Modeling large societies: Why countable additivity is necessary

Abstract

The economic literature with an atomless countably additive measure space to model the interaction of agents in a large society is enormous. However, there have been a number of attempts to relax the countable additivity assumption by working with a finitely (but non-countably) additive measure space (such as the set of natural numbers with a density charge). The main purpose of this paper is to demonstrate the necessity of countable additivity in modeling a large society in terms of the existence of equilibrium and its idealized limit property in both general equilibrium and game theory as illustrations. In addition, we point out that in the setting of atomless finitely additive agent spaces, even approximate equilibria may not exist in general, but do so only with additional assumptions.