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.

Generic technology R&D decision with technology spillover, cost difference, and bargaining power under oligopoly competition

ABSTRACT This study constructs an oligopoly model considering generic technology R&D, analyzes the market equilibrium results under R&D cooperation and non-cooperation, explores the impacts of technology spillover, cost difference, and firm’s bargaining power, and compares the boundary conditions between R&D cooperation and non-cooperation. The results show that when the degree of technology spillover is small, the R&D firm conducts drastic R&D, resulting in a complete monopoly. The market structure and equilibrium results are influenced by the degree of technology spillover, cost difference, bargaining power, and the number of following firms. Cooperation in R&D can be considered as a Nash equilibrium in certain cases. Compared with R&D non-cooperation, R&D cooperation can lead to higher profits for firms. When the degree of technology spillover is large and the number of following firms is small, R&D cooperation yields greater consumer surplus and social welfare.

Analyzing the blockchain mining strategy in Industrial Internet of Things: A game theoretical approach

Deploying Proof-of-Work (PoW) consensus based blockchain systems in Industrial Internet of Things (IIoT) scenarios that rely on wireless channels for data transmission faces new challenges. An IIoT device working in industrial production may participate in the operation of blockchain as a miner. These devices not only need to consume their own computation capability to compete for the right to generate blocks, but also need to transmit newly generated blocks to other devices through wireless channel to continuously maintain the global ledger. Asynchronous mining and data transmission by IIoT devices in wireless environments may significantly interfere with the propagation of blocks in the system and result the reduction of mining payoff, thereby affecting the willingness of IIoT devices to participate in maintaining blockchain system. This paper considers the behavior of independent mining between multiple IIoT devices as a non-cooperative game, which we call the Wireless Mining Game (WMG). Markov chain is adopted to derive the block propagation model under the wireless channel, and then the payoff model of IIoT device is formulated. IIoT devices can make reasonable mining decisions by evaluating the computation capability of other IIoT devices and the probability of successful block transmission. We analyze the Nash equilibrium of the WMG and the corresponding mining behavior set between two-players scenario through theoretical modeling and experiments. We also analyzed the expected mining payoff of the IIoT devices in the multiple devices scenario.

Public goods in networks with constraints on sharing

This paper considers incentives to provide goods that are partially shareable along social links. We introduce a model in which each individual in a social network not only decides how much of a shareable good to provide, but also decides which subset of neighbours to nominate as co-beneficiaries. An outcome of the model specifies an endogenously generated subnetwork and a public goods game occurring over the realised subnetwork. We prove the existence of specialised pure strategy Nash equilibria: those in which some individuals contribute while the remaining individuals free ride. We then consider how the set of efficient specialised equilibria vary as the constraints on sharing are relaxed and we show that, paradoxically, an increase in shareability may decrease efficiency.

Sequential multichannel allocation in cache-enabled multicasting small-cell networks

This paper investigates the optimization of multichannel allocation in cache-enabled multicasting small-cell networks (CMSNs). In the traditional fixed channel allocation scheme (FCA), small-cell base stations (SBSs) allocate a fix channel to each local cache popular content. In order to further increasing the utilization of allocated channels, we first propose a sequential multichannel allocation scheme (SMA), in which SBSs allocate a sequential channel set to their cached files dynamically. We formulate the problem of sequential multichannel allocation as two NP-hard non-linear programming problems with two optimization objectives of minimizing the global collision level (GCL) and maximizing the global file-downloading level (GFL), and demonstrate the advantages of the proposed DSCA over FCA. Then, two channel allocation potential games are proposed to achieve GCL and GFL optimization respectively. To achieve Nash equilibria (NEs) of the proposed games, we present a better response based SMA algorithm (BR-SMA) for both two games, and propose an improved better response based SMA algorithm (IBR-SMA) for the game for GCL maximization which have two exploration modes for each decision-maker.

Attack-Resilient Distributed Nash Equilibrium Seeking of Uncertain Multiagent Systems Over Unreliable Communication Networks.

This article investigates the distributed Nash equilibrium (NE) seeking problem of uncertain multiagent systems in unreliable communication networks. In this problem, the action of each agent is subject to a class of nonlinear systems with uncertain dynamics, and the communication network among agents will be affected by the nonperiodic denial of service (DoS) attacks. Note that, in this insecure network environment, the existence of DoS attacks will directly destroy the connectivity of the network, which leads to performance degradation or even failure of the most existing distributed NE seeking algorithms. To address this problem, we propose a two-stage distributed NE seeking strategy, including the attack-resilient distributed NE estimator and the neuroadaptive tracking controller. The estimator based on the projection subgradient method and the consensus protocol can converge exponentially to virtual NE against DoS attacks. Then, the neuroadaptive tracking controller is designed for uncertain multiagent systems with the output of the estimator as the reference signal such that the actual action of all agents can reach NE. Based on the Lyapunov stability theory and improved average dwell time automaton, the stability of the estimator and the controller is proven, and all signals in the closed-loop system are uniformly bounded. Numerical examples are presented to verify the effectiveness of the proposed strategy.

