Game-theoretic Problem Research Articles

Закон поражения целей и функция победы в бою (сражении, операции)

Закон поражения целей и функция победы в бою (сражении, операции) Having taken the law of target defeat of A. N. Kolmogorov as a basis and using the methods of probability theory, the transition from the models of application of individual and group weapons to the aggregated model of combat (battle, operation) is carried out. The victory function is one of the varieties of conflict functions, satisfies the axioms of S. Skaperdas and belongs to the class of models based on the relationship between the potentials of the parties. The scale parameter of the victory function is estimated according to the data of military statistics and substantively. It is shown that the uncertainties of the outcome of combat operations are significantly reduced when moving from the tactical level to the operational level, and from the operational level to the strategic level. The concept of a combat unit, which is basic in the theory of combat potentials, has been clarified. The justified victory function allows to analyze multi-scale combat actions from a unified position and is designed to solve the following problems: firstly, the assessment of combat capabilities of the parties in the preparation of combat actions, secondly, justification of the main elements of the commander's (commander's) plan for combat actions (determination of the directions of the main strike and concentration of the main forces as a result of solving game-theoretic problems), thirdly, justification of the composition and appearance of prospective combat formations.

Game-theoretic policy computing and simulation for blockchained buffering system via diffusion approximation

AbstractWe study 2-stage game-theoretic problem oriented 3-stage service policy computing, convolutional neural network (CNN) based algorithm design, and simulation for a blockchained buffering system with federated learning. More precisely, based on the game-theoretic problem consisting of both “win-lose” and “win-win” 2-stage competitions, we derive a 3-stage dynamical service policy via a saddle point to a zero-sum game problem and a Nash equilibrium point to a non-zero-sum game problem. This policy is concerning users-selection, dynamic pricing, and online rate resource allocation via stable digital currency for the system. The main focus is on the design and analysis of the joint 3-stage service policy for given queue/environment state dependent pricing and utility functions. The asymptotic optimality and fairness of this dynamic service policy is justified by diffusion modeling with approximation theory. A general CNN based policy computing algorithm flow chart along the line of the so-called big model framework is presented. Simulation case studies are conducted for the system with three users, where only two of the three users can be selected into the service by a zero-sum dual cost game competition policy at a time point. Then, the selected two users get into service and share the system rate service resource through a non-zero-sum dual cost game competition policy. Applications of our policy in the future blockchain based Internet (e.g., metaverse and web3.0) and supply chain finance are also briefly illustrated.

Peer-to-Peer Transactive Computation–Electricity Trading for Interconnected Virtual Power Plant Buildings

Advancements of the virtual power plant (VPP) concept have aggregated buildings as their power plants and/or service providers. This paper proposes a peer-to-peer transactive computation–electricity trading framework for multiple-building virtual power plants (BVPPs). In this framework, the interconnected BVPPs can proactively trade their available computation–electricity with each other. Multiple BVPP trading is an intractable optimization problem due to its strong computation–electricity decision-making couplings. Thus, the original problem is described as a game theoretic problem and resolved into the sequential subproblems of social computation–electricity allocation and payoff allocation. By considering the local decision-making of heterogeneous BVPPs, a fully distributed algorithm is further designed to optimize the trading problem by sharing only limited trading information. Finally, a three-BVPP system is used to verify the merits of system resource utilization and operational economy.

Game-theoretical problems for fractional-order nonstationary systems

Nonstationary fractional-order systems represent a new class of dynamic systems characterized by time-varying parameters as well as memory effect and hereditary properties. Differential game described by system of linear nonstationary differential equations of fractional order is treated in the paper. The game involves two players, one of which tries to bring the system’s trajectory to a terminal set, whereas the other strives to prevent it. Using the technique of set-valued maps and their selections, sufficient conditions for reaching the terminal set in a finite time are derived. Theoretical results are supported by a model example.

