Indivisible commodities and an equivalence theorem on the strong core

We consider a pure exchange economy with finitely many indivisible commodities that are available only in integer quantities. We prove that in such an economy with a sufficiently large number of agents, but finitely many agents, the strong core coincides with the set of expenditure-minimizing Walrasian allocations. Because of the indivisibility, the preference maximization does not imply the expenditure minimization. An expenditure-minimizing Walrasian equilibrium is a state where, under some price vector, all agents satisfy both the preference maximization and the expenditure minimization.     

Theorems on the core of an economy with infinitely many commodities and consumers

It is known that the classical theorems of Grodal [Grodal, B., 1972. A second remark on the core of an atomless economy. Econometrica 40, 581–583] and Schmeidler [Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometrica 40, 579–580] on the veto power of small coalitions in finite dimensional, atomless economies can be extended (with some minor modifications) to include the case of countably many commodities. This paper presents a further extension of these results to include the case of uncountably many commodities. We also extend Vind’s [Vind, K., 1972. A third remark on the core of an atomless economy. Econometrica 40, 585–586] classical theorem on the veto power of big coalitions in finite dimensional, atomless economies to include the case of an arbitrary number of commodities. In another result, we show that in the coalitional economy defined by an atomless individualistic model, core–Walras equivalence holds even if the commodity space is non-separable. The above-mentioned results are also valid for a differential information economy with a finite state space. We also extend Kannai’s [Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815] theorem on the continuity of the core of a finite dimensional, large economy to include the case of an arbitrary number of commodities. All of our results are applications of a lemma, that we prove here, about the set of aggregate alternatives available to a coalition. Throughout the paper, the commodity space is assumed to be an ordered Banach space which has an interior point in its positive cone.     

On top responsiveness and strict core stability

Alcalde and Revilla [Journal of Mathematical Economics 40 (2004) 869–887] introduce a top responsiveness condition on players’ preferences in hedonic games and show that it guarantees the existence of a core stable partition. In the present note we strengthen this observation by proving that under top responsiveness even the strict core is non-empty.     

Implementation of the recursive core for partition function form games

In partition function form games, the recursive core (r-core) is implemented by a modified version of Perry and Reny’s [Perry, M., Reny, P., 1994. A non-cooperative view of coalition formation and the core. Econometrica 62, 795-817] non-cooperative game. Specifically, every stationary subgame perfect Nash equilibrium (SSPNE) outcome is an r-core outcome. With the additional assumption of total r-balancedness, every r-core outcome is an SSPNE outcome.     

The Walras core of an economy and its limit theorem

The Walras core of an economy is the set of allocations that are attainable for the consumers when their trades are constrained to be based on some agreed price system, such that no alternative price system exists for any sub-coalition that allows all members to trade to something better. As compared with the Edgeworth core, both coalitional improvements and being a candidate allocation for the Walras core become harder. The Walras core may even contain allocations that violate the usual Pareto efficiency. Nevertheless, the competitive allocations are the same under the two theories, and the equal-treatment Walras core allocations converge under general conditions to the competitive allocations in the process of replication.     

Erratum to “Do pure indivisibilities prevent core equivalence? Core equivalence theorem in an atomless economy with purely indivisible commodities only” [J. Math. Econ. 41 (4–5) (2005) 571–601]

We consider a pure exchange economy with a continuum of agents and finitely many indivisible commodities. Every commodity can be consumed only in integer amounts. Thus, agents’ preferences are locally satiated and no commodity bundle has necessarily local cheaper points. We introduce a core which is an intermediate concept between the strong core and the weak core. In our economy, this core is the most natural concept in the sense that it coincides with the set of all exactly feasible Walras allocations.     

The core can be accessed with a bounded number of blocks

We show the existence of an upper bound for the number of blocks required to get from one imputation to another provided that accessibility holds. The bound depends only on the number of players in the TU game considered. For the class of games with non-empty cores this means that the core can be reached via a bounded sequence of blocks.     

Stable matchings and the small core in Nash equilibrium in the college admissions problem

Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes in truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium in truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how different are players' equilibrium strategies from the true preferences and how many other unstable matchings are there at that Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by the direct revelation game induced by a stable mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching. Received: 30 December 1998 / Accepted: 12 October 2001 This paper is a revision of the paper “Manipulation and Stability in a College Admissions Problem” circulated since 1994. I thank Rich McLean, Abraham Neyman, Mark Satterthwaite, Sang-Chul Suh, and Tetsuji Yamada for helpful discussions. I thank the associate editor and the two anonymous referees for their helpful comments that have greatly improved the paper. I am grateful to the Kellogg G.S.M. at the Northwestern University for the hospitality for my visit. Any errors are mine.     

A noncooperative approach to the Mas-Colell bargaining set

We propose an extensive form game to give a noncooperative interpretation of the Mas-Colell bargaining set for transferable utility games. Based on the objection and counterobjection in the sense of the Mas-Colell bargaining set, a bilateral negotiation is designed in our model. We show that the equilibrium outcomes of our extensive form game coincide with the Mas-Colell bargaining set in the supperadditive environments.     

The hybrid equilibria and core selection in exchange economies with externalities

This paper introduces a hybrid equilibrium concept that combines the elements of cooperative and non-cooperative behaviors in an exchange economy with externalities. For a fixed coalition structure (or a partition of traders), the hybrid equilibrium is a price and consumption bundle such that each coalition chooses a core solution from its budget set and the consumption bundles are feasible. It becomes the competitive equilibrium when the position is the finest, and it selects a core allocation when the partition is the coarsest. The paper provides sufficient conditions for the existence of a hybrid equilibrium for any coalition structure.     

