Existence Of Nash Equilibrium Research Articles

A back‐to‐back coordination‐based learning scheme for deceiving reactive jammers in distributed networks

AbstractReactive jammers select jamming strategies according to the users’ responses; thus, conventional anti‐jamming methods such as frequency hopping are inadequate to defeat the jamming attack. In this article, the authors propose a novel uncoupled deception scheme to trap the reactive jammer into attacking a decoy channel in distributed networks. Specifically, the authors design a multi‐functional network utility for every user to mislead the jammer with a minimum energy consumption while achieving the highest network throughput. Based on the network utility, the anti‐jamming problem is formulated as an exact potential game such that the existence of Nash equilibrium can be guaranteed theoretically. The authors further propose a back‐to‐back coordination‐based learning algorithm to reach the optimal channel selection and power adaption in a non‐cooperative way. To alleviate the lack of mutual information exchange, the back‐to‐back coordination mechanism derives all users to deceive the jammer by inferring others’ strategies based on a shared belief. Simulation results show that the proposed algorithm yields higher network throughput and efficiency‐cost ratio compared to the state‐of‐the‐art cooperative schemes.

Risk Assessment Edge Contract for Efficient Resource Allocation

The rapid growth of edge devices and mobile applications has driven the adoption of edge computing to handle computing tasks closer to end-users. However, the heterogeneity of edge devices and their limited computing resources raise challenges in the efficient allocation of computing resources to complete services with different characteristics and preferences. In this paper, we delve into an edge scenario comprising multiple Edge Computing Servers (ECSs), multiple Device-to-Device (D2D) Edge Nodes (ENs), and multiple edge devices. In order to address the resource allocation challenge among ECSs, ENs, and edge devices in high-workload environments, as well as the pricing of edge resources within the resource market framework, we propose a Risk Assessment Contract Algorithm (RACA) based on risk assessment theory. The RACA enables ECSs to assess risks associated with local users by estimating their future revenue potential and updating the contract autonomously at present and in the future. ENs acquire additional resources from ECSs to efficiently complete local users’ tasks. Simultaneously, ENs can also negotiate reasonable resource requests and pricing with ECSs by a Stackelberg game algorithm. Furthermore, we prove the unique existence of Nash equilibrium in the established game, implying that equilibrium solutions can stably converge through computational methods in heterogeneous environments. Finally, through simulation experiments on the dataset, we demonstrate that risk assessment can better enhance the overall profit capability of the system. Moreover, through multiple experiments, we showcase the stability of the contract’s autonomous update capability. The RACA exhibits better utility in terms of system profit capabilities, stability in high-workload environments, and energy consumption. This work provides a more dynamic and effective solution to the resource allocation problem in edge systems under high-workload environments.

Competition among Pairwise Lottery Contests

We investigate a two-stage competitive model involving multiple contests. In this model, each contest designer chooses two participants from a pool of candidate contestants and determines the biases. Contestants strategically distribute their efforts across various contests within their budget. We first show the existence of a pure strategy Nash equilibrium (PNE) for the contestants, and propose a fully polynomial-time approximation scheme to compute an approximate PNE. In the scenario where designers simultaneously decide the participants and biases, the subgame perfect equilibrium (SPE) may not exist. Nonetheless, when designers' decisions are made in two substages, the existence of SPE is established. In the scenario where designers can hold multiple contests, we show that the SPE always exists under mild conditions and can be computed efficiently.

Edge computing offloading strategy for space-air-ground integrated network based on game theory

The limited coverage of terrestrial networks makes it difficult to provide low-latency and high-reliable computing services for users and Internet of Thing (IoT) devices in remote areas such as mountainous regions and oceans. Space-Air-Ground Integrated Network (SAGIN) combined with Mobile Edge Computing (MEC) can provide seamless three-dimensional services for users by deploying edge servers on satellites, Unmanned Aerial Vehicles (UAVs) and ground infrastructures. However, due to the limited heterogeneous computing resources in satellites-UAV clusters network and the energy resources of IoT devices, it brings a significant challenge in determining how to offload the computing tasks generated by ground user devices to satellite edge nodes, UAV edge nodes or locally for processing. In this paper, we first propose a satellites-UAV clusters-ground three-layer edge computing network architecture consisting of a global controller, inter-domain controllers and MEC servers. We then model the task offloading problem as a Binary Integer Linear Programming (BILP) aiming at minimizing the offloading cost composed of delay and energy consumption and prove it is NP-hard. Next, the original offloading problem is transformed to a noncooperative strategic game and the existence of Nash equilibrium is proven using potential games. Finally, we propose a Nash Equilibrium Iteration Offloading algorithm based on Game theory (NEIO-G) to find the optimal offloading strategy. Compared with other baseline algorithms, simulation results demonstrate that the NEIO-G can significantly reduce the system offloading overhead in terms of delay and energy consumption.

