The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

A three way equivalence

In view of the well known core equivalence results in atomless economies, coincidence of market game equilibrium allocations with competitive allocations is tantamount to a three way equivalence between market game mechanisms, competitive equilibria and the core. Based on this idea we study equilibrium refinements of market games, which allow us to use the core equivalence machinery in order to provide an exact market game characterization of competitive equilibria.     

Bargaining, coalitions and competition

We study a Gale-like matching model in a large exchange economy, in which trade takes place through non-cooperative bargaining in coalitions of finite size. Under essentially the same conditions of core equivalence, we show that the strategic equilibrium outcomes of our model coincide with the Walrasian allocations of the economy. Our method of proof makes use of the theory of the core. With respect to previous work, our positive implementation result applies to a substantially larger class of economies: the model relaxes differentiability and convexity of preferences, and also admits an arbitrary number of divisible and indivisible goods.     

Implementations of the Nash solution based on its Walrasian characterization

Summary. The present paper provides three different support results for the Nash bargaining solution of -person bargaining games. First, for any bargaining game there is defined a non-cooperative game in strategic form, whose unique Nash equilibrium induces a payoff vector that coincides with the Nash solution of the bargaining game. Next this game is modified in such a way that the unique Nash equilibrium that supports the Nash solution is even in dominant strategies. After that an -stage game in extensive form is presented whose unique subgame perfect equilibrium supports the Nash solution of the bargaining game. Finally, the support results are shown to induce implementation results in the sense of mechanism theory. Received: October 3, 1999; revised version: October 26, 1999     

A two-stage core with applications to asset market and differential information economies

Summary. We introduce a new core concept, called the two-stage core, which is appropriate for economies with sequential trade. We prove a general existence theorem and present two applications of the two-stage core: (i) In asset markets economies where we extend our existence proof to the case of consumption sets with no lower bound, in order to capture the case of arbitrary short sales of assets. Further, we show that the two-stage core is non empty in the Hart (1975) example where a rational expectations equilibrium fails to exist. (ii) In differential information economies where we provide sufficient conditions for the incentive compatibility of trades. Namely, that no coalition of agents can misreport the true state and provide improvements to all its members, even by redistributing the benefits from misreporting. Received: December 20, 1995; revised version: December 6, 1996     

Cournot–Nash equilibria in continuum games with non-ordered preferences

In the usual framework of continuum games with externalities, we substantially generalize Cournot–Nash existence results [Balder, A unifying approach to existence of Nash equilibria, Int. J.Game Theory 24 (1995) 79–94; On the existence of Cournot–Nash equilibria in continuum games, J. Math. Econ. 32 (1999) 207–223; A unifying pair of Cournot–Nash equilibrium existence results, J. Econ. Theory 102 (2002) 437–470] to games with possibly non-ordered preferences, providing a continuum analogue of the seminal existence results by Mas-Colell [An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ. 1 (1974) 237–246], Gale and Mas-Colell [An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15], Shafer and Sonnenschein [Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348], Borglin and Keiding [Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econ. 3 (1976) 313–316] and Yannelis and Prabhakar [Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12 (1983) 233–245].     

Convergence of the Aumann–Davis–Maschler and Geanakoplos bargaining sets

Summary. Geanakoplos [17] defined a notion of bargaining set, and proved that his bargaining set is approximately competitive in large finite transferable utility (TU) exchange economies with smooth preferences. Shapley and Shubik [26] showed that the Aumann–Davis–Maschler bargaining set is approximately competitive in replica sequences of TU exchange economies with smooth preferences. We extend Geanakoplos result to nontransferable utility (NTU) exchange economies without smooth preferences, and we extend the Shapley and Shubik result to non-replica sequences of NTU exchange economies with smooth preferences.Received: November 11, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Information and size of coalitions

We consider a set of asymmetrically informed agents, where the information of each trader is susceptible of being altered when she becomes a member of a coalition. For this, we consider a general rule that depending on the coalition, a signal (or an information partition) is assigned to each member of the coalition. We set examples showing that Grodal’s (Econometrica 40:581–583, 1972), Schmeidler’s (Econometrica 40:579–580, 1972) and Vind’s (Econometrica 40:585–586, 1972) core characterizations of a continuum economy may fail in this general informational setting. However, under mild assumptions on the rule, we extend Schmeidler’s and Vind’s results to economies that allocate information to agents in each coalition according to the rule. We then focus on information mechanisms based on the size of coalitions and provide a general characterization result for the corresponding cores. Moreover, we pay close attention to the rule that assigns the shared information to each member of specific coalitions. We prove that the resulting cores are exactly the same independently of whether arbitrarily small or large coalitions share information.     

