arXiv subjects

Martin Bichler

Publications and source records attributed to Martin Bichler.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Equilibrium Computation in Colonel Blotto Games

Colonel Blotto is a fundamental resource-allocation game in which two players divide their budgets across battlefields, and the higher allocation on each battlefield wins its value. The problem of computing an equilibrium efficiently has remained unresolved for its classical continuous version with arbitrary budgets and with values that may vary across battlefields but are the same for both players. We give a fully polynomial-time approximation scheme that computes an $\varepsilon$-Nash equilibrium in time polynomial in the input length and $1/\varepsilon$, and nearly linear in the input length for fixed $\varepsilon$. For sufficiently balanced budgets and battlefield values, we also compute exact equilibria in linear time.

cs.GT↗

arXiv2026

Simultaneous Gradient Learning in First-Price Auctions

We study equilibrium learning in discretized Bayesian games, focusing on first-price auctions as a testbed for gradient-based learning dynamics under private information. While the symmetric independent private values model admits a unique Bayesian correlated equilibrium (BCE) in the continuous limit, standard convergence guarantees for gradient-based learners do not apply: classical no-regret theory predicts convergence only to Bayesian coarse correlated equilibria (BCCE), which can remain large and far from the competitive equilibrium even under fine discretizations. To bridge this gap, we introduce the notion of Bayesian semicoarse correlated equilibrium (BSCCE), an equilibrium concept that refines BCCE by restricting deviations to probability-preserving transformations consistent with projected gradient dynamics. We provide a linear programming characterization of BSCCE with polynomially many variables and constraints and establish the inclusion hierarchy $BCE \subseteq BSCCE \subseteq BCCE$. Our main finding is that the BSCCE set contracts in many cases as the discretization of the Bayesian game is refined. Even with priors for which BCCE fails to concentrate, the diameter of the BSCCE polytope shrinks toward zero, indicating convergence to the unique continuous-game Bayes-Nash equilibrium. These results provide a principled explanation for the observed robustness of gradient-based learning in discretized Bayesian games, showing that these dynamics select a smaller equilibrium set than standard no-regret theory predicts.

cs.GT↗

arXiv2026

How language model agents reshape price competition

Repeated Bertrand competition is a stringent benchmark for algorithmic pricing: in homogeneous-goods markets, standard learning procedures typically select the stage-game Nash outcome with prices near marginal cost. We show that LLM pricing agents can escape this benchmark, depending on deployment decisions. LLMs outperform standard online learners through faster adaptation, but this advantage largely disappears against full-information learners. An extracted surrogate algorithm illustrates how LLMs in specific deployments profit from regularly undercutting slower learners. Agents that post one price every period from raw market histories that include rival prices can sustain supracompetitive regimes in self-play. Agents that periodically emit probability distributions from summarized histories of their own profits instead move toward the competitive region in self-play. Markets mixing these interfaces can generate dynamic regimes in which LLMs establish a high-price anchor and regularly undercut it to capture demand. These regimes can keep both competitors' prices above marginal cost. Deployment choices shape these outcomes, shifting the focus from stationary equilibria to dynamic pricing regimes.

cs.GT↗

arXiv2026

Stability Beyond Minty: Numerical Lyapunov Certification for Equilibrium Learning in Bayesian Bertrand Competition

Computing Bayesian Nash equilibria with continuous types and actions requires solving for strategy functions. Learning methods provide a flexible approach, but standard variational conditions often fail to explain their convergence. We study symmetric Bayesian Bertrand competition with private marginal costs and piecewise-linear pricing strategies. We prove that Minty stability fails arbitrarily close to equilibrium in the continuous-strategy game. It also fails in sufficiently fine piecewise-linear approximations when sufficiently flat pricing segments are permitted. We develop a sum-of-squares semidefinite formulation for global Lyapunov functions of the projected Riemannian dynamics. Feasibility of the stated positive-rate conditions implies global asymptotic stability. A direct proximal Lyapunov argument yields almost-sure last-iterate convergence under persistent martingale noise and square-summable step sizes; a separate tracking result covers exact and vanishing-error gradients with more general diminishing steps. The computational design covers 243 systems with two to four strategy segments, varying firms, cost distributions, demand, and learning geometry. The search returns solver-feasible SOS representations throughout, including 111 systems with explicitly verified Minty failure. Of these 243 outputs, 237 have a positive requested decrease rate; six have a rate rounded to zero and are assessed separately. Numerical-certificate acceptance additionally requires reconstruction and Lyapunov checks at stated tolerances. Euclidean candidates are quadratic throughout, entropic candidates are almost always quadratic, and Burg candidates sometimes have higher degree. The results demonstrate the computational reach of the Lyapunov approach beyond Minty and reveal substantial differences in search complexity across learning geometries.