Momentum Ranking Function of Z-Numbers and its Application to Game Theory

After Zadeh introduced the concept of z-number scientists in various fields have shown keen interest in applying this concept in various applications. In applications of z-numbers, to compare two z-numbers, a ranking procedure is essential. While a few ranking functions have been already proposed in the literature there is a need to evolve some more good ranking functions. In this paper, a novel ranking function for z-numbers is proposed- "the Momentum Ranking Function"(MRF). Also, game theoretic problems where the payoff matrix elements are z-numbers are considered and the application of the momentum ranking function in such problems is demonstrated.

Controller Design for Game Theoretic Steady-State Control: An LMI Approach

We consider a set of LTI agents subject to constant external disturbances seeking to minimize coupled cost functions in steady-state, modelled as a game-theoretic problem. Using a novel framework, the Nash equilibrium seeking problem has been shown to reduce to the design of a set of decentralized stabilizing controllers. Herein, we consider two such controller designs, based off LMI approaches. The first employs a diagonal stability argument, and the second relies on H∞ design coupled with the small gain theorem. Applications to sensor networks are provided and the trade-offs between the two methods are discussed.

Adaptive learning with artificial barriers yielding Nash equilibria in general games

Abstract Artificial barriers in Learning Automata (LA) is a powerful and yet under-explored concept although it was first proposed in the 1980s. Introducing artificial non-absorbing barriers makes the LA schemes resilient to being trapped in absorbing barriers, a phenomenon which is often referred to as lock in probability leading to an exclusive choice of one action after convergence. Within the field of LA and reinforcement learning in general, there is a sacristy of theoretical works and applications of schemes with artificial barriers. In this paper, we devise a LA with artificial barriers for solving a general form of stochastic bimatrix game. Classical LA systems possess properties of absorbing barriers and they are a powerful tool in game theory and were shown to converge to game’s of Nash equilibrium under limited information. However, the stream of works in LA for solving game theoretical problems can merely solve the case where the Saddle Point of the game exists in a pure strategy and fail to reach mixed Nash equilibrium when no Saddle Point exists for a pure strategy. Furthermore, we provide experimental results that are in line with our theoretical findings.

Expensive and Ineffective Regulation – Theoretical Problems of Preventive Regulation of Money Laundering

Based on estimates, it can be assumed that proceeds from crime (‘dirty money’) that circulate in the global economy total annually approximately USD 1,900–4,800 billion. It is therefore understandable that significant investments have been made in combating money laundering. A key development target, also in the EU, has been the preventive regulation of money laundering, the first aim of which is to prevent money laundering and to promote revealing and investigating it. As a result of constant increase in preventive regulation, the costs incurred in combating money laundering have grown considerably. The volume of money laundering has not, however, been reported to have decreased. It does seem that the costs incurred from preventive regulation of money laundering do not correlate with the benefit received from the regulation. This article addresses the following research question: which theoretical factors could explain why preventive regulation does not seem to function as it should? The question is answered by analysing the operational logic of the expansion of preventive regulation, the objectives of the regulation and the game-theoretical problem involved in preventive regulation.

Game Theory–Based Parameter Tuning for Energy-Efficient Path Planning on Modern UAVs

Present-day path planning algorithms for UAVs rely on various parameters that need to be tuned at runtime to be able to plan the best possible route. For example, for a sampling-based algorithm, the number of samples plays a crucial role. The dimension of the space that is being searched to plan the path, the minimum distance for extending a path in a direction, and the minimum distance that the drone should maintain with respect to obstacles while traversing the planned path are all important variables. Along with this, we have a choice of vision algorithms, their parameters, and platforms. Finding a suitable configuration for all these parameters at runtime is very challenging because we need to solve a complicated optimization problem, and that too within tens of milliseconds. The area of theoretical exploration of the optimization problems that arise in such settings is dominated by traditional approaches that use regular nonlinear optimization techniques often enhanced with AI-based techniques such as genetic algorithms. These techniques are sadly rather slow, have convergence issues, and are typically not suitable for use at runtime. In this article, we leverage recent and promising research results that propose to solve complex optimization problems by converting them into approximately equivalent game-theoretic problems. The computed equilibrium strategies can then be mapped to the optimal values of the tunable parameters. With simulation studies in virtual worlds, we show that our solutions are 5-21% better than those produced by traditional methods, and our approach is 10× faster.

