Similarity Solutions of a Replicator Dynamics Equation Associated to a Continuum of Pure Strategies

Abstract

We introduce a nonlinear degenerate parabolic equation containing a nonlocal term. The equation serves as a replicator dynamics model where the set of strategies is a continuum. In our model the payoff operator (which is the continuous analog of the payoff matrix) is nonsymmetric and, also, evolves with time. We are interested in solutions $u(t, x)$ of our equation which are positive and their integral (with respect to $x$) over the whole space is $1$, for any $t > 0$. These solutions, being probability densities, can serve as time-evolving mixed strategies of a player. We show that for our model there is an one-parameter family of self-similar such solutions $u(t, x)$, all approaching the Dirac delta function $\delta(x)$ as $t \to 0^+$.