An adaptive differential evolution algorithm based on relaxation strategy to realize the Nash equilibrium of multi-oligopoly game

. Inspired by the idea of game learning, this paper studies the Cournot game model of the multi-oligopoly market and studies the realization of the Nash equilibrium by using adaptive differential evolution algorithm based on relaxation strategies. The realization path of Nash equilibrium is attained through an adaptive differential evolution based on a relaxation strategy procedure applied to an objective function with best response dynamics. Among them, the objective function is derived from the sum of the payoff changes of each oligopoly under best response decisions. And theoretically proved that the Nash equilibrium is the optimal solution of the objective function.

A Collaborative Neurodynamic Optimization Approach to Distributed Nash-Equilibrium Seeking in Multicluster Games With Nonconvex Functions.

In this article, we propose a collaborative neurodynamic optimization (CNO) method for the distributed seeking of generalized Nash equilibriums (GNEs) in multicluster games with nonconvex functions. Based on an augmented Lagrangian function, we develop a projection neural network for the local search of GNEs, and its convergence to a local GNE is proven. We formulate a global optimization problem to which a global optimal solution is a high-quality local GNE, and we adopt a CNO approach consisting of multiple recurrent neural networks for scattering searches and a metaheuristic rule for reinitializing states. We elaborate on an example of a price-bidding problem in an electricity market to demonstrate the viability of the proposed approach.

Distributed Optimal Solutions for Multiagent Pursuit-Evasion Games for Capture and Formation Control

In this paper, distributed optimal solutions are designed for networked multiagent pursuit-evasion (MPE) games for capture and formation control. In the games, the pursuers aim to minimize the distance from their target evaders while the evaders attempt to maximize it, and at the same time, all players desire to maintain cohesion with their teammates. The goals of agents are obviously reflected in the obtained optimal control strategies which consist of an attracting term and/or a repelling term. Nash equilibrium is obtained by means of optimal strategies using the solutions of the HJI equations. Furthermore, three scenarios are considered in the MPE game: one-pursuer-one-evader, multiple-pursuer-one-evader, and multiple-pursuer-multiple-evader, where sufficient conditions are given for pursuers in achieving exponential capture or formation control with ultimate zero or bounded errors. It is shown that the conditions depend on the structure of the communication graph, the parameters in the controllers, and the expected formation configurations. Finally, both simulations and real flight experiments successfully demonstrate the effectiveness of the proposed strategies.

Non-additive network pricing with non-cooperative mobility service providers

This study addresses a mobility network pricing problem in a competitive environment. We consider a multimodal transportation network where the links are operated by multiple profit-maximizing, mobility service providers (MSPs). We take the perspective of a network regulator that aims to increase ridership in a target mobility network by providing non-additive, path-based subsidies to travelers. We model paths’ attractiveness using generalized cost functions that combine path travel time and path cost, and we use linear elastic travel demand functions to capture the proportion of demand served by a path. MSPs are non-cooperative and adjust link fares according to the subsidy policy implemented by the regulator. The goal of the network regulator is to solve a budget-constrained mobility network pricing problem under MSP competition. This game-theoretical framework is modeled as a single-leader multi-follower game (SLMFG) wherein the leader player represents the network regulator and multiple follower players represent the MSPs. We conduct a theoretical analysis of this SLMFG by identifying necessary and sufficient conditions for the existence of solutions to the parameterized generalized Nash equilibrium problem (GNEP) that is played amongst MSPs. We show that this GNEP is jointly convex and we use this property to develop an exact numerical approach to solve the SLMFG based on customized branch-and-bound algorithms. Numerical results reveal the impact of MSP competition in this mobility network pricing problem and shed novel insights into the design of optimal path-based subsidy policies.

Safe tracking in games: Achieving optimal control with unknown dynamics and constraints

AbstractThis paper introduces mix‐zero‐sum differential (MZSD) game theory to address multi‐player tracking systems, offering a better understanding of the coexistence of cooperation and competition among players. Within this framework, we present an optimal safety tracking control (OSTC) method, which incorporates a control barrier function (CBF) into the value function to ensure that the tracking error remains within a specified range, thus guaranteeing safety while achieving optimization. Simultaneously, to eliminate the need for system dynamics, we propose a novel approach leveraging off‐policy integral reinforcement learning (IRL) technology to obtain the Nash equilibrium solution of the MZSD games. We establish a unique critics–actors neural network (NN) structure that updates concurrently. Furthermore, we analyze stability and convergence using the Lyapunov method. We conduct two simulations to demonstrate the effectiveness of the proposed algorithm.

