On the existence of approximated equilibria in discontinuous economies.

In this paper, we prove an existence theorem for approximated equilibria in a class of discontinuous economies. The existence result is a direct consequence of a discontinuous extension of Brouwer’s fixed point Theorem (1912), and is a refinement of several classical results in the standard general equilibrium with incomplete markets (GEI) model. [Brouwer, L.E.J., 1912. Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen 71, 97–115.] As a by-product, we get the first existence proof of an approximated equilibrium in the GEI model, without perturbing the asset structure nor the endowments. Our main theorem rests on a new topological structure result for the asset equilibrium space and may be of interest by itself.General equilibrium; Incomplete markets; Discontinuity; Fixed point;

Nash equilibrium existence for some discontinuous games

Answering to an open question of Herings et al. (see [3]), one extends their fixed point theorem to mappings defined on convex compact subset of Rn, and not only polytopes. Such extension is important in non-cooperative game theory, where typical strategy sets are convex and compact. An application in game theory is given.Discontinuous game, Nash equilibrium, fixed point theorem.

On the orientability of the asset equilibrium manifold

This paper addresses partly an open question raised in the Handbook of Mathematical Economics about the orientability of the pseudo-equilibrium manifold in the basic two-period General Equilibrium with Incomplete markets (GEI) model. For a broad class of explicit asset structures, it is proved that the asset equilibrium space is an orientable manifold if S-J is even, where S is the number of states of nature and J the number of assets. This implies, under the same conditions, the orientability of the pseudo-equilibrium manifold. By a standard homotopy argument, it also entails the index theorem for S-J even. A particular case is Momi's result, i.e the index theorem for generic endowments and real asset structures if S-J is even.incomplete markets, equilibria manifold, orientability, index theorem

An answer to a question of herings et al

One answers to an open question of Herings et al. (2008), by proving that their fixed point theorem for discontinuous functions works for mappings defined on convex compact subset of $\R^n$, and not only polytopes. This fixed point theorem can be applied to the problem of Nash equilibrium existence in discontinuous games.fixed point theorem; discontinuity; nash equilibrium

On the orientability of the asset equilibrium manifold.

This paper addresses partly an open question raised in the Handbook of Mathematical Economics about the orientability of the pseudo-equilibrium manifold in the basic two-period General Equilibrium with Incomplete Markets (GEI) model. For a broad class of explicit asset structures, it is proved that the asset equilibrium space is an orientable manifold if S − J is even. This implies, under the same conditions, the orientability of the pseudo-equilibrium manifold. By a standard homotopy argument, it also entails the index theorem for S − J even. A particular case is Momi's result, i.e. the index theorem for generic endowments and real asset structures if S − J is even.General equilibrium; Incomplete markets; Index theorem; Orientability;

On the continuous representation of quasi-concave mappings by their upper level sets

We provide a continuous representation of quasi-concave mappings by their upper level sets. A possible motivation is the extension to quasi-concave mappings of a result by Ulam and Hyers, which states that every approximately convex mapping can be approximated by a convex mapping.Quasi-concave, upper level set.

On the existence of approximated equilibria in discontinuous economies

In this paper, we prove an existence theorem for approximated equilibria in a class of discontinuous economies. The existence result is a direct consequence of a discontinuous extension of Brouwer's fixed point Theorem (1912), and is a refinement of several classical results in the standard General Equilibrium with Incomplete markets (GEI) model (e.g., Bottazzi (1995), Duffie and Shafer (1985), Husseini et al. (1990), Geanakoplos and Shafer (1990), Magill and Shafer (1991)). As a by-product, we get the first existence proof of an approximated equilibrium in the GEI model, without perturbing the asset structure nor the endowments. Our main theorem rests on a new topological structure result for the asset equilibrium space and may be of interest by itself.general equilibrium, incomplete markets, approximated equilibrium

An answer to a question of herings et al.

One answers to an open question of Herings et al. (2008), by proving that their fixed point theorem for discontinuous functions works for mappings defined on convex compact subset of a Euclidean space, and not only polytopes. This rests on a fixed point result of ToussaintNash equilibrium, fixed point, discontinuity

Existence of pure Nash equilibria in discontinuous and non quasiconcave games

In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoff functions. In this paper, we prove that the quasiconcavity assumption in Reny's theorem can be weakened : we introduce a measure allowing to localize the lack of quasiconcavity, which allows to refine the analysis of equilibrium existence.Nash equilibrium, discontinuity, quasiconcavity.

Some fixed point theorems for discontinuous mappings

This paper provides a fixed point theorem à la Schauder, where the mappings considered are possibly discontinuous. Our main result generalizes and unifies several well-known results.Schauder fixed point theorem, Brouwer fixed point theorem, discontinuity.