Two Remarks on the Infinite Approximation of a Finite World in Economic Models

Abstract

While the assumption of infinity is prevalent in almost every area of economics, for two well-known frameworks in decision theory, we note some fundamental differences between the finite versions and their infinite counterpart. The first is on the usage of mixed strategies in finite games, while the second is on a characterization of the truth axiom in models of information and knowledge, where the properties of belief can be related to the properties of preference.

On a Fixed Point Theorem for General Multivalued Mappings on Finite Sets with Applications in Game Theory

“…[2] recently showed, using a counterexample, that it is not possible to characterize the fixed points of F using Abian's three set partitioning method and as a result, an Abian-like theorem cannot be given for multivalued mappings on finite sets. e authors of [2] also went one step further to show that the theorem does not generalize even if the number of partitions is increased. In fact, it was shown that only the trivial partition (singletons) always works for multivalued maps.…”

“…(It is important to note that the generalization of the problem to in nite spaces with a richer topological structure does not solve the pure strategy equilibrium problem when the space is nite and mixed strategies are not allowed). Nevertheless, as observed by [2], the latter work do not give a complete characterization of equilibrium points in nite games as they only provide su cient conditions and hence, there are no known characterizations of pure strategy equilibria for nite games.…”

“…e nite version of Abian's [1] xed point theorem states that given a function from a nite set into itself, the set cannot be partitioned into three sets so that the intersection of each of these sets with their image is empty if and only if the mapping has a xed point. Although this result can be applied to all functions that have a nite domain, it can be shown that it does not generalize to multivalued mappings which are often referred to as correspondences (1 see [2] for a proof of this claim). As a result, Abian's theorem cannot be applied to problems in game theory where the best response map is multivalued.…”

“…It was argued by [2] that it is the lack of a topological structure when the strategy space (without mixed extension) is a nite set which renders the problem at hand di cult. [2] also observes that Abian's partitioning method does not work for multivalued maps.…”

“…It was argued by [2] that it is the lack of a topological structure when the strategy space (without mixed extension) is a nite set which renders the problem at hand di cult. [2] also observes that Abian's partitioning method does not work for multivalued maps. In this paper, we solve this problem by using an equivalent xed point result due to the author [33] who relates the idea of cycle to Abian's fixed point theorem.…”