arXiv subjects

Tomáš Kroupa

Publications and source records attributed to Tomáš Kroupa.

13 recordsLinked to original sources

arXiv2026

STON'R Converges to First-Order Nash~Equilibria of Multiplayer Games

Nonconcave games present a unique challenge, as neither pure Nash equilibria nor local Nash equilibria (LNE) are guaranteed to exist, even in zero-sum settings. Additionally, computing approximate LNE in smooth multiplayer games over bounded regions is PPAD-hard. These challenges, coupled with the inherent complexity, have driven recent research toward broader equilibrium concepts, such as min-max critical points, and first-order Nash equilibria (FONE), which correspond to solutions of specific non-monotone variational inequalities. This paper addresses general-sum multiplayer games with compact convex strategy sets and smooth, nonconcave utility functions. Daskalakis et al. introduced the STON'R algorithm for solving variational inequality problems and established convergence under smoothness assumptions. They further showed that the algorithm's limit points correspond to equilibria in specific classes of games, namely local minimax equilibria in two-player zero-sum games and Nash equilibria in smooth concave games. In this work, we extend the convergence result to multiplayer general-sum games and show that the variational inequality solutions targeted by STON'R correspond to first-order Nash equilibria (FONE), a general game-theoretic solution concept that unifies these previously studied cases. We demonstrate the effectiveness and robustness of the algorithm on various examples from recent literature.

cs.GT↗

arXiv2026

A Tractable Class of Cooperative Games Defined by Directed Networks: Unanimity Decomposition and Shapley Value

We introduce a class of cooperative games induced by weighted directed graphs. Specifically, the coalitional value combines an internal interaction term given by the induced subgraph game with an external component based on minimal incoming edges from outside the coalition. The resulting game has a convenient representation in terms of unanimity games. This representation enables closed-form polynomial-time formulas for the Shapley and Banzhaf values. We further establish that the game has a nonempty core and is totally balanced. The class of such games therefore provides an analytically and computationally tractable example of structured network- induced cooperative games in which stability-based allocations and fairness-based solution concepts do not coincide.

cs.GT↗

arXiv2024

Globally Optimal Inverse Kinematics as a Quadratic Program

We show how to compute globally optimal solutions to inverse kinematics (IK) by formulating the problem as an indefinite quadratically constrained quadratic program. Our approach makes it feasible to solve IK instances of generic redundant manipulators. We demonstrate the performance on randomly generated designs and on real-world robots with up to ten revolute joints. The same technique can be used for manipulator design by introducing kinematic parameters as variables.

cs.RO↗

arXiv2023

Catch Me If You Can: Improving Adversaries in Cyber-Security With Q-Learning Algorithms

The ongoing rise in cyberattacks and the lack of skilled professionals in the cybersecurity domain to combat these attacks show the need for automated tools capable of detecting an attack with good performance. Attackers disguise their actions and launch attacks that consist of multiple actions, which are difficult to detect. Therefore, improving defensive tools requires their calibration against a well-trained attacker. In this work, we propose a model of an attacking agent and environment and evaluate its performance using basic Q-Learning, Naive Q-learning, and DoubleQ-Learning, all of which are variants of Q-Learning. The attacking agent is trained with the goal of exfiltrating data whereby all the hosts in the network have a non-zero detection probability. Results show that the DoubleQ-Learning agent has the best overall performance rate by successfully achieving the goal in $70\%$ of the interactions.

cs.AI↗

arXiv2022

Values of Games for Information Decomposition

The information decomposition problem requires an additive decomposition of the mutual information between the input and target variables into nonnegative terms. The recently introduced solution to this problem, Information Attribution, involves the Shapley-style value measuring the influence of predictors in the coalitional game associated with the joint probability distribution of the input random vector and the target variable. Motivated by the original problem, we consider a general setting of coalitional games where the players form a boolean algebra, and the coalitions are the corresponding down-sets. This enables us to study in detail various single-valued solution concepts, called values. Namely, we focus on the classes of values that can represent very general alternatives to the solution of the information decomposition problem, such as random-order values or sharing values. We extend the axiomatic characterization of some classes of values that were known only for the standard coalitional games.

cs.IT↗

arXiv2022

A finite axiomatization of positive MV-algebras

Positive MV-algebras are the subreducts of MV-algebras with respect to the signature $\{\oplus, \odot, \lor, \land, 0, 1\}$. We provide a finite quasi-equational axiomatization for the class of such algebras.

