This paper presents scalable algorithms for computing pure Nash equilibria (PNEs) in large-scale integer programming games (IPGs), where existing exact methods typically handle only small numbers of players. Motivated by a county-level aquatic invasive species (AIS) prevention problem with 84 decision makers, we develop and analyze random-restart best-response dynamics (RR-BRD), a randomized search framework for PNEs. For IPGs with finite action sets, we model RR-BRD as a Markov chain on the best-response state graph and show that, whenever a PNE exists and the restart law has positive probability of reaching a PNE within the round cap, RR-BRD finds a PNE almost surely. We also propose a Monte Carlo sampling-and-simulation procedure to estimate success behavior under a fixed round cap, which informs our instance-dependent performance characterization. We then embed RR-BRD as a randomized local-search subroutine within the zero-regret (ZR) framework, yielding BRD-incorporated zero-regret (BZR). Using solver callbacks, RR-BRD searches for and supplies PNEs, while ZR separates and adds equilibrium inequalities to tighten the formulation. We introduce edge-weighted budgeted maximum coverage (EBMC) games to model AIS prevention and establish PNE existence results for both selfish and locally altruistic utilities. Computational experiments on synthetic EBMC and knapsack problem game instances show that RR-BRD and BZR scale equilibrium computation up to $n \le 30$ players. We further solve a real-world EBMC game derived from the Minnesota AIS dataset with $n = 84$ county players.
We present ZERO, a modular and extensible C++ library interfacing Mathematical Programming and Game Theory. ZERO provides a comprehensive toolkit of modeling interfaces and algorithms for Reciprocally Bilinear Games (RBGs), i.e., simultaneous non-cooperative games where each player solves a mathematical program with a linear objective in the player's variable and bilinear in its opponents' variables. This class of games generalizes the classical problems of Operations Research to a multi-agent setting. ZERO modular structure gives users all the elementary ingredients to design new game-theoret…
Designing efficient algorithms to compute Nash equilibria poses considerable challenges in Algorithmic Game Theory and Optimization. In this work, we employ integer programming techniques to compute Nash equilibria in Integer Programming Games, a class of simultaneous and non-cooperative games where each player solves a parametrized integer program. We introduce ZERO Regrets, a general and efficient cutting plane algorithm to compute, enumerate, and select Nash equilibria. Our framework leverages the concept of equilibrium inequality, an inequality valid for any Nash equilibrium, and the assoc…
This paper uses value functions to characterize the pure-strategy subgame-perfect equilibria of an arbitrary, possibly infinite-horizon game. It specifies the game's extensive form as a pentaform (Streufert 2023p, arXiv:2107.10801v4), which is a set of quintuples formalizing the abstract relationships between nodes, actions, players, and situations (situations generalize information sets). Because a pentaform is a set, this paper can explicitly partition the game form into piece forms, each of which starts at a (Selten) subroot and contains all subsequent nodes except those that follow a subse…
We explore the power of semidefinite programming (SDP) for finding additive $epsilon$-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium (NE) problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact NE can be recovered. We show that for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on iterative linearization of smooth nonconvex objective functions …
Computing Equilibria in Integer Programming Games with Shared Constraints
Chang’an University（长安大学）；Grado Department of Industrial and Systems Engineering, Virginia Tech（弗吉尼亚理工大学格雷多工业与系统工程系）；Department of Industrial and Systems Engineering, University of Wisconsin–Madison（威斯康星大学麦迪逊分校工业与系统工程系）
This paper defines a general class of cooperative games for which the nucleolus is efficiently computable. This class includes new members for which the complexity of computing their nucleolus was not previously known. We show that when the minimum excess coalition problem of a cooperative game can be formulated as a hypergraph dynamic program its nucleolus is efficiently computable. This gives a general technique for designing efficient algorithms for computing the nucleolus of a cooperative game. This technique is inspired by a recent result of Pashkovich (2018) on weighted voting games. How…
Simple stochastic games are turn-based 2.5-player zero-sum graph games with a reachability objective. The problem is to compute the winning probability as well as the optimal strategies of both players. In this paper, we compare the three known classes of algorithms -- value iteration, strategy iteration and quadratic programming -- both theoretically and practically. Further, we suggest several improvements for all algorithms, including the first approach based on quadratic programming that avoids transforming the stochastic game to a stopping one. Our extensive experiments show that these im…
