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.     

A decentralized and informationally efficient mechanism realizing fair outcomes in economies with public goods

We present a decentralized mechanism (called Lindahl Egalitarian), which yields Pareto efficient and envy free allocations (i.e. fair outcomes). We show that the mechanism is informationally efficient in general production economies with an arbitrary, but finite, number of private and public goods, and a finite number of agents. The mechanism reduces to the Walrasian mechanism starting from equal wealth when no agent cares about public goods. We also prove that the set of Public Competitive equilibrium allocations (from equal endowments and proportional taxation), and the set of the Lindahl Egalitarian equilibrium allocations are the same. We are grateful to Xavier Calsamiglia and Albert Marcet for helpful conversations, and to A. de la Fuente, I. Macho, and an anonimous referee for useful suggestions. A. Manresa’s research has been supported by the grant CICYT PB90-0172. J. Aizpurua acknowledges financial support from the Government of Navarra.     

The bargaining set and the core in mixed markets with atoms and an atomless sector

We show that for a large class of both finite and mixed market games, the Bargaining Set coincides with the core, and is strictly larger than the set of all Walrasian allocations. This is to be constrasted with the recently established equivalence between the three sets in atomless markets.     

The veto mechanism in atomic differential information economies

We establish new characterizations of Walrasian expectations equilibria based on the veto mechanism in the framework of differential information economies with a complete finite measure space of agents. We show that it is enough to consider the veto power of a single coalition, consisting of the entire set of agents, to obtain the Aubin private core. Moreover, we investigate on the veto power of arbitrarily small and big coalitions, providing an extension to mixed markets of well known Schmeidler (1972) and Vind’s (1972) results in terms of Aubin private core allocations.     

The equilibrium set of infinite dimensional Walrasian economies and the natural projection

The natural projection plays a fundamental role to understand the behavior of the Walrasian economies. In this paper, we extend this method to analyze the behavior of infinite dimensional economies. We introduce the definition of the social equilibrium set, and we show that there exists a bijection between this set and the Walrasian equilibrium set of an infinite dimensional economy. In order to describe the main topological characteristics of both sets, we analyze the main differential characteristics of the excess utility function and then, we extend the method of the natural projection as suggested by Y. Balasko.     

Consistent collusion-proofness and correlation in exchange economies

We present a feasible strategic market mechanism with finitely many agents whose Nash, semi-strong Nash and coalition-proof Nash equilibria fully implement the Walrasian equilibria. We define a strategic equilibrium concept, called correlated semi-strong equilibrium, and show that the Walrasian equilibria can be implemented by these equilibria, and also by the coalition-proof correlated equilibria of our mechanism. We show that these two concepts, suitably modified with transfers, fully implement the Pareto optimal allocations.     

Replica core equivalence theorem: An extension of the Debreu–Scarf limit theorem to double infinity monetary economies

An overlapping generations model with the double infinity of commodities and agents is the most fundamental framework to introduce outside money into a static economic model. In this model, competitive equilibria may not necessarily be Pareto-optimal. Although Samuelson (1958) emphasized the role of fiat money as a certain kind of social contract, we cannot characterize it as a cooperative game-theoretic solution like the core. In this paper, we obtained a finite replica core characterization of Walrasian equilibrium allocations under non-negative wealth transfer and a core-limit characterization of Samuelson’s social contrivance of money. Preferences are not necessarily assumed to be ordered.     

Characterization of the Walrasian equilibria of the assignment model

We study the assignment model where a collection of indivisible goods are sold to a set of buyers who want to buy at most one good. We characterize the extreme and interior points of the set of Walrasian equilibrium price vectors for this model. Our characterizations are in terms of demand sets of buyers. Using these characterizations, we also give a unique characterization of the minimum and the maximum Walrasian equilibrium price vectors. Also, necessary and sufficient conditions are given under which the interior of the set of Walrasian equilibrium price vectors is non-empty. Several of the results are derived by interpreting Walrasian equilibrium price vectors as potential functions of an appropriate directed graph.     

