Yeneng Sun

The exact law of large numbers via Fubini extension and characterization of insurable risks

A Fubini extension is formally introduced as a probability space that extends the usual product probability space and retains the Fubini property. Simple measure-theoretic methods are applied to this framework to obtain various versions of the exact law of large numbers and their converses for a continuum of random variables or stochastic processes. A model for a large economy with individual risks is developed; and insurable risks are characterized by essential pairwise independence. The usual continuum product based on the Kolmogorov construction together with the Lebesgue measure as well as the usual finitely additive measure-theoretic framework is shown further to be not suitable for modeling individual risks. Measurable processes with essentially pairwise independent random variables that have any given variety of distributions exist in a rich product probability space that can also be constructed by extending the usual continuum product.

Existence of independent random matching

This paper shows the existence of independent random matching of a large (continuum) population in both static and dynamic systems, suitable for applications of the exact law of large numbers for random matching that have been popular in the economics and genetics literatures. We construct a joint agent-probability space, and randomized mutation, partial matching, and match-induced type-changing functions that satisfy appropriate independence conditions. The proofs are achieved via nonstandard analysis. The proof for the dynamic setting relies on a new Fubini-type theorem for Loeb transition probabilities and their products, based on which a continuum of independent Markov chains is derived from random mutation, random partial matching and random type changing.

Pure strategies in games with private information

Abstract Pure strategy equilibria of finite player games with informational constraints have been discussed under the assumptions of finite actions, and of independence and diffuseness of information. We present a mathematical framework, based on the notion of a distribution of a correspondence, that enables us to handle the case of countably infinite actions. In this context, we extend the Radner-Rosenthal theorems on the purification of a mixed-strategy equilibrium, and present a direct proof, as well as a generalized version of Schmeidlers large games theorem, on the existence of a pure strategy equilibrium. Our mathematical results pertain to the set of distributions induced by the measurable selections of a correspondence with a countable range, and rely on the Bollobas-Varopoulos extension of the marriage lemma.

The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games

In 1951, Dvoretzky, Wald and Wolfowitz (henceforth DWW) showed that corresponding to any mixed strategy into a finite action space, there exists a pure-strategy with an identical integral with respect to a finite set of atomless measures. DWW used their theorem for purification: the elimination of randomness in statistical decision procedures and in zero-sum two-person games. In this short essay, we apply a consequence of their theorem to a finite-action setting of finite games with incomplete and private information, as well as to that of large games. In addition to simplified proofs and conceptual clarifications, the unification of results offered here re-emphasizes the close connection between statistical decision theory and the theory of games.

Individual risk and Lebesgue extension without aggregate uncertainty

Many economic models include random shocks imposed on a large number (continuum) of economic agents with individual risk. In this context, an exact law of large numbers and its converse is presented in Sun (2006) to characterize the cancelation of individual risk via aggregation. However, it is well known that the Lebesgue unit interval is not suitable for modeling a continuum of agents in the particular setting. The purpose of this note is to show that an extension of the Lebesgue unit interval does work well as an agent space with various desirable properties associated with individual risk.

On a private information game without pure strategy equilibria1

Abstract We present an example of a two-person game of private information in which there is no equilibrium in pure strategies. Our example satisfies all the hypotheses of the existence theorems present in the literature on the subject of pure strategy equilibria, except for the fact that the action set of each player is given by the interval [−1,1]. As such, it illustrates the limitations that pertain to the purification of equilibria in a standard setting.

Large Games with a Bio-Social Typology

We present a comprehensive theory of large games in which players have names and determinate social-types and/or biological traits, and identify through four decisive examples, essentially based on a matching-pennies type game, pathologies arising from the use of a Lebesgue interval for playerʼs names. In a sufficiently general context of traits and actions, we address this dissonance by showing a saturated probability space as being a necessary and sufficient name-space for the existence and upper hemi-continuity of pure-strategy Nash equilibria in large games with traits. We illustrate the idealized results by corresponding asymptotic results for an increasing sequence of finite games.

Chapter 46 Non-cooperative games with many players

In this survey article, we report results on the existence of pure-strategy Nash equilibria in games with an atomless continuum of players, each with an action set that is not necessarily finite. We also discuss purification and symmetrization of mixed-strategy Nash equilibria, and settings in which private information, anonymity and idiosyncratic shocks are given particular prominence.

Perfect competition in asymmetric information economies: compatibility of efficiency and incentives

The idea of perfect competition for an economy with asymmetric information is formalized via an idiosyncratic signal process in which the private signals of almost every individual agent can influence only a negligible group of agents, and the individual agents’ relevant signals are essentially pairwise independent conditioned on the true states of nature. Thus, there is no incentive for an individual agent to manipulate her private information. The existence of incentive compatible, ex post Walrasian allocations is shown for such a perfectly competitive asymmetric information economy with or without “common values”. Consequently, the conflict between incentive compatibility and Pareto eciency is resolved exactly, and its asymptotic version is derived for a sequence of large, but finite private information economies.

On a Reformulation of Cournot-Nash Equilibria'

Abstract We present variations on a theme of Mas-Colell and report results on the existence of Cournot-Nash equilibrium distributions in which individual actions and the payoffs are represented by relations that are not necessarily complete or transitive.