arXiv subjects

Christian Kroer

Publications and source records attributed to Christian Kroer.

At least 19 recordsLinked to original sources

arXiv2026

Logarithmic Regret for Alternating Mirror Descent

Alternating online mirror descent (AltOMD) is a simple learning rule for zero-sum games, which improves upon simultaneous OMD numerically and theoretically. Previous work obtained $O(T^{1/3})$ and $O(T^{1/5})$ regret bounds for AltOMD, by leveraging modified-energy-function-based analyses. However, settling whether AltOMD achieves near-constant regret has remained an open question. In this paper we resolve the question of whether AltOMD achieves near-constant regret in the affirmative. In particular, we show an $O(log(T))$ regret bound on the sum of the players' external regrets for any two-player zero-sum bilinear game with compact convex strategy sets, under mild regularity assumptions on the regularizers. We verify that our assumptions hold for the negative entropy regularizer, thereby obtaining logarithmic regret for the alternating multiplicative weights updates algorithm. The regret bound also yields an $O(log(T)/T)$ duality gap bound for the averaged strategies.

cs.LG↗

arXiv2026

Impossibility of No-Regret in Multi-Agent Bidding

We consider $T$ repeated second-price auctions across $n$ pacing bidders with i.i.d. item arrivals, and show that it is computationally impossible for all bidders to achieve no-regret simultaneously while satisfying their respective budget or return-on-investment constraints. Therefore, in the prominent internet auction systems where buyers simultaneously deploy pacing algorithms or automated bidding agents, we cannot hope to achieve joint no-regret guarantees in general. We establish a deterministic relationship that high variance in online bidding systems implies high regret. By showing that one can extract a pacing equilibrium from a low-variance system, we prove PPAD-hardness of attaining joint no-regret. For several widely used bidding algorithms, we also show evidence from dynamical systems and numerical simulations that they can indeed display high-variance behavior through cycling, even on simple market instances.

cs.GT↗

arXiv2026

Efficient Exploration for Iterative Nash Preference Optimization

Preference alignment is central to improving large language models (LLMs), but reward-based formulations can be restrictive when human preferences are non-transitive. Nash learning from human feedback (NLHF) addresses this limitation by modeling alignment as a preference game and seeking a Nash equilibrium. However, the learning-theoretic foundations of scalable NLHF remain limited: existing regret guarantees rely on explicit preference-model estimation and minimax oracles, whereas simpler iterative methods lack such guarantees. We study online iterative NLHF and identify exploration as a key obstacle. First, we show that standard iterative NLHF can incur an exponential dependence on the inverse KL-regularization parameter, demonstrating that implicit exploration through policy updates can be insufficient. We then propose Exploratory Nash Preference Optimization (ENPO), which combines a SFT-type regularization with adversarial policy exploration. ENPO eliminates this exponential dependence without requiring minimax oracles or explicit preference-model estimation. We further introduce Bonus-Explorer ENPO (BENPO), which uses additional oracles to achieve an $O(\log T)$ regret bound. Finally, we develop Direct ENPO (DENPO), a practical variant of ENPO for fine-tuning LLMs. Experiments with Llama-3-8B-Instruct demonstrate consistent improvements over the evaluated RLHF and NLHF baselines across multiple benchmarks.

cs.LG↗

arXiv2026

Computing Lindahl Equilibrium for Public Goods with and without Funding Caps

Lindahl equilibrium is a solution concept for allocating a fixed budget across several divisible public goods. It always lies in the weak core, meaning that the equilibrium allocation satisfies desirable stability and proportional fairness properties. We consider a model where agents have separable linear utility functions over the public goods, and the output assigns to each good an amount of spending, summing to at most the available budget. In the uncapped setting, each of the public goods can absorb any amount of funding. In this case, Lindahl equilibrium is known to be equivalent to maximizing Nash social welfare, and can be computed by a public-goods variant of the proportional response dynamics. We introduce a new convex programming formulation for computing this solution and show that it is related to Nash welfare maximization through double duality and reformulation. We then show that the proportional response dynamics is equivalent to running mirror descent on our new formulation. Our new formulation has similarities to Shmyrev's convex program for Fisher markets. In the capped setting, each public good has an upper bound on the amount of funding it can receive, which is a type of constraint that appears in fractional committee selection and participatory budgeting. In this setting, existence of Lindahl equilibrium was only known via fixed-point arguments. The existence of an efficient algorithm computing one has been a long-standing open question. We prove that our new convex program continues to work when the cap constraints are added, and its optimal solutions are Lindahl equilibria. Thus, we establish that approximate Lindahl equilibrium can be efficiently computed in the capped setting to any desired accuracy. Our result also implies that approximately core-stable allocations can be computed for the class of separable piecewise-linear concave (SPLC) utilities.

