Choice Paralysis in Evolutionary Games
Brendon G. Anderson
TL;DR
We study when finite-strategy approximations of infinite-strategy evolutionary dynamics faithfully reproduce the true dynamics. The paper introduces choice mobility, a sufficient condition ensuring that finite-dimensional trajectories converge to the infinite-dimensional dynamics on finite time horizons, and shows that, under mild regularity, these approximations converge uniformly on compact time in the $d_{BL}$ metric. It also defines choice paralysis, proving that if strategy-switching rates decay as more strategies are added, the long-time behavior of finite approximations may diverge from the infinite-strategy game, as demonstrated by an explicit example. The results justify using sufficiently fine finite approximations to capture short- and medium-term dynamics while highlighting potential failures for long-run predictions, and they motivate the development of new analysis techniques for the infinite-strategy setting.
Abstract
In this paper, we consider finite-strategy approximations of infinite-strategy evolutionary games. We prove that such approximations converge to the true dynamics over finite-time intervals, under mild regularity conditions which are satisfied by classical examples, e.g., the replicator dynamics. We identify and formalize novel characteristics in evolutionary games: choice mobility, and its complement choice paralysis. Choice mobility is shown to be a key sufficient condition for the long-time limiting behavior of finite-strategy approximations to coincide with that of the true infinite-strategy game. An illustrative example is constructed to showcase how choice paralysis may lead to the infinite-strategy game getting "stuck," even though every finite approximation converges to equilibrium.