The core as the set of eventually stable outcomes: A note

As a justification of the core as a set of stable social states, Sengupta and Sengupta [1996. A property of the core. Games Econ. Behav. 12, 266–273] show that for any transferable utility (TU) cooperative game with non-empty core, for every imputation outside the core there is an element in the core that indirectly dominates the imputation in a desirable way. In this note we show that this appealing property of the core no longer holds even for the class of hyperplane games, an immediate generalization of TU games into the environments without side payments.     

Market Games in Large Economies with a Finite Number of Types

We study market games derived from an exchange economy with a continuum of agents, each having one of finitely many possible types. The type of agent determines his initial endowment and utility function. It is shown that, unlike the well-known Shapley–Shubik theorem on market games (Shapley and Shubik in J Econ Theory 1:9–25, 1969), there might be a (fuzzy) game in which each of its sub-games has a non-empty core and, nevertheless, it is not a market game. It turns out that, in order to be a market game, a game needs also to be homogeneous. We also study investment games – which are fuzzy games obtained from an economy with a finite number of agents cooperating in one or more joint projects. It is argued that the usual definition of the core is inappropriate for such a model. We therefore introduce and analyze the new notion of comprehensive core. This solution concept seems to be more suitable for such a scenario. We finally refer to the notion of feasibility of an allocation in games with a large number of players. Some of the results in this paper appear in a previous draft distributed by the name “Cooperative investment games or Population games”. An anonymous referee of Economic Theory is acknowledged for his/her comments     

Extendable Cooperative Games

A (TU) cooperative game is extendable if every core allocation of each subgame can be extended to a core allocation of the game. It is strongly extendable if any minimal vector in the upper core of any of its subgames can be extended to a core allocation. We prove that strong extendability is equivalent to largeness of the core. Further, we characterize extendability in terms of an extension of the balanced cover of the game. It is also shown how this extension can unify the analysis of many families of games under one roof.     

Cost allocation in spanning network enterprises with stochastic connection costs

This paper considers the problem of network formation and cost allocation that arises when customers need to be connected to a single service provider through a (communication) network. The cost of a connection consists of both the construction costs and the maintenance costs. Since the latter costs are unknown ex ante connection costs are represented by random variables. It is shown how an ‘optimal’ network is determined and that the core of the corresponding stochastic spanning tree game is non-empty. A two stage Bird allocation is defined and shown to be a core allocation of stochastic spanning tree games.     

Stable Provision vs. Cournot–Nash Equilibria in Pure Public Good Economies

Shitovitz and Spiegel (1998, Journal of Economic Theory 83 , pp. 1–18) demonstrated that in pure public good economies, for some consumers the Lindahl consumption bundles can be inferior (utility-wise) to their Cournot–Nash allocation. In this paper, we prove that in any finite pure public good economy there exists a core allocation that is unanimously preferred, utility-wise, by all consumers over their Cournot–Nash consumption bundles.     

A weak bargaining set for contract choice problems

In this paper, we consider the problem of choosing a set of multi-party contracts, where each coalition of agents has a non-empty finite set of feasible contracts to choose from. We call such problems, contract choice problems. The main result in this paper states that every contract choice problem has a non-empty weak bargaining set. The need for such a solution concept which is considerably weaker than the core arises, since it is well-known that even for very simple contract choice problems, the core may be empty. We also show by means of an example that an analog of the bargaining set due to Mas-Colell [Mas-Colell, A., 1989. An equivalence theorem for a bargaining set. Journal of Mathematical Economics 18, 129–139], as well as the natural analog of the bargaining set due to Aumann and Mashler [Aumann, R., Maschler, M., 1964. The bargaining set for cooperative games. In: Dresher, M., Shapley, L., Tucker, A. (Eds.), Advances in Game Theory. Princeton University Press, Princeton, NJ] may be empty for contract choice problems.     