cs.GT↗

arXiv2026

Online Fair Allocation with Best-of-Many-Worlds Guarantees

We investigate the online fair allocation problem with sequentially arriving items under various input models, with the goal of balancing fairness and efficiency. We propose the unconstrained PACE (Pacing According to Current Estimated utility) algorithm, a parameter-free allocation dynamic that requires no prior knowledge of the input while using only integral allocations. PACE attains near-optimal convergence or approximation guarantees under stationary, stochastic-but-nonstationary, and adversarial input types, thereby achieving the first best-of-many-worlds guarantee in online fair allocation. Beyond theoretical bounds, PACE is highly simple, efficient, and decentralized, and is thus likely to perform well on a broad range of real-world inputs. Numerical results support the conclusion that PACE works well under a variety of input models. We find that PACE performs very well on two real-world datasets even under the true temporal arrivals in the data, which are highly nonstationary.

cs.GT↗

arXiv2026

Online Generalized-Mean Welfare Maximization: Achieving Near-Optimal Regret from Samples

We study online fair allocation of $T$ sequentially arriving items among $n$ agents with heterogeneous preferences, with the objective of maximizing generalized-mean welfare, defined as the $p$-mean of agents' time-averaged utilities, with $p\in (-\infty, 1)$. We first consider the i.i.d. arrival model and show that the pure greedy algorithm -- which myopically chooses the welfare-maximizing integral allocation -- achieves $\widetilde{O}(1/T)$ average regret. Importantly, in contrast to prior work, our algorithm does not require distributional knowledge and achieves the optimal regret rate using only the online samples. We then go beyond i.i.d. arrivals and investigate a nonstationary model with time-varying independent distributions. In the absence of additional data about the distributions, it is known that every online algorithm must suffer $Ω(1)$ average regret. We show that only a single historical sample from each distribution is sufficient to recover the optimal $\widetilde{O}(1/T)$ average regret rate, even in the face of arbitrary non-stationarity. Our algorithms are based on the re-solving paradigm: they assume that the remaining items will be the ones seen historically in those periods and solve the resulting welfare-maximization problem to determine the decision in every period. Finally, we also account for distribution shifts that may distort the fidelity of historical samples and show that the performance of our re-solving algorithms is robust to such shifts.

cs.GT↗

arXiv2026

Throttling Equilibria in Auction Markets

Throttling is a popular method of budget management for online ad auctions in which the platform modulates the participation probability of an advertiser in order to smoothly spend her budget across many auctions. In this work, we investigate the setting in which all of the advertisers simultaneously employ throttling to manage their budgets, and we do so for both first-price and second-price auctions. We analyze the structural and computational properties of the resulting equilibria. For first-price auctions, we show that a unique equilibrium always exists, is well-behaved and can be computed efficiently via tatonnement-style decentralized dynamics. In contrast, for second-price auctions, we prove that even though an equilibrium always exists, the problem of finding even an approximate equilibrium is PPAD-complete, there can be multiple equilibria, and it is NP-hard to find the revenue maximizing one. We also compare the equilibrium outcomes of throttling to those of multiplicative pacing, which is the other most popular and well-studied method of budget management. Finally, we characterize the Price of Anarchy of these equilibria for liquid welfare by showing that it is at most 2 for both first-price and second-price auctions, and demonstrating that our bound is tight.

