How to study the evolution of parametrized games?

I consider a family of games with corresponding payoff matrices, parametrized by the parameter The informal question of interest is: what game is the best? Instead of attempting to suggest a more or less arbitrary definition of the best game, I allow natural selection to find it through honest competition between games. Specifically, I consider a model of a community composed of populations with different payoff matrices and study the natural selection between the populations. The outcome of the community evolution determines the best game. I develop here a mathematical toolbox, that allows the study of evolution of parametrized games. The developed approach is applied to three examples of parametrized games from the literature: local replicator dynamics, pairwise competition and the game of alleles in diploid genomes.