On a class of operator equations arising in infinite dimensional replicator dynamics

On a Degenerate Nonlocal Parabolic Problem Describing Infinite Dimensional Replicator Dynamics

“…The abstract form of equation (5.2) does not actually allow us to obtain insight on the form of its solutions and thus a better understanding of the evolutionary dynamics of the corresponding game. In order to have a better overview of the evolutionary game, following the approach in [23,24], we restrict our attention to measures Q(t) which, for each t > 0, are absolutely continuous with respect to the Lebesgue measure, with probability density u(x, t). Then the replicator dynamics equation (5.2) can be reduced to the following integro-differential equation…”

Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory

“…The analogue for a setting involving finitely many strategies goes back to the works of Taylor and Jonker [22] and Maynard Smith [17], and infinite-dimensional variants have been treated in [3] and [18]. Building on these, for steep payoff kernels of Gaussian type, in [10,11] the PDE (1.2) was introduced. For additional details concerning the modeling background, we refer to [9, Appendix A], [11], and the references therein.…”

“…Building on these, for steep payoff kernels of Gaussian type, in [10,11] the PDE (1.2) was introduced. For additional details concerning the modeling background, we refer to [9, Appendix A], [11], and the references therein.…”

Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity

“…Problem (1) arises in the context of evolutionary game dynamics ( [15,4]), more precisely in the framework of replicator dynamics ( [22,19]), applied to an infinite dimensional continuous setting. For more information on the modelling, we refer to the appendix of [8] and to [10,11] and the references therein. In this setting, anyhow, Ω is a set of "strategies" and, in the manner of a probability density, u gives the relative frequency with which they are pursued in the population of "players" considered -and accordingly the case of highest interest from an applications viewpoint is that of Ω u 0 = 1.…”

“…Let us mention some results concerning explicit examples of solutions to (1) and solutions of a particular form: In [11] and [17], self-similar solutions to (1) were constructed in Ω = R and Ω = R n , respectively, and [16] investigates self-similar solutions in Ω = R for a related model, where the Laplacian is perturbed by a time-dependent term involving first derivatives as well. In [10,9], stationary solutions of (1) were studied; however, it remained open if those accurately capture the long-term behaviour of solutions to (1). A discussion of existence and qualitative properties of solutions to (1) for rather general initial data in bounded domains can be found in the previous work [8], where the existence of solutions to (1) has been shown as limit of solutions to the actually non-degenerate parabolic problems…”