Spatial ecology, optimal control and game theoretical fishing problems.

Of paramount importance in both ecological systems and economic policies are the problems of harvesting of natural resources. A paradigmatic situation where this question is raised is that of fishing strategies. Indeed, overfishing is a well-known problem in the management of live-stocks, as being too greedy may lead to an overall dramatic depletion of the population we are harvesting. A closely related topic is that of Nash equilibria in the context of fishing policies. Namely, two players being in competition for the same pool of resources, is it possible for them to find an equilibrium situation? The goal of this paper is to provide a detailed analysis of these two queries (i.e optimal fishing strategies for single-player models and study of Nash equilibria for multiple players games) by using a basic yet instructive mathematical model, the logistic-diffusive equation. In this framework, the underlying model simply reads [Formula: see text] where K accounts for natural resources, [Formula: see text] for the density of the population that is being harvested and [Formula: see text] encodes either the single player fishing strategy or, when dealing with Nash equilibria, a combination of the fishing strategies of both players. This article consists of two main parts. The first one gives a very fine characterisation of the optimisers for the single-player game where one aims at solving [Formula: see text], under [Formula: see text] and [Formula: see text] constraints on the fishing strategies [Formula: see text]. In particular, we show that, depending on the value of these constraints, this optimal control problem may behave like a convex or, conversely, concave problem. We also provide a detailed analysis of the large diffusivity limit of this problem. In the case where two players are involved, we rather write [Formula: see text] as [Formula: see text] where [Formula: see text], the fishing strategy of the i-th player, also satisfies [Formula: see text] and [Formula: see text] constraints. Defining [Formula: see text] we aim at finding a Nash equilibrium. We prove the existence of Nash equilibria in several different regimes and investigate several related qualitative queries, for instance providing examples of the well-known tragedy of commons. Our study is completed by a variety of numerical simulations that illustrate our results and allow us to formulate open questions and conjectures.

Efficient Game-Theoretic Planning With Prediction Heuristic for Socially-Compliant Autonomous Driving

Planning under social interactions with other agents is an essential problem for autonomous driving. As the actions of the autonomous vehicle in the interactions affect and are also affected by other agents, autonomous vehicles need to efficiently infer the reaction of the other agents. Most existing approaches formulate the problem as a generalized Nash equilibrium problem solved by optimization-based methods. However, they demand too much computational resource and easily fall into the local minimum due to the non-convexity. Monte Carlo Tree Search (MCTS) successfully tackles such issues in game-theoretic problems. However, as the interaction game tree grows exponentially, the general MCTS still requires a huge amount of iterations to reach the optima. In this letter, we introduce an efficient game-theoretic trajectory planning algorithm based on general MCTS by incorporating a prediction algorithm as a heuristic. On top of it, a social-compliant reward and a Bayesian inference algorithm are designed to generate diverse driving behaviors and identify the other driver's driving preference. Results demonstrate the effectiveness of the proposed framework with datasets containing naturalistic driving behavior in highly interactive scenarios.

Rolling horizon games of resilient networks with non-uniform horizons

A two-player game-theoretic problem on resilient graphs is formulated. An attacker is capable to disable some of the edges of the network with the objective to divide the agents into clusters by emitting jamming signals while, in response, the defender recovers some of the edges by increasing the transmission power for the communication signals. We consider repeated games between the attacker and the defender where the optimal strategies for the two players are derived in a rolling horizon fashion by taking account of the sizes of the clusters. The players’ actions at each discrete-time step are constrained by their energy for transmissions of signals. We derive several theoretical results to characterize the properties of the two-player game under some specific conditions of the agents’ communication network and the players’ energy parameters. In order to investigate more general cases, we provide some numerical evaluations to show the effects of the values of horizon lengths and game periods on the players’ performance.