Game of banks - biform game theoretical framework for ATM network cost sharing

Automated teller machines (ATM) play a major role in the world economy as they enable financial transactions and hence good exchanges and consumption. ATM transaction fees are incurred to cover the cost of running the network and these are often settled among the members including banks and cash machine operators. In this paper, we develop a novel biform game theoretic model for members to optimally invest in the ATM network and to share the cost. This biform game includes both a cooperative game theory mechanism for interchange fee sharing and a non-cooperative counterpart to model the fact that members also wish to maximize their utilities. While the proposed coopetition framework is applicable to general ATM networks, we focus the case study on the UK ATM network thanks to the accessibility of the data in addition to the notable stability issues that the network is currently experiencing as has been widely featured by the mainstream media. On the technical side, we prove the existence of a pure Nash equilibrium, which can be computed efficiently. We also show that, under some settings, the Shapley allocation belongs to the core and hence it is not only fair to all members but also leads to a stable ATM network. In addition, we show that the Shapley value allocation dominates the current mechanism in terms of social welfare. Finally, we provide numerical analysis and managerial insights using real data on the complete UK ATM network.

On the Hotelling Game with Close Players

The principle of minimum differentiation stated by Hotelling in his pioneer paper was proved to be invalid 50 years later, since there is a region where the players are so close that Nash equilibrium does not exist. Some authors have contributed with amendments to the original game in the aim of ensuring the existence of Nash equilibrium independently of the location of the players. However, this work analyzes the less frequently studied case of the Hotelling game with close players under different implementations of the game, considering elastic and inelastic demand (conventional Hotelling game, the Hotelling game with reservation cost and Hotelling-Smithies game). The study is supported by means of numerical simulation of the game.

Frequency Dynamics With Inverters: Proof of Stabilizability and Existence of Nash Equilibrium

Frequency Dynamics With Inverters: Proof of Stabilizability and Existence of Nash Equilibrium

Demand Side Management in Smart Grid: A Dynamic-Price-Based Demand Response Model

Demand side management (DSM) is an important way to achieve smart energy management. Herein, a dynamic price (DP)-based demand response (DR) model is developed for DSM in smart grid. The proposed DR model can shift the peak electricity demand, thereby improving the power system stability and reliability. In a district-scale smart grid with a high photovoltaic power penetration, the energy service provider (ESP) optimizes the DP to maximize its utilities and reduce the load fluctuation while minimizing the bills and dissatisfaction of electricity consumers (ECs). The game theory model is used to explore the interaction between ESP and ECs, and the existence of Nash equilibrium is proved. The proposed DR model is validated with real-world data from a commercial and residential cluster in Suzhou City, Jiangsu Province, China. The results show that the peak electricity demands of commercial and residential ECs decreased by 4.99% and 9.99%, respectively, through the proposed DR model. Meanwhile, the ESP's net profits increased by 7.13% and 2.37%, respectively, while ensuring the ECs’ benefits. The results also demonstrate that the proposed DR model is robust in different scenarios. This article contributes to the effectiveness and efficiency of energy engineering management.

Precise Tracking Control for Articulating Crane: Prescribed Performance, Adaptation, and Fuzzy Optimality by Nash Game.

