Francesca Busetto

Cournot-Walras equilibrium as a subgame perfect equilibrium

In this paper, we investigate the problem of the strategic foundation of the Cournot–Walras equilibrium approach. To this end, we respecify à la Cournot–Walras the mixed version of a model of simultaneous, noncooperative exchange, originally proposed by Lloyd S. Shapley. We show, through an example, that the set of the Cournot–Walras equilibrium allocations of this respecification does not coincide with the set of the Cournot–Nash equilibrium allocations of the mixed version of the original Shapley’s model. As the nonequivalence, in a one-stage setting, can be explained by the intrinsic two-stage nature of the Cournot–Walras equilibrium concept, we are led to consider a further reformulation of the Shapley’s model as a two-stage game, where the atoms move in the first stage and the atomless sector moves in the second stage. Our main result shows that the set of the Cournot–Walras equilibrium allocations coincides with a specific set of subgame perfect equilibrium allocations of this two-stage game, which we call the set of the Pseudo–Markov perfect equilibrium allocations.

Reconsidering two-agent Nash implementation

In this paper, we reconsider the full characterization of two-agent Nash implementation provided in the celebrated papers by Moore and Repullo (Econometrica 58:1083–1099, 1990) and Dutta and Sen (Rev Econ Stud 58:121–128, 1991), since we are able to show that the characterizing conditions are not logically independent. We prove that an amended version of the conditions proposed in these papers is still necessary and sufficient for Nash implementability. Then, by using our necessary and sufficient condition, we show that Maskin’s impossibility result can be avoided under restrictions on the outcomes and the domain of preferences much weaker than those previously imposed by Moore and Repullo (Econometrica 58:1083–1099, 1990) and Dutta and Sen (Rev Econ Stud 58:121–128, 1991).

“Very Nice” trivial equilibria in strategic market games

Abstract Following Shapley [Theory of Measurement of Economic Externalities, Academic Press, New York, 1976], we study the problem of the existence of a Nash Equilibrium (NE) in which each trading post is either active or “legitimately” inactive, and we call it a Shapley NE. We consider an example of an exchange economy, borrowed from Cordella and Gabszewicz [Games Econ. Behav. 22 (1998) 162–169], which satisfies the assumptions of Dubey and Shubik [J. Econ. Theory 17 (1978) 1–20], and we show that the trivial equilibrium, the unique NE of the associated strategic market game, is not “very nice,” in the sense that it is not “legitimately” trivial. This result has the more general implication that, under the Dubey and Shubiks assumptions, a Shapley NE may fail to exist.

Asymptotic equivalence between Cournot–Nash and Walras equilibria in exchange economies with atoms and an atomless part

In this paper, we consider an exchange economy à la Shitovitz (Econometrica 41:467–501, 1973), with atoms and an atomless set. We associate with it a strategic market game of the kind first proposed by Lloyd S. Shapley, known as the Shapley window model. We analyze the relationship between the set of the Cournot–Nash allocations of the strategic market game and the Walras allocations of the exchange economy with which it is associated. We show, with an example, that even when atoms are countably infinite, any Cournot–Nash allocation of the game is not a Walras allocation of the underlying exchange economy. Accordingly, in the original spirit of Cournot (Recherches sur les principes mathématiques de la théorie des richesses. Hachette, Paris, 1838), we partially replicate the mixed exchange economy by increasing the number of atoms, without affecting the atomless part, and ensuring that the measure space of agents remains finite. Our main theorem shows that any sequence of Cournot–Nash allocations of the strategic market games associated with the partial replications of the exchange economy has a limit point for each trader and that the assignment determined by these limit points is a Walrasian allocation of the original economy.

Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities.

We show the existence of a Cournot–Nash equilibrium for a mixed version of the Shapley window model, where large traders are represented as atoms and small traders are represented by an atomless part. Previous existence theorems for the Shapley window model, provided by Sahi and Yao (1989) in the case of economies with a finite number of traders and by Busetto et al. (2011) in the case of mixed exchange economies, are essentially based on the assumption that there are at least two atoms with strictly positive endowments and indifference curves contained in the strict interior of the commodity space. Our result does not require this restriction. It relies on the characteristics of the atomless part of the economy and exploits the fact that traders belonging to the atomless part have an endogenous “Walrasian” behavior.

Nondictatorial Arrovian Social Welfare Functions: An Integer Programming Approach

In the line opened by Kalai and Muller (1977), we explore new con- ditions on preference domains which make it possible to avoid Arrows impossibility result. In our main theorem, we provide a complete char- acterization of the domains admitting nondictatorial Arrovian social welfare functions with ties (i.e. including indi erence in the range) by introducing a notion of strict decomposability. In the proof, we use integer programming tools, following an approach rst applied to so- cial choice theory by Sethuraman, Teo and Vohra ((2003), (2006)). In order to obtain a representation of Arrovian social welfare functions whose range can include indi erence, we generalize Sethuraman et al.s work and specify integer programs in which variables are allowed to assume values in the set indeed, we show that there exists a one-to-one correspondence between the solutions of an integer program de ned on this set and the set of all Arrovian social welfare functions - without restrictions on the range