Multi-Cluster Aggregative Games: A Linearly Convergent Nash Equilibrium Seeking Algorithm and Its Applications in Energy Management

Multi-Cluster Aggregative Games: A Linearly Convergent Nash Equilibrium Seeking Algorithm and Its Applications in Energy Management

Dynamic bargaining game DEA carbon emissions abatement allocation and the Nash equilibrium

Allocating carbon emissions abatement (CEA) is crucial for reducing carbon footprint of organizations (or decision-making units, DMUs) and mitigating global warming. However, discrepancies often arise among DMUs regarding CEA allocation. This study introduces a dynamic bargaining game approach to address the issue under the nonparametric production frontier analysis framework. Our approach employs an iterative process, allowing each DMU to propose its individual preferred CEA allocation proposal in each iteration. Subsequently, dynamic negotiations among the DMUs occur, leading to the eventual consensus on the CEA allocation result. Our theoretical analysis demonstrates that, at the end of the dynamic bargaining game, all DMUs will converge on the same CEA allocation result, termed as the consensus CEA allocation result. This agreement is reached despite each DMU customizing its individual CEA allocation proposal to suit its own interests. Moreover, we prove that this consensus CEA allocation result represents a Nash equilibrium solution, thereby ensuring its stability and acceptance among all DMUs. Finally, we provide a simple numerical example and a case study of CEA allocation across 27 European Union countries to illustrate the usefulness of our approach and compare it with prior CEA allocation approaches.

A Reinforcement Learning-Based Pantograph Control Strategy for Improving Current Collection Quality in High-Speed Railways.

In high-speed railways, the pantograph-catenary system (PCS) is a critical subsystem of the train power supply system. In particular, when the double-PCS (DPCS) is in operation, the passing of the leading pantograph (LP) causes the contact force of the trailing pantograph (TP) to fluctuate violently, affecting the power collection quality of the electric multiple units (EMUs). The actively controlled pantograph is the most promising technique for reducing the pantograph-catenary contact force (PCCF) fluctuation and improving the current collection quality. Based on the Nash equilibrium framework, this study proposes a multiagent reinforcement learning (MARL) algorithm for active pantograph control called cooperative proximity policy optimization (Coo-PPO). In the algorithm implementation, the heterogeneous agents play a unique role in a cooperative environment guided by the global value function. Then, a novel reward propagation channel is proposed to reveal implicit associations between agents. Furthermore, a curriculum learning approach is adopted to strike a balance between reward maximization and rational movement patterns. An existing MARL algorithm and a traditional control strategy are compared in the same scenario to validate the proposed control strategy's performance. The experimental results show that the Coo-PPO algorithm obtains more rewards, significantly suppresses the fluctuation in PCCF (up to 41.55%), and dramatically decreases the TP's offline rate (up to 10.77%). This study adopts MARL technology for the first time to address the coordinated control of double pantographs in DPCS.

Broadcast Gossip Algorithm for Nash Equilibrium Seeking of Nonaggregative Games

Broadcast Gossip Algorithm for Nash Equilibrium Seeking of Nonaggregative Games

An Adaptive Dual-Mode Task-Oriented Resource Management Strategy for GEO Relay Systems

With the fierce global competition on satellite networks, the building of satellite constellations grows explosively. Sharply increasing on-orbit data will face the challenge of satellite-ground data transmission. GEO satellites become the top choice for satellite data relay due to their stable satellite-ground link. Most existing spectrum resource management for GEO relays is equipment-oriented and benefit priority, which may lead to a waste of spectrum resources. In this paper, we propose a real-time task-oriented resource allocation strategy for GEO relay systems. We model the spectrum allocation problem as a distributed non-cooperative Stackelberg game process. We prove that when both sides of the game pursue the maximization of personal revenue, the system will enter a Nash equilibrium state, whereas spectrum resources are not fully used. Based on the maximization of individual utilities (U-prior) and spectrum utilization (S-prior) methods, we design an adaptive dual-mode pricing mode to maximize the spectrum resources within a certain loss of revenue. The simulation results show that the S-prior and U-prior have better performance than the baseline method and existing optimization methods. Our proposed dual-mode strategy is making more throughputs and has less delay with little loss of utility values than that of individual utility maximization.

An Underwater Source Location Privacy Protection Scheme Based on Game Theory in a Multi-Attacker Cooperation Scenario.

