Common knowledge and state-dependent equilibria

Many puzzling social behaviors, such as avoiding eye contact, using innuendos, and insignificant events that trigger revolutions, seem to relate to common knowledge and coordination, but the exact relationship has yet to be formalized. Herein, we present such a formalization. We state necessary and sufficient conditions for what we call state-dependent equilibria - equilibria where players play different strategies in different states of the world. In particular, if everybody behaves a certain way (e.g. does not revolt) in the usual state of the world, then in order for players to be able to behave a different way (e.g. revolt) in another state of the world, it is both necessary and sufficient for it to be common p-believed that it is not the usual state of the world, where common p-belief is a relaxation of common knowledge introduced by Monderer and Samet [16]. Our framework applies to many player r-coordination games - a generalization of coordination games that we introduce - and common (r,p)-beliefs - a generalization of common p-beliefs that we introduce. We then apply these theorems to two particular signaling structures to obtain novel results. © 2012 Springer-Verlag

Bayesian games with a continuum of states

We show that every Bayesian game with purely atomic types has a measurable Bayesian equilibrium when the common knowl- edge relation is smooth. Conversely, for any common knowledge rela- tion that is not smooth, there exists a type space that yields this common knowledge relation and payoffs such that the resulting Bayesian game will not have any Bayesian equilibrium. We show that our smoothness condition also rules out two paradoxes involving Bayesian games with a continuum of types: the impossibility of having a common prior on components when a common prior over the entire state space exists, and the possibility of interim betting/trade even when no such trade can be supported ex ante

Deep Reinforcement Learning from Self-Play in Imperfect-Information Games

Many real-world applications can be described as large-scale games of imperfect information. To deal with these challenging domains, prior work has focused on computing Nash equilibria in a handcrafted abstraction of the domain. In this paper we introduce the first scalable end-to-end approach to learning approximate Nash equilibria without prior domain knowledge. Our method combines fictitious self-play with deep reinforcement learning. When applied to Leduc poker, Neural Fictitious Self-Play (NFSP) approached a Nash equilibrium, whereas common reinforcement learning methods diverged. In Limit Texas Holdem, a poker game of real-world scale, NFSP learnt a strategy that approached the performance of state-of-the-art, superhuman algorithms based on significant domain expertise.Comment: updated version, incorporating conference feedbac

Moral Hazard in a model of Bank Run with Noisy Signals

We show that multiple equilibria exist in a model of bank run with moral hazard. Furthermore, this is true even with noisy signals on the economic fundamentals. We argue that the conditions under which this happens can arise naturally in models of banking with moral hazard problem.banking, fundamentals, signals, equilibria