Nash equilibrium in a pay-as-bid electricity market Part 2 - best response of a producer

We consider a multi-leader-common-follower model of a pay-as-bid electricity market in which the producers provide the regulator with either linear or quadratic bids. We prove that for a given producer only linear bids can maximize his profit. Such linear bids are referred as the ‘best response’ of the given producer. They are obtained assuming the demand is known and some estimate of the bids of the other producers is available. Nevertheless we also show that whenever no best response exists, the optimal profit can be asymptotically attained by a sequence of quadratic bids converging to the so-called ‘limiting best response’. An explicit formula for such a sequence is provided.     

On generalized Nash games and variational inequalities

We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.     

Generalized Nash equilibrium problem, variational inequality and quasiconvexity

It is well known that the generalized Nash equilibrium problem, a model for multi-leader–follower games, can be reformulated as a quasivariational inequality. We show that, in fact, a reformulation in terms of a variational inequality can be obtained in the general setting of quasiconvex nondifferentiable decision functions. An existence result is deduced.     

Generalized Nash equilibrium problems

The Generalized Nash equilibrium problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields. In this survey paper we aim at discussing its main properties and solution algorithms, pointing out what could be useful topics for future research in the field. The work of Christain Kanzow has been partially supported by the program “Identification, Optimization and Control with Applications in Modern Technologies” of the Elite Network of Bavaria, Germany.     

A best-response approach for equilibrium selection in two-player generalized Nash equilibrium problems

ABSTRACT

In this paper, we propose a best-response approach to select an equilibrium in a two-player generalized Nash equilibrium problem. In our model we solve, at each of a finite number of time steps, two independent optimization problems. We prove that convergence of our Jacobi-type method, for the number of time steps going to infinity, implies the selection of the same equilibrium as in a recently introduced continuous equilibrium selection theory. Thus the presented approach is a different motivation for the existing equilibrium selection theory, and it can also be seen as a numerical method. We show convergence of our numerical scheme for some special cases of generalized Nash equilibrium problems with linear constraints and linear or quadratic cost functions.     

An equilibrium model for urban transit assignment based on game theory

The urban public transport system is portrayed as a special commodity market where passenger is consumer, transit operator is producer and the special goods is the service for passenger’s trip. The generalized Nash equilibrium game is applied to describe how passengers adjust their route choices and trip modes. We present a market equilibrium model for urban public transport system as a series of mathematical programmings and equations, which is to describe both the competitions among different transit operators and the interactive influences among passengers. The proposed model can simultaneously predict how passengers choose their optimal routes and trip modes. An algorithm is designed to obtain the equilibrium solution. Finally, a simple numerical example is given and some conclusions are drawn.     

Mathematical programs with multiobjective generalized Nash equilibrium problems in the constraints

This paper considers a class of mathematical programs that include multiobjective generalized Nash equilibrium problems in the constraints. Little research can be found in the literature although it has some interesting applications. We present a single level reformulation for this kind of problems and show their equivalence in terms of global and local minimizers. We find that the reformulation is a special case of the so-called mathematical program with equilibrium constraints which is extensively studied in the literature.     

On solving generalized Nash equilibrium problems via optimization

This paper deals with the generalized Nash equilibrium problem (GNEP), i.e. a noncooperative game in which the strategy set of each player, as well as his payoff function, depends on the strategies of all players. We consider an equivalent optimization reformulation of GNEP using a regularized Nikaido–Isoda function so that solutions of GNEP coincide with global minima of the optimization problem. We then propose a derivative-free descent type method with inexact line search to solve the equivalent optimization problem and we prove that our algorithm is globally convergent. The convergence analysis is not based on conditions guaranteeing that every stationary point of the optimization problem is a solution of GNEP. Finally, we present the performance of our algorithm on some examples.     

On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables

ABSTRACT

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it.     

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.     

An improved two-step method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is a noncooperative game in which the strategy set of each player, as well as his payoff function, depend on the rival players strategies. As a generalization of the standard Nash equilibrium problem (NEP), the GNEP has recently drawn much attention due to its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. In this paper, we propose an improved two-step (a prediction step and a correction step) method for solving the quasi-variational inequality (QVI) formulation of the GNEP. Per iteration, we first do a projection onto the feasible set defined by the current iterate (prediction) to get a trial point; then, we perform another projection step (correction) to obtain the new iterate. Under certain assumptions, we prove the global convergence of the new algorithm. We also present some numerical results to illustrate the ability of our method, which indicate that our method outperforms the most recent projection-like methods of Zhang et al. (2010).     

Strategic bidding in continuous electricity auctions: an application to the Spanish electricity market

In this paper we introduce an asymmetric model of continuous electricity auctions with limited production capacity and bounded supply functions. The strategic bidding is studied with this model by means of an electricity market game. We prove that for every electricity market game with continuous cost functions a mixed-strategy Nash equilibrium always exists. In particular, we focus on the behavior of producers in the Spanish electricity market. We consider a very simple form for the Spanish electricity market: an oligopoly consisting just of independent hydro-electric power production units in a single wet period. We show that a pure-strategy Nash equilibrium for the Spanish electricity market game always exists.     

