The Temporal Logic of Coalitional Goal Assignments in Concurrent Multiplayer Games

Abstract

We introduce and study a natural extension of the Alternating time temporal logic ATL , called Temporal Logic of Coalitional Goal Assignments (TLCGA). It features one new and quite expressive coalitional strategic operator, called the coalitional goal assignment operator ⦉ γ ⦊, where γ is a mapping assigning to each set of players in the game its coalitional goal , formalised by a path formula of the language of TLCGA, i.e., a formula prefixed with a temporal operator X , U , or G , representing a temporalised objective for the respective coalition, describing the property of the plays on which that objective is satisfied. Then, the formula ⦉ γ ⦊ intuitively says that there is a strategy profile Σ for the grand coalition Agt such that for each coalition C , the restriction Σ | C of Σ to C is a collective strategy of C that enforces the satisfaction of its objective γ (C) in all outcome plays enabled by Σ | C . We establish fixpoint characterizations of the temporal goal assignments in a μ-calculus extension of TLCGA, discuss its expressiveness and illustrate it with some examples, prove bisimulation invariance and Hennessy–Milner property for it with respect to a suitably defined notion of bisimulation, construct a sound and complete axiomatic system for TLCGA, and obtain its decidability via finite model property.