Generalized Nash Equilibrium Problem Research Articles

Best Response Dynamics Convergence for Generalized Nash Equilibrium Problems: An Opportunity for Autonomous Multiple Access Design in Federated Learning

Best Response Dynamics Convergence for Generalized Nash Equilibrium Problems: An Opportunity for Autonomous Multiple Access Design in Federated Learning

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.

A note on the Finite intersection property in “Existence of Nash equilibria for generalized multiobjective games through the vector extension of Weierstrass and Berge maximum theorems”

This note qualifies some results in Cotrina and Flores-Bazán (2024) on the ‘Finite intersection property’. We remind the essential role of this property for proving our results. In this note, we recall the statement of this property. We then specify the restrictive conditions that should be provided. An example proves the interest of this note revising some points in Cotrina and Flores-Bazán. Furthermore, it is showed that a class of generalized Nash equilibrium problems can be viewed as a particular case of a vector optimization problem, whose vector-valued function always possesses the Finite intersection property on its natural domain. Thus, the existence of a generalized Nash equilibrium will be a consequence of our Berge-type theorem and the Kakutani fixed point theorem, being it more general than those existing in recent literature, as shown by our examples.

Approximate weak optimality conditions in multiobjective generalized Nash equilibrium problems

A multiobjective generalized Nash equilibrium problem (MGNEP) is a Nash equilibrium problem with constraints that include multiobjective games. We focus in this paper on examining the approximate Karush–Kuhn–Tucker (KKT) conditions for MGNEPs and their impact on the global convergence of algorithms. To begin, we define standard approximate weak KKT (standard-AWKKT) conditions for MGNEPs. We demonstrate that every weak efficient solution meets the standard-AWKKT condition. As these optimality conditions are strong, we propose a new set of approximate weak KKT conditions, specifically tailored for MGNEPs, and show their convergence towards weak KKT points provided a cone-continuity property is satisfied. It is important to note that while these newly introduced approximate weak KKT conditions do not serve as optimality conditions for general MGNEPs, we illustrate that they hold true for the special case of a weak variational equilibrium point in a jointly convex MGNEP. Finally, we propose an enhanced Lagrangian-type algorithm for estimating an approximate weak KKT of an MGNEP and demonstrate its global convergence.

Generalized Nash equilibrium problems with mixed-integer variables

AbstractWe consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido–Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

On generalized Nash equilibrium problems in infinite-dimensional spaces using Nikaido–Isoda type functionals

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE.

A Game-Theoretic Approach for Electric Vehicle Aggregators Participating in Phase Balancing Considering Network Topology

With the increasing popularity of electric vehicles (EVs), uncontrolled charging of single-phase plug-in EVs will result in severe three-phase unbalance problem in distribution networks. To solve the problem, a phase balancing (PB) scheme performed by EV aggregators (EVAs) based on the linear multiphase power flow model is proposed in this paper. In the scheme, the PB compensations are provided to encourage EVAs to help the distribution system operator reduce the phase unbalance of the entire grid. Considering all EVAs are selfish and rational, the PB scheme is formulated as a game problem, in which each EVA competes with other EVAs to determine its charging strategy to minimize the charging cost minus the PB compensation. The consideration of coupling voltage constraints leads this game model to a challenging generalized Nash equilibrium problem (GNEP). By employing the theory of variational inequality, the properties of solutions for the GNEP are investigated. Furthermore, a two-level distributed algorithm is designed to find the unique variational solution, a fair and stable solution. Finally, comprehensive case studies are conducted on the IEEE-13 and IEEE-123 test systems to corroborate that the proposed PB game model can effectively mitigate voltage unbalance and cut EV users’ costs.

Personalized Incentives as Feedback Design in Generalized Nash Equilibrium Problems

Personalized Incentives as Feedback Design in Generalized Nash Equilibrium Problems

A note on jointly convex Nash games

In this note we are interested in a relevant generalized Nash equilibrium problem, which was proposed by Rosen in 1965. An existence result is established in the general setting of quasiconvexity, by relaxing the continuity hypothesis of the objective functions, which is independent from the one given by Aussel and Dutta in 2008.

