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We study the Bochner and Gel fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting… (More)
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 liter-atures. We construct a joint agent-probability space, and randomized mutation, partial matching and match-induced type-changing functions that satisfy appropriate… (More)
We present results on the existence of pure strategy Nash equilibria in nonatomic games. We also show by means of counterexamples that the stringent conditions on the cardinality of action sets cannot be relaxed, and thus resolve questions which have remained open since Schmeidler's 1973 paper.
We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large non-anonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover… (More)
This paper provides micro-foundations for independent random matching of a large population , as widely used in the economics literature. We consider both static and dynamic systems with random mutation, partial matching arising from search, and type changes induced by matching. Under independence assumptions at each randomization step, we show that there… (More)
The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is… (More)
We present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a Loeb space, which include closed-ness, convexity, compactness, purification, and semicontinuity. We also note that all the properties reported in the main theorems are not satisfied by some correspondences on the unit Lebesgue… (More)