We study pure Nash equilibria in games on graphs with an imperfect monitoring based on a public signal. In such games, deviations and players responsible for those deviations can be hard to detect and track. We propose a generic epistemic game abstraction, which conveniently allows to represent the knowledge of the players about these deviations, and give a characterization of Nash equilibria in terms of winning strategies in the abstraction. We then use the abstraction to develop algorithms for some payoff functions. * This work has been supported by ERC project EQualIS (FP7-308087).1 This can be expressed by payoff player i = Rwhere γ i is the signal-to-interferenceand-noise ratio for player i, R is the rate at which the wireless system transmits the information and L is the size of the packets [29].Recently imperfect information has also been introduced in the setting of multi-agent temporal logics [20,21,2,3], and the main decidability results assume hierarchical information. However, while those logics allow to express rich interactions, it can somehow only consider qualitative properties. Furthermore, no tight complexity bounds are provided.In [11], a deviator game abstraction is proposed. It twists the suspect game abstraction [9] to allow for more general solution concepts (so-called robust equilibria), but it assumes visibility of actions (hence remove any kind of uncertainties). Relying on results of [13], this deviator game abstraction allows to compute equilibria with mean payoff functions. Our algorithms for mean payoff functions will also rely on the polyhedron problem of [13].
DefinitionsThroughout the paper, if S ⊆ R, we write S for S ∪ {−∞, +∞}.