Resource allocation and energy trading in the short blocklength regime for radio access networks

AbstractWith the large‐scale deployment of communication and computation units, the conventional radio access network has been gradually transformed into a sophisticated Cyber‐Physical System. Since the access network is put forward higher expectations for ultra‐reliable and low‐latency services, Short‐Packet Communications (SPC) based on Finite Blocklength Codes can be adopted to further decrease network latency and enhance communication reliability. Considering the interaction among the service provider, the network operator, and users, a three‐stage Stackelberg game is constructed to dynamically arrange communication and computation resources. In this paper, the tripartite interaction problem is modelled as a generalized Nash equilibrium problem, and the reverse induction method is employed to achieve the sub‐game equilibrium. To solve the problem of maximizing tripartite utilities, the authors propose an iterative algorithm for jointly managing resource allocation and tripartite pricing. Simulation results show that compared with the benchmark, the proposed scheme can effectively improve the utilities of users, the service provider, and the network operator and achieve a win–win situation.

Lipschitz Continuity Results for a Class of Parametric Variational Inequalities and Applications to Network Games

We consider a class of finite-dimensional variational inequalities where both the operator and the constraint set can depend on a parameter. Under suitable assumptions, we provide new estimates for the Lipschitz constant of the solution, which considerably improve previous ones. We then consider the problem of computing the mean value of the solution with respect to the parameter and, to this end, adapt an algorithm devised to approximate a Lipschitz function whose analytic expression is unknown, but can be evaluated in arbitrarily chosen sample points. Finally, we apply our results to a class of Nash equilibrium problems, and generalized Nash equilibrium problems on networks.

Competitive pricing of complementary telecommunication services with subscriber churn in a duopoly

Today, due to the widespread use of smartphones, numerous applications, and their advantages in a number of daily tasks, subscribers desire a high-speed connection more than ever before. There may be some subscribers who choose to subscribe to more than one telecommunications service provider to satisfy their needs. Operators in such a competitive environment try to take advantage of this diversity of consumer needs and preferences in order to provide different services and pricing options. In this paper, a duopoly market consisting of two mobile operators is examined from a game theory perspective. By offering and pricing various bundled call and data services, as complementary services, operators try to maximize their profits. Subscribers choose the bundle with the highest surplus that best suits their telecom needs. Subscribers from various market segments may churn if they discover that other operators offer service plans with higher surpluses. The problem is formulated as a Generalized Nash Equilibrium Problem (GNEP) where operators price service plans simultaneously while they are coupled through shared constraints. After reformulating the mathematical model, a heuristic algorithm is proposed to find an equilibrium solution. In addition to extensive numerical experiments, a thorough sensitivity analysis is conducted. Finally, this study provides some managerial insights that may assist telecom operators in making optimal decisions regarding the pricing and design of their bundles in a competitive environment.

An efficient hierarchical electric vehicle charging control strategy

AbstractElectric vehicle (EV) has emerged as a crucial component in addressing both energy and environmental problems, and has become an essential part of nowadays' intelligent transportation systems. However, the charging demands of large amounts of EVs can put substantial pressure on power grid systems and cause potential grid congestion problems. In this article, we propose a hierarchical EV charging control strategy that considers the network‐wide communication overheads, computational complexity, total energy cost, EV user preferences, and data privacy protection. The hierarchical charging structure contains two phases, that is, a centralized control for EV aggregators and a distributed control for EVs within an aggregator. We prove that the centralized control with the objective to minimize the total energy cost of the power system constitutes a convex quadratic programming problem. Then a unique global optimum for the energy consumption profiles of EV aggregators can be achieved. The distributed charging control for EVs within an aggregator is studied using a generalized Nash equilibrium problem (GNEP). We show that the solution of the GNEP can be obtained via a variational inequality. Then the Solodov and Svaiter hyperplane projection method is employed to iteratively approach the variational equilibrium while ensuring the protection of EV users' data privacy. Extensive simulation studies are conducted to verify the correctness and effectiveness of our proposed hierarchical charging control algorithm for EVs.

REGULARIZED JACOBI-TYPE ADMM-METHOD FOR FINDING SOLUTIONS TO GENERALIZED NASH EQUILIBRIUM PROBLEM

In this paper, we extended the well-known alternating direction method of multipliers (ADMM) for optimization problems to generalized Nash equilibrium problems (GNEP) with shared constraints. We developed an ADMM-type algorithm with fixed regularization to tackle the problem (GNEP) where an upper estimate for the operator norm is not known and then we apply a multiplier-penalty in order to get rid of the joint constraints. We equipped the Hilbert space with an appropriate weighted scalar product and it turns out to be weakly convergent under a lipschitz and monotonicity assumption. A proximal term is then added to improve the convergence properties. Furthermore, a comparative analysis of quasi-variational inequality method, interior point method, penalty method and the proposed method are discussed.