Optimal Mixed Strategies for Cost-Adversarial Planning Games

This paper shows that domain-independent tools from classical planning can be used to model and solve a broad class of game-theoretic problems we call Cost-Adversarial Planning Games (CAPGs). We define CAPGs as 2-player normal-form games specified by a planning task and a finite collection of cost functions. The first player (a planning agent) strives to solve a planning task optimally but has limited knowledge about its action costs. The second player (an adversary agent) controls the actual action costs. Even though CAPGs need not be zero-sum, every CAPG has an associated zero-sum game whose Nash equilibrium provides the optimal randomized strategy for the planning agent in the original CAPG. We show how to find the Nash equilibrium of the associated zero-sum game using a cost-optimal planner via the Double Oracle algorithm. To demonstrate the expressivity of CAPGs, we formalize a patrolling security game and several IPC domains as CAPGs.

Learning Automata with Artificial Reflecting Barriers in Games with Limited Information

This paper deals with the problem of solving stochastic games (which have numerous business and economic applications), using the interesting tools of Learning Automata (LA), the precursors to Reinforcement Learning (RL). Classical LA systems that possess properties of absorbing barriers, have been used as powerful tools in game theory to devise solutions that converge to the game's Nash equilibrium under limited information(Sastry, Phansalkar, and Thathachar 1994). Games with limited information are intrinsically hard because the player does not know the actions chosen of other players, neither their outcomes. The player might not be even aware of the fact that he/she is playing against an opponent.
 With the state-of-the-art, the numerous works in LA applicable for solving game theoretical problems, can merely solve the case where the game possesses a Saddle Point in a pure strategy.
 They are unable to reach mixed Nash equilibria when a Saddle Point is non-existent in pure strategies. Additionally, within the field of LA and RL in general, the theoretical and applied schemes of LA with artificial barriers are scarce, even though incorporating artificial barriers in LA has served as a powerful and yet under-explored concept, since its inception in the 1980’s. More recently, the phenomenon of introducing artificial non-absorbing barriers was pioneered, and this renders the LA schemes to be resilient to being trapped in absorbing barriers. In this paper, we devise a LA with artificial barriers for solving a general form of stochastic bimatrix games. The problem’s complexity has been augmented with the scenario that we consider games in which there is no Saddle Point. By resorting to the above-mentioned powerful concept of artificial reflecting barriers, we propose a LA that converges to an optimal mixed Nash equilibrium even though there may be no Saddle Point when a pure strategy is invoked.

Special Section on Mathematical Modeling, Analysis, and Control of Epidemics

This special section of the SIAM Journal on Control and Optimization (SICON) addresses the fundamental challenges inherent in the mathematical modeling, analysis, and control of epidemics. The ongoing COVID-19 pandemic has brought into the spotlight the critical importance of understanding complex epidemic processes. Yet the modeling, analysis, estimation, and control of these processes presents several formidable challenges. Epidemics inherently exhibit nonlinear dynamics as they spread through populations, requiring analytical techniques that can accurately capture both the transient and the steady-state aspects of the disease. Furthermore, careful attention must be paid to the resolution at which to model the epidemics, ranging from coarse mass-action models to metapopulation and individual-level models. Each of these approaches provides different insights and challenges for analysis and control. Models that operate at finer resolutions will also come with an increase in complexity, necessitating tractable abstractions ranging from mean-field models to deterministic and stochastic ODEs and PDEs. In the context of estimation of epidemics, the key challenge is to determine the important parameters of the epidemic (often in real time) based on data gathered from the affected population, in conjunction with dynamical models of the spreading process. Finally, formulating techniques to optimally control and address an ongoing epidemic (through either a centralized intervention, decentralized incentive mechanisms, or resource allocation policies) remains a critical challenge. The special section gathers contributions from the intersection of the fields of systems and control theory and the mathematical study of epidemic spread processes. A total of 16 papers represents a wide range of topics in modeling, identification, dynamic analysis, control, and optimization. Regarding modeling problems, the section covers contributions on Polya contagion networks, networked bivirus epidemic models, and age-differentiated compartmental models. For dynamic analysis, the section highlights contributions on explicit solutions and control design for simplified models, convergence and equilibria analysis, as well as the role of delays and saturations in closed loop settings. For identification and estimation problems, contributions are offered on the identifiability of model parameters and on parameter estimation using limited measurements. Most of the contributed articles focus on control design, optimization, and game theoretic problems. Centralized and distributed strategies are proposed. Actuation mechanisms include edge deletion, test allocation, optimal incentives, optimal switching between lockdown and opening the economy, screening, and curing policies. We believe this special section presents an excellent cross section of current research and hope that it will be of great interest to the broad readership of SICON. We offer our deepest thanks to all authors who submitted their work and to all reviewers who contributed their time and energy to the peer-review process. We would also like to thank the proficient and timely editorial support provided by Brian Fauth and the insightful and gracious advice that we received from the SICON Editor-in-Chief George Yin. Carolyn Beck University of Illinois at Urbana-Champaign Francesco Bullo University of California, Santa Barbara Giacomo Como Politecnico di Torino Kimon Drakopoulos University of Southern California Dang H. Nguyen University of Alabama Cameron Nowzari George Mason University Victor M. Preciado University of Pennsylvania Shreyas Sundaram Purdue University, Guest editors