cs.GT↗

arXiv2026

AltGDA Achieves Global $O(1/T)$ Ergodic Convergence in Matrix Games

Alternating gradient descent-ascent (AltGDA) is a simple and practically effective method for solving finite two-player zero-sum matrix games. However, the theory of AltGDA remains limited: existing results either apply only to unconstrained settings or require restrictive assumptions on the equilibrium in constrained settings. We show that AltGDA converges globally at an $O(1/T)$ ergodic rate in every finite two-player zero-sum matrix game. Unlike prior results, our guarantee holds for every initialization and every horizon $T$: the uniform averages of the AltGDA iterates satisfy an $O(1/T)$ duality-gap bound. Our proof is inspired by numerical results obtained using a novel performance estimation programming (PEP) framework for Lyapunov function search over compact convex sets. Additionally, we provide simple counterexamples showing that the last-iterate duality gap of AltGDA does not converge to zero. This justifies why averaging of iterates is indeed necessary to achieve an $O(1/T)$ rate. We have formalized and machine-checked our global ergodic convergence result in Lean 4.

math.OC↗

arXiv2026

Pacing Equilibria in Abstract Mechanisms

Digital platforms increasingly rely on automated budget-management systems to regulate participation across allocation opportunities. A common tool is multiplicative pacing, which scales buyers' bids so that campaign-level budgets are spent gradually. Although pacing is operationally simple, its aggregate behavior depends on the underlying mechanism. In first-price single-item auction markets, pacing enjoys strong structural and computational properties that often fail in second-price auctions. It is unclear whether these properties extend to richer platform mechanisms. We develop a unified theory of pacing equilibria across a hierarchy of mechanisms. We first show that, even in first-price position auctions, the market-equilibrium interpretation of the single-item benchmark can fail. We then identify bid-maximizing pay-your-bid mechanisms as a broad class in which pacing equilibria exist, are unique, and admit an Eisenberg-Gale-type convex-program characterization. This implies efficient computation, Pareto-efficiency, and liquid-welfare guarantees, enabling pacing integration into large-scale optimization-based allocation routines. Finally, for abstract mechanisms satisfying a payment monotonicity condition, we prove equilibrium existence by smoothing discontinuities. With additional conditions and appropriate tie-breaking, we obtain uniqueness, revenue maximality among budget-feasible pacing vectors, and shill-proof implementability, and provide convergent budget-adjustment dynamics for approximate equilibria. Overall, many favorable properties of first-price pacing extend beyond single-item auctions under appropriate mechanism-level conditions.

cs.GT↗

arXiv2026

Ex-Post Equilibria: Structure and Computation

This paper studies ex-post equilibria (EPEs) in simultaneous-move games with parameter uncertainty. We first compare EPEs with existing notions of robust equilibrium and provide a general foundation for EPEs as a solution concept for games with parameter uncertainty, and we show that EPEs are fully characterized by two game-theoretic properties (monotonicity and set-consistency). Since ex-post equilibria may fail to exist, we introduce the notion of an optimal approximate ex-post equilibrium, in which players adopt approximate best responses while minimizing the degree of suboptimality. We study the problem of computing EPEs and optimal approximate EPEs, focusing on two important classes of games: zero-sum games and concave potential games. We provide several hardness results, as well as a general class of computational approaches based on auxiliary minimax formulations.

cs.GT↗

arXiv2026

Fully Distributed Tâtonnement for Chores Markets

We study price-adjustment dynamics for computing competitive equilibria (CE) in Fisher markets with chores. Unlike in classical goods markets, prices in chores markets are payments for taking on undesirable tasks, and natural excess-demand dynamics can fail; even the naïve analogue of Walrasian tâtonnement may diverge. Recent work of Chaudhury et al. [2025] overcomes this obstacle via relative tâtonnement, which subtracts the average excess-demand signal from the excess demand vector. This recovers convergence, but at the cost of coupling the price updates across all chores. This leaves open whether such global coupling is inherent, or whether convergent tâtonnement can be recovered through a genuinely local update in which each chore reacts only to its own excess demand. We answer this question affirmatively through multiplicative tâtonnement, a fully distributed dynamics in which each chore price is updated using only its current price and its own excess-demand signal. Although the update contains no explicit normalization term, Walras' law and the multiplicative form of the update implicitly preserve the relevant aggregate price geometry. We prove that multiplicative tâtonnement converges to a CE in any chores Fisher market with continuous, convex, and $1$-homogeneous (CCH) disutilities. For convex CES disutilities, we further prove an approximate-CE convergence rate with the same $O(1/\varepsilon^2)$ dependence as relative tâtonnement, but with improved dependence on problem constants. Experiments on real-world and simulated instances show that multiplicative tâtonnement is substantially faster in practice, often by an order of magnitude.