cs.GT↗

arXiv2026

The Complexity of Multiplayer Colonel Blotto Games with Player-Specific Values

We study equilibrium computation in discrete multiplayer Colonel Blotto games with player-specific battlefield values. In the two-player model with common battlefield values, equilibria can be computed in polynomial time. We show that this tractability breaks down in the multiplayer model with player-specific values under the standard uniform tie-breaking rule. In particular, computing a $(c/n)$-approximate Nash equilibrium is PPAD-hard for some constant $c>0$, even when every player has three resources, where $n$ is the number of players. The main technical step is PPAD-hardness for computing a constant-approximate well-supported Nash equilibrium. In contrast, under uniform tie-breaking, a pure Nash equilibrium can be computed in polynomial time when every player has one resource. We also prove PPAD membership for computing $\varepsilon$-approximate Nash equilibria for inverse-exponentially small $\varepsilon$. Finally, for non-uniform monotone tie-breaking, we show PPAD-hardness even when every player has one resource and all players have identical battlefield values.

cs.GT↗

arXiv2026

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↗

arXiv2026

Distributionally Robust Contract Design with Deferred Inspection

We study a robust contract design problem with deferred inspection, in which a principal allocates a scarce resource to an agent, observes the agent's realized outcome ex post at negligible cost, and conditions transfers on this information through rewards. The principal faces ambiguity about the agent's value distribution and seeks to maximize worst-case expected revenue subject to incentive compatibility and limited liability. In contrast to existing work on inspection mechanisms, which relies on common-prior assumptions, we adopt a distributionally robust approach based on moment information. Our main contribution is a complete characterization of the robust contract design problem with a single agent. When the ambiguity set is defined by the first moment, we identify a robustly optimal contract with a concave allocation rule and a linear payment rule. We further show that robustness does not uniquely pin down transfers: we construct a Pareto robustly optimal contract that preserves the same allocation while extracting maximal feasible payments from all types, yielding strictly higher expected revenue under non-worst-case distributions. We also derive structural results for multi-agent extensions. For ambiguity sets defined by the first $N$ moments, we show that robust optimality requires aggregate payments to be lower bounded by a multi-dimensional polynomial of degree $N$. However, unlike the single-agent case, robust multi-agent mechanisms are substantially more complex: dominant-strategy incentive compatibility becomes necessary, simple monotone mechanisms are no longer tractable, and worst-case distributions may involve correlated types or degenerate to a Dirac distribution at the mean. These results highlight a sharp contrast between robust contract design and robust multi-agent mechanism design with inspection.

econ.TH↗

arXiv2025

Algorithmic Predation: Equilibrium Analysis in Dynamic Oligopolies with Smooth Market Sharing

Predatory pricing -- where a firm strategically lowers prices to undermine competitors -- is a contentious topic in dynamic oligopoly theory, with scholars debating practical relevance and the existence of predatory equilibria. Although finite-horizon dynamic models have long been proposed to capture the strategic intertemporal incentives of oligopolists, the existence and form of equilibrium strategies in settings that allow for firm exit (drop-outs following loss-making periods) have remained an open question. We focus on the seminal dynamic oligopoly model by Selten (1965) that introduces the subgame perfect equilibrium and analyzes smooth market sharing. Equilibrium can be derived analytically in models that do not allow for dropouts, but not in models that can lead to predatory pricing. In this paper, we leverage recent advances in deep reinforcement learning to compute and verify equilibria in finite-horizon dynamic oligopoly games. Our experiments reveal two key findings: first, state-of-the-art deep reinforcement learning algorithms reliably converge to equilibrium in both perfect- and imperfect-information oligopoly models; second, when firms face asymmetric cost structures, the resulting equilibria exhibit predatory pricing behavior. These results demonstrate that predatory pricing can emerge as a rational equilibrium strategy across a broad variety of model settings. By providing equilibrium analysis of finite-horizon dynamic oligopoly models with drop-outs, our study answers a decade-old question and offers new insights for competition authorities and regulators.

cs.GT↗

arXiv2025

Assignment Markets with Budget Constraints

This paper studies markets where a set of indivisible items is sold to bidders with quasilinear, unit-demand valuations, subject to a hard budget constraint. Without financial constraints the well-known assignment market model of Shapley and Shubik (1971) allows for a simple ascending auction format that is incentive-compatible, and strongly Pareto-optimal. However, this auction model does not capture the possibility that bidders face hard budget constraints. We design an iterative auction that depends on demand queries and an easily verifiable additional condition to maintain the properties in the presence of budget constraints. If instead this additional condition does not hold, incentive compatibility and core stability are at odds, and we cannot hope to achieve strong Pareto optimality in a simple ascending auction even with truthful bidding. Moreover, even in a complete information model where the auctioneer has access to valuations and budget constraints, the problem is NP-hard.

cs.GT↗

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

Semicoarse Correlated Equilibria and LP-Based Guarantees for Gradient Dynamics in Normal-Form Games

