The Temporal Logic of Coalitional Goal Assignments in Concurrent Multiplayer Games

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.