Variational Inequality Approaches to

Generalized Nash Equilibrium Problems

Guidance

Professor Masao FUKUSHIMA

Koichi NABETANI

2006 Graduate Course

in

Department of Applied Mathematics and Physics

Graduate School of Informatics

Kyoto University

KY

OTO

UNIVERSITY

FO

UN DED 1897KYOTO JAPAN

February 2008

Abstract

We consider the generalized Nash equilibrium problem (GNEP), in which each player’s strategyset may depend on the rivals’ strategies. The GNEP can be formulated as a quasi-variationalinequality (QVI). However, unlike the standard variational inequality (VI), there are only a fewmethods available for solving a QVI efficiently. A practical approach to find a GNE is to solvea related VI rather than solving the corresponding QVI directly, and it has been receiving muchattention recently. From the viewpoint of game theory, it is important to find GNEs as manyas possible, or all of them if possible. We propose the price-directed and the resource-directedparametrization methods that construct parametrized families of VIs related to the GNEP. Wethen show that those VIs yield all GNEs under suitable conditions. Moreover, by means of somenumerical examples, we show that GNEs obtained by the proposed VI approaches are widelydistributed in the GNEP solution set.

Contents

1 Introduction 1

2 Problem Formulation and Assumptions 2

3 Parameterized VI Approaches to GNE with Shared Constraints 43.1 Price-Directed Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Resource-Directed Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Relation between some GNE concepts and VI approaches . . . . . . . . . . . . . . 10

4 Numerical Experiments 134.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Concluding Remarks 21

1 Introduction

The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equi-librium problem (NEP), in which each player’s strategy set may depend on the rivals’ strategies[1, 23, 8]. Recently, the GNEP has attracted much attention [5, 14] because there are manyinteresting applications in the fields of economics, mathematics and engineering. For example,Robinson [20, 21] discussed a two-sided game model of combat as an application of GNEP. Weiand Smeers [26] and Hobbs [11] formulated oligopolistic electricity models as GNEPs.

It is well known that NEP where each player solves a convex programming problem can beformulated as a finite-dimensional variational inequality (VI) [9, 6]. The VI has a long history andthe state-of-the-art and various computational methods are presented in the recent monograph[6]. On the other hand, GNEP can be formulated as a quasi-variational inequality (QVI) [8, 19].However, unlike the VI, there are only a few methods available for computing a generalized Nashequilibrium (GNE) or solving a QVI efficiently [17, 19, 18]. A practical approach to GNEP is tofind a GNE via a VI rather than solving a QVI directly [8, 26, 18, 19, 3, 4] and it has receivedmuch attention recently.

Pang and Fukushima [19] propose a penalty method for GNEP, which solves a sequence ofpenalized NEPs, and establish its convergence under some assumptions. Pang [18] formulatesGNEP as a partitioned VI via the equivalent Karush-Kuhn-Tucker (KKT) system and proposes toapply the Josephy-Newton method that solves a linearized VI at each iteration. Local superlinearconvergence is established under convexity and regularity assumptions.

Wei and Smeers [26] formulate an oligopolistic electricity model as a GNEP with “sharedconstraints”, which means the constraint functions that depend on rivals’ strategies are identicalamong all players. They present a VI formulation such that its solution is GNE, and establish theuniqueness of GNE under some restrictive assumptions. Facchinei et al. [3] also consider a GNEPwith shared constraints and propose to apply Newton-type methods to solve its VI formulation.This approach seems to be promising because we can find a GNE by solving a single VI. However,the obtained GNE is a particular equilibrium such that the multipliers of the shared constraintsare identical [3, 4]. In general, GNEP has multiple, or even infinitely many, solutions [8]. Inpractical situations, the players usually have different objective functions, and the multipliers ofthe shared constraints need not be identical. Therefore, the VI formulation considered in [26, 3, 4]may fail to identify some important GNEP solutions.

Another approach for GNEP is the algorithms [24, 13, 10] based on the Nikaido–Isoda function[16]. Uryasev and Krawczyk [24, 13] develop the relaxation method and establish the globalconvergence of the algorithm under some assumptions, which are, however, rather restrictive. Ina recent paper, von Heusinger and Kanzow [10] consider the regularization of the Nikaido–Isodafunction and reformulate a GNEP with shared constraints as a smooth optimization problem.However, it is shown that these methods also find only a part of GNEP solutions in general.

