Existence Of Nash Equilibrium Research Articles

Dynamic Game for Strategy Selection in Hardware Trojan Attack and Defense

The offshore outsourcing introduces serious threats to semiconductor suppliers and integrated circuit(IC) users for the possibility of hardware trojans (HTs). To alleviate this threat, IC designers use the active defenses against HTs which are implanted by the malicious manufacturer. In this paper, a game-theoretic framework based on fuzzy theory is proposed to obtain the optimal strategy. It analyzes the interactions between the active defense designer and the malicious manufacturer. The attack and defense on IC is formulated as a noncooperative dynamic game. The active defense strategy is decided in IC design. And the overall payoff including design costs and losses avoided is optimized. Subsequently, the HT is implanted considering the implantation cost as well as the damage caused by it. To solve the problem of uncertain payoff caused by insufficient information, fuzzy variable is used to represent the influence of the defense coverage rate on the payoff. In order to verify the applicability of fuzzy variable in dynamic game framework, the existence of pure strategy Nash equilibrium in the game is proved. A solution algorithm for pure strategy Nash Equilibrium is proposed to obtain the optimal strategy of attacker and defender. Thanks to the case study of Field Programmable Gate Array (FPGA), the proposed framework is feasible for HT attack and defense game.

Admission Control and Power Allocation for NOMA-Based Satellite Multi-Beam Network

This paper investigates the admission control problem on the satellite multi-beam networks with non-orthogonal multiple access (NOMA). The goal is to maximize the number of supported users on the premise of ensuring the quality of service (QoS) by optimizing the subchannel and power allocation. We provide the system model and then formulate the admission control problem as a mixed integer non-convex optimization problem. The non-convexity and existence of integer variable make the optimal solution difficult to get. Therefore, we propose a joint subchannel matching and power allocation algorithm to obtain the suboptimal solution so as to reduce the computation complexity. The proposed algorithm can be used for both NOMA and orthogonal frequency division multiplexing access (OFDMA). Specifically, the subchannel matching problem is solved by a two-stage matching process where users are accessed to subchannel dynamically. The power allocation problem is modeled as a super-modular game where the existence and uniqueness of Nash equilibrium (NE) are analyzed. Moreover, an iterative power allocation algorithm is proposed based on the NE searching method. Finally, simulation results are provided for demonstrating the effectiveness and feasibility of the proposed algorithm.

Verification and Design of Zero-Sum Potential Games

Zero-sum game is a class of game where one player’s gain is equivalent to another’s loss, which can be used in competitive situation. But pure Nash equilibrium maybe not exist in general zero-sum games. Potential games have nice properties, such as existence of pure Nash equilibrium. To combine advantages of zero-sum games and potential games, zero-sum potential game is proposed in this paper. Verification for a finite non-cooperative game being a zero-sum potential game is considered. Conversely, how to design a zero-sum potential game is also studied when the potential function is given. We show that verification and design of zero-sum potential game can be realized by solving linear equations. Furthermore, we find that if any two players play the zero-sum potential game in a network, then the networked game is also a zero-sum potential game.

Weighted-Directed-Hypergraph-Based Spectrum Access for Energy Harvesting Cognitive Radio Sensor Network

In this paper, we investigate the spectrum access for energy harvesting cognitive radio sensor network (EH-CRSN), where sensor nodes (SNs) connected to different sinks share several channels. To maximize the sum of system energy efficiency (EE) and spectrum efficiency (SE), different from traditional works modeling the interference relationship by binary graph and unweighted hypergraph, we apply a novel weighted directed hypergraph (WDH) to accurately characterize the degree of interference among neighbor SNs and formulate the channel access problem in multi-sink network as a WDH game. We analyze the existence of Nash equilibrium (NE) and design a spectrum access algorithm to achieve the optimal solution which is one of NE. Moreover, simulation results are presented to verify the effectiveness of the proposed algorithm.

Operation Mechanism and Strategies for Transactive Electricity Market With Multi-Microgrid in Grid-Connected Mode

Under the policy support of electricity market reform and the promotion of microgrids in grid-connected operation mode construction in China, the sales side of electricity market is gradually liberalized. Microgrids are developing rapidly through participating in the electricity market competition. In order to find the multi-player game relationship of real-time electricity market with multi-microgrid in grid-connected operation mode and provide an incentive for electricity trading among microgrids, we propose a transactive real-time market trading mechanism based on the principle of proportional distribution. Through the analysis of multi-player game in real-time market, we build cost-benefit models of microgrids with power surplus and those with power shortage. Based on the game theory, non-cooperative game models with multi-buyer and multi-seller of microgrids with power surplus and those with power shortage are established. The existence of Nash equilibrium for the game models of microgrids with power surplus and those with power shortage has been proved and the Nash equilibrium is solved to obtain their optimal bid and selling strategies. The feasibility and validity of the proposed mechanism and models are proved through case study. The result demonstrates that all microgrids can obtain the best profits with their optimal strategies.