Inequalities and Nash equilibria

This paper concentrates on the problem of the existence of equilibrium points for non-cooperative generalized N-person games, N-person games of normal form and their related inequalities. We utilize the K-K-M lemma to obtain a theorem and then use it to obtain a new Fan-type inequality and minimax theorems. Various new equilibrium point theorems are derived, with the necessary and sufficient conditions and with strategy spaces with no fixed point property. Examples are given to demonstrate that these existence theorems cover areas where other existence theorems break down.     

Market power analysis in electricity markets using supply function equilibrium model

This paper presents a surveillance method based on the gametheory which is used by the ISO to find whether a power supplierin an electricity market has market power. The paper uses thesupply function equilibrium model to analyse the generationsuppliers’ bidding behaviour and models the ISO's marketpower monitoring problem as a bi-level multi-objective problem.The outer sub-problem is a multi-objective problem which maximizessuppliers’ payoffs, while the inner one is the ISO's marketclearing problem based on the locational marginal pricing mechanism.A discrete method is adopted to find ‘good enough’solutions, in a continuous bidding strategy space, which arethe intersection of all suppliers’ optimal response spacesaccording to Nash equilibrium. The paper utilizes the IEEE 118-bussystem to illustrate the application of the proposed methodwith three suppliers as price setters in the energy market andthe other generators as price takers. The numerical resultsshow that the transmission congestion may enhance the suppliers’ability to exercise market power. Likewise, suppliers’gaming behaviour could relieve the transmission congestion.It is shown that applying price caps is an efficient way ofmitigating market power.     

Nash equilibrium and the law of large numbers

Pascoa (1993a) showed that the failure of the law of large numbers for a continuum of independent randomizations implies that Schmeidler's (1973) concept of a measure-valued profile function in equilibrium might not coincide with the concept of mixed strategies equilibrium of a nonatomic game. The latter should be defined as a probability measure on pure strategies profiles which is induced by the product measure of players' mixed strategies. This paper addresses existence and approximate purification of the latter and presents an assumption on continuity of payoffs that guarantees the equivalence between the two equilibrium concepts.     

Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game

Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line.     

A variational inequality approach for the determination of oligopolistic market equilibrium

This paper presents an alternative approach to that by Murphy, Sherali and Soyster [13] for computing market equilibria with mathematical programming methods. This approach is based upon a variational inequality representation of the problem and the use of a diagonalization/relaxation algorithm.     

Maximal element with applications to Nash equilibrium problems in Hadamard manifolds

ABSTRACT

The purpose of this paper is to study the existence of maximal elements with applications to Nash equilibrium problems for generalized games in Hadamard manifolds. By employing a KKM lemma, we establish a new maximal element theorem in Hadamard manifolds. As applications, some existence results of Nash equilibria for generalized games are derived. The results in this paper unify, improve and extend some known results from the literature.     

Finding a Nash equilibrium in noncooperativeN-person games by solving a sequence of linear stationary point problems

In this paper we present an algorithm for finding a Nash equilibrium in a noncooperative normal formN-person game. More generally, the algorithm can be applied for solving a nonlinear stationary point problem on a simplotope, being the Cartesian product of several simplices. The algorithm solves the problem by solving a sequence of linear stationary point problems. Each problem in the sequence is solved in a finite number of iterations. Although the overall convergence cannot be proved, the method performs rather well. Computational results suggest that this algorithm performs at least as good as simplicial algorithms do.For the special case of a bi-matrix game (N=2), the algorithm has an appealing game-theoretic interpretation. In that case, the problem is linear and the algorithm always finds a solution. Furthermore, the equilibrium found in a bi-matrix game is perfect whenever the algorithm starts from a strategy vector at which all actions are played with positive probability.This research is part of the VF-program Co-operation and Competition, which has been approved by the Netherlands Ministery of Education and Sciences.     

Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions

We consider the generalized Nash equilibrium problem which, in contrast to the standard Nash equilibrium problem, allows joint constraints of all players involved in the game. Using a regularized Nikaido-Isoda-function, we then present three optimization problems related to the generalized Nash equilibrium problem. The first optimization problem is a complete reformulation of the generalized Nash game in the sense that the global minima are precisely the solutions of the game. However, this reformulation is nonsmooth. We then modify this approach and obtain a smooth constrained optimization problem whose global minima correspond to so-called normalized Nash equilibria. The third approach uses the difference of two regularized Nikaido-Isoda-functions in order to get a smooth unconstrained optimization problem whose global minima are, once again, precisely the normalized Nash equilibria. Conditions for stationary points to be global minima of the two smooth optimization problems are also given. Some numerical results illustrate the behaviour of our approaches.