Теоретико-ігрові та оптимізаційні моделі і методи підвищення безпеки кіберінфраструктур

Критична інфраструктура взаємозалежних сучасних секторів все більше покладається на кіберсистеми та кіберінфраструктури, які характеризуються зростанням ризиків їх кіберкомпонентів, у тому числі кіберфізичних підсистем. Тому кібербезпека є важливою для захисту критичної інфраструктури. Пошук економічно ефективних шляхів підвищення або підвищення безпеки кіберінфраструктури базується на оптимізаційних моделях і методах стабільності, безпеки та надійності кіберінфраструктури. Ці моделі та методи мають різні сфери застосування та різні напрямки, не обов’язково орієнтовані на стійкість кіберінфраструктури. Зростання ролі інформаційно-комунікаційних технологій вплинуло на концепцію безпеки та характер війни. Багато критичних інфраструктур (аеропорти, лікарні, нафтопроводи) стали потенційно вразливими для організованих кібератак. Сьогодні здійснення головної державної функції оборони і безпеки значною мірою залежить від успішного застосування інформаційно-комунікаційних технологій як сучасних конкурентоспроможних (кінцевих і проміжних) продуктів подвійного призначення, які використовують різні особи з різними цілями. Теорію ігор усе більше застосовують для оцінювання стратегічних взаємодій між нападниками й оборонцями у кіберпросторі. Для дослідження безпеки кіберпростору поєднуються підходи теорії ігор і моделювання. У кіберпросторі арсенал зброї будується шляхом знаходження більшої кількості уразливіших місць у захисті цілі. Вразливість — це слабкість у процедурах безпеки системи, проєкті системи чи його реалізації, а також в організації внутрішнього контролю, якими може скористатися джерело загрози. Динамічний характер вразливостей означає, що вони постійно змінюються з часом. Виявлення вразливості оборонцем знижує ефективність кіберзброї нападника, яка користується даною вразливістю, і підвищує захист цілі. Теорія ігор застосовувалася для вирішення багатьох проблем, включаючи розподіл ресурсів, безпеку мережі, кооперацію осіб. У кіберпросторі часто зустрічається гра розміщення, де нападник і оборонець приймають рішення, куди розподіляти свої відповідні ресурси. Ресурсами оборонця можуть бути інфраструктура безпеки (брандмауери), фінанси, підготовка кадрів. Наприклад, адміністратор мережі може шукати таке розміщення ресурсів, яке мінімізує ризики кібератак (нападів) і водночас витрати захисту від кібератак. Нападник має обмежені ресурси і зазнає ризику бути відстеженим і покараним. Проблему розподілу ресурсів у кіберпросторі можна сформулювати як теоретико-ігрову задачу з урахуванням поняття загального знання і проблеми невизначеної спостережуваності.