cs.GT↗

arXiv2026

Competitive Equilibrium in Labor Economies through the Lens of Goods and Chores Fisher Markets

In this paper, we study a two-sided labor market that couples the classical Fisher market with goods and the Fisher market with bads into a single unified framework. In our model, users demand tasks in order to derive utility, while workers supply labor to perform these tasks in exchange for earnings. Each task thus plays a dual role: it is a good for the user side of the market and a chore for the worker side. Given prices for tasks, users choose utility-maximizing bundles subject to budgets, while workers choose disutility-minimizing task bundles subject to earning requirements; the resulting choices induce demand and supply endogenously for each task, and a CE corresponds to prices at which these coincide. We show that such markets are guaranteed to admit a CE in a very general setting, and the first and second welfare theorems hold for our labor market model. We next study the computation of equilibria under linear preferences. We show that, similar to the chores setting, equilibria correspond to KKT points of an Eisenberg-Gale-like non-convex program. Despite the non-convex characterization, we go on to show a set of surprisingly positive results. First, we show that there exists a polynomial-time combinatorial algorithm for computing CE, which relies on a natural Walrasian scheme for updating prices. In the "CEEI-like" case, this yields a strongly polynomial-time algorithm. We next show that our market admits a natural dual program, and this non-convex labor-market program admits a change of variables that transforms it into a linear program (albeit with irrational coefficients). Finally, leveraging this LP, we give yet another polynomial-time algorithm while deriving an approach for addressing the irrational coefficients in an efficient manner. We note that, even for goods-only linear Fisher markets, obtaining such an LP formulation remains open.

cs.GT↗

arXiv2026

On the $O(1/T)$ Convergence of Alternating Gradient Descent-Ascent in Bilinear Games

We study the alternating gradient descent-ascent (AltGDA) algorithm in two-player zero-sum games. Alternating methods, where players take turns to update their strategies, have long been recognized as simple and practical approaches for learning in games, exhibiting much better numerical performance than their simultaneous counterparts. However, our theoretical understanding of alternating algorithms remains limited, and results are mostly restricted to the unconstrained setting. We show that for two-player zero-sum games that admit an interior Nash equilibrium, AltGDA converges at an $O(1/T)$ ergodic convergence rate when employing a small constant stepsize. This is the first result showing that alternation improves over the simultaneous counterpart of GDA in the constrained setting. For games without an interior equilibrium, we show an $O(1/T)$ local convergence rate with a constant stepsize that is independent of any game-specific constants. In a more general setting, we develop a performance estimation programming (PEP) framework to jointly optimize the AltGDA stepsize along with its worst-case convergence rate. The PEP results indicate that AltGDA may achieve an $O(1/T)$ convergence rate for a finite horizon $T$, whereas its simultaneous counterpart appears limited to an $O(1/\sqrt{T})$ rate.

cs.GT↗

arXiv2025

A Direct Second-Order Method for Solving Two-Player Zero-Sum Games

We introduce, to our knowledge, the first direct second-order method for computing Nash equilibria in two-player zero-sum games. To do so, we construct a Douglas-Rachford-style splitting formulation, which we then solve with a semi-smooth Newton (SSN) method. We show that our algorithm enjoys local superlinear convergence. In order to augment the fast local behavior of our SSN method with global efficiency guarantees, we develop a hybrid method that combines our SSN method with the state-of-the-art first-order method for game solving, Predictive Regret Matching$^+$ (PRM$^+$). Our hybrid algorithm leverages the global progress provided by PRM$^+$, while achieving a local superlinear convergence rate once it switches to SSN near a Nash equilibrium. Numerical experiments on matrix games demonstrate order-of-magnitude speedups over PRM$^+$ for high-precision solutions.

