Minimal stable sets in tournaments

We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, encompassing the top cycle, the uncovered set, the Banks set, the minimal covering set, and the tournament equilibrium set. Moreover, the hierarchy includes a new tournament solution, the minimal extending set, which is conjectured to refine both the minimal covering set and the Banks set.     

Approximate cores of games and economies with clubs

We introduce the framework of parameterized collections of games with and without sidepayments and provide three nonemptiness of approximate core theorems. The parameters bound (a) the number of approximate types of players and the size of the approximation and (b) the size of nearly effective groups of players and their distance from exact effectiveness. Our theorems are based on a new notion of partition-balanced profiles and approximately partition-balanced profiles. The results are applied to a new model of an economy with clubs. In contrast to the extant literature, our approach allows both widespread externalities and uniform results. A fundamental new mathematical result, the partition-balancing effect of large numbers, which promises to have further applications, is contained in Appendix A.     

Null or nullifying players: The difference between the Shapley value and equal division solutions

A famous solution for cooperative transferable utility games is the Shapley value. Most axiomatic characterizations of this value use some axiom related to null players, i.e. players who contribute zero to any coalition. We show that replacing null players with nullifying players characterizes the equal division solution distributing the worth of the ‘grand coalition’ equally among all players. A player is nullifying if every coalition containing this player earns zero worth. Using invariance we provide similar characterizations of the equal surplus division solution assigning to every player its own worth, and distributing the remaining surplus equally among all players.     

Potential,value, and random partitions

We provide a new interpretation of the potential of the Shapley value as the expected worth of some random partition of the player set. Using this insight, we advocate the potential as an index of power concentration in simple monotonic games.     

Equivalence and axiomatization of solutions for cooperative games with circular communication structure

We study cooperative games with transferable utility and limited cooperation possibilities. The focus is on communication structures where the set of players forms a circle, so that the possibilities of cooperation are represented by the connected sets of nodes of an undirected circular graph. Single-valued solutions are considered which are the average of specific marginal vectors. A marginal vector is deduced from a permutation on the player set and assigns as payoff to a player his marginal contribution when he joins his predecessors in the permutation. We compare the collection of all marginal vectors that are deduced from the permutations in which every player is connected to his immediate predecessor with the one deduced from the permutations in which every player is connected to at least one of his predecessors. The average of the first collection yields the average tree solution and the average of the second one is the Shapley value for augmenting systems. Although the two collections of marginal vectors are different and the second collection contains the first one, it turns out that both solutions coincide on the class of circular graph games. Further, an axiomatization of the solution is given using efficiency, linearity, some restricted dummy property, and some kind of symmetry.     

Reputation with equal discounting in repeated games with strictly conflicting interests

We analyze reputation effects in two-player repeated games of strictly conflicting interests. In such games, player 1 has a commitment action such that a best reply to it gives player 1 the highest individually rational payoff and player 2 the minmax payoff. Players have equal discount factors. With positive probability player 1 is a type who chooses the commitment action after every history. We show that player 1's payoff converges to the maximally feasible payoff when the discount factor converges to one. This contrasts with failures of reputation effects for equal discount factors that have been demonstrated in the literature.     

ANTI‐DUAL OF ECONOMIC COALITIONAL TU GAMES*

Some well‐known coalitional TU (transferable utility) games applied to specific economic problems are shown to be connected through the relation defined as the anti‐dual. Solutions such as the core, the Shapley value and the nucleolus of anti‐dual games are obtained straightforwardly from original games.     

Coalitions, agreements and efficiency

If agents negotiate openly and form coalitions, can they reach efficient agreements? We address this issue within a class of coalition formation games with externalities where agents’ preferences depend solely on the coalition structure they are associated with. We derive Ray and Vohra's [Equilibrium binding agreements, J. Econ. Theory 73 (1997) 30-78] notion of equilibrium binding agreements using von Neumann and Morgenstern [Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944] abstract stable set and then extend it to allow for arbitrary coalitional deviations (as opposed to nested deviations assumed originally). We show that, while the extended notion facilitates the attainment of efficient agreements, inefficient agreements can nevertheless arise, even if utility transfers are possible.     

Cores of combined games

This paper studies the core of combined games, obtained by summing different coalitional games when bargaining over multiple independent issues. It is shown that the set of balanced transferable utility games can be partitioned into equivalence classes of component games to determine whether the core of the combined game coincides with the sum of the cores of its components.     

Median stable matching for markets with wages

We define the median stable matching for two-sided matching markets with wages and prove constructively that it exists.     

Equilibrium binding agreements: a comment

To construct their Equilibrium Binding Agreements, Ray and Vohra (J. Econ. Theory, 73 (1997) 30-78) define a concept of an equilibrium between coalitions and prove its existence for any coalition structure. We show that this result crucially depends on the quasi-concavity of the utility functions, which in turn depends on the type of mixed strategies used by the coalitions. When coalitions use uncorrelated mixed strategies utility functions may not be quasi-concave and an equilibrium may not exist. However, if coalitions use correlated strategies, an equilibrium always exist.     

The nucleolus of large majority games

Members of a shareholder meeting or legislative committee have greater or smaller voting power than meets the eye if the nucleolus of the induced majority game differs from the voting weight distribution. We establish a new sufficient condition for the weight and power distributions to be equal, and we characterize the limit behavior of the nucleolus in case all relative weights become small.     

Coalition formation as a dynamic process

We study coalition formation as an ongoing, dynamic process, with payoffs generated as coalitions form, disintegrate, or regroup. A process of coalition formation (PCF) is an equilibrium if a coalitional move to some other state can be “justified” by the expectation of higher future value, compared to inaction. This future value, in turn, is endogenous: it depends on coalitional movements at each node. We study existence of equilibrium PCFs. We connect deterministic equilibrium PCFs with unique absorbing state to the core, and equilibrium PCFs with multiple absorbing states to the largest consistent set. In addition, we study cyclical as well as stochastic equilibrium PCFs.     

A strategic implementation of the Average Tree solution for cycle-free graph games

In this note we provide a strategic implementation of the Average Tree solution for zero-monotonic cycle-free graph games. That is, we propose a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the average hierarchical outcome of the game. This mechanism takes into account that a player is only able to communicate with other players (i.e., to make proposals about a division of the surplus of cooperation) when they are connected in the graph.     

Characterizations of bargaining solutions in production economies with unequal skills

In production economies with unequal skills, this paper characterizes bargaining solutions by using axioms on allocation rules rather than axioms on classical bargaining solutions. We introduce a new axiom, consistency w.r.t. technological innovations, so that the non-welfaristic characterizations of bargaining solutions in the production economies are provided. By the characterizations, we can classify the three bargaining solutions (the Nash, the Kalai-Smorodinsky, and the Egalitarian solutions) from the viewpoint of responsibility and compensation discussed by Dworkin.     

Allocation rules for land division

This paper studies the classical land division problem formalized by Steinhaus (Econometrica 16 (1948) 101-104) in a multi-profile context. We propose a notion of an allocation rule for this setting.  We discuss several examples of rules and properties they may satisfy. Central among these properties is division independence: a parcel may be partitioned into smaller parcels, these smaller parcels allocated according to the rule, leaving a recommended allocation for the original parcel. In conjunction with two other normative properties, division independence is shown to imply the principle of utilitarianism.