Limit-Computable Grains of Truth for Arbitrary Computable Extensive-Form (Un)Known Games

A Bayesian player acting in an infinite multi-player game learns to predict the other players' strategies if his prior assigns positive probability to their play (or contains a grain of truth). Kalai and Lehrer's classic grain of truth problem is to find a reasonably large class of strategies that contains the Bayes-optimal policies with respect to this class, allowing mutually-consistent beliefs about strategy choice that obey the rules of Bayesian inference. Only small classes are known to have a grain of truth and the literature contains several related impossibility results. In this paper we present a formal and general solution to the full grain of truth problem: we construct a class of strategies wide enough to contain all computable strategies as well as Bayes-optimal strategies for every reasonable prior over the class. When the "environment" is a known repeated stage game, we show convergence in the sense of [KL93a] and [KL93b]. When the environment is unknown, agents using Thompson sampling converge to play $\varepsilon$-Nash equilibria in arbitrary unknown computable multi-agent environments. Finally, we include an application to self-predictive policies that avoid planning. While these results use computability theory only as a conceptual tool to solve a classic game theory problem, we show that our solution can naturally be computationally approximated arbitrarily closely.