Cooperate Delegation of Computation for Rational Party Using Zero-Determinant Strategy Approach

Delegating calculation is an important approach to solve the correctness and reliability of task subcontracting and delegating calculation results in cloud computing environment. However, the dynamic, complexity and openness of cloud computing bring unprecedented risks to the security and reliability of computing tasks. To solve the problem the new approach is presented. Firstly, the game theory is introduced into the delegating calculation, and the single-client and multi-server rational delegating calculation game model is given. Secondly, the Zero-determinant Strategy scheme based on delegating calculation is constructed through implementing the single-client and single-server rational delegating calculation protocol and Zero-determinant Strategy, and the conditions of the existence of Nash equilibrium of Zero-determinant Strategy is analyzed. Then, through the iterative implementation of the Zero-determinant Strategy, the participants will cooperate with each other actively. Finally, the performance analysis results show that the entrusting party can regulate the betrayers in the computing party through Zero-determinant Strategy to ensure the interests of honest people in cloud computing.

The three kinds of degree distributions and nash equilibrium on the limiting random network

A generalized dynamically evolving random network and a game model taking place on the evolving network are presented. We show that there exists a high-dimensional critical curved surface of the parameters related the probabilities of adding or removing vertices or edges such that the evolving network may exhibit three kinds of degree distributions as the time goes to infinity when the parameters belong to the super-critical, critical and sub-critical curved surfaces, respectively. Some sufficient conditions are given for the existence of a regular Nash equilibrium which depends on the three kinds of degree distributions in the game model on the limiting random network.

Game‐theoretic power allocation algorithm for downlink NOMA cellular system

A new power allocation algorithm is proposed based on the Glicksberg game for cellular downlink non-orthogonal multiple access (NOMA) networks. First, a price-based user's utility function is proposed, and shown that it is effective and restrictive. Secondly, the Hessian matrix is used to derive an expression for power price based on the restricted transmission power and number of the served users in the cell. Then, the existence of a unique Nash equilibrium is proven and the optimum solution that maximises the utility function is presented. Finally, simulation results show that the proposed power allocation mechanism outperforms existing algorithms in terms of sum data rate and average data rate of the users.

Joint Computation Offloading and Coin Loaning for Blockchain-Empowered Mobile-Edge Computing

The blockchain-empowered mobile-edge computing (MEC) is a promising solution for enhancing the computation capabilities of mobile equipments (MEs) to process computation-intensive tasks such as the real-time data processing tasks and mining tasks. However, because of the “cold start” and “long return” problems, efficient computation offloading cannot be achieved in blockchain-empowered MEC because the MEs do not always have enough coins to afford the offloading service cost. In this article, we study the joint computation-offloading and coin-loaning problem for blockchain-empowered MEC to minimize the total cost of all MEs. We introduce the banks that can provide loan services to the MEs to address the above two issues. We formulate the problem as a noncooperative game to model the competitions between the myopic MEs. By using a potential game method, we prove the existence of a pure-strategy Nash equilibrium (NE) and design a distributed algorithm to achieve the NE point with low computational complexity. We also provide an upper bound on the price of anarchy of the game by theoretical proof. Besides, two smart contracts are designed to automatically perform the computing resource trading and coin loaning processes. Lastly, our simulation results show that our proposed algorithm can significantly reduce the total cost of all MEs, has better performance compared with other solutions, and scales well as the number of MEs increases. Moreover, the financial cost for executing the two smart contracts on the Ethereum network is low.

Large tournament games

We consider a stochastic tournament game in which each player is rewarded based on her rank in terms of the completion time of her own task and is subject to cost of effort. When players are homogeneous and the rewards are purely rank dependent, the equilibrium has a surprisingly explicit characterization, which allows us to conduct comparative statics and obtain explicit solution to several optimal reward design problems. In the general case when the players are heterogenous and payoffs are not purely rank dependent, we prove the existence, uniqueness and stability of the Nash equilibrium of the associated mean field game, and the existence of an approximate Nash equilibrium of the finite-player game.

