arXiv subjects

Matthias Oberlechner

Publications and source records attributed to Matthias Oberlechner.

7 recordsLinked to original sources

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.

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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.

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arXiv2025

Characterizing the Convergence of Game Dynamics via Potentialness

Understanding the convergence landscape of multi-agent learning is a fundamental problem of great practical relevance in many applications of artificial intelligence and machine learning. While it is known that learning dynamics converge to Nash equilibrium in potential games, the behavior of dynamics in many important classes of games that do not admit a potential is poorly understood. To measure how ''close'' a game is to being potential, we consider a distance function, that we call ''potentialness'', and which relies on a strategic decomposition of games introduced by Candogan et al. (2011). We introduce a numerical framework enabling the computation of this metric, which we use to calculate the degree of ''potentialness'' in generic matrix games, as well as (non-generic) games that are important in economic applications, namely auctions and contests. Understanding learning in the latter games has become increasingly important due to the wide-spread automation of bidding and pricing with no-regret learning algorithms. We empirically show that potentialness decreases and concentrates with an increasing number of agents or actions; in addition, potentialness turns out to be a good predictor for the existence of pure Nash equilibria and the convergence of no-regret learning algorithms in matrix games. In particular, we observe that potentialness is very low for complete-information models of the all-pay auction where no pure Nash equilibrium exists, and much higher for Tullock contests, first-, and second-price auctions, explaining the success of learning in the latter. In the incomplete-information version of the all-pay auction, a pure Bayes-Nash equilibrium exists and it can be learned with gradient-based algorithms. Potentialness nicely characterizes these differences to the complete-information version.

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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.

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arXiv2023

Revenue in First- and Second-Price Display Advertising Auctions: Understanding Markets with Learning Agents

The transition of display ad exchanges from second-price auctions (SPA) to first-price auctions (FPA) has raised questions about its impact on revenue. Auction theory predicts the revenue equivalence between these two auction formats. However, display ad auctions are different from standard models in auction theory. First, automated bidding agents cannot easily derive equilibrium strategies in FPA because information regarding competitors is not readily available. Second, due to principal-agent problems, bidding agents typically maximize return-on-investment (ROI), not payoff. The literature on learning agents for real-time bidding is growing because of the practical relevance of this area; most research has found that learning agents do not converge to an equilibrium. Specifically, research on algorithmic collusion in display ad auctions has argued that FPA can induce symmetric Q-learning agents to tacitly collude, resulting in bids below equilibrium, leading to lower revenue compared to the SPA. Whether bids are in equilibrium cannot easily be determined from field data since the underlying values of bidders are unknown. In this paper, we draw on analytical modeling and numerical experiments and explore the convergence behavior of widespread online learning algorithms in both complete and incomplete information models. Contrary to prior results, we show that there are no systematic deviations from equilibrium behavior. We also explore the differences in revenue of the FPA and SPA, which have not been done for utility functions relevant to this domain, such as ROI. We show that learning algorithms also converge to equilibrium. Still, revenue equivalence does not hold, indicating that collusion may not be the explanation for lower revenue with FPA, and the change in auction format might have had substantial and non-obvious consequences for ad exchanges and advertisers.

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arXiv2023

On the Convergence of Learning Algorithms in Bayesian Auction Games

Equilibrium problems in Bayesian auction games can be described as systems of differential equations. Depending on the model assumptions, these equations might be such that we do not have a rigorous mathematical solution theory. The lack of analytical or numerical techniques with guaranteed convergence for the equilibrium problem has plagued the field and limited equilibrium analysis to rather simple auction models such as single-object auctions. Recent advances in equilibrium learning led to algorithms that find equilibrium under a wide variety of model assumptions. We analyze first- and second-price auctions where simple learning algorithms converge to an equilibrium. The equilibrium problem in auctions is equivalent to solving an infinite-dimensional variational inequality (VI). Monotonicity and the Minty condition are the central sufficient conditions for learning algorithms to converge to an equilibrium in such VIs. We show that neither monotonicity nor pseudo- or quasi-monotonicity holds for the respective VIs. The second-price auction's equilibrium is a Minty-type solution, but the first-price auction is not. However, the Bayes--Nash equilibrium is the unique solution to the VI within the class of uniformly increasing bid functions, which ensures that gradient-based algorithms attain the equilibrium in case of convergence, as also observed in numerical experiments.

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arXiv2023

Computing Bayes Nash Equilibrium Strategies in Auction Games via Simultaneous Online Dual Averaging

Auctions are modeled as Bayesian games with continuous type and action spaces. Determining equilibria in auction games is computationally hard in general and no exact solution theory is known. We introduce an algorithmic framework in which we discretize type and action space and then learn distributional strategies via online optimization algorithms. One advantage of distributional strategies is that we do not have to make any assumptions on the shape of the bid function. Besides, the expected utility of agents is linear in the strategies. It follows that if our optimization algorithms converge to a pure strategy, then they converge to an approximate equilibrium of the discretized game with high precision. Importantly, we show that the equilibrium of the discretized game approximates an equilibrium in the continuous game. In a wide variety of auction games, we provide empirical evidence that the approach approximates the analytical (pure) Bayes Nash equilibrium closely. This speed and precision is remarkable, because in many finite games learning dynamics do not converge or are even chaotic. In standard models where agents are symmetric, we find equilibrium in seconds. While we focus on dual averaging, we show that the overall approach converges independent of the regularizer and alternative online convex optimization methods achieve similar results, even though the discretized game neither satisfies monotonicity nor variational stability globally. The method allows for interdependent valuations and different types of utility functions and provides a foundation for broadly applicable equilibrium solvers that can push the boundaries of equilibrium analysis in auction markets and beyond.

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