Extended Graded Modalities in Strategy Logic

Benjamin Aminof
(Technische Universitat Wien, Austria)
Vadim Malvone
(Università degli Studi di Napoli Federico II, Italy)
Aniello Murano
(Università degli Studi di Napoli Federico II, Italy)
Sasha Rubin
(Università degli Studi di Napoli Federico II, Italy)

Strategy Logic (SL) is a logical formalism for strategic reasoning in multi-agent systems. Its main feature is that it has variables for strategies that are associated to specific agents with a binding operator. We introduce Graded Strategy Logic (GradedSL), an extension of SL by graded quantifiers over tuples of strategy variables, i.e., "there exist at least g different tuples (x_1,...,x_n) of strategies" where g is a cardinal from the set N union {aleph_0, aleph_1, 2^aleph_0}. We prove that the model-checking problem of GradedSL is decidable. We then turn to the complexity of fragments of GradedSL. When the g's are restricted to finite cardinals, written GradedNSL, the complexity of model-checking is no harder than for SL, i.e., it is non-elementary in the quantifier rank. We illustrate our formalism by showing how to count the number of different strategy profiles that are Nash equilibria (NE), or subgame-perfect equilibria (SPE). By analyzing the structure of the specific formulas involved, we conclude that the important problems of checking for the existence of a unique NE or SPE can both be solved in 2ExpTime, which is not harder than merely checking for the existence of such equilibria.

In Alessio Lomuscio and Moshe Y. Vardi: Proceedings of the 4th International Workshop on Strategic Reasoning (SR 2016), New York City, USA, 10th July 2016, Electronic Proceedings in Theoretical Computer Science 218, pp. 1–14.
Published: 10th July 2016.

ArXived at: https://dx.doi.org/10.4204/EPTCS.218.1 bibtex PDF
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