Articulating crane (AC) is used in various industrial activities. The articulated multisection arm exacerbates nonlinearities and uncertainties, making the precise tracking control challenging. This study proposes an adaptive prescribed performance tracking control (APPTC) for AC to robustly fulfill the task of precise tracking control, with adaptation to resist time-variant uncertainties, whose bounds are unknown but lie in prescribed fuzzy sets. Particularly, a state transformation is applied to simultaneously track the desired trajectory and satisfy the prescribed performance. Adopting the fuzzy set theory to describe uncertainties, APPTC does not invoke any IF-THEN fuzzy rules. There is no linearizations, or nonlinear cancelation for APPTC, thus making it approximation free. The performance of the controlled AC is twofold. First, deterministic performance in fulfilling the control task is ensured by the Lyapunov analysis using uniform boundedness and uniform ultimate boundedness. Second, fuzzy-based performance is further improved by an optimal design, which seeks the optima of control parameters by formulating a two-player Nash game. The existence of Nash equilibrium is theoretically proved, and its acquisition process is given. The simulation results are provided for validations. This is the first endeavor that explores the precise tracking control for fuzzy AC.

Strategic Production and Trading in Renewable Energy Certificate Markets: Existence, Uniqueness, and Efficiency of Market Equilibria

As an incentive policy to promote renewable generation, the renewable standard portfolio (RPS) scheme has been widely adopted to enforce utility companies to procure a certain fraction of their energy supply from renewable sources. We construct a non-cooperative game theoretic model to study the strategic behavior of utility companies in a renewable energy certificate (REC) market, where utility companies purchase REC from renewable producers to meet their RPS obligation. In the constructed game, each utility company makes the renewable production decisions so as to minimize its total cost (the sum of renewable production and RPS penalty costs). We establish mild conditions that guarantee the existence of a Nash equilibrium. We further prove that if a Nash equilibrium exists, it must be unique and minimize the system cost. When symmetric utility companies' energy loads are independent and identical Gaussian distributed, as the number of utility companies increases to infinity, we show that the expected cost (of a utility company) converges to a lower bound, the product of the levelized cost of electricity (LCOE) of renewable generation and its expected RPS obligation. Numerical studies on real-world (load, RPS requirement, penalty, and LCOE) data show that the REC trading can reduce 5%-10% of the system cost.

Topological Bounds on the Price of Anarchy of Clustering Games on Networks

We consider clustering games in which the players are embedded into a network and want to coordinate (or anti-coordinate) their strategy with their neighbors. The goal of a player is to choose a strategy that maximizes her utility given the strategies of her neighbors. Recent studies show that even very basic variants of these games exhibit a large Price of Anarchy: A large inefficiency between the total utility generated in centralized outcomes and equilibrium outcomes in which players selfishly maximize their utility. Our main goal is to understand how structural properties of the network topology impact the inefficiency of these games. We derive topological bounds on the Price of Anarchy for different classes of clustering games. These topological bounds provide a more informative assessment of the inefficiency of these games than the corresponding worst-case Price of Anarchy bounds. More specifically, depending on the type of clustering game, our bounds reveal that the Price of Anarchy depends on the maximum subgraph density or the maximum degree of the graph. Among others, these bounds enable us to derive bounds on the Price of Anarchy for clustering games on Erdős-Rényi random graphs. Depending on the graph density, these bounds stand in stark contrast to the known worst-case Price of Anarchy bounds. Additionally, we characterize the set of distribution rules that guarantee the existence of a pure Nash equilibrium or the convergence of best-response dynamics. These results are of a similar spirit as the work of Gopalakrishnan et al. [ 19 ] and complement work of Anshelevich and Sekar [ 4 ].

Data Synchronization in Vehicular Digital Twin Network: A Game Theoretic Approach

A fundamental issue of the vehicular digital twin (DT) is efficiently synchronizing the data between the DT and the vehicular user (VUE). In this paper, we consider the heterogeneous vehicular networks (HetVNets) in which a VUE can connect to the network through different networks. The HetVNets can improve the efficiency of communication by providing seamless connections. However, the uneven distribution of VUEs and the dynamics of HetVNets make the environment more complex. Therefore, we propose the network selection algorithm for data synchronization between VUEs and DTs in the HetVNets, where the behaviour between the VUEs is considered as a competition for wireless resources. A learning-based prediction model residing in the DT is developed where the DT can predict the waiting time of each relay and transmit the predicted results to the VUE for decision-making. We model the network selection problem as a potential game considering both the transmission time and the waiting time obtained from the prediction model and prove the existence of Nash equilibrium (NE). We analyze the performance of the proposed algorithm, and simulation results show that our approach can effectively find the optimal strategy while achieving a fast convergence speed and high-level performance compared to the baselines.

