Andreu Mas-Colell

Potential, Value, and Consistency

Let P be a real-valued function defined on the space of cooperative games with transferable utility, satisfying the following condition: In every game, the marginal contributions of all players (according to P) are efficient (i.e., add up to the worth of the grand coalition). It is proved that there exists just one such function P--called the potential--and moreover that the resulting payoff vector coincides with the Shapley value. The potential approach yields other characterizations for the value; in particular, in terms of a new internal consistency property. Further results deal with weighted values and with the nontransferable utility case. Copyright 1989 by The Econometric Society.

Bargaining and Value

The authors present and analyze a model of noncooperative bargaining among n participants, applied to situations describable as games in coalitional form. This leads to a unified solution theory for such games that have as special cases the Shapley value in the transferable utility case, the Nash bargaining solution in the pure bargaining case, and the recently introduced Maschler-Owen consistent value in the general nontransferable utility case. Moreover, the authors show that any variation (in a certain class) of their bargaining procedure which generates the Shapley value in the transferable utility setup must yield the consistent value in the general nontransferable utility setup. Copyright 1996 by The Econometric Society.

An equilibrium existence theorem for a general model without ordered preferences

In a recent paper the second author has shown that some of the usual hypotheses on consumers’ preferences are not needed for the proof of existence of a Walrasian General Equilibrium [Mas-Cole11 (1974)]. Specifically, it is not necessary that preferences come from a preference ordering. The only order property required is irreflexivity (meaning that a given commodity bundle is not preferred to itself). The properties of non-satiation, continuity and convexity of preferred sets turn out to be sufficient to obtain the existence result. The main purpose of the present note is to give a second proof of this fact which seems simpler than that of Mas-Cole11 (1974), and no more lengthy or complicated than the known equilibrium existence proofs which use ordered preferences. In two additional respects the model studied here generalizes the usual equilibrium model. The standard Walras, Arrow-Debreu theory treats what might be called the laissez-faire model in which each agent’s income is whatever he gets from selling goods plus his share of the profits of any firm in which he may own stock. In the present model the income of a consumer may be any continuous function of the prices, so the laissez-faire income function is included, but so also would any rule for assigning income to consumers (e.g., according to his ability or his need or the color of his eyes). Another way of saying this is that the model includes the possibility of arbitrary lump sum transfers of income among consumers, as might be achieved, for example, by a program of income taxes and subsidies. This substantial economic generalization requires no change whatever in the mathematical argument. The second generalization concerns production. The only requirement on our production set, besides the usual closure, convexity, and free disposal, is that it intersect the positive orthant in a boundedset.This means that the usual assumption

On a theorem of Schmeidler

Abstract Over a decade ago D. Schmeidler (1973) introduced a concept of non-cooperative equilibrium for games with a continuum of agents and, under a restriction on the payoff functions, established the existence of an equilibrium in pure strategies. The purpose of this note is to reformalize the model and the equilibrium notion of Schmeidler in terms of distributions rather than measurable functions. We shall see how once the definitions are available we get (pure strategy) equilibrium existence theorems quite effortlessly and under general conditions. A number of remarks contain applications to, among others, incomplete information games.

Uncoupled Dynamics Do Not Lead to Nash Equilibrium

We call a dynamical system uncoupled if the dynamic for each player does not depend on the payoffs of the other players. We show that there are no uncoupled dynamics that are guaranteed to converge to Nash equilibrium, even when the Nash equilibrium is unique.

A model of equilibrium with differentiated commodities

The equilibrium theory associated with the names of Arrow and Debreu [see Debreu (1959), Arrow and Hahn (1971)] contemplates a world with a finite number of homogeneous and perfectly divisible commodities where traders interact (exclusively) through a price system taken by each one of them as given. This perfectly competitive hypothesis has been justified by the ‘core equivalence theorem’ of Debreu and Scarf (1963) and Aumann (1964) [for a thorough account see Hildenbrand (1974)] : if no trader arrives to the market with a substantial amount, i.e., a ‘corner’, of any commodity, then unrestricted bargaining (in the cooperative sense of core theory) leads to perfect competition. In contrast, imperfect competition theory [in either Chamberlin (1956) or Robinson (1933) version] starts with a very different perception of the economic realm; commodities are not homogeneous but subject to differentiation and, consequently, traders enjoy a certain degree of monopoly with respect to the commodities they control. Still, the monopoly power of every single trader is limited by the existence of substitutability relations among commodities; it is a common contention of imperfect competition theory that in a large economy with a large number of mutually substitutive commodities and no ‘big’ trader, every commodity will be substitutable in the market with infinite elasticity and a perfectly competitive outcome will prevail [see, for an account, Samuelson (1969, p. 135)]. The analogy between the statements of the two previous paragraphs is clear.

Equilibrium theory in infinite dimensional spaces

Publisher Summary This chapter summarizes the account of the extension of the classical general equilibrium model to an infinite dimensional setting. The classical finite dimensional theory, the commodity space is the canonical finite dimensional linear space R n . By contrast, there is no canonical infinite dimensional linear space. Different economic applications require models involving different infinite dimensional linear spaces. The mathematical discipline of functional analysis has already been well developed as a tool for the abstract study of linear spaces. The chapter follows the methodology of functional analysis and attacks the existence problem. Advantage of this method is that it yields general results, capable of application in a wide variety of specific models. An important line of research in classical general equilibrium theory has been the relationship of the core to the set of competitive allocations. In the infinite dimensional setting, an extensive body of work has been developed, which centers around the infinite-dimensional version of the Debreu–Scarf core convergence theorem.

A General Class of Adaptive Strategies

We exhibit and characterize an entire class of simple adaptive strategies, in the repeated play of a game, having the Hannan-consistency property: in the long-run, the player is guaranteed an average payoff as large as the best-reply payoff to the empirical distribution of play of the other players; i.e., there is no “regret.” Smooth fictitious play (Fudenberg and Levine [1995, J. Econ. Dynam. Control19, 1065–1090]) and regret-matching (Hart and Mas-Colell [2000, Econometrica68, 1127–1150]) are particular cases. The motivation and application of the current paper come from the study of procedures whose empirical distribution of play is, in the long run, (almost) a correlated equilibrium. For the analysis we first develop a generalization of Blackwells (1956, Pacific J. Math.6, 1–8) approachability strategy for games with vector payoffs. Journal of Economic Literature Classification Numbers: C7, D7, C6.

Real indeterminacy with financial assets

The purpose of this paper, which takes up after D. Cass (1984a, 1984b) is to find the degree of real indeterminacy inherent in models with purely financial assets. We solve the problem for the case where there are enough traders (precisely, the number of traders is larger than the number of bonds) and the asset returns structure is in general position. We find that if the number of bonds is non-zero and fewer than the number of states then, generically, the number of dimensions of real indeterminacy is S-1, one less than the number of states. There is something of a surprise in the above result, namely the dimension of real indeterminacy does not depend on the number of bonds (except in the two limit cases). Indeed one initial conjecture was S-B. This points to an intriguing qualitative discontinuity at the complete market configuration. If markets are financially complete then the model is determinate. Let just one bond be missing and the model become highly indeterminate. Thus, in this sense, the complete markets hypothesis lacks robustness.

An equivalence theorem for a bargaining set

Abstract First, a modification of the Aumann-Maschler Bargaining Set is proposed. Then it is shown, under conditions of generality similar to the Core Equivalence Theorem, that the Bargaining Set and the set of Walrasian allocations coincide.