Projected gradient ascent is known to satisfy no-external regret as a learning algorithm. However, recent empirical work shows that projected gradient ascent often finds the Nash equilibrium in settings beyond two-player zero-sum interactions or potential games, including those where the set of coarse correlated equilibria is very large. We show that gradient ascent in fact satisfies a stronger class of linear $Φ$-regret in normal-form games; resulting in a refined solution concept which we dub semicoarse correlated equilibria. Our theoretical analysis of the discretised Bertrand competition mirrors those recently established for mean-based learning in first-price auctions. With at least two firms of lowest marginal cost, Nash equilibria emerge as the only semicoarse equilibria under concavity conditions on firm profits. In first-price auctions, the granularity of the bid space affects semicoarse equilibria, but finer granularity for lower bids also induces convergence to Nash equilibria. Unlike previous work that aims to prove convergence to a Nash equilibrium that often relies on epoch based analysis and probability theoretic machinery, our LP-based duality approach enables a simple and tractable analysis of equilibrium selection under gradient-based learning.

cs.GT↗

arXiv2025

Zonal vs. Nodal Pricing: An Analysis of Different Pricing Rules in the German Day-Ahead Market

The European electricity market is based on large pricing zones with a uniform day-ahead price. The energy transition leads to changes in supply and demand and increasing redispatch costs. In an attempt to ensure efficient market clearing and congestion management, the EU Commission has mandated the Bidding Zone Review (BZR) to reevaluate the configuration of European bidding zones. Based on a unique data set published in the context of the BZR for the target year 2025, we analyze the short-run effects of various pricing rules for the German-Luxembourgish bidding zone. We compare market clearing and pricing for different zonal models, including their generation and redispatch costs. In numerical experiments with this dataset, the differences in the average prices in different zones are low. The total costs across different configurations are similar and the reduction of standard deviations in prices is also small. This might be different with other load and generation scenarios, but the BZR data is important as it was created to make a decision about splits of the existing bidding zones. We can replicate several results from the BZR study, except the large cost savings when moving from one to two price zones in Germany and Luxembourg. In addition to the four zonal configurations analyzed in the BZR study, we compare these against a nodal pricing system. While the total cost savings after introducing zonal splits were less than 1%, nodal pricing led to savings of 5-6%. We also evaluate differences of nodal pricing rules with respect to the necessary uplift payments, which is relevant in the context of the discussion on non-uniform pricing in the EU. While the study focuses on Germany, the analysis is relevant beyond and feeds into the broader discussion about pricing rules in non-convex markets.

econ.GN↗

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↗

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.

cs.GT↗

arXiv2025

Exploring Competitive and Collusive Behaviors in Algorithmic Pricing with Deep Reinforcement Learning

Nowadays, a significant share of the business-to-consumer sector is based on online platforms like Amazon and Alibaba and uses AI for pricing strategies. This has sparked debate on whether pricing algorithms may tacitly collude to set supra-competitive prices without being explicitly designed to do so. Our study addresses these concerns by examining the risk of collusion when Reinforcement Learning (RL) algorithms are used to decide on pricing strategies in competitive markets. Prior research in this field focused on Tabular Q-learning (TQL) and led to opposing views on whether learning-based algorithms can result in supra-competitive prices. Building on this, our work contributes to this ongoing discussion by providing a more nuanced numerical study that goes beyond TQL, additionally capturing off- and on- policy Deep Reinforcement Learning (DRL) algorithms, two distinct families of DRL algorithms that recently gained attention for algorithmic pricing. We study multiple Bertrand oligopoly variants and show that algorithmic collusion depends on the algorithm used. In our experiments, we observed that TQL tends to exhibit higher collusion and price dispersion. Moreover, it suffers from instability and disparity, as agents with higher learning rates consistently achieve higher profits, and it lacks robustness in state representation, with pricing dynamics varying significantly based on information access. In contrast, DRL algorithms, such as PPO and DQN, generally converge to lower prices closer to the Nash equilibrium. Additionally, we show that when pre-trained TQL agents interact with DRL agents, the latter quickly outperforms the former, highlighting the advantages of DRL in pricing competition. Lastly, we find that competition between heterogeneous DRL algorithms, such as PPO and DQN, tends to reduce the likelihood of supra-competitive pricing.

econ.GN↗

arXiv2025

Challenges in Finding Stable Price Zones in European Electricity Markets: Aiming to Square the Circle?

The European day-ahead electricity market is split into multiple bidding zones with a uniform price. The increase in renewables leads to a growing number of interventions in the generation of energy sources and increasing redispatch costs. To ensure efficient congestion management, the EU Commission mandated a Bidding Zone Review (BZR) to reevaluate the configuration of European bidding zones. An integral part of this process was a locational marginal pricing study. Based on these prices, alternative bidding zone configurations were proposed. These bidding zones shall be stable and robust over time. For Germany, four configurations were suggested. We analyzed the proposed configurations considering different clustering algorithms and periods based on the publicly released data set in the context of the BZR, and found that the configurations do not reduce the price standard deviations within zones much, and the average prices across zones are similar. Other configurations identified based on clustering the prices lead to lower price variance but they are not geographically coherent. Independent of the clustering features and algorithms used, the resulting clusters are not stable over time. Interestingly, the effect of a split on prices would be low based on an analysis of the BZR data set.

econ.GN↗

arXiv2024

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.

cs.GT↗

arXiv2024

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.

cs.GT↗