Unpredictable dynamics in congestion games: memory loss can prevent chaos

TL;DR

This paper investigates how memory loss influences the complex dynamics of congestion games, showing that increased memory loss can stabilize the system and reduce chaos, with bifurcation phenomena depending on key parameters.

Contribution

It introduces a simple congestion game model with EWA dynamics, revealing how memory loss affects stability and chaos, and provides a detailed bifurcation analysis.

Findings

Memory loss can prevent chaos and stabilize dynamics.

High intensity of choice leads to destabilization and chaos.

Increasing memory loss simplifies dynamics, leading to predictable periodic orbits.

Abstract

We study the dynamics of simple congestion games with two resources where a continuum of agents behaves according to a version of Experience-Weighted Attraction (EWA) algorithm. The dynamics is characterized by two parameters: the (population) intensity of choice $a>0$ capturing the economic rationality of the total population of agents and a discount factor $\sigma\in [0,1]$ capturing a type of memory loss where past outcomes matter exponentially less than the recent ones. Finally, our system adds a third parameter $b \in (0,1)$, which captures the asymmetry of the cost functions of the two resources. It is the proportion of the agents using the first resource at Nash equilibrium, with $b=1/2$ capturing a symmetric network. Within this simple framework, we show a plethora of bifurcation phenomena where behavioral dynamics destabilize from global convergence to equilibrium, to limit cycles or even (formally proven) chaos as a function of the parameters $a$, $b$ and $\sigma$. Specifically, we show that for any discount factor $\sigma$ the system will be destabilized for a sufficiently large intensity of choice $a$. Although for discount factor $\sigma=0$ almost always (i.e., $b \neq 1/2$) the system will become chaotic, as $\sigma$ increases the chaotic regime will give place to the attracting periodic orbit of period 2. Therefore, memory loss can simplify game dynamics and make the system predictable. We complement our theoretical analysis with simulations and several bifurcation diagrams that showcase the unyielding complexity of the population dynamics (e.g., attracting periodic orbits of different lengths) even in the simplest possible potential games.