We present a comprehensive theory of large games in which players have names and determinate social-types and/or biological traits, and identify through four decisive examples, essentially based on a matching-pennies type game, pathologies arising from the use of a Lebesgue interval for player's names. In a sufficiently general context of traits and actions, we address this dissonance by showing a saturated probability space as being a necessary and sufficient name-space for the existence and upper hemi-continuity of pure-strategy Nash equilibria in large games with traits. We illustrate the idealized results by corresponding asymptotic results for an increasing sequence of finite games.(100 words)Under the assumption that a player's payoffs depend, in addition to her own action, on a statistical summary, be it an average or a distribution, of the plays of everyone else in the game, the basic thrust of the theory of large games is its focus on pure-strategy equilibria. 1 Indeed, this constitutes the raison d'etre of the theory and is easily justified by virtue of the fact that pure-strategy equilibria do not necessarily exist in games with a finite set of players. 2 The two distinguished, and defining, features of the theory are a player's numerical negligibility and her societal interdependence in the original Nash formulation being substituted by more composite and aggregate measures of the actions of everyone else in the game. Within such a rubric, results on the existence of pure-strategy equilibria, as well as their asymptotic implementability and invariance to permutations of names, have been established. 3 In addition, issues concerning measurability, purification and symmetrization have been identified and resolved in terms of decisive counterexamples and attendant theorems. The resulting theory has been shown to hinge on the cardinality of the underlying action set: if it is a finite, 4 or a countably infinite, set, 5 numerical negligibility of an individual player can be successfully formalized by an arbitrary atomless probability space; if, on the other hand, it is uncountableand compact, say even the unit interval, an arbitrary probability space does not suffice, and additional structure has to be put on the formalization of agent multiplicity. Initially, such a structure was invoked through the consideration of an atomless Loeb probability space as in [22], but recent work has identified a crucial property, namely saturation, and shown that agent multiplicity formalized by saturated probability spaces, 6 a more general class to which atomless Loeb probability spaces belong, is not only sufficient but also necessary for the results to hold. We have thereby a viable and robust theory of large games.A technical point of departure for the theory is the fact that the players' names do not have a natural "measure of closeness" defined on them, and that therefore the space of such names has to be conceived as an abstract rather than a topological measure space. It is by now well-appreciated that this space ...
