arXiv subjects

Jakub Černý

Publications and source records attributed to Jakub Černý.

14 recordsLinked to original sources

arXiv2026

Active Value Querying to Minimize Additive Error in Subadditive Set Function Learning

Subadditive set functions play a pivotal role in computational economics (especially in combinatorial auctions), combinatorial optimization or artificial intelligence applications such as interpretable machine learning. However, specifying a set function requires assigning values to an exponentially large number of subsets in general, a task that is often resource-intensive in practice, particularly when the values derive from external sources such as retraining of machine learning models. A~simple omission of certain values introduces ambiguity that becomes even more significant when the incomplete set function has to be further optimized over. Motivated by the well-known result about inapproximability of subadditive functions using deterministic value queries with respect to a multiplicative error, we study a problem of approximating an unknown subadditive (or a subclass of thereof) set function with respect to an additive error -- i. e., we aim to efficiently close the distance between minimal and maximal completions. Our contributions are threefold: (i) a thorough exploration of minimal and maximal completions of different classes of set functions with missing values and an analysis of their resulting distance; (ii) the development of methods to minimize this distance over classes of set functions with a known prior, achieved by disclosing values of additional subsets in both offline and online manner; and (iii) empirical demonstrations of the algorithms' performance in practical scenarios.

cs.LG↗

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

Market Equilibria With Buying Rights

We embed buying rights into a (repeated) Arrow-Debreu model to study the long-term effects of regulation through buying rights on arising inequality. Our motivation stems from situations that typically call for regulatory interventions, such as rationing, namely, distribution crises in which demand and supply are persistently misaligned. In such settings, scarce resources tend to become increasingly concentrated among more affluent individuals, while the needs of the broader population remain unmet. While fully centralized distribution may be logistically or politically unfeasible, issuing buying rights offers a more practical alternative: they can be implemented digitally, e.g., via tokens traded on online platforms, making them significantly easier to administer. We model a scenario in which a regulator periodically distributes buying rights with the aim of promoting a more equitable allocation. Our contributions include (i) the definition of the (iterated) market where in each round the buying rights are distributed and then traded alongside the resource, (ii) the approximation algorithm of the market-clearing prices in every round, and (iii) the upper bound on \textit{frustration} -- a notion conceptually similar to the Price of Anarchy, but for systems regulated through buying rights, defined as the arising loss in fairness the individual buyers have to take when the distribution is handled via the market.

cs.GT↗

arXiv2025

Contested Route Planning

We consider the problem of routing for logistics purposes, in a contested environment where an adversary attempts to disrupt the vehicle along the chosen route. We construct a game-theoretic model that captures the problem of optimal routing in such an environment. While basic robust deterministic routing plans are already challenging to devise, they tend to be predictable, which can limit their effectiveness. By introducing calculated randomness via modeling the route planning process as a two-player zero-sum game, we compute immediately deployable plans that are diversified and harder to anticipate. Although solving the game exactly is intractable in theory, our use of the double-oracle framework enables us to achieve computation times on the order of seconds, making the approach operationally viable. In particular, the framework is modular enough to accommodate specialized routing algorithms as oracles. We evaluate our method on real-world scenarios, showing that it scales effectively to realistic problem sizes and significantly benefits from explicitly modeling the adversary's capabilities, as demonstrated through ablation studies and comparisons with baseline approaches.

cs.GT↗

arXiv2025

Plausible Counterfactual Explanations of Recommendations

Explanations play a variety of roles in various recommender systems, from a legally mandated afterthought, through an integral element of user experience, to a key to persuasiveness. A natural and useful form of an explanation is the Counterfactual Explanation (CE). We present a method for generating highly plausible CEs in recommender systems and evaluate it both numerically and with a user study.

cs.LG↗

arXiv2025

GUARD: Constructing Realistic Two-Player Matrix and Security Games for Benchmarking Game-Theoretic Algorithms

Game-theoretic algorithms are commonly benchmarked on recreational games, classical constructs from economic theory such as congestion and dispersion games, or entirely random game instances. While the past two decades have seen the rise of security games -- grounded in real-world scenarios like patrolling and infrastructure protection -- their practical evaluation has been hindered by limited access to the datasets used to generate them. In particular, although the structural components of these games (e.g., patrol paths derived from maps) can be replicated, the critical data defining target values -- central to utility modeling -- remain inaccessible. In this paper, we introduce a flexible framework that leverages open-access datasets to generate realistic matrix and security game instances. These include animal movement data for modeling anti-poaching scenarios and demographic and infrastructure data for infrastructure protection. Our framework allows users to customize utility functions and game parameters, while also offering a suite of preconfigured instances. We provide theoretical results highlighting the degeneracy and limitations of benchmarking on random games, and empirically compare our generated games against random baselines across a variety of standard algorithms for computing Nash and Stackelberg equilibria, including linear programming, incremental strategy generation, and self-play with no-regret learners.

