A Modal Characterization of Nash Equilibrium

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 multimodal 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.

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