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

Evolution of distribution of strategies in game theory is an interesting question that has been studied only for specific cases. Here I develop a general method to extend analysis of the evolution of continuous strategy distributions given bi-linear and quadratic payoff functions for any initial distribution to answer the following question: given the initial distribution of strategies in a game, how will it evolve over time? I look at several specific examples, including normal distribution on the entire line, normal truncated distribution, as well as exponential, uniform and Gamma distributions. I show that the class of exponential distributions is invariant with respect to replicator dynamics in games with bi-linear payoff functions. I show also that the class of normal distributions is invariant with respect to replicator dynamics in games with quadratic payoff functions. The developed method can now be applied to a broad class of questions pertaining to evolution of strategies in games with different payoff functions and different initial distributions.