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     

Non-myopic learning in differential information economies: the core

Summary. We study the process of learning in a differential information economy, with a continuum of states of nature that follow a Markov process. The economy extends over an infinite number of periods and we assume that the agents behave non-myopically, i.e., they discount the future. We adopt a new equilibrium concept, the non-myopic core. A realized agreement in each period generates information that changes the underlying structure in the economy. The results we obtain serve as an extension to the results in Koutsougeras and Yannelis (1999) in a setting where agents behave non-myopically. In particular, we examine the following two questions: 1) If we have a sequence of allocations that are in an approximate non-myopic core (we allow for bounded rationality), is it possible to find a subsequence that converges to a non-myopic core allocation in a limit full information economy? 2) Given a non-myopic core allocation in a limit full information economy can we find a sequence of approximate non-myopic core allocations that converges to that allocation? Received: May 25, 1999; revised version: August 9, 1999     

Informational size and the incentive compatible coarse core in quasilinear economies

In this paper we show that in quasilinear economies the incentive compatible coarse core is non-empty if agents are informationally small in the sense of McLean and Postlewaite (2002) and the strict core in each state is non-empty.     

A Note on Viable Proposals*

Sengupta and Sengupta (“Viable Proposals,”International Economic Review 35 (1994), 347–59.) consider a payoff vector of a TU‐game as a viable proposal if it challenges each legitimate contender. They show that for each game the set of viable proposals is nonempty. Their proof, however, has a flaw. I present a proof based upon a result by Kalai and Schmeidler (“An Admissible Set Occurring in Various Bargaining Situations,”Journal of Economic Theory 14 (1977), 402–11) .     

Stochastic stability for roommate markets

We show that for any roommate market the set of stochastically stable matchings coincides with the set of absorbing matchings. This implies that whenever the core is non-empty (e.g., for marriage markets), a matching is in the core if and only if it is stochastically stable, i.e., stochastic stability is a characteristic of the core. Several solution concepts have been proposed to extend the core to all roommate markets (including those with an empty core). An important implication of our results is that the set of absorbing matchings is the only solution concept that is core consistent and shares the stochastic stability characteristic with the core.     

An elementary proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley Theorem

Summary This note provides an elementary short proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley (K-K-M-S) Theorem based on Brouwer's fixed point theorem. The usefulness of the K-K-M-S Theorem lies in the fact that it can be applied to prove directly Scarf's (1967) Theorem, i.e. any balanced game has a non-empty core. We also show that the K-K-M-S Theorem and the Gale-Nikaido-Debreu Theorem can be proved by the same arguments.We wish to thank Roko Aliprantis for useful comments.     

The core of some location games

An important problem in Location Theory is that of assigning plants to locations in an optimal manner. In the context of this problem, recognizing interplant transportation costs, Koopmans and Beckmann (1957) introduced the Quadratic Assignment Problem (QAP). It is shown in this paper that when the QAP is formulated as a cooperative location game, its core may be empty. By contrast, the core of the game corresponding to thelinear assignment problem (where transportation costs are disregarded) is assured to be non-empty. Some conditions under which the core is non-empty are discussed.I am grateful to two anonymous referees of this journal for their comments.     

Median-based extensions of an ordering over a set to the power set: An axiomatic characterization

In this paper we consider the problem of inducing an ordering over the set of all non-empty subsets of a finite set X of alternatives, given an ordering R over X. Assuming R to be antisymmetric and X to have at least six elements, we provide a set of independent, necessary, and sufficient conditions for the induced ordering to be “median-based” (so that every non-empty subset of X is “indifferent” to its own median set defined in terms of R).     