Manipulation of the Walrasian mechanism in production economies with unbounded short-selling

Hurwicz (1979) and Otani and Sicilian (1982, 1990) characterized the Nash equilibrium allocations of the Walrasian demand manipulation game in successively more general exchange environments. In this paper, I extend the analysis to production economies with short-selling. First, I generalize Hurwicz’s and Otani and Sicilian’s theorem that any allocation at which each agent’s consumption bundle lies above her true offer curve can be supported in Nash equilibrium. I then show that for finite economies of any size the set of such allocations is often topologically large.Received: 17 January 2003, Accepted: 4 April 2005, JEL Classification: D51, D82For comments on this and earlier versions of the paper, I wish to thank Rick Bond, Bhaskar Chakravorti, Tom Gresik, Costas Syropoulos and William Thomson. I would especially like to thank Mike Jerison for helping to overcome a difficulty with a previous version. Also, the comments of the anonymous referees are gratefully acknowledged.     

Walras degrees and probability of a blocking coalition at pareto allocations

We attempt to determine the probability of a blocking coalition from a notion of being non- Walrasian which does not rely on the explicit use of norms. The key concept introduced is that of Walras degrees. Theorem 1 gives the bounds of the probability of blocking in terms of these degrees. Theorem 2 gives the asymptotic conditional probability of a blocking coalition given that coalitions are losing. The relationship between the degrees of the conventional norms on allocations is also investigated.     

Convergence to competitive equilibria and elimination of no-trade (in a strategic market game with limit prices)

The strategic market games literature contains many results that predict Walrasian equilibria in the competitive limit. However, they usually come at the expense of ad hoc assumptions that rule out “pathological” no trade equilibria. This paper studies a strategic market game with limit prices. The set of Nash equilibrium allocations of this game converges to the set containing all competitive equilibria and no-trade, when players are replicated. Moreover, two rounds of iterated deletion of weakly dominated strategies eliminate the no-trade equilibria. Hence, replication paired with two rounds of iterated dominance gives a clean prediction of competitive equilibrium.     

Switching from complete to incomplete information

We construct an elementary mechanism [Dutta, B., Sen, A., Vohra, R., 1995. Nash implementation through elementary mechanisms in economic environments. Review of Economic Design 1, 173–203] that Nash implements the constrained Walrasian correspondence. We extend it to incomplete and non-exclusive information economies by enlarging the message space of agents. In addition, measurability restrictions on allocations with respect to prices proper to constrained rational expectations equilibria are imposed in the outcome function. We show that by imposing such restrictions, the mechanism Bayesian implements the constrained rational expectations equilibrium correspondence. This result shows game-theoretic connections between these two market equilibrium concepts. However, these connections are obtained at the price of strong restrictions on the behavior of agents.     

Fair compensation with different social concerns for forgiveness

Forgiveness is an ethical ideal that advocates that a fresh start should be conferred upon those individuals who have changed their preferences and regret their previous decisions. Despite the ethical debate that such an idea generates, only a few papers have dealt with this issue in depth, and they have just focused on the case of full compensation for regret. Therefore, based on efficiency, robustness, and ethical requirements, we characterise a social ordering function that formally connects the ideal of forgiveness to the problem of compensating individuals when they differ in both their preferences and their initial endowment. This social ordering allows us to rank allocations that may or may not be associated with different concerns for forgiveness. Specifically, it proposes reducing inequality between reference-comparable budget sets.     

Multilateral bargaining and Walrasian equilibrium

A class of bargaining games in which agents bargain over prices and maximum trading constraints is considered: It is proved that all the Stationary Subgame Perfect Equilibria of these games implement Walrasian allocations as the bargaining frictions vanish. The result holds for any number of agents and is robust to different specifications of the bargaining process.     

