Competitive outcomes and the inner core of NTU market games

We consider the inner core as a solution concept for cooperative games with non-transferable utility (NTU) and its relationship to payoffs of competitive equilibria of markets that are induced by NTU games. An NTU game is an NTU market game if there exists a market such that the set of utility allocations a coalition can achieve in the market coincides with the set of utility allocations the coalition can achieve in the game. In this paper, we introduce a new construction of a market based on a closed subset of the inner core which satisfies a strict positive separability. We show that the constructed market represents the NTU game and, further, has the given closed set as the set of payoff vectors of competitive equilibria. It turns out that this market is not uniquely determined, and thus, we obtain a class of markets. Our results generalize those relating to competitive outcomes of NTU market games in the literature.     

A necessary and sufficient condition for non-emptiness of the core of a non-transferable utility game

It is well-known that a transferable utility game has a non-empty core if and only if it is balanced. In the class of non-transferable utility games balancedness or the more general π-balancedness due to Billera (SIAM J. Appl. Math. 18 (1970) 567) is a sufficient, but not a necessary condition for the core to be non-empty. This paper gives a natural extension of the π-balancedness condition that is both necessary and sufficient for non-emptiness of the core.     

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.     

Stochastic Cooperative Games: Superadditivity, Convexity, and Certainty Equivalents

This paper extends the notions of superadditivity and convexity to stochastic cooperative games. It is shown that convex games are superadditive and have nonempty cores, and that these results also hold in the context of NTU games. Furthermore, a subclass of stochastic cooperative games to which one can associate a deterministic cooperative game is considered. It is shown that such a stochastic cooperative game satisfies properties like nonemptiness of the core, superadditivity, and convexity if and only if the corresponding deterministic game satisfies these properties.Journal of Economic LiteratureClassification Number: C71.     

On weighted Kalai–Samet solutions for non-transferable utility coalitional form games

We investigate the implications of the axiom of coalitional concavity for non-transferable utility coalitional form games. This axiom says that if the feasible set of some coalition is uncertain whereas the feasible sets of other coalitions are known, then all players in the coalition with the uncertain feasible set should (weakly) benefit from reaching a compromise before the uncertainty is resolved. By imposing this axiom, in addition to other minor axioms, we characterize the weighted Kalai–Samet [Econometrica 53 (1985) 307] solutions: these solutions coincide with the weighted egalitarian solutions on the domain of bargaining problems, and with the weighted Shapley values on the domain of transferable utility coalitional form games.     

On the Least Core and the Mas-Colell Bargaining Set

We show that the least core of a TU coalitional game with a finite set of players is contained in the Mas-Colell bargaining set. This result is extended to games with a measurable space of players in which the worth of the grand coalition is at least that of any other coalition in the game. As a consequence, we obtain an existence theorem for the Mas-Colell bargaining set in TU games with a measurable space of players. Journal of Economic Literature Classification Number: C71.     

Core-stable rings in auctions with independent private values

We propose a semi-cooperative game theoretic approach to check whether a given coalition is stable in a Bayesian game with independent private values. The ex ante expected utilities of coalitions, at an incentive compatible (noncooperative) coalitional equilibrium, describe a (cooperative) partition form game. A coalition is core-stable if the core of a suitable characteristic function, derived from the partition form game, is not empty. As an application, we study collusion in auctions in which the bidders? final utility possibly depends on the winner?s identity. We show that such direct externalities offer a possible explanation for cartels? structures (not) observed in practice.     

A bargaining approach to the Owen value and the Nash solution with coalition structure

The mechanism by Hart and Mas-Colell (1996) for non-transferable utility (NTU) games is generalized so that a coalition structure among players is taken into account. The new mechanism yields the Owen value for transferable utility (TU) games with coalition structure as well as the consistent value (Maschler and Owen 1989, 1992) for NTU games with trivial coalition structure. Furthermore, we obtain a solution for pure bargaining problems with coalition structure which generalizes the Nash (1950) bargaining solution.     

Information transmission in coalitional voting games

A core allocation of a complete information economy can be characterized as one that would not be unanimously rejected in favor of another feasible alternative by any coalition. We use this test of coalitional voting in an incomplete information environment to formalize a notion of resilience. Since information transmission is implicit in the Bayesian equilibria of such voting games, this approach makes it possible to derive core concepts in which the transmission of information among members of a coalition is endogenous. Our results lend support to the credible core of Dutta and Vohra [Incomplete information, credibility and the core, Math. Soc. Sci. 50 (2005) 148-165] and the core proposed by Myerson [Virtual utility and the core for games with incomplete information, Mimeo, University of Chicago, 2005] as two that can be justified in terms of coalitional voting.     

On the core of economies with multilateral environmental externalities

We revisit the cooperative model of coalition formation in economies with environmental externalities. Motivated by recent concerns over the true behavior and incentives of key players in international negotiations over the climate and the environment, we construct a cooperative game where the members of each coalition have uncertainty over the behavior of the nonmembers, and in particular they face uncertainty over their coalition structure. As a result, a coalition assigns various probability distributions over the set of partitions the outsiders can form. We compute the payoff of each coalition under this assumption and we derive conditions under which the core of the induced cooperative game is nonempty.     

