C. J. Himmelberg

Existence of

In this paper we prove the existence of p-equilibrium stationary strategies for non-zero-sum stochastic games when the reward functions and transitions satisfy certain separability conditions. We also prove some results for positive and discounted zero-sum stochastic games when the state space is infinite.

p

Let X be a metrizable space. A Vietoris-type topology, called the locally finite topology, is defined on the hyperspace 2x of all closed, nonempty subsets of X. We show that the locally finite topology coincides with the supremum of all Hausdorff metric topologies corresponding to equivalent metrics on X. We also investigate when the locally finite topology coincides with the more usual topologies on 2x and when the locally finite topology is metrizable.

-equilibrium and optimal stationary strategies in stochastic games

Abstract This paper is concerned with the existence of solutions to the initial value problem for generalized differential equations (orientor fields) when the right-hand side, F ( t , x ), may be unbounded. Two global and one local existence theorems are established when F satisfies Caratheodory type conditions involving weak integral boundedness conditions. The multiple-valued function F ( t , ·) is assumed to have closed graph and to be lower semicontinuous at each point x where F ( t , x ) is not convex.

The locally finite topology on 2

Abstract We construct measurable selections for closed set-valued maps into arbitrary complete metric spaces. We do not need to make any separability assumptions. We view the set-valued maps as point-valued maps into the hyperspace and our measurability assumptions arethe usual kinds of measurability of point-valued maps in this setting. We also discuss relationship of these measurability conditions to the ones usually considered in the theory of measurable selections.

Existence of solutions for generalized differential equations with unbounded right-hand side☆

SummaryThe main results are some very general theorems about measurable multifunctions on abstract measurable spaces with compact values in a separable metric space. It is shown that measurability is equivalent to the existence of a pointwise dense countable family of measurable selectors, and that the intersection of two compact-valued measurable multifunctions is measurable. These results are used to obtain a Filippov type implicit function theorem, and a general theorem concerning the measurability of y(t)=min f({t} × Γ(t)) when f is a real valued function and Γ a compact valued multifunction. An application to stochastic decision theory is given generalizing a result of Benes.

The hausdorff metric and measurable selections

SummaryThis note points out that the « only if » part of Theorem 1′ (i) of the above paper [Annali di Matematica pura ed applicata, (IV)101(1974), pp. 229–236] is false as is part (ii) of Theorem 2. Counterexamples are given and a weak form of the « only if » part of Theorem 1′ (i) is established.