  Game theory provides a well-established framework for the analysis of
concurrent and multi-agent systems. The basic idea is that concurrent processes
(agents) can be understood as corresponding to players in a game; plays
represent the possible computation runs of the system; and strategies define
the behaviour of agents. Typically, strategies are modelled as functions from
sequences of system states to player actions. Analysing a system in such a
setting involves computing the set of (Nash) equilibria in the concurrent game.
However, we show that, with respect to the above model of strategies (arguably,
the "standard" model in the computer science literature), bisimilarity does not
preserve the existence of Nash equilibria. Thus, two concurrent games which are
behaviourally equivalent from a semantic perspective, and which from a logical
perspective satisfy the same temporal logic formulae, may nevertheless have
fundamentally different properties (solutions) from a game theoretic
perspective. Our aim in this paper is to explore the issues raised by this
discovery. After illustrating the issue by way of a motivating example, we
present three models of strategies with respect to which the existence of Nash
equilibria is preserved under bisimilarity. We use some of these models of
strategies to provide new semantic foundations for logics for strategic
reasoning, and investigate restricted scenarios where bisimilarity can be shown
to preserve the existence of Nash equilibria with respect to the conventional
model of strategies in the computer science literature.
