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.
Keywords: Modal Logic, Dynamic Logic, Game Theory, Nash Equilibrium