Ensuring source location privacy is crucial for the security of underwater acoustic sensor networks amid the growing use of marine environmental monitoring. However, the traditional source location privacy scheme overlooks multi-attacker cooperation strategies and also has the problem of high communication overhead. This paper addresses the aforementioned limitations by proposing an underwater source location privacy protection scheme based on game theory under the scenario of multiple cooperating attackers (SLP-MACGT). First, a transformation method of a virtual coordinate system is proposed to conceal the real position of nodes to a certain extent. Second, through using the relay node selection strategy, the diversity of transmission paths is increased, passive attacks by adversaries are resisted, and the privacy of source nodes is protected. Additionally, a secure data transmission technique utilizing fountain codes is employed to resist active attacks by adversaries, ensuring data integrity and enhancing data transmission stability. Finally, Nash equilibrium could be achieved after the multi-round evolutionary game theory of source node and multiple attackers adopting their respective strategies. Simulation experiments and performance evaluation verify the effectiveness and reliability of SLP-MACGT regarding aspects of the packet forwarding success rate, security time, delay and energy consumption: the packet delivery rate average increases by 30%, security time is extended by at least 85%, and the delay is reduced by at least 90% compared with SSLP, PP-LSPP, and MRGSLP.

Deep Learning Pricing of Processing Firms in Agricultural Markets

The pricing behavior of agricultural processing firms in input markets has large impacts on farmers’ and processors’ prosperity as well as the overall market structure. Despite analytical approaches to food processors’ pricing in agricultural input markets, the need for models to represent complex market features is urgent. Agent-based models (ABMs) serve as computational laboratories to understand complex markets emerging from autonomously interacting agents. Yet, individual agents within ABMs must be equipped with intelligent learning algorithms. In this paper, we propose supervised and unsupervised learning agents to simulate the pricing behavior of firms in agricultural markets’ ABMs. Supervised learning firms are pre-trained to accurately best respond to their competitors and are deemed to result in the market Nash equilibria. Unsupervised learning firms play a course of pricing interaction with their competitors without any pre-knowledge but based on deep reinforcement learning. The simulation results show that unsupervised deep learning firms are capable of approximating the pricing equilibria obtained by the supervised firms in different spatial market settings. Optimal discriminatory and uniform delivery pricing emerges in agricultural input markets with the high and intermediary importance placed on space. Free on board pricing emerges in agricultural input markets with small importance placed on space.

Dynamic Price Dispersion of Seasonal Goods in Bertrand–Edgeworth Competition

The primary distinguishing characteristics of various revenue management systems include fixed capacities, uniform products, and customers' sensitivity to pricing when making purchase decisions. Even with only two or three competitors, the competition remains intense. This study examines the sub-game perfect Nash equilibrium of price competition within an oligopoly market where goods are perishable assets. Each seller possesses a single unit of a non-replenishable good and competes by setting prices to sell their inventory over a finite sales period. Each period, buyers seek to purchase one unit of the good, and the number of buyers entering is random. All sellers' prices are visible to buyers, and there is no cost associated with searching for goods. Through stochastic dynamic programming methods, sellers' optimal responses can be determined by treating the competition as a one-time price-setting game, considering the remaining sales periods and the current demand pattern. Assuming a binary demand model, it is demonstrated that the duopoly model exhibits a unique Nash equilibrium. In addition, it is shown that the oligopoly model does not result in price dispersion based on a discussed metric. Moreover, when a generalized demand model is considered, the duopoly model features a unique Nash equilibrium with mixed strategies, whereas the oligopoly model exhibits a unique symmetric Nash equilibrium with mixed strategies.

A comparative cost assessment of coalescing epidemic control strategies in heterogeneous social-contact networks

The outcome of an epidemic is contingent upon the mitigating control strategies deployed by policymakers. The deployment of control strategies is constrained by the cost of economic, social, and technological factors, which often depend on regional conditions. Our model facilitates a comparative assessment of costs between two interconnected subpopulations within a heterogeneous social-contact network, which may deploy either aligned or divergent control strategies, as is frequently the case for neighboring countries. Depending on their respective asymmetric resources, each subpopulation can deploy a contact confinement strategy, a vaccination strategy, or a hybrid of both. The multi-objective goal of minimizing both the epidemic burden and the economic cost from policy implementation and productivity losses is compounded using a scaled scalarization method. Due to the inter-dependence of policy outcomes, the optimal strategy is derived from the Nash equilibrium in a 2-player compounded pay-off matrix of the total cost. Our network-based model implements a probabilistic Susceptible-Exposed-Infectious-Recovered-Dead dynamics. It also includes the effects of ego-network support and waning immunity modulated by re-exposure to viral contaminants (‘SEIRSD’). Robust outcomes of repeated network simulations with COVID-19 parameters are examined. Our method aims to equip policymakers in a negotiation process with a comparative cost matrix for divergent mitigation policies and to identify the optimal Nash-based strategy profile.