Rational Generalized Nash Equilibrium Problems

Rational Generalized Nash Equilibrium Problems

Distributed GNE Seeking under Global-Decision and Partial-Decision Information over Douglas-Rachford Splitting Method

This paper develops an algorithm for solving the generalized Nash equilibrium problem (GNEP) in non-cooperative games. The problem involves a set of players, each with a cost function that depends on their own decision as well as the decisions of other players. The goal is to find a decision vector that minimizes the cost for each player. Unlike most of the existing algorithms for GNEP, which require full information exchange among all players, this paper considers a more realistic scenario where players can only communicate with a subset of players through a connectivity graph. The proposed algorithm enables each player to estimate the decisions of other players and update their own and others’ estimates through local communication with their neighbors. By introducing a network Lagrangian function and applying the Douglas-Rachford splitting method (DR), the GNEP is reformulated as a zero-finding problem. It is shown that the DR method can find the generalized Nash equilibrium (GNE) of the original problem under some mild conditions.

Distributed Computation of Stochastic GNE With Partial Information: An Augmented Best-Response Approach

In this paper, we focus on the stochastic generalized Nash equilibrium problem (SGNEP) which is an important and widely-used model in many different fields. In this model, subject to certain global resource constraints, a set of self-interested players aim to optimize their local objectives that depend on their own decisions and the decisions of others and are influenced by some random factors. We propose a distributed stochastic generalized Nash equilibrium seeking algorithm in a partial-decision information setting based on the Douglas-Rachford operator splitting scheme, which relaxes assumptions in the existing literature. The proposed algorithm updates players' local decisions through augmented best-response schemes and subsequent projections onto the local feasible sets, which comprise most of the computational workload. The projected stochastic subgradient method is applied to provide approximate solutions to the augmented best-response subproblems for each player. The Robbins-Siegmund theorem is leveraged to establish the main convergence results to a true Nash equilibrium using the proposed inexact solver. Finally, we illustrate the validity of the proposed algorithm via two numerical examples, i.e., a stochastic Nash-Cournot distribution game and a multi-product assembly problem with the two-stage model.

Generalized ordinal Nash games: Variational approach

It is known that the generalized Nash equilibrium problem can be reformulated as a quasivariational inequality. Our aim in this work is to introduce a variational approach to study the existence of solutions for generalized ordinal Nash games, that is, generalized games where the player preferences are binary relations that do not necessarily admit a utility representation.

Radio and Computing Resource Allocation in Co-Located Edge Computing: A Generalized Nash Equilibrium Model

Mobile Network Operators (MNO) can reduce their Capital and Operational Expenditure (CAPEX) and (OPEX) with the help of tower sharing approach by utilizing the physical infrastructure equipped by a third party tower provider to expand their network coverage. Moreover, Computing Resource Providers (CRP) are also setting up their micro-datacenters at tower stations to provide the Multi-access Edge Computing (MEC) services by cooperating with tower providers. Since both the communication and computing services contribute to the task offloading in MEC, the resource allocation has become a challenging problem. In this paper, we formulate the joint uplink, downlink, and computing resources allocation problem in which the objectives of both MNOs and CRP are to minimize their OPEX. The task offloading is modeled as a network of queues where the end-to-end latency is calculated based on the performance of the queue network. Then, the formulated problem is transformed into a Generalized Nash Equilibrium Problem (GNEP) to capture the conflicting interests in the resource allocation among MNOs and CRP. To solve the formulated GNEP efficiently, two decentralized algorithms are proposed by introducing the penalty parameters to the coupling constraints. In addition, the convergence and performance of the algorithms on different parameters are analyzed.

Online Learning for Rate-Adaptive Task Offloading Under Latency Constraints in Serverless Edge Computing

We consider the interplay between latency constrained applications and function-level resource management in a serverless edge computing environment. We develop a game theoretic model of the interaction between rate adaptive applications and a load balancing operator under a function-oriented pay-as-you-go pricing model. We show that under perfect information, the strategic interaction between the applications can be formulated as a generalized Nash equilibrium problem, and use variational inequality theory to prove that the game admits an equilibrium. For the case of imperfect information, we propose an online learning algorithm for applications to maximize their utility through rate adaptation and resource reservation. We show that the proposed algorithm can converge to equilibria and achieves zero regret asymptotically, and our simulation results show that the algorithm achieves good system performance at equilibrium, ensures fast convergence, and enables applications to meet their latency constraints.