The bargaining set of a large game

We study the equivalence between the Mas-Colell bargaining set and the core in the general context of TU games with a measurable space of players. In the first part of the paper, we study the problem without imposing any restriction on the class of games we consider. In the second part, we first introduce a new class of exact games, which we call thin games. For these games, we show not only that the Mas-Colell bargaining set is equal to the core, but also that it is the unique stable set in the sense of von Neumann and Morgenstern. We then study the relation between thin games, exact non-atomic market games and non-atomic convex games. Finally, by further developing “thinness” related ideas, we prove new equivalence results for a class of non-exact market games as well as a class of non-exact, non-market games.     

Edgeworth's conjecture with infinitely many commodities: commodity differentiation

Summary. Convergence of the cores of finite economies to the set of Walrasian allocations as the number of agents grows has long been taken as one of the basic tests of perfect competition. The present paper examines this test in the most natural model of commodity differentiation: the commodity space is the space of nonnegative measures, endowed with the topology of weak convergence. In Anderson and Zame [12], we gave counterexamples to core convergence in L ¹, a space in which core convergence holds for replica economies and core equivalence holds for continuum economies; in addition, we gave a core convergence theorem under the assumption that traders' utility functions exhibit uniformly vanishing marginal utility at infinity. In this paper, we provide two core convergence results for the commodity differentiation model. A key technical virtue of this space is that relatively large sets (in particular, closed norm-bounded sets) are compact. This permits us to invoke a version of the Shapley-Folkman Theorem for compact subsets of an infinite-dimensional space. We show that, for sufficiently large economies in which endowments come from a norm bounded set, preferences satisfy an equidesirability condition, and either (i) preferences exhibit uniformly bounded marginal rates of substitution or (ii) endowments come from an order-bounded set, core allocations can be approximately decentralized by prices. Received: July 29, 1996; revised version: January 14, 1997     

Existence and Uniqueness of Ordinal Nash Outcomes

In this paper we present necessary and sufficient conditions for existence and uniqueness of ordinal Nash outcomes. These outcomes are derived from the ordinal Nash solution—a reinterpretation and an extension of the Nash bargaining solution that allows bargainers to have preference relations that are more general than expected utility. Our task is undertaken by the construction of a new notion called “induced utilities”. Journal of Economic Literature Classification Number: C78.     

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.     

Tax systems and direct mechanisms in large finite economies

The equivalence of anonymous direct mechanisms and tax systems for continuum economies, which has been demonstrated by Hammond (1979) and Guesnerie (1981), is shown to have its counterparts in large finite economies.We are deeply indebted to Roger Guesnerie for stimulating our interest in this problem and for the inspiration gained through his work.     

Multilateral bargaining: conditional and unconditional offers

Summary. We present a game in which n persons split a cake, where a distinction between conditional and unconditional offers is made. This distinction sheds light on the contrasting results obtained in the previous literature of multilateral bargaining. By allowing the proposer to make both conditional and unconditional offers, we show that the game has a unique subgame perfect Nash equilibrium outcome. Received: March 14, 2000; revised version: March 13, 2001     

Strong Equilibrium in Congestion Games

[6]introduced the class of congestion games and proved that they always possess a Nash equilibrium in pure strategies. Here we obtain conditions for the existence of a strong equilibrium in this class of games, as well as for the equivalence of Nash and strong equilibria. We also give conditions for uniqueness and for Pareto optimality of the Nash equilibrium. Except for a natural monotonicity assumption on the utilities, the conditions are expressed only in terms of the underlying congestion game form. It turns out that avoiding a certain type of bad configuration in the strategy spaces is essential to positive results.Journal of Economic LiteratureClassification Numbers: C71, C72, D62.     

On the Interpretation of Nonsymmetric Bargaining Solutions and Their Extension to Nonexpected Utility Preferences

In a recent paper, Rubinstein, Safra, and Thomson reexamine the foundations of the axiomatic bargaining theory as formulated by Nash. In it they reinterpret the Nash bargaining solution and extend it to a family of nonexpected utility preferences, providing an interpretation of this solution concept and an axiomatization by means of a suitable translation of Nash axioms. This paper extends the work of these authors to the family, characterized by Kalai, of nonsymmetric solutions that appear from the Nash bargaining solution when the symmetry axiom is dropped. This extension requires an axiom relating the solution outcomes of isomorphic games. Journal of Economic Literature Classification Number: C78.     

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.     

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.     

Coalitional bargaining games with random proposers: Theory and application

We consider a non-cooperative coalitional bargaining game with random proposers in a general situation for which players differ in recognition probability and time preference. We characterize an efficient equilibrium as the generalized Nash bargaining solution that belongs to the core. The model is applied to wage bargaining between an employer and multiple workers. Although involuntary unemployment may occur in equilibrium, full employment emerges as players become sufficiently patient.