Haomiao Yu

Large Games with a Bio-Social Typology

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 playerʼs 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.

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.

On the closed-graph property of the Nash equilibrium correspondence in a large game: A complete characterization

We show that if every large game with a given player space and any given uncountable trait space (or action set) is a proper idealized limit, then the player space must be saturated. When the player space is allowed to be an arbitrary atomless probability space, even a non-saturated one such as the classical Lebesgue unit interval, we establish the following: (i) If a large game has a countable action set and a countable trait space, then the game has a closed Nash equilibrium correspondence, and is thus proper as an idealized limit; (ii) If every large game having a given action set and a given trait space is proper as an idealized limit, then both the action set and the trait space must be countable.

On the centipede game with a social norm

In this paper, we present a formulation of the centipede game where inter-subjectivity between players is formalized through random costs that are an increasing function, in the sense of first-order stochastic dominance, of the number of iterations for which the game is played. This injection of a time-dependent social norm, one that also parametrizes player proximity, as an element of each player’s introspective calculations formalizes a setting under which subgame perfection dictates cooperative outcomes.

On the equivalence of large individualized and distributionalized games: Large individualized and distributionalized games

The theory of large one-shot simultaneous-play games with a bio-social typology has been presented in both the individualized and distributionalized forms, LIG and LDG respectively. Using an example of an LDG with two actions and a single trait in which some Nash equilibrium distributions cannot be induced by the Nash equilibria of the representing LIG, this paper offers three equivalence results that delineate a relationship between the two game-forms. Our analysis also reveals the different roles that the Lebesgue unit interval and a saturated space play in the theory.