Games on Graphs with a Public Signal Monitoring

Abstract

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 ∪ {−∞, +∞}.

A Note on Game Theory and Verification

“…We have discussed in particular a general construction that can be made to compute Nash equilibria in games on graphs, and which gives some general understanding of how interaction between players can be understood. This construction has been refined in several respects (for other solution concepts [9,18], in some partial information contexts [2,7]), and might be useful in some more contexts.…”

Nash equilibria in games over graphs equipped with a communication mechanism

“…A strategy for a player a ∈ P from vertex v 0 is a mapping σ a : Hist G (v 0 ) → Act × {0, 1} * such that for every history h ∈ Hist G (v 0 ), σ a (h) [1] ∈ Allow(last(h), a), where the notation σ a (h) [1] denotes the first component of the pair σ(h). The value σ a (h) [1] represents the action that player a will do after h, while σ a (h) [2] is the message that she will append to her action and broadcast to all players b such that a b. The strategy σ a is said G-compatible if furthermore, for all histories h, h ∈ Hist(v 0 ), h ∼ a h implies σ a (h) = σ a (h ).…”

“…Given those reductions, we then propose the construction of a two-player game, which will track those normed profiles. This construction is inspired by the suspect-game construction of [4] and of the epistemic game of [3].…”

“…That is, at the first position where player d changes her strategy, it becomes public information that a deviation has occurred (even though some players know who deviated -all the players a with d a-, and some other don't know). It is furthermore called honest whenever for every [2] = id d . Somehow, player d admits she deviated, and does so immediately and forever.…”

“…Several series of works have followed. To roughly give an idea of the existing results, pure Nash equilibria always exist in turn-based games for ω-regular objectives [20] but not in concurrent games, even with simple objectives; they can nevertheless be computed [20,4,7,3] for large classes of objectives. The problem becomes harder with mixed (that is, stochastic) Nash equilibria, for which we often cannot decide the existence [19,5].…”