assumed in much research and many applications that people can express these probabilities. In contradiction to these assumptions Experiment 1 shows that the production of probability assessments biases decisions in an n-person game. Experiment 2 shows that the explicit assessment of numerical probabilities renders choices between gambles concerning future basketball events less optimal. These findings seem to be a result of overweighting the probability dimension relative to the payoff dimension given numerical judgments. Experiment 2 also suggests that without explicit numerical probability judgments subjects are less likely to violate the dominance principle. The theoretical and practical implications of the results are discussed.     

The Intrinsic Quantum Nature of Nash Equilibrium Mixtures

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics (QM), which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics.     

Three ancient problems solved by using the game theory logic based on the Shapley value

The ancient problems of bankruptcy, contested garment, and rights arbitration have generated many studies, debates, and controversy. The objective of this paper is to show that the Shapley value from game theory, measuring the power of each player in a game, may be consistently applied for getting the general one-step solution of all these three problems viewed as n-person games. The decision making is based on the same tool, namely the game theory logic based on the use of the Shapley value, but the specific games involved are slightly different in each problem. The kind of claims of the players, the relationship between the given claims and the given resources available, and the particular way of calculating the generalized characteristic function of the game determine the specific type of game which has to be solved in each of the three ancient problems mentioned. The iterative use of the Shapley value may also justify the well-known Aumann–Maschler step-by-step procedure for solving the bankruptcy problem.