On the space of players in idealized limit games

This paper demonstrates the class of atomless spaces that accurately models the space of players in a large game which represents an idealized limit of a sequence of finite-player games. Through two examples, we show that arbitrary atomless probability spaces, in particular, the Lebesgue unit interval, may not be appropriate to model the space of players of an idealized limit. This inappropriateness hinges on the fact there is a convergent sequence of exact pure-strategy Nash equilibria in the sequence of finite-player games, while the idealized limit game of the sequence does not have any equilibrium. Instead, a saturated probability space is shown to be not only sufficient but also necessary, to model the space of players in any proper idealized limit. This complements the study of large games with a bio-social typology in Khan et al. [10] as such a connection between finite-limiting and idealized continuum-limit games was not able to be obtained in their framework.