Family of chaotic maps from game theory

Abstract

From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such map f is usually repelling, it is globally Cesàro attracting on the diagonal, that is,for every x in the minimal invariant interval. This solves a known open question whether there exists a nontrivial smooth map other than x → axe −x with centers of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.

Fast and Furious Learning in Zero-Sum Games: Vanishing Regret with Non-Vanishing Step Sizes

“…[19] establish non-convergence for a continuous-time variant of MWU, known as the replicator dynamic, for a 2x2x2 game and show that as a result the system social welfare converges to states that dominate all Nash equilibria. [25,11] prove the existence of Li-Yorke chaos in MWU dynamics of 2x2 potential games. Our result add a new chapter in this area with new detailed understanding of the non-equilibrium trajectories of gradient descent in two-by-two zero-sum games and their implications to regret.…”

“…) is a strongly smooth function in the simplex, we expect for h * i (y i ) to be strongly convex [18] -at least when it's corresponding dual variable x i is positive. However, (11) is not strongly convex for all y t i ∈ R n i . This is because y t+1 i cannot appear anywhere in R n i .…”

“…This pessimism seems especially justified given recent results about the behavior of online dynamics with fixed step-sizes in other small games (e.g. two-by-two coordination/congestion games), where their behavior can be shown to become provably chaotic ( [25,11]). Nevertheless, we show that we can leverage this geometric information to provide the first to our knowledge sublinear regret guarantees for online gradient descent with fixed step-size in games.…”

Vortices Instead of Equilibria in MinMax Optimization: Chaos and Butterfly Effects of Online Learning in Zero-Sum Games

“…By comparing the above equality with (7), the summation in the RHS of above equality is analogous to [Ay] j . Similarly, M kj = ǫx j (B jk − ℓ∈J B ℓk • xℓ z∈J xz ), with the summation here analogous to [B T x] k in the MWU case.…”

“…We provide a theoretical underpinning for these phenomena for a wide spectrum of dynamics and games. Palaiopanos et al [22] and Chotibut et al [7] studied MWU and its variant in congestion games. While MWU with very small constant step-size converges to equilibrium in such games, they showed if we increase the step-size MWU becomes chaotic in a notion first defined by Li and Yorke [17].…”