Deriving Nash equilibria as the supercore for a relational system

In this paper, under a binary relation that refines the standard relation which only accounts for single profitable deviations, we obtain that the set of NE strategy profiles of every finite non-cooperative game in normal form coincides with the supercore (Roth, 1976) of its associated abstract system. Further, under the standard relation we show when these two solution concepts coincide.     

Instability in the labor market for researchers

R&D is both an activity which involves team effort, and with many of the features of a zero-sum game. This paper shows that these characteristics make the labor market for researchers unstable in the sense that firms have incentives to continuously change the composition of their research teams. Related results concerning the core of several cooperative games in characteristic form are also proved.     

Extreme points of two digraph polytopes: Description and applications in economics and game theory

In this paper, we introduce two polytopes that respect a digraph in the sense that for every vector in the polytope every component corresponds to a node and is at least equal to the component corresponding to each successor of this node. The sharing polytope is the set of all elements from the unit simplex that respect the digraph. The fuzzy polytope is the set of all elements of the unit cube respecting the digraph. The main results are characterizations of the extreme points of the above described two digraph polytopes. We also give an economic application of the result on the sharing polytope and a game-theoretical application for the fuzzy polytope.     

Public good provision and the Smith Process

An existing public good provision mechanism known as the Smith Process (SP) is extended to allow for non-zero fixed cost, non-constant marginal cost and imperfectly divisible output. Two versions of SP are considered: unrestricted (USP) and restricted with a unanimity rule (RSPU). USP implements efficient choice provided the gap between marginal and average cost is sufficiently low. RSPU relaxes the conditions for efficient implementation but increases the set of equilibria involving inefficient choice. Furthermore, if weakly dominated strategies are eliminated, then non-provision is no longer an equilibrium under RSPU but continues to be one under USP.     

Implementation of voting operators

We show that a theory of implementation can be developed in the Aizerman–Aleskerov framework, capturing the main ideas regarding Nash implementation in the Arrovian case. In fact we obtain the counterparts of the results of [Maskin, E., 1977. Nash Equilibrium and Welfare Optimality. MIT, Mimeo] and [Moore, J., Repullo, R., 1990. Nash implementation. A full characterization. Econometrica 58, 1038–1100] in the new framework.     

Cores of many-player games; nonemptiness and equal treatment

This paper provides sufficient conditions to ensure nonemptiness of approximate cores of many-player games and symmetry of approximate core payoffs (the equal treatment property). The conditions are: (a) essential superadditivity—an option open to a group of players is to partition into smaller groups and realize the worths of these groups and (b) small group effectiveness (SGE)—almost all gains to collective activities can be realized by cooperation only within members of some partition of players into relatively small groups. Another condition, small group negligibility (SGN), is introduced and shown to be equivalent to SGE. SGN dictates that small groups of players cannot have significant effects on average (i.e., per capita) payoffs of large populations; thus, SGN is a analogue, for games with a finite player set, of the condition built into models with a continuum of player that sets of measure zero can be ignored. SGE implies per capita boundedness (PCB), that the supremum of average or per capita payoffs is uniformly bounded above. Further characterization of SGE in terms of its relationship to PCB is provided. It is known that if SGE does not hold, then approximate cores of many-player games may be empty. Examples are developed to show that if SGE does not hold and if there are players of “scarce types” (in other works, players with scarce attributes) then even if there is only a finite number of types of players and approximate cores are non-empty, symmetry may be lost; moreover, even players of abundant types may be treated asymmetrically by the core.     

A nonparametric test of weak separability and consumer preferences

Two stochastic nonparametric procedures are developed to evaluate the significance of violations of weak separability. When the data have measurement error, we show that the necessary and sufficient weak separability conditions of Varian [Varian, H., 1983. Nonparametric tests of consumer behavior. Review of Economic Studies 50, 99–110] must also satisfy the Afriat inequalities. The tests detect weak separability with high probability for weakly separable data. In addition, the procedures correctly reject weak separability for both nonseparable and random utility simulated data sets. The tests also fail to reject weak separability for a monetary and consumption data set which suggests that measurement error may be the source of the observed violations.     

The foundation of stability in extensive games with perfect information

This paper studies the decision-theoretic foundation for the notion of stability in the dynamic context of strategic interaction. We formulate and show that common knowledge of rationality implies a “stable” pattern of behavior in extensive games with perfect information. In the “generic” case, our approach is consistent with Aumann’s [Aumann, R.J., 1995. Backward induction and common knowledge of rationality. Games and Economic Behavior 8, 6–19] result that common knowledge of rationality leads to the backward induction outcome.     

CUE with many weak instruments and nearly singular design

This paper analyzes many weak moment asymptotics under the possibility of similar moments. The possibility of highly related moments arises when there are many of them. Knight and Fu (2000) designate the issue of similar regressors as the “nearly singular” design in the least squares case. In the nearly singular design, the sample variance converges to a singular limit term. However, Knight and Fu (2000) assume that on the nullspace of the limit term, the difference between the sample variance and the singular matrix converges in probability to a positive definite matrix when multiplied by an appropriate rate. We consider specifically Continuous Updating Estimator (CUE) with many weak moments under nearly singular design. We show that the nearly singular design affects the form of the limit of the many weak moment asymptotics that is introduced by Newey and Windmeijer (2009a). However, the estimator is still consistent and the Wald test has the standard χ²${\chi }^2$ limit.