arXiv subjects

Julius Durmann

Publications and source records attributed to Julius Durmann.

4 recordsLinked to original sources

arXiv2026

Mean-based algorithms: A lower bound and regret

Mean-based algorithms are a class of online learning algorithms that assign low probability to actions with low average rewards. Recent work indicates these algorithms converge favorably to serially undominated actions, which approximate Nash equilibria in economic games. However, empirical studies also show slower convergence compared to established algorithms in bandit-feedback scenarios. We study mean-based algorithms when the time horizon is unknown and only bandit feedback is available. In this setting, we provide the first lower bound on the algorithm-defining sequence $\gamma_t$ that formally establishes a limit on how fast these algorithms can learn. Additionally, we propose two mean-based algorithms: one generalizes $\epsilon$-greedy, and the other extends the mean-based Exp3 to unknown horizons. Our experiments show that mean-based algorithms, although slightly slower, can perform competitively with other bandit-feedback algorithms. We further analyze the relationship to no-regret algorithms. Depending on the choice of $\gamma_t$, the intersection with no-regret algorithms is non-trivial, and we show that algorithms exist that are both mean-based and no-regret. This adds context to the "exploitability" of this class of algorithms that previous contributions suggest.

cs.LG↗

arXiv2025

Agentic Markets: Game Dynamics and Equilibrium in Markets with Learning Agents

Autonomous and learning agents increasingly participate in markets - setting prices, placing bids, ordering inventory. Such agents are not just aiming to optimize in an uncertain environment; they are making decisions in a game-theoretical environment where the decision of one agent influences the profit of other agents. While game theory usually predicts outcomes of strategic interaction as an equilibrium, it does not capture how repeated interaction of learning agents arrives at a certain outcome. This article surveys developments in modeling agent behavior as dynamical systems, with a focus on projected gradient and no-regret learning algorithms. In general, learning in games can lead to all types of dynamics, including convergence to equilibrium, but also cycles and chaotic behavior. It is important to understand when we can expect efficient equilibrium in automated markets and when this is not the case. Thus, we analyze when and how learning agents converge to an equilibrium of a market game, drawing on tools from variational inequalities and Lyapunov stability theory. Special attention is given to the stability of projected dynamics and the convergence to equilibrium sets as limiting outcomes. Overall, the paper provides mathematical foundations for analyzing stability and convergence in agentic markets driven by autonomous, learning agents.

cs.GT↗

arXiv2025

Algorithmic Pricing and Algorithmic Collusion

The rise of algorithmic pricing in online retail platforms has attracted significant interest in how autonomous software agents interact under competition. This article explores the potential emergence of algorithmic collusion - supra-competitive pricing outcomes that arise without explicit agreements - as a consequence of repeated interactions between learning agents. Most of the literature focuses on oligopoly pricing environments modeled as repeated Bertrand competitions, where firms use online learning algorithms to adapt prices over time. While experimental research has demonstrated that specific reinforcement learning algorithms can learn to maintain prices above competitive equilibrium levels in simulated environments, theoretical understanding of when and why such outcomes occur remains limited. This work highlights the interdisciplinary nature of this challenge, which connects computer science concepts of online learning with game-theoretical literature on equilibrium learning. We examine implications for the Business & Information Systems Engineering (BISE) community and identify specific research opportunities to address challenges of algorithmic competition in digital marketplaces.

cs.GT↗

arXiv2024

Online Optimization Algorithms in Repeated Price Competition: Equilibrium Learning and Algorithmic Collusion

This paper investigates whether online learning algorithms in pricing produce competitive outcomes or tacit collusion. This issue has drawn considerable attention from competition regulators as algorithmic pricing becomes more common in digital markets. Understanding when such algorithms lead to equilibrium or supra-competitive prices is critical for buyers, sellers, and policymakers. We study the behavior of multi-armed bandit (MAB) online learning algorithms in repeated price competition. These algorithms require little information to learn, making them realistic models of automated pricing. Our analysis shows that mean-based algorithms, a special variant of online learning algorithms, converge to correlated rationalizable actions. In the Bertrand environments considered, this implies convergence to the Nash equilibrium or adjacent prices. Numerical experiments reveal that most MAB algorithms, including those that are not mean-based, also converge. We observe supra-competitive prices only in specific cases where all sellers implement the same symmetric version of certain algorithms, such as UCB. This effect diminishes as the number of competitors increases. Our results suggest that, even in a stylized repeated Bertrand competition, sustained supra-competitive prices may be less of a concern when independent agents use different online learning algorithms. Our insights are relevant for regulators and managers considering the use of algorithmic pricing algorithms.

cs.GT↗