Measurable Selection for Purely Atomic Games

Abstract

A general selection theorem is presented constructing a measurable mapping from a state space to a parameter space under the assumption that the state space can be decomposed as a collection of countable equivalence classes under a smooth equivalence relation. It is then shown how this selection theorem can be used as a general purpose tool for proving the existence of measurable equilibria in broad classes of several branches of games when an appropriate smoothness condition holds, including Bayesian games with atomic knowledge spaces, stochastic games with countable orbits, and graphical games of countable degree—examples of a subclass of games with uncountable state spaces that we term purely atomic games. Applications to repeated games with symmetric incomplete information and acceptable bets are also presented.

Dense Orbits of the Bayesian Updating Group Action

“…• In Section 4, we discuss the relevance of the above results in relation to the concept of smoothness of the orbit relation, and through this, to the study of repeated games with (public) incomplete information. Smoothness had been used to study such repeated games in [Hellman and Levy, 2019]. In particular, findings in this paper limit the usefulness of those previous results with respect to repeated games, but also show that generically when the state space is inuitively 'quite large' relative to the signal and the action spaces, stationary equilibria (i.e., equilibria which condition actions solely on the current beliefs) do exist.…”

“…The implication 'no recurrent point implies smoothness' in Theorem 4.1 was proven (and in fact does not require continuity of the group action, that is, it holds for any group of Borel automorphisms 2 ) in Theorem 11.1 of [Hellman and Levy, 2019]. For the converse, Lemma 1.1 of [Sullivan et al, 1986] in particular says that the existence of a dense orbit in a perfect Polish space implies that each G-invariant Borel set is either meagre or co-meagre (a.k.a.…”

“…Relation to Smoothness. A related concept, used to formulate an array of results in [Hellman and Levy, 2019], is that of smoothness. A complete, separable, and metrisable space Ω is called a Polish space.…”

A Compact, Logical Approach to Large-Market Analysis

“…In addition to the fact that Hellman and Levy (2019) use (different) tools from mathematical logic, their paper is also conceptually related to ours. In a precise sense, their paper and ours complement each other: our paper gives a principled approach to lifting existence results (and beyond) from finite markets to countable ones, while Hellman and Levy (2019) give a principled approach to lifting existence results from infinitely-countable (but not from finite) markets to uncountable ones. Combining both approaches enables us to lift various existence results from finite to uncountable markets.…”

“…Once again, a better-behaved way to choose a Walrasian equilibrium for each connected component (once we have proven that such equilibria exist) is to use the result ofHellman and Levy (2019), which guarantees that the overall equilibrium be measurable. (See also the discussion of that paper in Section 1.1.…”