From the practical viewpoint of game theory, it is important to find as many GNEs as possible,or all of them if possible [25]. To the best of our knowledge, little attempt has been made todevelop a method to achieve this. The goal of the present paper is to construct, for GNEP withshared constraints, a parametrized family of VIs whose solutions include all GNEs. This extendsthe VI approaches studied in [26, 3, 4] and show that all GNEP solutions are contained in thesolution sets of those VIs. Moreover, we clarify the conditions that guarantee a solution of thoseVIs to be a GNE.

The paper is organized as follows. In the next section, we recall some definitions and basicconcepts. In Section 3, we present the parametrized family of VIs and show that those VIs yieldall GNEP solutions. We also show that the parameters involved in VIs can be restricted to a

1

bounded set. In Section 4, we present some numerical results with the proposed approach. Finally,Section 5 concludes the paper.

We use the following notations throughout the paper. For a set X, P(X) denotes the set of allsubsets of X. For a nonempty closed convex set X ⊂ �n, NX(x) = {d ∈ �n | dT (y − x) ≤ 0 ∀y ∈X} denotes the normal cone to X at x ∈ X. For a function f : �n × �m → �, f(x, ·) : �m → �denotes the function with x being fixed. We denote the nonnegative and positive orthants in �n

by �n+ and �n

++, respectively, that is,

�n+ := {x ∈ �n | x ≥ 0} and �n

++ := {x ∈ �n | x > 0}.

For vectors x, y ∈ �n, 〈x, y〉 denotes the inner product defined by 〈x, y〉 := xT y and x ⊥ y means〈x, y〉 = 0. For a vector x, ‖x‖ denotes the Euclidean norm defined by ‖x‖ :=

√〈x, x〉. A mappingF : �n → �n is said to be monotone on a nonempty closed convex set X ⊂ �n if

〈F (x) − F (y), x − y〉 ≥ 0 ∀x, y ∈ X,

strictly monotone on X if the above inequality is strict whenever x = y, and strongly monotoneon X if zero on the right-hand side of the above inequality is replaced by α‖x − y‖2 for someconstant α > 0.

2 Problem Formulation and Assumptions

The generalized Nash game with N players is to find a profile of strategies such that each player’sstrategy is an optimal response to the rival players’ strategies, where each player’s strategy setmay depend on the rival players’ strategies. For ν = 1, . . . , N , let xν ∈ �nν be a player ν’sstrategy, where nν is a positive integer. The vector formed by all these strategies is denotedx := (xν)Nν=1 ∈ �n, where n :=

∑Nν=1 nν , and the vector formed by all the players’ strategies

except those of player ν is denoted x−ν := (xν′)Nν′=1, ν′ �=ν ∈ �n−ν , where n−ν := n − nν . Forν = 1, . . . , N , let Kν be a given point-to-set mapping from �n−ν to �nν . Thus, for each fixed x−ν ,Kν(x−ν) is a subset of �nν , which is the strategy set of player ν with the other players’ strategiesgiven by x−ν .

Each player ν = 1, . . . , N , taking the other players’ strategies x−ν as exogenous variables,solves the minimization problem:

Pν(x−ν) : minimize θν(xν , x−ν)

subject to xν ∈ Kν(x−ν),(1)

where θν : �n → � is a given cost function of player ν. A vector x = (xν)Nν=1 is said to be feasibleto GNEP if xν ∈ Kν(x−ν) for each ν = 1, . . . , N .

The GNEP is to find a vector x∗ = (x∗ν)Nν=1 ∈ �n such that

x∗ν is an optimal solution of Pν(x∗

−ν) for all ν = 1, . . . , N. (2)

A vector x∗ satisfying (2) is called a generalized Nash equilibrium (GNE). The set of GNEs isdenoted by SOL∗.

In many practical applications, the strategy set Kν(x−ν) of player ν is represented by finitelymany inequality constraints. In particular, we assume that the feasible strategy set Kν(x−ν) ofplayer ν has the form

Kν(x−ν) = {xν ∈ Xν | gν(xν , x−ν) ≤ 0}, (3)

2

where gν = (gν,i)mνi=1 : �n → �mν and Xν ⊂ �nν , with mν a nonnegative integer. Thus, player

ν’s strategy is constrained in two ways; joint constraints that depend also on the other players’strategies, i.e., gν(x) ≤ 0, and individual constraints that depend only on player ν’s strategy, i.e.,xν ∈ Xν . In what follows, we let

X−ν :=N∏

ν′=1ν′ �=ν

Xν′ .

