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, which has 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 an infinite product of Loeb transition probabilities, based on which a continuum of independent Markov chains is derived from random mutation, random partial matching and random type changing.Comment: Published at http://dx.doi.org/10.1214/105051606000000673 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynamic Games with Almost Perfect Information

This paper aims to solve two fundamental problems on finite or infinite horizon dynamic games with perfect or almost perfect information. Under some mild conditions, we prove (1) the existence of subgame-perfect equilibria in general dynamic games with almost perfect information, and (2) the existence of pure-strategy subgame-perfect equilibria in perfect-information dynamic games with uncertainty. Our results go beyond previous works on continuous dynamic games in the sense that public randomization and the continuity requirement on the state variables are not needed. As an illustrative application, a dynamic stochastic oligopoly market with intertemporally dependent payoffs is considered

Monte Carlo Simulation of Macroeconomic Risk with a Continuum Agents : The General Case

In large random economies with heterogeneous agents, a standard stochastic framework presumes a random macro state, combined with idiosyncratic micro shocks. This can be formally represented by a ran-dom process consisting of a continuum of random variables that are conditionally independent given the macro state. However, this process satisfies a standard joint measurability condition only if there is essentially no idiosyncratic risk at all. Based on iteratively complete product measure spaces, we characterize the validity of the standard stochastic framework via Monte Carlo simulation as well as event-wise measurable conditional probabilities. These general characterizations also allow us to strengthen some earlier results related to exchangeability and independence.large economy ; event-wise measurable conditional probabilities ; ex-changeability ; conditional independence ; Monte Carlo convergence ; Monte Carlo-algebra ; stochastic macro structure

Characterization of Risk : A Sharp Law of Large Numbers

An extensive literature in economics uses a continuum of random variables to model individual random shocks imposed on a large population. Let H denote the Hilbert space of square-integrable random variables. A key concern is to characterize the family of all H-valued functions that satisfy the law of large numbers when a large sample of agents is drawn at random. We use the iterative extension of an infinite product measure introduced in [6] to formulate a “sharp” law of large numbers. We prove that an H-valued function satisfies this law if and only if it is both Pettis-integrable and norm integrably bounded.

The one-way Fubini property and conditional independence : an equivalence result

A general parameter process defined by a continuum of random variables is not jointly measurable with respect to the usual product σ-algebra. For the case of independent random variables, a one-way Fubini extension of the product space was constructed in [11] to satisfy a limited form of joint measurability. For the general case we show that this extension exists if and only if there is a countably generated σ-algebra given which the random variables are essentially pairwise conditionally independent, while their joint conditional distribution also satisfies a suitable joint measurability condition. Applications include new characterizations of essential pairwise independence and essential pairwise exchangeability through regular conditional distributions with respect to the usual product σ-algebra in the framework of a one-way Fubini extension