A stochastic behavioral model and a `microscopic' foundation of   evolutionary game theory
  A stochastic model for behavioral changes by imitative pair interactions of individuals is developed. `Microscopic' assumptions on the specific form of the imitative processes lead to a stochastic version of the game dynamical equations. That means, the approximate mean value equations of these equations are the game dynamical equations of evolutionary game theory.   The stochastic version of the game dynamical equations allows the derivation of covariance equations. These should always be solved along with the ordinary game dynamical equations. On the one hand, the average behavior is affected by the covariances so that the game dynamical equations must be corrected for increasing covariances. Otherwise they may become invalid in the course of time. On the other hand, the covariances are a measure for the reliability of game dynamical descriptions. An increase of the covariances beyond a critical value indicates a phase transition, i.e. a sudden change in the properties of the considered social system.   The applicability and use of the introduced equations are illustrated by computational results for the social self-organization of behavioral conventions. 
Dirk Helbing