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Evolution of distribution of strategies in game theory is an interestingquestion that has been studied only for specific cases. Here I develop ageneral method to extend analysis of the evolution of continuous strategydistributions given bi-linear and quadratic payoff functions for any initialdistribution to answer the following question: given the initial distributionof strategies in a game, how will it evolve over time? I look at severalspecific examples, including normal distribution on the entire line, normaltruncated distribution, as well as exponential, uniform and Gammadistributions. I show that the class of exponential distributions is invariantwith respect to replicator dynamics in games with bi-linear payoff functions. Ishow also that the class of normal distributions is invariant with respect toreplicator dynamics in games with quadratic payoff functions. The developedmethod can now be applied to a broad class of questions pertaining to evolutionof strategies in games with different payoff functions and different initialdistributions.

Dynamics of Strategy Distribution in a One-Dimensional Continuous Trait Space with a Bi-linear and Quadratic Payoff Functions