arXiv subjects

Maximilian Fichtl

Publications and source records attributed to Maximilian Fichtl.

7 recordsLinked to original sources

arXiv2026

Threshold Dynamics and Correlated Prophet Inequalities

Prophet inequalities have become a central tool for analyzing the performance of online algorithms. However, most existing results assume that input random variables are independent, which limits their applicability. Motivated by this gap, we study prophet inequalities under two correlation models induced by a latent state of the world variable $Z$. In the common-base model, the algorithm observes the sequence $Z+X_1,\dots,Z+X_n$. We analyze single-threshold algorithms with the constraint that they always accept the final item, guaranteeing a reward of at least $Z$. When $Z$ is chosen adversarially, we characterize the optimal deterministic algorithm of this form, achieving a competitive ratio of $0.381$. We then show that randomizing improves the guarantee to $0.4$. By a minimax argument, the same ratio is achievable when $Z$ is random. We depart from standard techniques by establishing a stronger lower bound of $0.41$ and an upper bound of $0.475$, ruling out the possibility that this class of algorithms attains the $1/2$ ratio known for independent inputs. The core technical contribution is a new analytical framework that captures the reward dynamics of single-threshold algorithms. We introduce a differential equation characterizing the expected reward of a threshold in the worst-case instance, parameterized by the distribution of the maximum. This equation admits a closed-form and unifies known single-threshold prophet inequalities, yielding a simple threshold-optimality condition applicable to the common-base model. Finally, we study the common-scale model, where inputs take the form $Z\cdot X_1,\dots,Z\cdot X_n$. We show that this minimal multiplicative correlation yields strong impossibility results: no algorithm can achieve a competitive ratio exceeding $1/n$.

cs.GT↗

arXiv2026

Calibeating for general proper losses: A Bregman divergence approach

This work introduces a general framework for calibeating based on regret minimization. As compared to Foster and Hart's seminal calibeating work which had specialized treatments of Brier score (squared loss) and log loss, we consider a large family of proper losses that includes $\alpha$-Tsallis losses (for $\alpha \in [1, 2]$) and Lipschitz losses. Our results for Tsallis losses also hold for an unscaled version of Tsallis loss that recovers log loss. Our analysis is oriented around the Bregman divergence view of a proper loss. Technically, our results for the family of Tsallis losses that we consider are U-calibration results, simultaneously obtaining logarithmic regret for all losses in this family while having a weaker dependence on the dimension compared to previous results. Of potential independent interest, we also show a new regret equality for the regret of Be The Regularized Leader. This regret equality holds for general proper losses and itself is based on two results related to online updating formulas for the generalized variance, the latter being a previously introduced generalization of variance based on Bregman divergences.

cs.LG↗

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↗

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.

cs.GT↗

arXiv2022

Computing Candidate Prices in Budget-Constrained Product-Mix Auctions

We study the problem of computing optimal prices for a version of the Product-Mix auction with budget constraints. In contrast to the ``standard'' Product-Mix auction, the objective is to maximize revenue instead of social welfare. We prove correctness of an algorithm proposed by Paul Klemperer and DotEcon which is sufficiently efficient in smaller markets.

cs.GT↗

arXiv2021

On the Expressiveness of Assignment Messages

In this note we prove that the class of valuation functions representable via integral assignment messages is a proper subset of strong substitutes valuations. Thus, there are strong substitutes valuations not expressible via assignment messages.

cs.GT↗

arXiv2021

Equilibrium Learning in Combinatorial Auctions: Computing Approximate Bayesian Nash Equilibria via Pseudogradient Dynamics

Applications of combinatorial auctions (CA) as market mechanisms are prevalent in practice, yet their Bayesian Nash equilibria (BNE) remain poorly understood. Analytical solutions are known only for a few cases where the problem can be reformulated as a tractable partial differential equation (PDE). In the general case, finding BNE is known to be computationally hard. Previous work on numerical computation of BNE in auctions has relied either on solving such PDEs explicitly, calculating pointwise best-responses in strategy space, or iteratively solving restricted subgames. In this study, we present a generic yet scalable alternative multi-agent equilibrium learning method that represents strategies as neural networks and applies policy iteration based on gradient dynamics in self-play. Most auctions are ex-post nondifferentiable, so gradients may be unavailable or misleading, and we rely on suitable pseudogradient estimates instead. Although it is well-known that gradient dynamics cannot guarantee convergence to NE in general, we observe fast and robust convergence to approximate BNE in a wide variety of auctions and present a sufficient condition for convergence

cs.GT↗