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     

Preference for safety under the Choquet model: in search of a characterization

Victor prefers safety more than Ursula if whenever Ursula prefers a constant to an uncertain act, so does Victor. This paradigm, whose expected utility (EU) version is Arrow and Pratt’s more risk aversion concept, will be studied in the Choquet expected utility (CEU) model. Necessary condition Pointwise inequality between a function of the utility functions and another of the capacities is necessary and sufficient for the preference by Victor of safety over a dichotomous act whenever such is the preference of Ursula. However, increased preference for safety versus dichotomous acts does not imply preference by Victor of safety over a general act whenever such is the preference of Ursula. A counterexample will be provided, via the casino theory of Dubins and Savage. Sufficient condition Separation of the two functions by some convex function is sufficient for Victor to prefer safety more than Ursula, over general acts. Furthermore, a condition on the capacities will be presented for simplicity seeking, the preference by Victor over any act for some dichotomous act that leaves Ursula indifferent. This condition is met in particular if Victor’s capacity is a convex function of Ursula’s capacity. For these cases, the pointwise inequality (necessary) condition is a characterization of greater preference for safety, extending the Arrow–Pratt notion from EU to CEU and rank-dependent utility (RDU). These inequalities preserve the flavor of the “more pessimism than greediness” characterization of monotone risk aversion by Chateauneuf, Cohen and Meilijson in the RDU model and its extension by Grant and Quiggin to CEU. Preferences between safety and dichotomous acts are at the core of the biseparable preferences model of Ghirardato and Marinacci.     

On industry equilibrium under uncertainty

The paper considers an industry consisting of numerous firms that produce a homogeneous output, the demand for which is a random variable. Each firm belongs to one of K possible types, and each type is characterized by a U-shaped average cost curve. It is shown that: (i) the first-order necessary conditions for efficient investment and output are sufficient; accordingly, the set of competitive equilibria is non-empty and coincides with the set of efficient allocations; (ii) a dynamic process of free entry and exit of firms, guided by expected profits, is quasistable and every limit point is a competitive equilibrium. The paper also defines a sufficient condition for uniqueness of the competitive equilibrium, in which case it is stable.     

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).     

Aggregation and the existence of a social utility function

In a well-known paper Gorman (Econometrica21 (1953)) established that the necessary and sufficient condition for the existence of an aggregate, or social, utility function, independent of the distribution of income, is that all individuals' income consumption paths be parallel straight lines. Recently Chipman (J. Econ. Theory8 (1974)), building on the paper of Hurwicz and Uzawa (in “Preference Utility and Demand”) has shown that if the distribution of income is proportional and individual preferences are homothetic, aggregate consumption behavior obeys the necessary integrability conditions. It is shown here that the consistency of aggregate behavior can be derived from more general conditions than the ones used by Chipman and Gorman. Examples of demand systems from which aggregate behavior implies a social utility function are provided. It is then shown that if individual demand functions are linear in income—a form employed by both Gorman and Chipman—it is not necessary that the distribution of income be fixed.     

Robust implementation in general mechanisms

A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation.Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.     

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).     

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.     

The minimal dominant set is a non-empty core-extension

A set of outcomes for a transferable utility game in characteristic function form is dominant if it is, with respect to an outsider-independent dominance relation, accessible and closed. This outsider-independent dominance relation is restrictive in the sense that a deviating coalition cannot determine the payoffs of those coalitions that are not involved in the deviation. Each game generates a unique minimal (for inclusion) dominant set. This minimal dominant set is non-empty and returns the coalition structure core in case this core is non-empty. We provide an algorithm to find the minimal dominant set.     

The relative interior of the base polyhedron and the core

Summary. This note provides two necessary and sufficient conditions for the relative interior of the core (and the base polyhedron) to be non-empty: (i) the second largest excess of the prenucleolus is negative; (ii) the grand coalition's payoff is greater than the minimum no-blocking payoff. Such conditions imply an intuitive method in proving core existence, they also imply results on the sensitivity of the base polyhedron and the core. Received: June 25, 1999; revised version: May 8, 2000     

On the Existence of Undominated Nash Equilibria in Normal Form Games

In this paper the existence problem of undominated Nash equilibrium in normal form games is analyzed. It is shown that an undominated Nash equilibrium exists, if (a) strategy sets are convex polytopes inRⁿand (b) utility functions are affine with respect to each player's own strategy. It is shown by counterexamples that, first, it is not sufficient to have concave utility functions instead of affine under condition (b) even when condition (a) is satisfied, and, second, it is not sufficient to have just compact and convex strategy sets instead of polytopes in condition (a) even when condition (b) is satisfied.     

Recursive preferences and balanced growth

We study a class of utility functions that are defined recursively by an aggregator W(x,y) where u_t=W(c_t,u_t+1). In single-agent economies it is known that a sufficient condition for the existence of a balanced growth path is that utility should be homogenous of degree γ. In the context of a multi-agent economy we show that this restriction implies that either a balanced growth equilibrium fails to exist or all agents have the same constant discount factor. We suggest a generalization of recursive preferences wherein the intertemporal utility function is time dependent. Within this class we establish that there may exist a balanced growth equilibrium even if agents are different.     

