Existence Of Nash Equilibrium Research Articles

Dynamic environment UAV deployment algorithm based on potential game theory

AbstractTo address the issue of low coverage resulting from the challenge of acquiring the optimal deployment position in commonly used distributed deployment algorithms, this study presents a three‐dimensional deployment algorithm for Unmanned Aerial Vehicles (UAVs) based on potential games. First, a local mutually beneficial game model is designed to demonstrate the existence of exact potential games and Nash equilibrium. The Nash equilibrium solution corresponds to the maximum coverage. Next, drawing inspiration from exploration, a solution method called Exploration Spatial Adaptive Play is proposed. It utilizes the maximum utility function value from multiple step sizes in the exploration direction to update the action selection probability, thereby ensuring the optimal deployment position in each decision cycle. To address the issue of sensor position error, a method for processing sensor position errors is proposed. The simulation results demonstrate that the proposed distributed deployment algorithm achieves higher coverage compared to commonly used methods.

Individual and cluster demand response in retail electricity trading with end-users in differentiated oligopoly market: A game-theoretical approach

The characterization of the competitive interdependence among power consumers in power trading system contributes to pursuing efficient power supply and suppressing peak–valley load differences. Even though Stackelberg–Nash game approach has been applied to retail electricity trading with a single utility company and a group of end-users under competitive interdependence, the extension to multi-cluster retail electricity markets is not focused yet and the best strategies of end-users are still decoupled to each other. To bridge this gap this paper investigates the price-based demand response problem in multi-cluster retail electricity markets to coordinate the energy consumption behavior of the end-users and the retail price behavior of the utility companies. Adopting the structure of hierarchical game theory, the cluster retail price is controlled by the utility company to affect the energy consumption demand of end-users within the corresponding cluster and hence increase the income of both the utility companies and the end-users. In order to describe the coupled competition relationship among the bottom end-users in the system, the Cournot competition model of the differentiated oligopoly market is constructed by taking energy consumption as the individual strategy of end-users. The existence and uniqueness of Nash equilibrium and Stackelberg Nash equilibrium in such a hierarchical game system are proved, and a seeking algorithm is proposed to seek the Stackelberg Nash equilibrium, which is regarded as the extension from the existing algorithm for multi-cluster setup. This work gives an insight to theoretically check whether the utility companies need to redesign the contract with the power generator to ask for larger maximum supply.

The existence of a pure-strategy Nash equilibrium in a discrete ponds dilemma

In a variety of economic situations discrete agents choose one resource among several available resources and, once admitted to the resource of choice, divide it among fellow agents admitted there. The amount of the resource an agent gets is proportional to her relative ability to acquire this particular resource, what we refer to as an agent's weight at the resource. The relevant applications include students self-selecting into colleges, politicians self-selecting into races, and athletes self-selecting into teams. We find that this game has a pure-strategy Nash equilibrium in at least three special cases: 1) when agents have the same weight at each resource, 2) when all resources are the same, 3) when there are only two resources. We also show that this game always has an approximate Nash equilibrium when the number of players is large. Existence in the general case remains an open problem.

Edge computing collaborative offloading strategy for space‐air‐ground integrated networks

SummaryDue to geographical factors, it is impossible to build large‐scale communication network infrastructure in remote areas, which leads to poor network communication quality in these areas and a series of delay‐sensitive tasks cannot be timely processed and responded. Aiming at the problem of limited coverage in remote areas, the space‐air‐ground integrated networks (SAGIN) combined with mobile edge computing (MEC) can provide low latency and high reliability transmission for offloading delay‐sensitive tasks for users in remote areas. Considering the strong limitation of satellite resources in the space‐ground integrated network and insufficient energy of local user equipment, firstly, a satellite‐UAV cluster‐ground three‐layer edge computing network architecture is proposed in this paper. Under the condition that the delay requirements of various ground tasks are met, the task offloading problem is transformed into a Stackelberg game between ground user equipment and edge servers. In addition, it is proved that the existence of Nash equilibrium in non‐cooperative game between ground user equipment by using potential game. Finally, a Nash equilibrium iterative offloading algorithm based on Stackelberg game (NEIO‐SG) is proposed to find the optimal offloading strategy for tasks to minimize the system offloading cost and the optimal forwarding percentage strategy for offloading tasks to maximize the utility function of the edge server. Simulation results show that compared to other baseline algorithms, NEIO‐SG reduces the total system latency during task offloading by about 13 and the energy consumption of the edge server by about 35.

Joint optimization of deployment, user association, channel, and resource allocation for fairness‐aware multi‐UAV network

AbstractThis paper studies the problem of joint deployment, user association, channel, and resource allocation in unmanned aerial vehicle‐enabled access network. Since different user equipments performing different tasks and have different data rate requirements, the priority‐based traffic fairness problem is investigated. This problem, however, is a mixed integer nonlinear programming problem with NP‐hard complexity, making it challenging to be solved. To address this issue, a self‐organized and distributed framework “sense‐as‐you‐fly” based on the decomposition process, which divides the original problem into several subproblems, is proposed. Assuming without central controller, we derive the closed‐form resource allocation scheme and propose distributed many‐to‐one matching to optimize user association subproblem. Considering the coupled characteristics, the multi‐unmanned aerial vehicle deployment and channel allocation subproblems are modelled as a local altruistic game. The existence of Nash equilibrium is proved with the aid of exact potential game and efficient best response learning‐based algorithm is proposed. The original problem is finally addressed by solving the sub‐problems alternately and iteratively. Simulation results verify its effectiveness. By jointly optimizing multidimensional variables, the proposed algorithm unlocks network performance gains, especially in resource‐limited regimes.

Bayesian Nash equilibrium in all-pay auctions with interdependent types

We prove the existence of a behavioral-strategy Bayesian Nash equilibrium in all-pay auctions with statistically interdependent types (signals) under quite general assumptions on the values, costs and tie-breaking rules. Moreover, the set of equilibria is shown to be the same for any tie-breaking rule used in the auction.

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.

Hierarchical Control for the Oldroyd Equation in Memoriam to Professor Luiz Adauto Medeiros.

This manuscript deals with a hierarchical control problem for Oldroyd equation under the Stackelberg-Nash strategy. The Oldroyd equation model is defined by non-regular coefficients, that is, they are bounded measurable functions. We assume that we can act in the dynamic of the system by a hierarchy of controls, where one main control (the leader) and several additional secondary control (the followers) act in order to accomplish their given tasks: controllability for the leader and optimization for followers. We obtain the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability with respect to the leader control, and the optimality system for leader control.

Nash equilibria in risk-sensitive Markov stopping games under communication conditions

<p>This paper analyzes the existence of Nash equilibrium in a discrete-time Markov stopping game with two players. At each decision point, Player Ⅱ is faced with the choice of either ending the game and thus granting Player Ⅰ a final reward or letting the game continue. In the latter case, Player Ⅰ performs an action that affects transitions and receives a running reward from Player Ⅱ. We assume that Player Ⅰ has a constant and non-zero risk sensitivity coefficient, while Player Ⅱ strives to minimize the utility of Player Ⅰ. The effectiveness of decision strategies was measured by the risk-sensitive expected total reward of Player Ⅰ. Exploiting mild continuity-compactness conditions and communication-ergodicity properties, we found that the value function of the game is described as a single fixed point of the equilibrium operator, determining a Nash equilibrium. In addition, we provide an illustrative example in which our assumptions hold.</p>

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.