arXiv subjects

Jakub Bielawski

Publications and source records attributed to Jakub Bielawski.

7 recordsLinked to original sources

arXiv2026

Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos

We study the long-term behavior of the Multiplicative Weights Update (MWU) algorithm in game settings where learning dynamics frequently fail to converge to Nash equilibria and instead exhibit Li-Yorke chaos. While such chaos precludes the prediction of specific long-term strategy profiles, it does not imply a lack of statistical structure. We demonstrate that natural invariant measures - a fundamental concept from ergodic theory - provide the rigorous framework necessary to find order within this chaos. Focusing on a two-strategy congestion game, we prove that these measures allow for a comprehensive statistical characterization of the dynamics. Crucially, we show that this framework extends beyond simple strategy frequencies to \emph{general observables}, enabling the precise calculation of long-term time averages for broad classes of economic metrics - including payoffs, social cost, and regret - despite chaos. Our results reveal that this simple learning algorithm captures the full spectrum of behaviors found in one-dimensional dynamical systems, from unique or multiple absolutely continuous invariant measures to complex periodic attractors as well as coexisting chaotic and stable (periodic) behaviors. By bridging game theory and dynamical systems, we show that statistical predictability is attainable even in the absence of pointwise convergence.

math.DS↗

arXiv2026

Strange bifurcation diagrams

We investigate a family of one dimensional maps for which the bifurcation diagram looks differently than the usual ones. We describe and exemplify various unique and interesting phenomena arising for this family of maps.

math.DS↗

arXiv2025

The emergence of chaos in population game dynamics induced by comparisons

Precise description of population game dynamics introduced by revision protocols - an economic model describing the agent's propensity to switch to a better-performing strategy - is of importance in economics and social sciences in general. In this setting innovation or imitation of others is the force which drives the evolution of the economic system. As the continuous-time game dynamics is relatively well understood, the same cannot be said about revision driven dynamics in the discrete time. We investigate the behavior of agents in a $2\times 2$ anti-coordination game with symmetric random matching and a unique mixed Nash equilibrium. In continuous time the Nash equilibrium is attracting and induces a global evolutionary stable state. We show that in the discrete time one can construct (either innovative or imitative) revision protocol and choose a level of the time step, under which the game dynamics is Li-Yorke chaotic, inducing complex and unpredictable behavior of the system, precluding stable predictions of equilibrium. Moreover, we reveal that this unpredictability is encoded into any imitative revision protocol. Furthermore, we show that for any such game there exists a perturbed pairwise proportional imitation protocol introducing chaotic behavior of the agents for sufficiently large time step.

math.DS↗

arXiv2024

Interval maps mimicking circle rotations

We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We concentrate on a specific two-parameter family, describing the dynamics arising from models in game theory, mathematical biology and machine learning. If one parameter is a rational number, $k/n$, with $k,n$ coprime, and the second one is large enough, we prove that there is a periodic orbit of period $n$. It behaves like an orbit of the circle rotation by an angle $2πk/n$ and attracts trajectories of Lebesgue almost all starting points. We also discover numerically other interesting phenomena. While we do not give rigorous proofs for them, we provide convincing explanations.

math.DS↗

arXiv2022

Unpredictable dynamics in congestion games: memory loss can prevent chaos

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 $σ\in [0,1]$ 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 $σ$. Specifically, we show that for any discount factor $σ$ the system will be destabilized for a sufficiently large intensity of choice $a$. Although for discount factor $σ=0$ almost always (i.e., $b \neq 1/2$) the system will become chaotic, as $σ$ 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.

cs.GT↗

arXiv2021

Follow-the-Regularized-Leader Routes to Chaos in Routing Games

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 any fixed learning rate, unless the game is fully symmetric, increasing the population size or the scale of costs causes learning dynamics to become unstable and eventually chaotic, in the sense of Li-Yorke and positive topological entropy. Furthermore, we show 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 exact equilibrium for any choice of learning rate and any scale of costs.

cs.GT↗

arXiv2019

The interplay between migrants and natives as a determinant of migrants' assimilation: A coevolutionary approach

We study the migrants' assimilation, which we conceptualize as forming human capital productive on the labor market of a developed host country, and we link the observed frequent lack of assimilation with the relative deprivation that the migrants start to feel when they move in social space towards the natives. In turn, we presume that the native population is heterogenous and consists of high-skill and low-skill workers. The presence of assimilated migrants might shape the comparison group of the natives, influencing the relative deprivation of the low-skill workers and, in consequence, the choice to form human capital and become highly skilled. To analyse this interrelation between assimilation choices of migrants and skill formation of natives, we construct a coevolutionary model of the open-to-migration economy. Showing that the economy might end up in a non-assimilation equilibrium, we discuss welfare consequences of an assimilation policy funded from tax levied on the native population. We identify conditions under which such costly policy can bring the migrants to assimilation and at the same time increase the welfare of the natives, even though the incomes of the former take a beating.

econ.TH↗