Research on Resource Allocation of Autonomous Swarm Robots Based on Game Theory

To address the issue of resource allocation optimization in autonomous swarm robots during emergency situations, this paper abstracts the problem as a two-stage extended game. In this game, participants are categorized as either resource-providing robots or resource-consuming robots. The strategies of the resource-providing robots involve resource production and pricing, whereas the strategies of the resource-consuming robots consist of determining the quantity to be purchased based on resource pricing. In the first stage of the game, the resource-providing robots use the Cournot game to determine the resource production according to market supply and demand conditions; in the second stage of the game, the resource-providing robots and the resource-consuming robots play the price game and establish the utility function of the swarm robots to seek the optimal pricing and the optimal purchasing strategy of the swarm robots. After the mathematical derivation, this paper demonstrates the existence of a single Nash equilibrium in the constructed game. Additionally, the inverse distributed iterative search algorithm solves the game’s optimal strategy. Finally, simulation verifies the game model’s validity. This study concludes that the designed game mechanism enables both sides to reach equilibrium and achieve optimal resource allocation.

Learning to Mitigate Epidemic Risks: A Dynamic Population Game Approach

We present a dynamic population game model to capture the behavior of a large population of individuals in presence of an infectious disease or epidemic. Individuals can be in one of five possible infection states at any given time: susceptible, asymptomatic, symptomatic, recovered and unknowingly recovered, and choose whether to opt for vaccination, testing or social activity with a certain degree. We define the evolution of the proportion of agents in each epidemic state, and the notion of best response for agents that maximize long-run discounted expected reward as a function of the current state and policy. We further show the existence of a stationary Nash equilibrium and explore the transient evolution of the disease states and individual behavior under a class of evolutionary learning dynamics. Our results provide compelling insights into how individuals evaluate the trade-off among vaccination, testing and social activity under different parameter regimes, and the impact of different intervention strategies (such as restrictions on social activity) on vaccination and infection prevalence.

Stochastic bidding for VPPs enabled ancillary services: A case study

Strategic bidding which aims to optimally harvest the price difference in the wholesale electricity market can efficiently allocate VPPs’ aggregated resources to provide large flexibility for energy and frequency regulation (FR), aiding in the real-time rebalancing of supply and demand. However, the uncertainty of renewable energy results in substantial imbalances between supply and demand, which causes high randomness of system frequency deviation and significant price fluctuations in the electricity market, making bidding challenging. To tackle these issues, a risk-averse optimal bidding strategy is proposed for VPPs to participate in both energy and FR markets. Moreover, a payment recovery mechanism is designed to recover the cost of FR undersupply. Specifically, each VPP submits bids to maximize profit, while the market operator clears the market and penalizes undersupply through the payment recovery mechanism. The existence of Nash equilibrium is proved, and a distributed best response algorithm is implemented to calculate the risk-averse optimal bidding strategy, taking into account Denial-of-Service attacks. The mean-convergence and variance-convergence of our proposed algorithm are derived. A case study of Australian National Electricity Market (NEM) validates the effectiveness of the proposed method in reducing overbidding risk, strengthening FR service reliability and improving profits for VPPs.

On the Existence and Computation of Nash Equilibrium in Network Competitive Location Under Delivered Pricing and Price Sensitive Demand

On the Existence and Computation of Nash Equilibrium in Network Competitive Location Under Delivered Pricing and Price Sensitive Demand

Secure IoU in Adversarial Environments: User Authentication and Performance Optimization

The emerging UAV network suffers from severe security risks from attackers. If the attackers sneak into an Internet of UAV (IoU) network, they would occupy the scarce resources which should be allocated among authenticated users, degrading the communication performance. Therefore, a two-stage scheme is proposed to both enhance IoU security and optimize network resource allocation. In the first stage, a distributed radio-frequency fingerprinting-based authentication model with consensus voting is designed to enhance the IoU security. Moreover, in the second stage, a performance optimization problem in consideration of authentication accuracy is formulated to maximize the data rate. It is proved that the problem can be transformed into a potential game, which implies that the optimization of the overall network's utility equals the optimization of each individual's utility. Also, the existence of this game's Nash equilibrium (NE) is proved and an iterative algorithm is used to achieve its NE. Finally, simulation results show that that our proposed consensus-based authentication model improves recognition accuracy by 40\%compared to the conventional Adaboost method under typical parameter settings, and our proposed algorithm can improve the network utility significantly. The effect of various parameters on the utility is also numerically verified.