Approximate Nash Solutions for Multiplayer Mixed-Zero-Sum Game With Reinforcement Learning

Inspired by Nash game theory, a multiplayer mixed-zero-sum (MZS) nonlinear game considering both two situations [zero-sum and nonzero-sum (NZS) Nash games] is proposed in this paper. A synchronous reinforcement learning (RL) scheme based on the identifier-critic structure is developed to learn the Nash equilibrium solution of the proposed MZS game. First, the MZS game formulation is presented, where the performance indexes for players 1 to ${N}$ − 1 and ${N}$ NZS Nash game are presented, and another performance index for players ${N}$ and ${N}$ + 1 zero-sum game is presented, such that player ${N}$ cooperates with players 1 to ${N}$ − 1, while competes with player ${N}$ + 1, which leads to a Nash equilibrium of all players. A single-layer neural network (NN) is then used to approximate the unknown dynamics of the nonlinear game system. Finally, an RL scheme based on NNs is developed to learn the optimal performance indexes, which can be used to produce the optimal control policy of every player such that Nash equilibrium can be obtained. Thus, the widely used actor NN in RL literature is not needed. To this end, a recently proposed adaptive law is used to estimate the unknown identifier coefficient vectors, and an improved adaptive law with the error performance index is further developed to update the critic coefficient vectors. Both linear and nonlinear simulations are presented to demonstrate the existence of Nash equilibrium for MZS game and performance of the proposed algorithm.

Relaying-Assisted Communications for Demand Response in Smart Grid: Cost Modeling, Game Strategies, and Algorithms

The electricity costs of the Utilities are increased with the deviations from the forecasted demand, which are caused by forecast errors and demand fluctuations. The forecast errors are deterministic and can be avoided by the Utilities with long-term operations in electricity markets, whereas the demand fluctuations are stochastic and unavoidable. Demand response can be used for mitigating the demand fluctuations by measuring the electricity usage of consumers and publishing control commands periodically. The performance of demand response is dependent on the quality of communications between the control center and the consumers. The two-way communications are established based on the data aggregator units (DAU) deployed by the Utilities. To improve the quality of communications, we utilize the base stations in telecom networks to forward the metering and control data for the DAUs. The telecom operators maximize their profits and decide the fractions of transmission power allocated for relaying, and then the Utilities select the payments to obtain the corresponding transmission power allocated for relaying. We characterize the electricity costs of the Utilities based on Taguchi loss function and establish a Stackelberg game between the telecom operators and the Utilities. We prove the existence and uniqueness of Nash equilibrium for the follower-level payment selection game and develop an iterative algorithm to search for the equilibrium. Then, the Stackelberg equilibrium can be obtained by a backward induction method. Numerical results show that the cost of the Utilities can be reduced and the profits of the telecom operators are increased.

Refinements of Nash equilibrium in a Pentagonal Fuzzy Bimatrix Game

In this paper, we introduce the concept of symmetric pentagonal fuzzy bimatrix and constant pentagonal fuzzy matrix for which we define the existence of Nash equilibrium in both pure and mixed strategies along with some numerical illustrations. By applying Ranking to the pay-offs, we convert the fuzzy valued game problem to crisp valued game problem. Therefore, Solution concept of perfect equilibrium for normal form game is reviewed and a new concept of proper equilibrium of pentagonal fuzzy bimatrix game is discussed.

Existence of solutions of set-valued equilibrium problems in topological vector spaces with applications

In this paper, we first introduce the new notions of $$\overline{{\mathbb {R}}}_+$$-local-intersection property, $$\overline{{\mathbb {R}}}_+$$-local-inclusion property, $$\overline{{\mathbb {R}}}_+$$-strongly transfer lower semicontinuity, $${\mathbb {R}}_-$$-weakly transfer lower semicontinuity and $${\mathbb {R}}_-$$-strongly transfer lower semicontinuity for a set-valued mapping $$\Phi $$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.

Hybrid spectrum access model using game theory approach for multi-channel heterogeneous mobile cognitive radio wireless sensor network

<p>An efficient spectrum access and medium access control (MAC) design is required to enable wireless sensor network (WSN) with cognitive Internet of Things (IoT) that enables presence of sensor network with existing wireless infrastructure. Designing spectral, energy efficient and fair spectrum access design for IoT in CR-WSN is challenging especially for heterogeneous mobile CR-WSN. Existing model induces high collision and energy overhead. As a result, they are not efficient in utilizing spectrum. To overcome the research issues, this paper proposes a hybrid spectrum access model (HSAM) for multi-channel heterogeneous mobile CR-WSN. Next it elaborates on an accurate feasible channel accessible likelihood estimation. Further, we present a game theory model for HSAM for existence of fine-grained Nash equilibrium. Experiments are conducted to evaluate the performance of HSAM-CRWSN over existing model. The outcome shows HSAM-CRWSN attains significant performance improvement over state-of-art technique in terms of energy efficiency, throughput, successful packet transmission and packet collision. The HSAM-CRWSN improves the overall spectrum efficiency of CR-WSN.</p><p><span style="font-size: small;"><br /></span></p>