cs.GT↗

arXiv2025

Distribution through Repeated Market with Buying Rights

Resource distribution is a fundamental problem in economic and policy design, particularly when demand and supply are not naturally aligned. Without regulation, wealthier individuals may monopolize this resource, leaving the needs of others unsatisfied. While centralized distribution can ensure fairer division, it can struggle to manage logistics efficiently, and adapt to changing conditions, often leading to shortages, surpluses, and bureaucratic inefficiencies. Building on previous research on market-based redistribution, we examine a repeated hybrid market that incorporates buying rights. These rights, distributed iteratively by a central authority (for instance, as digital tokens), are intended to enhance fairness in the system - a unit of right is required to acquire a unit of the resource, but the rights themselves can also be traded alongside the resource in the market. We analyze how this regulatory mechanism influences the distribution of the scarce resource in the hybrid market over time. Unlike past works that relied on empirical methods, we explore the exact analytical properties of a system in which traders optimize over multiple rounds. We identify its market equilibrium, which is a natural generalization of the free market equilibrium, and show that it is coalition-proof. To assess the fairness in the system, we use the concept of frustration, which measures the gap between the resources a buyer is entitled to through their buying rights and what they actually obtain through trading. Our main theoretical result shows that using buying rights reduces the frustration by at least half compared to the free market. Empirical evaluations further support our findings, suggesting the system performs well even beyond the theoretically studied assumptions.

cs.GT↗

arXiv2024

Commitment to Sparse Strategies in Two-Player Games

While Nash equilibria are guaranteed to exist, they may exhibit dense support, making them difficult to understand and execute in some applications. In this paper, we study $k$-sparse commitments in games where one player is restricted to mixed strategies with support size at most $k$. Finding $k$-sparse commitments is known to be computationally hard. We start by showing several structural properties of $k$-sparse solutions, including that the optimal support may vary dramatically as $k$ increases. These results suggest that naive greedy or double-oracle-based approaches are unlikely to yield practical algorithms. We then develop a simple approach based on mixed integer linear programs (MILPs) for zero-sum games, general-sum Stackelberg games, and various forms of structured sparsity. We also propose practical algorithms for cases where one or both players have large (i.e., practically innumerable) action sets, utilizing a combination of MILPs and incremental strategy generation. We evaluate our methods on synthetic and real-world scenarios based on security applications. In both settings, we observe that even for small support sizes, we can obtain more than $90\%$ of the true Nash value while maintaining a reasonable runtime, demonstrating the significance of our formulation and algorithms.

cs.GT↗

arXiv2024

Reducing Optimism Bias in Incomplete Cooperative Games

Cooperative game theory has diverse applications in contemporary artificial intelligence, including domains like interpretable machine learning, resource allocation, and collaborative decision-making. However, specifying a cooperative game entails assigning values to exponentially many coalitions, and obtaining even a single value can be resource-intensive in practice. Yet simply leaving certain coalition values undisclosed introduces ambiguity regarding individual contributions to the collective grand coalition. This ambiguity often leads to players holding overly optimistic expectations, stemming from either inherent biases or strategic considerations, frequently resulting in collective claims exceeding the actual grand coalition value. In this paper, we present a framework aimed at optimizing the sequence for revealing coalition values, with the overarching goal of efficiently closing the gap between players' expectations and achievable outcomes in cooperative games. Our contributions are threefold: (i) we study the individual players' optimistic completions of games with missing coalition values along with the arising gap, and investigate its analytical characteristics that facilitate more efficient optimization; (ii) we develop methods to minimize this gap over classes of games with a known prior by disclosing values of additional coalitions in both offline and online fashion; and (iii) we empirically demonstrate the algorithms' performance in practical scenarios, together with an investigation into the typical order of revealing coalition values.

