Generalized Nash Equilibrium Research Articles

Gap functions for quasi-variational inequalities via duality

This paper deals with an application of duality theory in optimization to the construction of gap functions for quasi-variational inequalities. The same approach was investigated for variational inequalities and equilibrium problems in (Pac. J. Optim. 2(3): 667-678, 2006; Asia-Pac. J. Oper. Res. 24(3): 353-371, 2007), and the study shows that we can obtain some previous results for variational inequalities as special cases. Moreover, some applications dealing with the generalized Nash equilibrium problems and mixed variational inequalities are presented.

Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers

AbstractIn this paper, we propose a new cybersecurity investment supply chain game theory model, assuming that the demands for the product are known and fixed and, hence, the conservation law of each demand market is fulfilled. The model is a generalized Nash equilibrium model with nonlinear budget constraints for which we define the variational equilibrium, which provides us with a variational inequality formulation. We construct an equivalent formulation, enabling the analysis of the influence of the conservation laws and the importance of the associated Lagrange multipliers. We find that the marginal expected transaction utility of each retailer depends on this Lagrange multiplier and its sign. Finally, numerical examples with reported equilibrium product flows, cybersecurity investment levels, and Lagrange multipliers, along with individual firm vulnerability and network vulnerability, illustrate the obtained results.

Catchability coefficient influence on the fishermen's net economic revenues

The present paper describes a prey-predator type fishery model with two predators in competition. The aim of the paper is to maximize the net economic revenue earn from the fishery through implementing the sustainable properties of the fishery to keep the ecological balance. The existence of the steady states and the stability of the interior equilibrium point is studied using Routh Hurwitz criterion. The problem of determining the fishing effort that maximizes the net economic revenue of each fisherman results in a Generalized Nash Equilibrium Problem. More precisely, we are interested in equilibrium of mathematical game given by the situation where all fishermen try to optimize their strategies according to the strategies of all other fishermen. The importance of marine reserve is analyzed through the obtained results of the numerical simulations of proposed model system. The results depict that reserves will be most effective when the coefficient of catchability decreases.

Investigating the effect of Arctic sea routes on the global maritime container transport system via a generalized Nash equilibrium model

This paper proposes a generalized Nash equilibrium model to investigate if Arctic routes can be used as a “relief valve” for current intercontinental sea routes. This model is presented as a Stacke...

Cognitive Hierarchy Theory for Distributed Resource Allocation in the Internet of Things

In this paper, the problem of distributed resource allocation is studied for an Internet of Things (IoT) system, composed of a heterogeneous group of nodes compromising both machine-type devices (MTDs) and human-type devices (HTDs). The problem is formulated as a noncooperative game between the heterogeneous IoT devices which seek to find the optimal time allocation so as to meet their quality-of-service (QoS) requirements in terms of energy, rate, and latency. Since the strategy space of each device is dependent on the actions of the other devices, the generalized Nash equilibrium (GNE) solution is first characterized, and the conditions for uniqueness of the GNE are derived. Then, to explicitly capture the heterogeneity of the devices, in terms of resource constraints and QoS needs, a novel and more realistic game-theoretic approach, based on the behavioral framework of cognitive hierarchy (CH) theory, is proposed. This approach is then shown to enable the IoT devices to reach a CH equilibrium (CHE), a concept that takes into account the various levels of rationality corresponding to the heterogeneous computational capabilities and the information accessible for each one of the MTDs and HTDs. Simulation results show that the CHE solution maintains a stable performance. In particular, the proposed CHE solution keeps the percentage of devices with satisfied QoS constraints above 96% for IoT networks containing up to 10000 devices without considerably degrading the overall system performance in terms of the total utility. Simulation results also show that the proposed CHE solution brings a two-fold increase in the total rate of HTDs and deceases the total energy consumed by MTDs by 78% compared with the equal time policy.

A nonmonotone trust-region method for generalized Nash equilibrium and related problems with strong convergence properties

The generalized Nash equilibrium problem (GNEP) is often difficult to solve by Newton-type methods since the problem tends to have locally nonunique solutions. Here we take an existing trust-region method which is known to be locally fast convergent under a relatively mild error bound condition, and modify this method by a nonmonotone strategy in order to obtain a more reliable and efficient solver. The nonmonotone trust-region method inherits the nice local convergence properties of its monotone counterpart and is also shown to have the same global convergence properties. Numerical results indicate that the nonmonotone trust-region method is significantly better than the monotone version, and is at least competitive to an existing software applied to the same reformulation used within our trust-region framework. Additional tests on quasi-variational inequalities (QVI) are also presented to validate efficiency of the proposed extension.

