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A Modal Characterization of Nash Equilibrium

Harrenstein P., Van der Hoek W., Meyer J.-J., Witteveen C.

Fundamenta Informaticae

2003

Vol. 57, nr 2-4

281--321

Multi-agent systems comprise entities whose individual decision making behavior may depend on one another's. Game-theory provides apposite concepts to reason in a mathematically precise fashion about such interactive and interdependent situations. This paper concerns a logical analysis of the game-theoretical notions of Nash equilibrium and its subgame perfect variety as they apply to a particular class of extensive games of perfect information. Extensive games are defined as a special type of labelled graph and we argue that modal languages can be employed in their description. We propose a logic for a multi-modal language and prove its completeness with respect to a class of frames that correspond with a particular class of extensive games. In this multi-modal language (subgame perfect) Nash equilibria can be characterized. Finally, we show how this approach can formally be refined by using Propositional Dynamic Logic ( PDL), though we leave completeness as an open question.