Reasoning about graded strategy quantifiers

Abstract

In this paper we introduce and study Graded Strategy Logic (GSL), an extension of Strategy Logic (SL) with graded quantifiers. SL is a powerful formalism that allows to describe useful game concepts in multi-agent settings by explicitly quantifying over strategies treated as first-order citizens. In GSL, by means of the existential construct 《x≥g》φ, one can enforce that there exist at least g strategies x satisfying φ. Dually, via the universal construct 〚x<g〛φ, one can ensure that all but less than g strategies x satisfy φ.Strategies in GSL are counted semantically. This means that strategies inducing the same outcome, even though looking different, are counted as one. While this interpretation is natural, it requires a suitable machinery to allow for such a counting, as we do. Precisely, we introduce a non-trivial equivalence relation over strategy profiles based on the strategic behavior they induce.To give an evidence of GSL usability, we investigate some basic questions about the Vanilla GSL[1g] fragment, that is the vanilla restriction of the well-studied One-Goal Strategy Logic fragment of SL augmented with graded strategy quantifiers. We show that the model-checking problem for this logic is PTime-complete. We also report on some positive results about the determinacy.