Stability of Possibly Nonisolated Solutions of Constrained Equations, with Applications to Complementarity and Equilibrium Problems

We present a new covering theorem for a nonlinear mapping on a convex cone, under the assumptions weaker than the classical Robinson’s regularity condition. When the latter is violated, one cannot expect to cover the entire neighborhood of zero in the image space. Nevertheless, our covering theorem gives rise to natural conditions guaranteeing stability of a solution of a cone-constrained equation subject to wide classes of perturbations, and allowing for nonisolated solutions, and for systems with the same number of equations and variables. These features make these results applicable to various classes of variational problems, like nonlinear complementarity problems. We also consider the related stability issues for generalized Nash equilibrium problems.

Existence results of the system of generalized variational inequalities with multi-valued mappings

In this article, we study a generalized variational inequality problem with multi-valued mappings over product sets and the system of generalized variational inequalities with multi-valued mappings which are equivalent problems. By developing the idea of generalized densely relatively pseudomonotone mappings, and by using well-known Fan-KKM theorem and fixed point theorem, we prove existence results of our problem. We construct an example of generalized Nash equilibrium problem in the context of our problem, and as an application of our results, we establish an existence of coincident point result.

The noncooperative transportation problem and linear generalized Nash games

We extend the classical transportation problem from linear optimization and introduce several competing forwarders. This results in a noncooperative game which is commonly known as linear generalized Nash equilibrium problem. We show the existence of Nash equilibria and present numerical methods for their efficient computation. Furthermore, we discuss several equilibrium selection concepts that are applicable to this particular Nash game.

Semi-Decentralized Nash Equilibrium Seeking in Aggregative Games With Separable Coupling Constraints and Non-Differentiable Cost Functions

We study the Nash equilibrium seeking problem for noncooperative agents whose decision making process can be modeled as a generalized aggregative game. Specifically, we consider players with convex local cost functions, convex local constraints, and convex separable coupling constraints, and we extend the literature on generalized aggregative games by handling possibly non-differentiable cost functions. We recast the Nash equilibrium seeking problem as the problem to find a zero of a set-valued monotone operator and show that the variational Nash equilibria correspond to KKT solutions of the original game where the shared constraints have the same Lagrange multipliers for all the players. Finally, we design a semi-decentralized algorithm with global convergence guarantee for generalized Nash equilibrium seeking.

Game-Based Generation Scheduling Optimization for Power Plants Considering Long-Distance Consumption of Wind-Solar-Thermal Hybrid Systems

With the increasing penetration of renewable energy in power systems, fluctuation of renewable energy power plants has great influence on stability of the system, and renewable power curtailment is also becoming more and more serious due to the insufficient consumptive ability of local power grid. In order to maximize the utilization of renewable energy, this paper focuses on the generation scheduling optimization for a wind-solar-thermal hybrid system considering that the produced energy will be transmitted over a long distance to satisfy the demands of the receiving end system through ultra-high voltage (UHV) transmission lines. Accordingly, a bilevel optimization based on a non-cooperative game method is proposed to maximize the profit of power plants in the hybrid system. Users in the receiving end system are at the lower level of the bilevel programming, and power plants in the transmitting end system are at the upper level. Competitive behavior among power plants is formulated as a non-cooperative game and the profit of power plant is scheduled by adjusting generation and bidding strategies in both day-ahead markets and intraday markets. In addition, generation cost, wheeling cost, and carbon emissions are all considered in the non-cooperative game model. Moreover, a distributed algorithm is presented to obtain the generalized Nash equilibrium solution, which realizes the optimization in terms of maximizing profit. Finally, several simulations are implemented and analyzed to verify the effectiveness of the proposed optimization method.

Distributed Nash equilibrium seeking for aggregative games with coupled constraints

In this paper, we study a distributed continuous-time design for aggregative games with coupled constraints in order to seek the generalized Nash equilibrium by a group of agents via simple local information exchange. To solve the problem, we propose a distributed algorithm based on projected dynamics and non-smooth tracking dynamics, even for the case when the interaction topology of the multi-agent network is time-varying. Moreover, we prove the convergence of the non-smooth algorithm for the distributed game by taking advantage of its special structure and also combining the techniques of the variational inequality and Lyapunov function.

A generalized Nash equilibrium approach for optimal control problems of autonomous cars

SummaryWe consider optimal control problems with ordinary differential equations that are coupled by shared, possibly nonconvex, constraints. For these problems, we use the generalized Nash equilibrium approach and provide a reformulation of normalized Nash equilibria as solutions to a single optimal control problem. By this reformulation, we are able to prove existence, and in some settings, exploiting convexity properties, we also get a limited number or even uniqueness of the normalized Nash equilibria. Then, we use our approach to discuss traffic scenarios with several autonomous vehicles, whose dynamics is described through differential equations, and the avoidance of collisions couples the optimal control problems of the vehicles. For the solution to the discretized problems, we prove strong convergence of the states and weak convergence of the controls. Finally, using existing optimal control software, we show that the generalized Nash equilibrium approach leads to reasonable results for a crossing scenario with different vehicle models.