Parallel demand side auction mechanism for dynamic and efficient resource allocation

Effective allocation mechanisms are required for infinitesimally divisible demand side resources capturing self-interest and non-cooperation among the users who consume the resources, in order to achieve efficient resource allocation maximising the social welfare with decentralisation of information sharing and decision making. The authors study an auction based mechanism to allocate the demand side resource, where a set of users who consume resources game with each other by bidding. The proposed auction mechanism fits into the progressive second price (PSP) mechanism, which follows a Vickrey-Clarke-Groves type payment. These kind of mechanisms admit incentive compatibility and existence of the efficient Nash equilibrium (NE). Furthermore, considering the fairness and privacy of the bidders as well as limitation of sequential algorithms in practical applications, the authors present a parallel updating algorithm in which all the bidders synchronously update their bids to achieve the efficient NE. By rigorous theoretical analyses, the proposed algorithm is guaranteed to converge to a bid profile which is shown to be the efficient NE. Numerical simulations demonstrate the performance of the proposed auction mechanism and parallel algorithm.

Cooperative and Distributed Computation Offloading for Blockchain-Empowered Industrial Internet of Things

Offloading computation-intensive blockchain mining tasks to the edge servers (ESs) is a promising solution for blockchain-empowered Industrial Internet of Things (IIoT) because the computing capabilities in IIoT are usually limited, whereas the blockchain mining tasks are computationally intensive. However, the computation offloading solutions for data processing tasks and for blockchain mining tasks have been studied separately. Moreover, most of the existing solutions for offloading assume that all IIoT devices can directly connect to the ESs or cloud data centers. To address these issues, in this paper, we propose a multihop cooperative and distributed computation offloading algorithm that considers the data processing tasks and the mining tasks together for blockchain-empowered IIoT. First, we study the multihop computation offloading problem for both the data processing tasks and the mining tasks to minimize the economic cost of IIoT devices. Second, we formulate the offloading problem as a potential game in which the IIoT devices can make their decisions autonomously and prove the existence of Nash equilibrium (NE) for the game. Third, we design an efficient distributed algorithm based on exchanging messages between IIoT devices to achieve the NE with low computational complexity. Lastly, our experimental results demonstrate that our distributed algorithm scales well as the number of IIoT devices increases and has the minimum system cost compared with other approaches.

Characterization of Nash equilibria of large games

For a large game with traits, this paper characterizes its trait–action distributions in equilibrium by using an inequality under the following two sets of conditions: (i) both trait space and action space of the game contain at most countably many elements; (ii) the agent space is nowhere equivalent to the characteristic type space. Two examples are presented to show the failure of the characterization result when the conditions are not satisfied. Then, we also show the existence of the characterization inequality and the necessity of the two sets of conditions to the characterization results. Finally, the existence of Nash equilibria under the two sets of conditions comes naturally as a corollary.

A dynamic model of the limit order book

We consider an equilibrium model of the Limit Order Book in a stock market, where a large number of competing agents post buy or sell orders. For the one-shot game, it is shown that the two sides of the LOB are determined by the distribution of the random size of the incoming order, and by the maximum price accepted by external buyers (or the minimum price accepted by external sellers). We then consider an iterated game, where more agents come to the market, posting both market orders and limit orders. Equilibrium strategies are found by backward induction, in terms of a value function which depends on the current sizes of the two portions of the LOB. The existence of a unique Nash equilibrium is proved under a natural assumption, namely: the probability that the external order is so large that it wipes out the entire LOB should be sufficiently small.

Joint Power and Sub-Channel Allocation for Secure Transmission in NOMA-Based mMTC Networks

In this paper, we consider the physical layer security for non-orthogonal multiple access (NOMA)-based uplink massive machine type communication (mMTC) networks. Aiming at the maximization of the system secrecy capacity, with the presence of eavesdroppers, we propose a joint power and sub-channel allocation for secrecy capacity (JPSASC) algorithm to obtain the suboptimal solution of the joint problem. Particularly, the power allocation problem is modeled as a non-cooperative game with a distributed perspective, where each MTC device selfishly optimizes its power allocation over multi-channels to maximize its own secrecy capacity. The existence of Nash equilibrium (NE) is proved and a sufficient condition to ensure the uniqueness of NE is given. Moreover, the distributed power allocation and preference secrecy capacity maximum (PSCM) algorithms are proposed for power allocation and sub-channel allocation problem, respectively. Simulation results verify that JPSASC algorithm outperforms other algorithms in maximizing secrecy capacity. Furthermore, the secrecy capacity in NOMA-based mMTC is improved compared with that in orthogonal multiple access schemes.