  Continuous-time game dynamics are typically first order systems where payoffs
determine the growth rate of the players' strategy shares. In this paper, we
investigate what happens beyond first order by viewing payoffs as higher order
forces of change, specifying e.g. the acceleration of the players' evolution
instead of its velocity (a viewpoint which emerges naturally when it comes to
aggregating empirical data of past instances of play). To that end, we derive a
wide class of higher order game dynamics, generalizing first order imitative
dynamics, and, in particular, the replicator dynamics. We show that strictly
dominated strategies become extinct in n-th order payoff-monotonic dynamics n
orders as fast as in the corresponding first order dynamics; furthermore, in
stark contrast to first order, weakly dominated strategies also become extinct
for n>1. All in all, higher order payoff-monotonic dynamics lead to the
elimination of weakly dominated strategies, followed by the iterated deletion
of strictly dominated strategies, thus providing a dynamic justification of the
well-known epistemic rationalizability process of Dekel and Fudenberg (1990).
Finally, we also establish a higher order analogue of the folk theorem of
evolutionary game theory, and we show that con- vergence to strict equilibria
in n-th order dynamics is n orders as fast as in first order.