cs.GT↗

arXiv2024

Layered Graph Security Games

Security games model strategic interactions in adversarial real-world applications. Such applications often involve extremely large but highly structured strategy sets (e.g., selecting a distribution over all patrol routes in a given graph). In this paper, we represent each player's strategy space using a layered graph whose paths represent an exponentially large strategy space. Our formulation entails not only classic pursuit-evasion games, but also other security games, such as those modeling anti-terrorism and logistical interdiction. We study two-player zero-sum games under two distinct utility models: linear and binary utilities. We show that under linear utilities, Nash equilibrium can be computed in polynomial time, while binary utilities may lead to situations where even computing a best-response is computationally intractable. To this end, we propose a practical algorithm based on incremental strategy generation and mixed integer linear programs. We show through extensive experiments that our algorithm efficiently computes $ε$-equilibrium for many games of interest. We find that target values and graph structure often have a larger influence on running times as compared to the size of the graph per se.

cs.GT↗

arXiv2023

Price of Anarchy in a Double-Sided Critical Distribution System

Measures of allocation optimality differ significantly when distributing standard tradable goods in peaceful times and scarce resources in crises. While realistic markets offer asymptotic efficiency, they may not necessarily guarantee fair allocation desirable when distributing the critical resources. To achieve fairness, mechanisms often rely on a central authority, which may act inefficiently in times of need when swiftness and good organization are crucial. In this work, we study a hybrid trading system called Crisdis, introduced by Jedličková et al., which combines fair allocation of buying rights with a market - leveraging the best of both worlds. A frustration of a buyer in Crisdis is defined as a difference between the amount of goods they are entitled to according to the assigned buying rights and the amount of goods they are able to acquire by trading. We define a Price of Anarchy (PoA) in this system as a conceptual analogue of the original definition in the context of frustration. Our main contribution is a study of PoA in realistic complex double-sided market mechanisms for Crisdis. The performed empirical analysis suggests that in contrast to market free of governmental interventions, the PoA in our system decreases.

cs.MA↗

arXiv2021

Computing Ex Ante Coordinated Team-Maxmin Equilibria in Zero-Sum Multiplayer Extensive-Form Games

Computational game theory has many applications in the modern world in both adversarial situations and the optimization of social good. While there exist many algorithms for computing solutions in two-player interactions, finding optimal strategies in multiplayer interactions efficiently remains an open challenge. This paper focuses on computing the multiplayer Team-Maxmin Equilibrium with Coordination device (TMECor) in zero-sum extensive-form games. TMECor models scenarios when a team of players coordinates ex ante against an adversary. Such situations can be found in card games (e.g., in Bridge and Poker), when a team works together to beat a target player but communication is prohibited; and also in real world, e.g., in forest-protection operations, when coordinated groups have limited contact during interdicting illegal loggers. The existing algorithms struggle to find a TMECor efficiently because of their high computational costs. To compute a TMECor in larger games, we make the following key contributions: (1) we propose a hybrid-form strategy representation for the team, which preserves the set of equilibria; (2) we introduce a column-generation algorithm with a guaranteed finite-time convergence in the infinite strategy space based on a novel best-response oracle; (3) we develop an associated-representation technique for the exact representation of the multilinear terms in the best-response oracle; and (4) we experimentally show that our algorithm is several orders of magnitude faster than prior state-of-the-art algorithms in large games.

cs.GT↗

arXiv2020

Complexity and Algorithms for Exploiting Quantal Opponents in Large Two-Player Games

Solution concepts of traditional game theory assume entirely rational players; therefore, their ability to exploit subrational opponents is limited. One type of subrationality that describes human behavior well is the quantal response. While there exist algorithms for computing solutions against quantal opponents, they either do not scale or may provide strategies that are even worse than the entirely-rational Nash strategies. This paper aims to analyze and propose scalable algorithms for computing effective and robust strategies against a quantal opponent in normal-form and extensive-form games. Our contributions are: (1) we define two different solution concepts related to exploiting quantal opponents and analyze their properties; (2) we prove that computing these solutions is computationally hard; (3) therefore, we evaluate several heuristic approximations based on scalable counterfactual regret minimization (CFR); and (4) we identify a CFR variant that exploits the bounded opponents better than the previously used variants while being less exploitable by the worst-case perfectly-rational opponent.

cs.AI↗

arXiv2012

Improvement on the decay of crossing numbers

We prove that the crossing number of a graph decays in a continuous fashion in the following sense. For any epsilon>0 there is a delta>0 such that for a sufficiently large n, every graph G with n vertices and m > n^{1+epsilon} edges, has a subgraph G' of at most (1-delta)m edges and crossing number at least (1-epsilon)cr(G). This generalizes the result of J. Fox and Cs. Toth.

math.CO↗