Large games with a bio-social typology

Abstract

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 ...

Strategic uncertainty and the ex post Nash property in large games

“…In terms of future work and direction, note that the notions of a MSE and a RSED admit direct translation to a more elaborate model of a large game, one in which agent names and agent traits are disentangled, as considered in Khan et al (2013). We leave it to the interested reader to check that Theorems 1 and 2 are still valid for such a model, and, indeed, virtually the exact proofs carry over to this more elaborate setting.…”

“…12 See the survey chapter in Khan and Sun (2002) and its references. 13 See Khan et al (2013) for a detailed discussion and bibliographic details.…”

“…In keeping with the discussion in Section 2 of the measurability problem pertaining to simultaneous independence and measurability, it is unclear whether and how games with arbitrary actions, saturated spaces are used to characterize the existence of pure strategy Nash equilibria; see Keisler and Sun (2009) and Khan et al (2013). 21 Since the classical law of large numbers in the sequential case can be restated to the exact setting as an integral with respect to a purely finitely additive measure on a countable player space, a natural question arises as to whether one can work meaningfully with a MSE on such a countable player space.…”

Strict Nash equilibria in non-atomic games with strict single crossing in players (or types) and actions

“…are countable and compact, conditions for the existence of pure Nash equilibrium are given in Khan and Sun (1995) and then generalized in Yu and Zhang (2007). When the action space is an uncountable compact metric space, saturated probability spaces can be used to guarantee the existence of a pure-strategy Nash equilibrium, as shown in Keisler and Sun (2009) and Khan et al (2013a). 17…”

“…Consider the following game, which is an instance of the class of games considered in Khan et al (2013a) (see discussion at p. 1130), and that represents a slight generalization of a static population game (see Sandholm 2010, for a formal definition 21 Alternative notions of evolutionary stability have been proposed in the literature (Vickers and Cannings 1987;Bomze and Pötscher 1989;Riedel 2001, 2002). of population games).…”