Risk-Sensitive Learning in Population Games under Extreme Events: Bifurcations and Chaotic Dynamics
Abstract
Inspired by nonequilibrium phenomena in game dynamics and behavioral evidence on the impact of extreme events on decision making, we investigate the nonlinear dynamics of a discrete-time multiagent learning rule in population congestion games under extreme events affecting one of the actions. The population state, following a risk-sensitive variant of the Multiplicative Weights Update (MWU), is coupled with a belief variable capturing the agents perceived risk and updated through an adaptive expectation rule. We perform a two-parameter bifurcation analysis with respect to the agents controlled parameters, identifying regions of qualitatively distinct behavior. Equilibria are studied first from both game-theoretic and dynamical perspectives. The resulting two-dimensional system exhibits complex behavior, including multi-stability among fixed points, invariant curves, periodic and chaotic attractors. Despite this complexity, the attractors can be grouped into distinct families, while the Cesàro averages of the trajectories are shown to converge to the stationary equilibrium. The incorporation of risk associated with the extreme event leads to new dynamical phenomena: attracting invariant curves arise and give rise to phase-locking Arnold tongues, within which the dynamics is qualitatively similar. In this setting, codimension-two resonances are identified as organizing centers, both within individual tongues and along the bifurcation curves associated with the fixed-point family. Chaotic attractors emerge and are destroyed through Feigenbaum cascades and forward or reverse boundary crises, with interior and merging crises also observed, along with transient chaos and narrow periodic windows. For each qualitatively distinct region, representative phase portraits and the associated basins of attraction are examined.
Related
- Extrapolating learned dynamics across bifurcations based on generalized synchronization
- Simulation-Free Learning of Population Dynamics with Wasserstein Lagrangian Residuals
- Polymerase-mediated quasispecies dynamics: bifurcations, complementation, and robustness of τ-tipping
- Global Dynamics and Periodic-Annulus Bifurcations in a Reversible Quintic Liénard System
- Robustness against lying of the confidence interval for a fraction of population with a sensitive characteristic
BibTeX
@misc{metaxas2026risksensitive,
title = {{Risk-Sensitive Learning in Population Games under Extreme Events: Bifurcations and Chaotic Dynamics}},
author = {Konstantinos Metaxas and Themistoklis P. Sapsis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.29967},
eprint = {2606.29967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.29967},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Risk-Sensitive Learning in Population Games under Extreme Events: Bifurcations and Chaotic Dynamics
AU - Konstantinos Metaxas
AU - Themistoklis P. Sapsis
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2606.29967
UR - https://arxiv.org/abs/2606.29967
AB - Inspired by nonequilibrium phenomena in game dynamics and behavioral evidence on the impact of extreme events on decision making, we investigate the nonlinear dynamics of a discrete-time multiagent learning rule in population congestion games under extreme events affecting one of the actions. The population state, following a risk-sensitive variant of the Multiplicative Weights Update (MWU), is coupled with a belief variable capturing the agents perceived risk and updated through an adaptive expectation rule. We perform a two-parameter bifurcation analysis with respect to the agents controlled parameters, identifying regions of qualitatively distinct behavior. Equilibria are studied first from both game-theoretic and dynamical perspectives. The resulting two-dimensional system exhibits complex behavior, including multi-stability among fixed points, invariant curves, periodic and chaotic attractors. Despite this complexity, the attractors can be grouped into distinct families, while the Cesàro averages of the trajectories are shown to converge to the stationary equilibrium. The incorporation of risk associated with the extreme event leads to new dynamical phenomena: attracting invariant curves arise and give rise to phase-locking Arnold tongues, within which the dynamics is qualitatively similar. In this setting, codimension-two resonances are identified as organizing centers, both within individual tongues and along the bifurcation curves associated with the fixed-point family. Chaotic attractors emerge and are destroyed through Feigenbaum cascades and forward or reverse boundary crises, with interior and merging crises also observed, along with transient chaos and narrow periodic windows. For each qualitatively distinct region, representative phase portraits and the associated basins of attraction are examined.
ER -