Fryderyk Falniowski

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

We examine the impact of a change in diversity introduced by a new product on the evolution of an economic system. Modeling Schumpeterian competition as a population game with a unique, attracting, evolutionarily stable state (the Schumpeterian state), in which both innovators and imitators coexist, we examine how the Schumpeterian state evolves depending on the properties of the new product developed by innovators. This way, we demonstrate that the change in diversity is one of the spiritus movens of innovation, influencing the process of creative destruction.

Inspired by the challenges at the intersection of Evolutionary Game Theory and Machine Learning, we investigate a class of discrete-time multiagent reinforcement learning (MARL) dynamics in population/nonatomic congestion games, where agents have diverse beliefs and learn at different rates. These congestion games, a well-studied class of potential games, are characterized by individual agents having negligible effects on system performance, strongly aligned incentives, and well-understood advantageous properties of Nash equilibria. Despite the presence of static Nash equilibria, we demonstrate that MARL dynamics with heterogeneous learning rates can deviate from these equilibria, exhibiting instability and even chaotic behavior and resulting in increased social costs. Remarkably, even within these chaotic regimes, we show that the time-averaged macroscopic behavior converges to exact Nash equilibria, thus linking the microscopic dynamic complexity with traditional equilibrium concepts. By employing dynamical systems techniques, we analyze the interaction between individual-level adaptation and population-level outcomes, paving the way for studying heterogeneous learning dynamics in discrete time across more complex game scenarios.

Recent studies have raised concerns on the inevitability of chaos in congestion games with large learning rates. We further investigate this phenomenon by exploring the learning dynamics in simple two-resource congestion games, where a continuum of agents learns according to a simplified experience-weighted attraction algorithm. The model is characterized by three key parameters: a population intensity of choice (learning rate), a discount factor (recency bias or exploration parameter), and the cost function asymmetry. The intensity of choice captures agents’ economic rationality in their tendency to approximately best respond to the other agent’s behavior. The discount factor captures a type of memory loss of agents, where past outcomes matter exponentially less than the recent ones. Our main findings reveal that while increasing the intensity of choice destabilizes the system for any discount factor, whether the resulting dynamics remains predictable or becomes unpredictable and chaotic depends on both the memory loss and the cost asymmetry. As memory loss increases, the chaotic regime gives place to a periodic orbit of period 2 that is globally attracting except for a countable set of points that lead to the equilibrium. Therefore, memory loss can suppress chaotic behaviors. The results highlight the crucial role of memory loss in mitigating chaos and promoting predictable outcomes in congestion games, providing insights into designing control strategies in resource allocation systems susceptible to chaotic behaviors.