  Under what conditions do the behaviors of players, who play a game
repeatedly, converge to a Nash equilibrium? If one assumes that the players'
behavior is a discrete-time or continuous-time rule whereby the current mixed
strategy profile is mapped to the next, this becomes a problem in the theory of
dynamical systems. We apply this theory, and in particular the concepts of
chain recurrence, attractors, and Conley index, to prove a general
impossibility result: there exist games for which any dynamics is bound to have
starting points that do not end up at a Nash equilibrium. We also prove a
stronger result for $\epsilon$-approximate Nash equilibria: there are games
such that no game dynamics can converge (in an appropriate sense) to
$\epsilon$-Nash equilibria, and in fact the set of such games has positive
measure. Further numerical results demonstrate that this holds for any
$\epsilon$ between zero and $0.09$. Our results establish that, although the
notions of Nash equilibria (and its computation-inspired approximations) are
universally applicable in all games, they are also fundamentally incomplete as
predictors of long term behavior, regardless of the choice of dynamics.