Mean-Payoff Games with ω-Regular Specifications

Multi-player mean-payoff games are a natural formalism for modelling the behaviour of concurrent and multi-agent systems with self-interested players. Players in such a game traverse a graph, while attempting to maximise a (mean-)payoff function that depends on the play generated. As with all games, the equilibria that could arise may have undesirable properties. However, as system designers, we typically wish to ensure that equilibria in such systems correspond to desirable system behaviours, for example, satisfying certain safety or liveness properties. One natural way to do this would be to specify such desirable properties using temporal logic. Unfortunately, the use of temporal logic specifications causes game theoretic verification problems to have very high computational complexity. To address this issue, we consider ω-regular specifications. These offer a concise and intuitive way of specifying system behaviours with a comparatively low computational overhead. The main results of this work are characterisation and complexity bounds for the problem of determining if there are equilibria that satisfy a given ω-regular specification in a multi-player mean-payoff game in a number of computationally relevant game-theoretic settings.

Consumer-centric decarbonization framework using Stackelberg game and Blockchain

Energy policy is too often not designed for energy consumers in a low-cost and consumer-friendly manner. This paper proposes a novel Stackelberg game and Blockchain-based framework that enables consumer-centric decarbonization by automating iterative negotiations between policy makers and consumers or generators to reduce carbon emissions. This iterative negotiation is modeled as a Stackelberg game-theoretic problem, and securely facilitated by Blockchain technologies. The policy maker formulates carbon prices and monetary compensation rates to dynamically incentivize the carbon reduction, whereas consumers and generators schedule their power profiles to minimize bills and maximize profits of generation, respectively. The negotiating agreement is yielded by reaching a Stackelberg equilibrium. The exchanged information and controlling functions are realized by using smart contracts of Blockchain technologies. Case studies of GB power systems show that the proposed framework can incentivize 9% more bill savings for consumers and 45.13% more energy generation from renewable energy sources. As a consumer-centric decarbonization framework, it can at least reduce carbon emissions by 40%.

Game-theoretic models of battle action

The main types of combined arms combat operations are offensive and defense. Using the function of victory in battle, which is an extension of the function of conflict by G. Tullock, the following game-theoretic problems have been solved. First, the extended Gross-Germeier ''attack-defense'' model, which is a special case of a more general ''offensive-defense'' model, and describing the solution by the parties of the nearest tactical tasks, is investigated. Secondly, it has been proved that in the problem of breaking through points of defense (the closest tactical task), the criteria ``breaking through the weakest point'' and ``breaking through at least one point'' are equivalent. Thirdly, in the model of resource distribution of attackers and defenders between tactical tasks (echelons), the use of two criteria: 1) the product of the probabilities of solving the nearest and subsequent tactical tasks, 2) the minimum value of the named probabilities, - gives two fundamentally different solutions. Fourthly, the results of decisions were checked for compliance with the principles of military art and the practice of battles, battles and operations.

Lattice approximations of the first-order mean field type differential games

The theory of first-order mean field type differential games examines the systems of infinitely many identical agents interacting via some external media under assumption that each agent is controlled by two players. We study the approximations of the value function of the first-order mean field type differential game using solutions of model finite-dimensional differential games. The model game appears as a mean field type continuous-time Markov game, i.e., the game theoretical problem with the infinitely many agents and dynamics of each agent determined by a controlled finite state nonlinear Markov chain. Given a supersolution (resp. subsolution) of the Hamilton–Jacobi equation for the model game, we construct a suboptimal strategy of the first (resp. second) player and evaluate the approximation accuracy using the modulus of continuity of the reward function and the distance between the original and model games. This gives the approximations of the value function of the mean field type differential game by values of the finite-dimensional differential games. Furthermore, we present the way to build a finite-dimensional differential game that approximates the original game with a given accuracy.