math.LO↗

arXiv2020

On Decomposition of Solutions for Coalitional Games

A solution concept on a class of transferable utility coalitional games is a multifunction satisfying given criteria of economic rationality. Every solution associates a set of payoff allocations with a coalitional game. This general definition specializes to a number of well-known concepts such as the core, Shapley value, nucleolus etc. In this note it is shown that in many cases a solution factors through a set of games whose members can be viewed as elementary building blocks for the solution. Two factoring maps have a very simply structure. The first decomposes a game into its elementary components and the second one combines the output of the first map into the respective solution outcome. The decomposition is then studied mainly for certain polyhedral cones of zero-normalized games.

math.CO↗

arXiv2020

Double Oracle Algorithm for Computing Equilibria in Continuous Games

Many efficient algorithms have been designed to recover Nash equilibria of various classes of finite games. Special classes of continuous games with infinite strategy spaces, such as polynomial games, can be solved by semidefinite programming. In general, however, continuous games are not directly amenable to computational procedures. In this contribution, we develop an iterative strategy generation technique for finding a Nash equilibrium in a whole class of continuous two-person zero-sum games with compact strategy sets. The procedure, which is called the double oracle algorithm, has been successfully applied to large finite games in the past. We prove the convergence of the double oracle algorithm to a Nash equilibrium. Moreover, the algorithm is guaranteed to recover an approximate equilibrium in finitely-many steps. Our numerical experiments show that it outperforms fictitious play on several examples of games appearing in the literature. In particular, we provide a detailed analysis of experiments with a version of the continuous Colonel Blotto game.

cs.GT↗

arXiv2020

The two-sorted algebraic theory of states, and the universal states of MV-algebras

We introduce a two-sorted algebraic theory whose models are states of MV-algebras and, to within a categorical equivalence that extends Mundici's well-known one, states of Abelian lattice-groups with (strong order) unit. We discuss free states, and their relation to the universal state of an~MV-algebra. We clarify the relationship of such universal states with the theory of affine representations of lattice-groups. Main result: The universal state of any locally finite MV-algebra---in particular, of any Boolean algebra---has semisimple codomain.

math.LO↗

arXiv2019

Facets of the Cone of Totally Balanced Games

The class of totally balanced games is a class of transferable-utility coalitional games providing important models of cooperative behavior used in mathematical economics. They coincide with market games of Shapley and Shubik and every totally balanced game is also representable as the minimum of a finite set of additive games. In this paper we characterize the polyhedral cone of totally balanced games by describing its facets. Our main result is that there is a correspondence between facet-defining inequalities for the cone and the class of special balanced systems of coalitions, the so-called irreducible min-balanced systems. Our method is based on refining the notion of balancedness introduced by Shapley. We also formulate a conjecture about what are the facets of the cone of exact games, which addresses an open problem appearing in the literature.

math.CO↗

arXiv2018

The cone of supermodular games on finite distributive lattices

In this article we study supermodular functions on finite distributive lattices. Relaxing the assumption that the domain is a powerset of a finite set, we focus on geometrical properties of the polyhedral cone of such functions. Specifically, we generalize the criterion for extremality and study the face lattice of the supermodular cone. An explicit description of facets by the corresponding tight linear inequalities is provided.

math.CO↗

arXiv2016

Representing Strategic Games and Their Equilibria in Many-Valued Logics

We introduce the notion of logical A-games for a fairly general class of algebras A of real truth-values. This concept generalizes the Boolean games of Harrenstein et al. as well as the recently defined Lukasiewicz games of Marchioni and Wooldridge. We demonstrate that a wide range of strategic n-player games can be represented as logical A-games. Moreover we show how to construct, under rather general conditions, propositional formulas in the language of A that correspond to pure and mixed Nash equilibria of logical A-games.

math.LO↗

arXiv2015

Modal Extensions of Łukasiewicz Logic for Modeling Coalitional Power

Modal logics for reasoning about the power of coalitions capture the notion of effectivity functions associated with game forms. The main goal of coalition logics is to provide formal tools for modeling the dynamics of a game frame whose states may correspond to different game forms. The two classes of effectivity functions studied are the families of playable and truly playable effectivity functions, respectively. In this paper we generalize the concept of effectivity function beyond the yes/no truth scale. This enables us to describe the situations in which the coalitions assess their effectivity in degrees, based on functions over the outcomes taking values in a finite Łukasiewicz chain. Then we introduce two modal extensions of Łukasiewicz finite-valued logic together with many-valued neighborhood semantics in order to encode the properties of many-valued effectivity functions associated with game forms. As our main results we prove completeness theorems for the two newly introduced modal logics.

math.LO↗