Basic programming and algorithm learning is one of the compulsory subjects required for students majoring in computers. As this lesson is knowledge base, it is very important and essential that before learn programmings languages students must be encourages to learn it to avoid difficulties that by using the algorithm learning games application with Near Isometric Projection, Students or prospective students become more interested in learning algorithms and programming. In this study, basic learning algorithms focused on the material Sequencing, Overloading, Procedures, Recursive Loops and Con…
Simple stochastic games are turn-based 2.5-player zero-sum graph games with a reachability objective. The problem is to compute the winning probability as well as the optimal strategies of both players. In this paper, we compare the three known classes of algorithms -- value iteration, strategy iteration and quadratic programming -- both theoretically and practically. Further, we suggest several improvements for all algorithms, including the first approach based on quadratic programming that avoids transforming the stochastic game to a stopping one. Our extensive experiments show that these im…
Matter, especially DNA, is now programmed to carry out useful processes at the nanoscale. As these programs and processes become more complex and their envisioned safety-critical applications approach deployment, it is essential to develop methods for engineering trustworthiness into molecular programs. Some of this can be achieved by adapting existing software engineering methods, but molecular programming also presents new challenges that will require new methods. This paper presents a method for dealing with one such challenge, namely, the difficulty of ascertaining how robust a molecular p…
Cut games are among the most fundamental strategic games in algorithmic game theory. It is well-known that computing an exact pure Nash equilibrium in these games is PLS-hard, so research has focused on computing approximate equilibria. We present a polynomial-time algorithm that computes $2.7371$-approximate pure Nash equilibria in cut games. This is the first improvement to the previously best-known bound of $3$, due to the work of Bhalgat, Chakraborty, and Khanna from EC 2010. Our algorithm is based on a general recipe proposed by Caragiannis, Fanelli, Gravin, and Skopalik from FOCS 2011 an…
On Dynamic Programming Theory for Leader-Follower Stochastic Games
University of Groningen（格罗宁根大学）；ENS de Lyon（里昂高等师范学院）；CNRS（法国国家科学研究中心）；University Claude Bernard Lyon 1（里昂第一大学克洛德·贝尔纳大学）；Inria（法国国家信息与自动化研究所）；Univ Rennes（雷恩大学）
This paper addresses one of the most challenging issues in designing an efficient and sustainable ridesharing service: ridesharing market design. We formulate it as a fair cost allocation problem through the lens of the cooperative game theory. A special property of the cooperative ridesharing game is that its characteristic function values are calculated by solving an optimization problem. Several concepts of fairness are investigated and special attention is paid to a solution concept named nucleolus, which aims to minimize the maximum dissatisfaction in the system. Due to its computational …
This document focuses on modeling a complex situations to achieve an advantage within a competitive context. Our goal is to devise the characteristics of games to teach and exercise non-easily quantifiable tasks crucial to the math-modeling process. A computerized game to exercise the math-modeling process and optimization problem formulation is introduced. The game is named The Formula 1 Championship, and models of the game were developed in the computerized simulation platform MoNet. It resembles some situations in which team managers must make crucial decisions to enhance their racing cars …
We study the problem of finding equilibrium strategies in multi-agent games with incomplete payoff information, where the payoff matrices are only known to the players up to some bounded uncertainty sets. In such games, an ex-post equilibrium characterizes equilibrium strategies that are robust to the payoff uncertainty. When the game is one-shot, we show that in zero-sum polymatrix games, an ex-post equilibrium can be computed efficiently using linear programming. We further extend the notion of ex-post equilibrium to stochastic games, where the game is played repeatedly in a sequence of stag…
University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Univ. Grenoble Alpes（格勒诺布尔大学）；INRIA（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；LIG（格勒诺布尔信息与信号处理实验室）；Max-Planck Institute for Software Systems (MPI-SWS)（马克斯·普朗克软件系统研究所）；University of California, Berkeley（加利福尼亚大学伯克利分校）
We describe an approximate dynamic programming (ADP) approach to compute approximations of the optimal strategies and of the minimal losses that can be guaranteed in discounted repeated games with vector-valued losses. Such games prominently arise in the analysis of regret in repeated decision-making in adversarial environments, also known as adversarial online learning. At the core of our approach is a characterization of the lower Pareto frontier of the set of expected losses that a player can guarantee in these games as the unique fixed point of a set-valued dynamic programming operator. Wh…
