Browsing by Author "Falniowski, Fryderyk"

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    Family of chaotic maps from game theory
    (Taylor & Francis, 2021) Chotibut, Thiparat; Falniowski, Fryderyk; Misiurewicz, Michał; Piliouras, Georgios; Mathematical Sciences, School of Science
    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, limn→∞1n∑n−1k=0fk(x)=b for every x∈(0,1). This solves a known open question whether there exists a ‘natural’ nontrivial smooth map other than x↦axe−x with centres of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.
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    Follow-the-Regularized-Leader Routes to Chaos in Routing Games
    (Proceedings of Machine Learning Research, 2021) Bielawski, Jakub; Chotibut, Thiparat; Falniowski, Fryderyk; Kosiorowski, Grzegorz; Misiurewicz, Michał; Piliouras, Georgios; Mathematical Sciences, School of Science
    We study the emergence of chaotic behavior of Follow-the-Regularized Leader (FoReL) dynamics in games. We focus on the effects of increasing the population size or the scale of costs in congestion games, and generalize recent results on unstable, chaotic behaviors in the Multiplicative Weights Update dynamics to a much larger class of FoReL dynamics. We establish that, even in simple linear non-atomic congestion games with two parallel links and \emph{any} fixed learning rate, unless the game is fully symmetric, increasing the population size or the scale of costs causes learning dynamics to becomes unstable and eventually chaotic, in the sense of Li-Yorke and positive topological entropy. Furthermore, we prove the existence of novel non-standard phenomena such as the coexistence of stable Nash equilibria and chaos in the same game. We also observe the simultaneous creation of a chaotic attractor as another chaotic attractor gets destroyed. Lastly, although FoReL dynamics can be strange and non-equilibrating, we prove that the time average still converges to an \emph{exact} equilibrium for any choice of learning rate and any scale of costs.