Paper discusses the evolutionary game theory applied to the analysis of the social interaction dynamics under presence of the arbitrary start dependence. General setup of the game model with n players and n strategies is presented. Differences between the best reply related adaptive mechanisms with alternative adaptive mechanisms, including the natural selection, are examined. When it comes to the incentives problem, there are set forth conditions under which the final convention depends on the initial strategy distribution. The number experiment results are presented. For the coordination problem, varieties of the initial conditions are presented that determine asymptotic game dynamics.