High Probability Guarantees in Repeated Games: Theory and Applications in Information Theory

TL;DR

This paper develops a high probability framework for repeated games with incomplete information, extending classical results and applying to information transmission systems with low error probabilities.

Contribution

It introduces a high probability guarantee framework for repeated games, providing a novel counterpart to classical expectation-based results and applying it to communication channels.

Findings

High probability guarantees can be strictly stronger than expectation guarantees.

The framework applies to compound arbitrarily varying channels.

Guaranteed payoffs differ between players in the high probability setting.

Abstract

We introduce a "high probability" framework for repeated games with incomplete information. In our non-equilibrium setting, players aim to guarantee a certain payoff with high probability, rather than in expected value. We provide a high probability counterpart of the classical result of Mertens and Zamir for the zero-sum repeated games. Any payoff that can be guaranteed with high probability can be guaranteed in expectation, but the reverse is not true. Hence, unlike the average payoff case where the payoff guaranteed by each player is the negative of the payoff by the other player, the two guaranteed payoffs would differ in the high probability framework. One motivation for this framework comes from information transmission systems, where it is customary to formulate problems in terms of asymptotically vanishing probability of error. An application of our results to a class of compound arbitrarily varying channels is given.