A Multi-Period Charging Service Pricing Game for Public Charging Network Operators Considering the Dynamics of Coupled Traffic-Power Systems

The proliferation of electric vehicles (EVs) couples the operations of traffic systems and power systems, necessitating the interdependent traffic-power modeling to optimize the on-road EV charging decisions. In this paper, the multi-period charging service pricing interactions between multiple charging network operators (CNOs) are discussed considering the dynamics of traffic-power systems. A multi-period user equilibrium model considering mixed vehicle types is formulated to describe the possible vehicle trip transitions between different periods in urban transportation network. For the power distribution network, a convexified multi-period AC optimal power flow model is adopted for local electricity market clearing. With the dynamics of traffic-power systems, the pricing interaction between multiple CNOs is a non-cooperative game. We analyze the existence of Nash Equilibrium with fixed-point theory, and propose an iterative method based on the best response strategy to find <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon$</tex-math> </inline-formula> -Nash Equilibrium. Numerical studies based on an interdependent power-traffic system consisting of two 18-node power distribution networks and one revised Nguyen-Dupuis transportation network demonstrate that with the proposed CNOs’ charging pricing strategy, the CNOs’ service profits can increase by 1%-3% while the PDNs’ operation costs decrease about 1%-2% for the interdependent traffic-power system.

Age-Dependent Distributed MAC for Ultra-Dense Wireless Networks

We consider an ultra-dense wireless network with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$</tex-math> </inline-formula> channels and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M = N$</tex-math> </inline-formula> devices. Messages with fresh information are generated at each device according to a random process and need to be transmitted to an access point. The value of a message decreases as it ages, so each device searches for an idle channel to transmit the message as soon as it can. However, each channel probing is associated with a fixed cost (energy), so a device needs to adapt its probing rate based on the “age” of the message. At each device, the design of the optimal probing strategy can be formulated as an infinite horizon Markov Decision Process (MDP) where the devices compete with each other to find idle channels. While it is natural to view the system as a Bayesian game, it is often intractable to analyze such a system. Thus, we use the Mean Field Game (MFG) approach to analyze the system in a large-system regime, where the number of devices is very large, to understand the structure of the problem and to find efficient probing strategies. We present an analysis based on the MFG perspective. We begin by characterizing the space of valid policies and use this to show the existence of a Mean Field Nash Equilibrium (MFNE) in a constrained set for any general increasing cost functions with diminishing rewards. Further we provide an algorithm for computing the equilibrium for any given device, and the corresponding age-dependent channel probing policy.

Game Model of Optimal Quality Experience Strategy for Internet of Vehicles Bandwidth Service Based on DAG Blockchain

With the rapid development of the Internet of Vehicles (IoV), vehicle networks will generate large amounts of data. Vehicle-to-Vehicle (V2V) data sharing can improve driving experience and quality of service, and Vehicle-to-Roadside unit (V2R) communication can improve road transport efficiency. A certain amount of bandwidth is consumed in the process of data sharing between V2V and V2R, and when multiple vehicles request data from the RSU at the same time, there may be network delay or congestion. Therefore, the vehicle obtains data faster by requesting high bandwidth from the RSU. However, since the bandwidth resources of RSUs are limited, optimal bandwidth service strategies need to be investigated. In addition, in order to ensure the safety, efficiency and traceability of data sharing between V2V and V2R, a blockchain-supported IoV needs to be established. Therefore, we design a blockchain-based framework for V2V and V2R transaction framework that integrates DAG blockchain and modern cryptography. The framework ensures security and scalability during the execution of data exchange and completion of bandwidth transactions, improves the service quality of vehicles and realizes decentralized payment. In this work, a two-stage Stackelberg game is established under two pricing schemes of uniform pricing and discriminatory pricing to jointly maximize the benefits of vehicles and RSUs. In Stage I, the RSU sets the corresponding bandwidth service and pricing strategy. In Stage II, the vehicle decides the service demand according to the observed pricing strategy. The optimal bandwidth allocation and pricing strategy between the RSU and the vehicle is determined through the game of both parties, and the existence and uniqueness of the two-stage subgame Nash equilibrium is analyzed by applying backward induction. The correctness of the model is verified by experiments, and the comparative analysis shows that the proposed model has high throughput.