Existence and Welfare Properties of Equilibrium in an Exchange Economy with Multiple Divisible and Indivisible Commodities and Linear Production Technologies

In this paper we consider a class of economies with a finite number of divisible commodities, linear production technologies, and indivisible goods and a finite number of agents. This class contains several well-known economies with indivisible goods and money as special cases. It is shown that if the utility functions are continuous on the divisible commodities and are weakly monotonic both on one of the divisible commodities and on all the indivisible commodities, if each agent initially owns a sufficient amount of one of the divisible commodities, and if a “no production without input”-like assumption on the production sector holds, then there exists a competitive equilibrium for any economy in this class. The usual convexity assumption is not needed here. Furthermore, by imposing strong monotonicity on one of the divisible commodities we show that any competitive equilibrium is in the core of the economy and therefore the first theorem of welfare also holds. We further obtain a second welfare theorem stating that under some conditions a Pareto efficient allocation can be sustained by a competitive equilibrium allocation for some well-chosen redistribution of the total initial endowments. Journal of Economic Literature Classification Numbers: D4, D46, D5, D51, D6, D61.     

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.     

The Dynamic Vickrey Auction

We study the efficient allocation of a single object over a finite time horizon. Buyers arrive randomly over time, are long-lived, and have independent private values. The valuation of a buyer may depend on the time of the allocation in an arbitrary way. We construct an incentive compatible mechanism in which (A) there is a single financial transaction (with the buyer), (B) ex-post participation constraints are fulfilled, (C) there is no positive transfer to any agent and (D) payments are determined online. We exploit that under the efficient allocation rule, there is a unique potential winning period for each buyer. This reduces the multidimensional type to one dimension and the payment of the winner can be defined as the lowest valuation for the potential winning period, with which the buyer would have won the object. In a static model, this payment rule coincides with the payment rule of the Vickrey Auction.     

Von Neumann-Morgenstern stable sets in matching problems

The following properties of the core of a one-to-one matching problem are well-known: (i) the core is non-empty; (ii) the core is a distributive lattice; and (iii) the set of unmatched agents is the same for any two matchings belonging to the core. The literature on two-sided matching focuses almost exclusively on the core and studies extensively its properties. Our main result is the following characterization of (Von Neumann-Morgenstern) stable sets in one-to-one matching problems. We show that a set V of matchings is a stable set of a one-to-one matching problem only if V is a maximal set satisfying the following properties: (a) the core is a subset of V; (b) V is a distributive lattice; and (c) the set of unmatched agents is the same for all matchings belonging to V. Furthermore, a set is a stable set if it is the unique maximal set satisfying properties (a), (b), and (c).     

The core in an oligopoly market with indivisibility

Summary. This paper studies the core in an oligopoly market with indivisibility. It provides necessary and sufficient conditions for core existence in a general m-buyer n-seller market with indivisibility. When costs are dominated by opportunity costs (i.e., a firm's variable costs are sufficiently small), the core condition can be characterized by the primitive market parameters. In a 3-2 market with opportunity cost, the core is non-empty if and only if the larger seller's opportunity cost is either sufficiently large or sufficiently small. Received: June 9, 1999; revised version: October 22, 1999     

Robustness of sunspot equilibria

Summary. A sunspot equilibrium (SSE) is based on some extrinsic randomizing device (RD). We analyze the robustness of SSE. (1) We say that an SSE allocation is robust to refinements if it is also an SSE allocation based on any refinement of its RD. (2) We introduce two core concepts for analyzing the robustness of SSE in the face of cooperative-coalition formation. In the first, the blocking allocations are based on the RD that defines the SSE. In the second (stronger) core concept, coalitions select their own RDs. For the convex economy with restricted market participation, SSE allocations are robust under each of the definitions and the cores converge on replication of the economy to the set of SSE allocations. For the economy with an indivisible good, SSE allocations are not always robust. We provide examples of each of the following: (i) an SSE allocation that is not robust to refinement, (ii) an SSE allocation that is in neither core, (iii) an SSE allocation that is in the first core, but not in the second, and (iv) a core that does not converge upon replication to the set of SSE allocations. Received: July 31, 1995; revised version August 30, 1996