Heterogeneity and chaos in congestion games

We analyze a class of congestion games where agents use resources to send a finite amount of goods from an initial location to a terminal one. The resources are costly and costs are load dependent. In this context we concentrate on the heterogeneity because we not only assume that agents have limited computational capability but also they differ in the quantities they must send and the reactivity. We introduce an appropriate dynamical system, which has the steady state exactly at the unique Nash equilibrium of the static congestion game, and we investigate the dynamical behavior of the game. We provide a closed form characterization on the unique Nash equilibrium in the underlying static congestion game and we prove that the equilibrium crucially depends on the aggregate congestion. Not only do we provide a local stability condition in terms of the agents’ reactivity and the nonlinearity of the cost functions but also we study the role of the heterogeneity in this context. We show analytically that heterogeneity can be destabilizing with a cascade of flip-bifurcation leading to periodic cycles and finally to chaos but also that mastering the degree of heterogeneity can be used to tame and control complex dynamics; moreover in relation to the reactivity levels their product must roughly small. However if both reactivity levels are near but outside the stability region then it is sufficient to act on just one to restore stability. Finally, if both levels are far form the stability region acting on one will not restore stability.