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We extend a result on existence of Walrasian equilibria in He and Yannelis (Econ Theory 61:497-513, 2016) by replacing the compactness assumption on consumption sets made there by the standard assumption that these sets are closed and bounded from below. This provides a positive answer to a question explicitly raised in He and Yannelis (Econ Theory 61:497–513, 2016). Our new equilibrium existence theorem generalizes many results in the literature as we do not require any transitivity or completeness or continuity assumption on preferences, initial endowments need not be in the interior of the consumption sets, preferences may be interdependent and price-dependent, and no monotonicity or local non satiation is needed for any of the agents.

We characterize Nash equilibria of games with a continuum of players in terms of approximate equilibria of large finite games. This characterization precisely describes the relationship between the equilibrium sets of the two classes of games. In particular, it yields several approximation results for Nash equilibria of games with a continuum of players, which roughly state that all finite-player games that are sufficiently close to a given game with a continuum of players have approximate equilibria that are close to a given Nash equilibrium of the non-atomic game.

In the context of anonymous games (i.e., games where the payoff of a player is, apart from his/her own action, determined by the distribution of the actions made by the other players), we present a model in which, generically (in a precise sense), finite-player games have strict pure strategy Nash equilibria if the number of agents is large. A key feature of our model is that payoff functions have differentiability properties. A consequence of our existence result is that, in our model, equilibrium distributions of non-atomic games are asymptotically implementable by pure strategy Nash equilibria of large finite-player games.

We provide conditions guaranteeing the existence of Nash equilibrium in games in which players' preferences can be arbitrary binary relations. Our main result generalizes Reny's (Economic Theory, forthcoming) existence result for games with ordered preferences and He and Yannelis' (Economic Theory, forthcoming) existence result for abstract economies with non-ordered preferences.

A new proof of the existence of a Walrasian equilibrium with an infinite dimensional commodity space is provided, which allows agents’ preferences to be discontinuous. The new theorems include as corollaries the existence results of Mas-Collel, Yannelis and Zame, Araujo and Monteiro, and Mas-Collel and Richard, among others.

This note presents new results on existence of rich Fubini extensions. The notion of a rich Fubini extension was recently introduced by Sun (J Econ Theory 126: 31-69, 2006) and shown by him to provide the proper framework to obtain an exact law of large numbers for a continuum of random variables. In contrast to the existence results for rich Fubini extensions established by Sun, the arguments in this note do not use constructions from nonstandard analysis.

We present a result on convexity and weak compactness of the range of a vector measure with values in a Banach space, based on the Maharam classification of measure spaces. Our result extends a recent result of Khan and Sagara (Illinois J. Math. 2013 ). We apply our result to integration of Banach space valued correspondences and to the core-Walras equivalence problem in coalitional exchange economies with an infinite-dimensional commodity space.

This paper presents new methods to obtain purification results for continuum games, which don't make use of the \"many more players than strategies\" assumption (Yannelis in Econ Theory (in press) 2007) or of Loeb spaces (Loeb and Sun in Illinois J Math 50, 747-762, 2006). The approach presented doesn't use nonstandard analysis; it is based on standard measure theory and in particular on the super-nonatomicity notion introduced in Podczeck (J Math Econ (in press) 2007).

We answer the question asked by Robert Aumann as to whether core equivalence depends on there being \"many more agents than commodities.\" We show that for a large class of commodity spaces, which might be infinite-dimensional and even non-separable, core equivalence is indeed equivalent to the presence of \"many more agents than commodities\" when allocations are Bochner integrable. By contrast, we show that in a classical model of an atomless economy with an infinite-dimensional commodity space, the model where the commodity space is L∞(μ) with the Mackey topology and allocations are Gelfand integrable, core equivalence holds in full generality, even when there are \"many more commodities than agents.\" The assumptions we make on economies are much weaker than what is commonly used in core equivalence results for infinite-dimensional commodity spaces and reduce to Aumann's original assumptions when there are finitely many commodities.

Random matching models with a continuum population are widely used in economics to study environments where agents interact in small coalitions. This paper provides foundations to such models. In particular, the paper establishes an existence result for random matchings that are universal in the sense that certain desirable properties are satisfied for any assignment of types to agents. The result applies to infinitely many types of agents, thus covering random matching models which are currently used in the literature without a foundation. Furthermore, the paper provides conditions guaranteeing uniqueness of random matching.