Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2412.06037

The emergence of chaos in population game dynamics induced by comparisons

Jakub Bielawski·Łukasz Cholewa·Fryderyk Falniowski
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.DScs.GTecon.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jakub Bielawski, Łukasz Cholewa, Fryderyk Falniowski. 2025-04-21. The emergence of chaos in population game dynamics induced by comparisons. https://arxiv.org/abs/2412.06037

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Asymmetry of a class of Mellin transforms

We introduce the quantity $μ_η$, defined for every complex $s$ in the critical strip, as a transformation of the Mellin transform associated to the functions $η$. We establish a sufficient condition on $η$ under which $μ_η(s)$ and $μ_η(1-s)$ cannot both vanish outside the critical line. An application is given to the case in which $η$ is the fractional part function, and the zeros of $μ_η$ coincide with the zeros of the Riemann zeta function.

math.DS↗
arXiv2026

Infinite Set of Resonances in the Linear Damped Oscillator Subject to Harmonic Forcing with Non-standard Frequency Modulation

It is shown that harmonic signals incorporating a type of weak non-standard frequency modulation (wNSFM) have interesting spectral properties, namely, time-dependent bandwidths that become increasingly broader with increasing time. As such, they represent a class of signals with frequency-time coupling in their spectra. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are persistent to changes in the parameters of the wNSFM. Our results reveal interesting infinite sets of resonances in linear SDOF resonators under frequency-modulated excitations.

math.DS↗