Algorithms for generalized potential games with mixed-integer variables

We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss–Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches.

Quasi-variational inequality problems with non-compact valued constraint maps

Quasi-variational inequality problems correspond to variational inequality problems in which the constraint set depends on the variable. They are playing nowadays an increasing role in the modelization of real life problem, in particular, because they provide a perfect framework for the reformulation of generalized Nash equilibrium problems. Our aim in this work is to establish the existence of solutions for quasi-variational inequalities defined by a non-monotone map and a constraint map which possibly admits unbounded values. The key tools are the use of coercivity conditions and Himmelberg fixed point theorem. Applications to existence of generalized Nash equilibrium is also considered.

A game-theoretic model of car ownership and household time allocation

The explosive growth of private cars in China and other developing countries has attracted a great deal of renewed research interest in car ownership. This paper investigates households’ car ownership decision-making process from the perspective of household time allocation. Applying the game-theoretic approach to capturing household members’ interactive decision-making mechanism, we propose a two-stage model that links household members’ short-term time allocation decisions to long-term car ownership decisions. The first stage models the bargaining of household members (e.g., husband and wife) over the car ownership decision, taking into consideration of government policies for regulating car ownership; and the second stage is a generalized Nash equilibrium model for activity-travel pattern analysis incorporating individuals’ interactions concerning activity participation. The existence and uniqueness of the generalized Nash equilibrium solution is examined, and a heuristic procedure that combines backwards induction and method of exhaustion is adopted to solve the two-stage game. The proposed model is applied to an empirical case study in Beijing, which demonstrates the applicability of the model in predicting car ownership and examining interactions between car ownership and household time allocation. The empirical model is applied to assess the impacts of plate-number-based vehicle usage rationing policies on car ownership and time allocation to travel and daily activities. Results show that the model can be applied to evaluate the car ownership impacts of car usage rationing policies.

A half-space projection method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is an n-person noncooperative game in which each player’s strategy set depends on the rivals’ strategy set. In this paper, we presented a half-space projection method for solving the quasi-variational inequality problem which is a formulation of the GNEP. The difference from the known projection methods is due to the next iterate point in this method is obtained by directly projecting a point onto a half-space. Thus, our next iterate point can be represented explicitly. The global convergence is proved under the minimal assumptions. Compared with the known methods, this method can reduce one projection of a vector onto the strategy set per iteration. Numerical results show that this method not only outperforms the known method but is also less dependent on the initial value than the known method.

A Game Theoretical Approach Based Bidding Strategy Optimization for Power Producers in Power Markets with Renewable Electricity

In a competitive electricity market with substantial involvement of renewable electricity, maximizing profits by optimizing bidding strategies is crucial to different power producers including conventional power plants and renewable ones. This paper proposes a game-theoretic bidding optimization method based on bi-level programming, where power producers are at the upper level and utility companies are at the lower level. The competition among the multiple power producers is formulated as a non-cooperative game in which bidding curves are their strategies, while uniform clearing pricing is considered for utility companies represented by an independent system operator. Consequently, based on the formulated game model, the bidding strategies for power producers are optimized for the day-ahead market and the intraday market with considering the properties of renewable energy; and the clearing pricing for the utility companies, with respect to the power quantity from different power producers, is optimized simultaneously. Furthermore, a distributed algorithm is provided to search the solution of the generalized Nash equilibrium. Finally, simulation results were performed and discussed to verify the feasibility and effectiveness of the proposed non-cooperative game-based bi-level optimization approach.

A Bridge Between Bilevel Programs and Nash Games

We study connections between optimistic bilevel programming problems and generalized Nash equilibrium problems. We remark that, with respect to bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a Nash game that, transforming the so-called implicit value function constraint into an explicitly defined constraint function, incorporates some taste of hierarchy and turns out to be related to the bilevel programming problem. We provide a complete theoretical analysis of the relationship between the vertical bilevel problem and our “uneven” horizontal model: in particular, we define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of our game. Furthermore, by referring to some applications in economics, we show that our “uneven” horizontal model, in some sense, lies between the vertical bilevel model and a “pure” horizontal game.

Distributed Learning for Stochastic Generalized Nash Equilibrium Problems

This work examines a stochastic formulation of the generalized Nash equilibrium problem (GNEP) where agents are subject to randomness in the environment of unknown statistical distribution. We focus on fully-distributed online learning by agents and employ penalized individual cost functions to deal with coupled constraints. Three stochastic gradient strategies are developed with constant step-sizes. We allow the agents to use heterogeneous step-sizes and show that the penalty solution is able to approach the Nash equilibrium in a stable manner within $O(\mu_\text{max})$, for small step-size value $\mu_\text{max}$ and sufficiently large penalty parameters. The operation of the algorithm is illustrated by considering the network Cournot competition problem.