Diversification, convex preferences and non-empty core in the Choquet expected utility model

Summary. We show, in the Choquet expected utility model, that preference for diversification, that is, convex preferences, is equivalent to a concave utility index and a convex capacity. We then introduce a weaker notion of diversification, namely “sure diversification.” We show that this implies that the core of the capacity is non-empty. The converse holds under concavity of the utility index, which is itself equivalent to the notion of comonotone diversification, that we introduce. In an Anscombe-Aumann setting, preference for diversification is equivalent to convexity of the capacity and preference for sure diversification is equivalent to non-empty core. In the expected utility model, all these notions of diversification are equivalent and are represented by the concavity of the utility index. Received: July 27, 1999; revised version: November 7, 2000     

Subgame perfect implementation: A full characterization

Moore and Repullo [Subgame perfect implementation, Econometrica 56 (1988) 1191-1220], and Abreu and Sen [Subgame perfect implementation: a necessary and almost sufficient condition, J. Econ. Theory 50 (1990) 285-299] introduce distinct necessary and sufficient conditions for SPE implementation, when the number of players is at least three. This paper closes the gap between the conditions—a complete characterization of the SPE implementable choice rules is provided. The characterization consists of α^*, which strengthens α of Abreu-Sen by adding it a restricted veto-power condition, and the unanimity condition. Under strict preferences α^* is equal to α.     

Minimum Memory for Equivalence between Ex Ante Optimality and Time-Consistency

I give a necessary and sufficient condition on memory of a player in an extensive game for equivalence between ex ante optimality and time-consistency (for all payoff assignments). The condition is called A-loss recall and requires that each loss of a player's memory can be traced back to some loss of memory of his own action. A-loss recall is also shown to be a necessary and sufficient condition for the existence of a time-consistent strategy (for all payoff assignments) if the player is conscious-minded. Journal of Economic Literature Classification Numbers: C72, D80.     

An Afriat Theorem for the collective model of household consumption

We provide a nonparametric ‘revealed preference’ characterization of rational household behavior in terms of the collective consumption model, while accounting for general individual preferences that can be non-convex. Our main result is the Collective Afriat Theorem, which parallels the well-known Afriat Theorem for the unitary model. First, it provides a characterization of collectively rational consumption behavior in terms of collective Afriat inequalities. Next, it implies the Collective Axiom of Revealed Preference (CARP) as a testable necessary and sufficient condition for data consistency with collective rationality. Finally, the theorem has some interesting testability implications. With only a finite set of observations, the nature of consumption externalities (positive or negative) in the intra-household allocation process is non-testable. The same non-testability conclusion holds for privateness (with or without externalities) or publicness of consumption. By contrast, concavity of individual utility functions (representing convex preferences) turns out to be testable. In addition, monotonicity is testable for the model that assumes all household consumption is public.     

Endogenous Verifiability and Optimality in Agency

This paper investigates the optimal contract design in a principal-agent model where verification of an agent's action is endogenously determined through strategic interactions between contracting parties. We derive a necessary and sufficient condition for the first best outcome to be implemented as an equilibrium. The equilibrium has the following features: (i) The action level that the agent chooses is not verified even if it is possible. (ii) Nevertheless, the first best can be attained by making a contract contingent on the unverified action. Journal of Economic Literature Classification Numbers: D20, K40.     

When is multidimensional screening a convex program?

A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis, quasi-linear utilities, and that agents can choose only pure strategies, we identify a structural condition on the value b(x,y) of product type y to agent type x — and on the principal?s costs c(y) — which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal?s problem theoretically and computationally tractable; in particular, it allows us to derive uniqueness and stability of the principal?s optimal strategy — and similarly of the strategy maximizing the expected welfare of the agents when the principal?s profitability is constrained. We call this condition non-negative cross-curvature: it is also (i) necessary and sufficient to guarantee convexity of the set of b-convex functions, (ii) invariant under reparametrization of agent and/or product types by diffeomorphisms, and (iii) a strengthening of Ma, Trudinger and Wang?s necessary and sufficient condition (A3w) for continuity of the correspondence between an exogenously prescribed distribution of agents and of products. We derive the persistence of economic effects such as the desirability for a monopoly to establish prices so high they effectively exclude a positive fraction of its potential customers, in nearly the full range of non-negatively cross-curved models.     

Stability and Largeness of the Core

In general, there are examples of TU games where the core is stable but is not large. In this paper, we show that the extendability condition introduced by Kikuta and Shapley (1986, “Core Stability in n-Person Games,” Mimeo) is sufficient for the core to be stable as well as large, for TU games with five or fewer players. We provide a counter example when the number of players is six. We then introduce a stronger extendability condition and show that it is necessary and sufficient for the core to be large. Our proof makes use of a well-known result from the theory of convex sets. Journal of Economic Literature Classification Number: C71.     

Job Matching and Coalition Formation with Utility or Disutility of Co-workers

This paper studies the job matching market in Kelso and Crawford (1982), Econometrica50, 1483–1504, with one exception, that co-workers may generate utility or disutility in the workplace. We provide a simple idea to show how a great number of sufficient conditions for a nonempty core in the literature can be extended to this labor market. A generalized competitive equilibrium with a tax/subsidy system is also discussed. Journal of Economic Literature Classification Number: D71.