Deviator Detection under Imperfect Monitoring

TL;DR

This paper investigates the synthesis of strategies to detect deviators in iterated games with imperfect monitoring, establishing decidability results and providing an effective method under certain information conditions.

Contribution

It introduces a formal framework for synthesizing deviator detection strategies in finite-state games with imperfect information, and proves decidability in specific monitoring scenarios.

Findings

Undecidability in the general case with unmonitorable global state

Decidability when the global state is perfectly monitored

An effective synthesis procedure for private monitoring scenarios

Abstract

Grim-trigger strategies are a fundamental mechanism for sustaining equilibria in iterated games: the players cooperate along an agreed path, and as soon as one player deviates, the others form a coalition to play him down to his minmax level. A precondition to triggering such a strategy is that the identity of the deviating player becomes common knowledge among the other players. This can be difficult or impossible to attain in games where the information structure allows only imperfect monitoring of the played actions or of the global state. We study the problem of synthesising finite-state strategies for detecting the deviator from an agreed strategy profile in games played on finite graphs with different information structures. We show that the problem is undecidable in the general case where the global state cannot be monitored. On the other hand, we prove that under perfect monitoring of the global state and imperfect monitoring of actions, the problem becomes decidable, and we present an effective synthesis procedure that covers infinitely repeated games with private monitoring.