cs.GT↗

arXiv2025

Tâtonnement Dynamics for Fisher Markets with Chores

In this paper, we initiate the study of tâtonnement dynamics in markets with chores. Tâtonnement is a fundamental market dynamics, capturing how prices evolve when they are adjusted in proportion of their excess demand. While its convergence to a competitive equilibrium (CE) is well understood in goods markets for broad classes of utilities, no analogous results are known for chore markets. Analyzing tâtonnement in the chores market presents new challenges. Several elegant structural properties that facilitate convergence in goods markets-such as convexity of the equilibrium price set and monotonicity of excess demand under the tâtonnement price updates-fail to hold in the chore setting. Consistent with these difficulties, we first show that naïve tâtonnement diverges. To overcome this, we propose a modified process called relative tâtonnement, where prices are updated according to normalized excess demand. We prove its convergence to a CE under suitable step-size choices for a broad class of disutility functions, namely continuous, convex, and 1-homogeneous (CCH) disutilities. This class includes many standard forms such as linear and convex CES disutilities. Our proof proceeds by introducing a nonsmooth, nonconvex yet regular objective function-a generalization of the objective in the Eisenberg-Gale-type dual program introduced by [CKMN24]. For convex CES disutilities, where disutility is the weighted $p$-norm of the individual chore disutilities for $p \in (1, \infty)$, we show that relative tâtonnement converges to an $\varepsilon$-CE in $O(1/\varepsilon^2)$ iterations. This quadratic convergence rate is established by leveraging the polar gauge (or gauge dual) of the disutility function. Finally, following the framework of [AH58], we analyze the stability of CE and provide a complete characterization of local stability.

cs.GT↗

arXiv2025

Spatial Branch-and-Bound for Computing Multiplayer Nash Equilibrium

Equilibria of realistic multiplayer games constitute a key solution concept both in practical applications, such as online advertising auctions and electricity markets, and in analytical frameworks used to study strategic voting in elections or assess policy impacts in integrated assessment models. However, efficiently computing these equilibria requires games to have a carefully designed structure and satisfy numerous restrictions; otherwise, the computational complexity becomes prohibitive. In particular, finding even approximate Nash equilibria in general-sum normal-form games with two or more players is known to be PPAD-complete. Current state-of-the-art algorithms for computing Nash equilibria in multiplayer normal-form games either suffer from poor scalability due to their reliance on non-convex optimization solvers, or lack guarantees of convergence to a true equilibrium. In this paper, we propose a formulation of the Nash equilibrium computation problem as a polynomial complementarity problem and develop a complete and sound spatial branch-and-bound algorithm based on this formulation. We provide a qualitative analysis arguing why one should expect our approach to perform well, and show the relationship between approximate solutions to our formulation and that of computing an approximate Nash equilibrium. Empirical evaluations demonstrate that our algorithm substantially outperforms existing complete methods.

cs.GT↗

arXiv2025

Colonel Blotto with Battlefield Games

We study a class of two-player zero-sum Colonel Blotto games in which, after allocating soldiers across battlefields, players engage in (possibly distinct) normal-form games on each battlefield. Per-battlefield payoffs are parameterized by the soldier allocations. This generalizes the classical Blotto setting, where outcomes depend only on relative soldier allocations. We consider both discrete and continuous allocation models and examine two types of aggregate objectives: linear aggregation and worst-case battlefield value. For each setting, we analyze the existence and computability of Nash equilibrium. The general problem is not convex-concave, which limits the applicability of standard convex optimization techniques. However, we show that in several settings it is possible to reformulate the strategy space in a way where convex-concave structure is recovered. We evaluate the proposed methods on synthetic and real-world instances inspired by security applications, suggesting that our approaches scale well in practice.

cs.GT↗

arXiv2025

The Bias-Variance Tradeoff in Data-Driven Optimization: A Local Misspecification Perspective

Data-driven stochastic optimization is ubiquitous in machine learning and operational decision-making problems. Sample average approximation (SAA) and model-based approaches such as estimate-then-optimize (ETO) or integrated estimation-optimization (IEO) are all popular, with model-based approaches being able to circumvent some of the issues with SAA in complex context-dependent problems. Yet the relative performance of these methods is poorly understood, with most results confined to the dichotomous cases of the model-based approach being either well-specified or misspecified. We develop the first results that allow for a more granular analysis of the relative performance of these methods under a local misspecification setting, which models the scenario where the model-based approach is nearly well-specified. By leveraging tools from contiguity theory in statistics, we show that there is a bias-variance tradeoff between SAA, IEO, and ETO under local misspecification, and that the relative importance of the bias and the variance depends on the degree of local misspecification. Moreover, we derive explicit expressions for the decision bias, which allows us to characterize (un)impactful misspecification directions, and provide further geometric understanding of the variance.

stat.ML↗