The monoclus of a coalitional game

The analysis of single-valued solution concepts, providing payoffs to players for the grand coalition only, has a long tradition. Opposed to most of this literature we analyze allocation scheme rules, which assign payoffs to all players in all coalitions. We introduce several closely related allocation scheme rules, each resulting in a population monotonic allocation scheme (PMAS) whenever the underlying coalitional game with transferable utilities has a PMAS. Monotonicities, which measure the payoff difference for a player between two nested coalitions, are the driving force. These monotonicities can best be compared with the excesses in the definition of the (pre-)nucleolus. Variants are obtained by considering different domains and/or different collections of monotonicities. We deal with nonemptiness, uniqueness, and continuity, followed by an analysis of conditions for (some of) the rules to coincide. We then focus on characterizing the rules in terms of subbalanced weights. Finally, we deal with computational issues.     

Balance and discontinuities in infinite games with type-dependent strategies

Under study are games in which players receive private signals and then simultaneously choose actions from compact sets. Payoffs are measurable in signals and jointly continuous in actions. Stinchcombe (2011) [19] proves the existence of correlated equilibria for this class of games. This paper is a study of the information structures for these games, the discontinuous expected utility functions they give rise to, and the notion of a balanced approximation to an infinite game with discontinuous payoffs.     

A note on equivalence of consistency and bilateral consistency through converse consistency

In the framework of (set-valued or single-valued) solutions for coalitional games with transferable utility, the three notions of consistency, bilateral consistency, and converse consistency are frequently used to provide axiomatic characterizations of a particular solution (like the core, prekernel, prenucleolus, Shapley value). Our main equivalence theorem claims that a solution satisfies consistency (with respect to an arbitrary reduced game) if and only if the solution satisfies both bilateral consistency and converse consistency (with respect to the same reduced game). The equivalence theorem presumes transitivity of the reduced game technique as well as difference independence on payoff vectors for two-person reduced games.     

The least core, kernel and bargaining sets of large games

Summary. We study the least core, the kernel and bargaining sets of coalitional games with a countable set of players. We show that the least core of a continuous superadditive game with a countable set of players is a non-empty (norm-compact) subset of the space of all countably additive measures. Then we show that in such games the intersection of the prekernel and the least core is non-empty. Finally, we show that the Aumann-Maschler and the Mas-Colell bargaining sets contain the set of all countably additive payoff measures in the prekernel. Received: June 6, 1996; revised version: March 1, 1997     

Pillage and property

This paper introduces a class of coalitional games, called pillage games, as a model of Hobbesian anarchy. Any coalition can pillage, costlessly and with certainty, any less powerful coalition. Power is endogenous, so a pillage game does not have a characteristic function, but pillage provides a domination concept that defines a stable set, which represents an endogenous balance of power. Every stable set contains only finitely many allocations, and can be represented as a farsighted core. Additional results are obtained for particular games, including the game in which the power of each coalition is determined by its total wealth.     

A Property of the Core

For any transferable utility game in coalitional form with nonempty core we show that, given any allocation outside the core, there is an allocation in the core that indirectly dominates it.Journal of Economic LiteratureClassification Number: C71.     

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.     

Core tâtonnement

Two discrete time tâtonnement processes—one featuring successive tâtonnement, the other featuring simultaneous tâtonnement—for the core of coalitional games with transferable utility are introduced. For totally balanced games, the successive core tâtonnement process corresponds to the standard simultaneous price tâtonnement process of competitive equilibrium theory via the Shapley-Shubik (market game-direct market) correspondence. The simultaneous core tâtonnement process is based entirely on the intuition behind the definition of the core for games with transferable utility, and it does not correspond to any evident competitive equilibrium tâtonnement process. Both processes are proven to be globally stable. The two processes offer easily implementable algorithms for approximately computing core points.     

The Partnered Core of a Game without Side Payments

A payoff for a game is partnered if it admits no asymmetric dependencies. We introduce the partnered core of a game without side payments and show that the partnered core of a balanced game is nonempty. The result is a strengthening of Scarf's Theorem on the nonemptiness of the core of a balanced game without side payments. In addition, it is shown that if there are at most a countable number of points in the partnered core of a game then at least one core point isminimallypartnered, meaning that no player requires any other player in particular to obtain his part of the core payoff.Journal of Economic LiteratureClassification Number: C71.     

Convex Games and Stable Sets

It is well known that the core of a convex coalitional game with a finite set of players is the unique von Neumann–Morgenstern stable set of the game. We extend the definition of a stable set to coalitional games with an infinite set of players and give an example of a convex simple game with a countable set of players which does not have a stable set. But if a convex game with a countable set of players is continuous at the grand coalition, we prove that its core is the unique von Neumann–Morgenstern stable set. We also show that a game with a countable (possibly finite) set of players which is inner continuous is convex iff the core of each of its subgames is a stable set.Journal of Economic LiteratureClassification Numbers: C70, C71.