  The game dynamical equations are derived from Boltzmann-like equations for
individual pair interactions by assuming a certain kind of imitation behavior,
the so-called proportional imitation rule. They can be extended to a stochastic
formulation of evolutionary game theory which allows the derivation of
approximate and corrected mean value and covariance equations. It is shown
that, in the case of phase transitions (i.e. multi-modal probability
distributions), the mean value equations do not agree with the game dynamical
equations. Therefore, their exact meaning is carefully discussed. Finally, some
generalizations of the behavioral model are presented, including effects of
expectations, other kinds of interactions, several subpopulations, or memory
effects.