Discrete representations of a continuum economy

We consider an atomless economy in which the continuum of agents is represented by a real interval. By dividing the interval and associating to every agent in each subinterval the same initial endowments and preferences, we define sequences of discrete economies as approximations to the initial continuum economy. We obtain convergence results for the core (or, alternatively, for the set of Walrasian allocations) of the continuum economy in terms of the cores of the approximating discrete economies. Finally, we state some counterexamples which provide a boundary for more general results in this framework. Received: 29 December 2000 / Accepted: 28 April 2002 C. Hervés and E. Moreno acknowledge support by Research Grant BEC2000-1388-C04-01 from the Dirección General de Investigación Científica y Técnica (DGICYT), Spanish Ministry of Education.?E. Moreno acknowledges support from the Spanish Ministry of Education through a post-doctoral fellowship in 1997 and from a Training and Mobility of Researchers (E.C.) fellowship in 1998 while visiting Universidade Nova de Lisboa.     

Power and efficiency in production pillage games

A pillage game is a formal model of Hobbesian anarchy as a coalitional game. The technology of pillage is specified by a power function that determines the power of each coalition as a function of its members and their wealth. A coalition can despoil any other coalition less powerful than itself. The present paper studies the problem of achieving an efficient allocation of resources when the required reallocation changes the distribution of power. For example, land redistribution may increase total production, but may also deprive the original owners of the power they need to compel compensation. In this case the original owners would block the redistribution. Previous work on pillage games has focused on the stable set (von Neumann–Morgenstern solution) as a representation of a stable balance of power. However, the balance of power is typically too delicate to support all efficient allocations. The present paper shows that for a large class of power functions, a recently developed extension of the stable set, called the legitimate set, can be rich enough to support all efficient allocations.      

Representation of non-transferable utility games by coalition production economies

We prove that every compactly generated non-transferable utility (NTU) game can be generated by a coalition production economy. The set of Walrasian payoff vectors for our induced coalition production economy coincides with the inner core of the balanced cover of the original game. This equivalence depends heavily on our representation. We exemplify that this equivalence need not hold in other representations. We also give a sufficient condition for the existence of a Walrasian equilibrium for our induced coalition production economy.     

Cournot vs. Walras: A reappraisal through simulations

Best-reply behavior in Cournot oligopolies generally leads to Cournot-Nash equilibrium, but imitative behavior selects the Walrasian equilibrium as the unique stochastically stable state. Previous work (Alós-Ferrer, 2004) showed that in the presence of memory, imitative behavior leads to a non-trivial dynamics selecting all quantities between the Cournot and Walrasian outcomes. However, the scope of previous results was limited to specific assumptions on demand and cost functions, and did not provide information on the shape of the distribution of outcomes. We use computational simulations to address these limitations. We show that the selection result for non-trivial memory holds beyond the set of well-behaved Cournot games previously analyzed. Further, we find that, in Cournot games, the limit distribution of long-run outcomes is highly skewed towards the Walrasian quantity. Although longer memory increases the importance of the Cournot equilibrium, the competitive outcome remains the dominant prediction.     

Learning and stability of the Bayesian–Walrasian equilibrium

Under the Bayesian–Walrasian Equilibrium (BWE) (see Balder and Yannelis, 2009), agents form price estimates based on their own private information, and in terms of those prices they can formulate estimated budget sets. Then, based on his/her own private information, each agent maximizes interim expected utility subject to his/her own estimated budget set. From the imprecision due to the price estimation it follows that the resulting equilibrium allocation may not clear the markets for every state of nature, i.e., exact feasibility of allocations may not occur. This paper shows that if the economy is repeated from period to period and agents refine their private information by observing the past BWE, then in the limit all agents will obtain the same information and market clearing will be reached. The converse is also true. The analysis provides a new way of looking at the asymmetric equilibrium which has a statistical foundation.     

Convergence to Walrasian equilibrium with minimal information

We consider convergence to Walrasian equilibrium in a situation where firms know only market price and their own cost function. We term this a situation of minimal information. We model the problem as a large population game of Cournot competition. The Nash equilibrium of this model is identical to the Walrasian equilibrium. We apply the best response (BR) dynamic as our main evolutionary model. This dynamic can be applied under minimal information as firms need to know only the market price and the their own cost to compute payoffs. We show that the BR dynamic converges globally to Nash equilibrium in an aggregative game like the Cournot model. Hence, it converges globally to the Walrasian equilibrium under minimal information. We extend the result to some other evolutionary dynamics using the method of potential games.