Non-Atomic Games and an Application to Jet Lag

Abstract

A region in the brain contains a large network of neuronal oscillators which maintain circadian rhythms. By modeling these oscillators with a game-theoretic approach, the equilibrium dynamics of the oscillators arise endogenously from first principles. The size of the network suggests an approximation by a continuum player approach, necessitating a theory for limits of finite-player network games.

First, we consider static finite-player network games and their continuum analogs, graphon games. Existence and uniqueness results are provided, as well as convergence of the finite-player network game optimal strategy profiles to their analogs for the graphon games. We also show that equilibrium strategy profiles of a graphon game provide approximate Nash equilibria for the finite-player games.

Next, we model a large population of circadian oscillators. Lacking the technical tools to approach a graphon game in the dynamic setting, we opt for a mean field game approach. Our analysis is driven by two goals: 1) to understand the long time behavior of the oscillators when an individual remains in the same time zone, and 2) to quantify the costs from jet lag recovery when the individual has traveled across time zones. Numerical results are benchmarked against explicit solutions derived for a special case. The numerics suggest that the cost the oscillators accrue while recovering is larger for eastward travel, which is consistent with the widely admitted

wisdom that jet lag is worse after traveling east than west.

An alternative to our game-theoretic approach is to consider a central planner optimizing the total cost of the population. The discrepancy between the two approaches is aptly named the price of anarchy. In order to derive explicit computations in the setting of circadian oscillators, we instead focus on a class of linear quadratic extended mean field games, for which explicit computations are possible. A sufficient and necessary condition to have no price of anarchy is presented. Limiting behaviors of the price of anarchy are proved for various model coefficients.