Graded modalities in Strategy Logic

Abstract

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 using a binding operator. In this paper 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 ∪ {ℵ 0 , ℵ 1 , 2 ℵ0 }. 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 Graded N SL, the complexity of model-checking is no harder than for SL, i.e., it is non-elementary in the quantifier-block rank. We illustrate our formalism by showing how to count the number of different strategy profiles that are Nash equilibria (NE). By analysing the structure of the specific formulas involved, we conclude that the important problem of checking for the existence of a unique NE can be solved in 2ExpTime, which is not harder than merely checking for the existence of such an equilibrium.

Improving parity games in practice

“…For our tests, we set n = 2 k where 8 ≤ k ≤ 19, and report our experiments in Table 3 whose trend is drawn in Fig. 5, where the seconds are limited to [0,2]. As the figure shows, there is a better performance for CRO than IRS using low-scaled (up to 2 13 ) values as input parameter.…”

“…Parity games [22,52] are abstract infinite-round games that represent a powerful mathematical framework to address fundamental questions in computer science. They are intimately related to other infinite-round games, such as mean and discounted payoff, stochastic, and multi-agent games [2,9,[12][13][14][15]38].…”

Reasoning about Quality and Fuzziness of Strategic Behaviours

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Strategic coalitions in stochastic games

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