We distinguish these two types of constraints since our parametrization will involve only the jointconstraints. Throughout this paper, we make the following blanket assumption on the smoothnessand convexity of functions involved in the GNEP.

Assumption A. For ν = 1, . . . , N , the set Xν is nonempty, closed, convex, and, for each fixedx−ν ∈ X−ν , the functions θν(·, x−ν) and gν,i(·, x−ν), i = 1, . . . ,mν, are differentiable and convex.

By Assumption A, problem (1) is a differentiable convex programming problem. Thus a nec-essary and sufficient condition for x∗

ν ∈ Kν(x∗−ν) to be optimal for (1) is that the inequalities

〈∇xν θν(x∗ν , x

∗−ν), xν − x∗

ν〉 ≥ 0 ∀xν ∈ Kν(x∗−ν)

hold. Thus, by defining

F (x) := (∇xνθν(xν , x−ν))Nν=1 , (4)

K(x) :=N∏

ν=1

Kν(x−ν),

it follows that x∗ is a GNE if and only if x∗ ∈ K(x∗) and

〈F (x∗), x − x∗〉 ≥ 0 ∀x ∈ K(x∗).

The latter problem is a QVI, which we denote by QVI (F,K).Suppose that x∗ is a solution of GNEP. Then, for each ν = 1, . . . , N , x∗

ν is an optimal solutionof the convex programming problem:

Pν(x∗−ν) : minimize θν(xν , x∗−ν)

subject to gν(xν , x∗−ν) ≤ 0, xν ∈ Xν .

Under a suitable CQ at x∗ (see, e.g., [2, Section 5.4], [22]), there exists for each ν = 1, . . . , N avector λ∗

ν ∈ �mν satisfying the Karush-Kuhn-Tucker (KKT) condition:

0 ∈ ∇xν Lν(xν , x∗−ν , λν) + NXν (xν),

0 ≤ λν ⊥ gν(xν , x∗−ν) ≤ 0, xν ∈ Xν ,(5)

where the Lagrangian function Lν is defined by

Lν(x, λν) := θν(x) + 〈gν(x), λν〉.

The vector λ∗ = (λ∗ν)Nν=1 is called a Lagrange multiplier vector. Under Assumption A, if (x∗, λ∗)

satisfies (5), then x∗ is a GNE. A well-known CQ at x is the Mangasarian-Fromovitz CQ (MFCQ)[22, page 198]:{

0 ∈ ∇xνgν(x)λν + NXν (xν),0 ≤ λν ⊥ gν(x) ≤ 0, xν ∈ Xν

}=⇒ λν = 0, ν = 1, . . . , N,

3

where

∇xνgν(x) :=(∇xν gν,1(x) · · · ∇xνgν,mν (x)

).

Another useful CQ at x is the Linear Independence CQ (LICQ):{

0 ∈ ∇xνgν(x)λν + NXν (xν) + (−NXν (xν)),λν ⊥ gν(x) ≤ 0, xν ∈ Xν

}=⇒ λν = 0, ν = 1, . . . , N. (6)

This CQ implies uniqueness of the multiplier vector λ∗ for each x∗.

3 Parameterized VI Approaches to GNE with Shared Constraints

In this section, we consider the case of “shared constraints”, that is, all players share commonconstraints that depend on all the players’ strategies. Specifically, we make the following blanketassumption.

Assumption B. For some m and g = (gi)mi=1 : �n → �m, we have m1 = · · · = mN = m andg1 = · · · = gN = g. Moreover, gi : �n → � is differentiable and convex for all i = 1, . . . ,m.

Then, (x∗, λ∗) satisfies the KKT condition (5) if and only if (x∗, λ∗) satisfies

0 ∈ ∇xνθν(x) + ∇xνg(x)λν + NXν (xν), ν = 1, . . . , N,

0 ≤ λν ⊥ g(x) ≤ 0, xν ∈ Xν , ν = 1, . . . , N.(7)

Note from the complementarity condition that gi(x) < 0 implies that λν,i = 0 for all ν = 1, . . . , Nand λν,i > 0 for some ν implies that gi(x) = 0.

The VI approach to finding a GNE [26, 3, 4] is to define

X := {x ∈ X1 × · · · × XN | g(x) ≤ 0},

which is a closed convex (possibly empty) set, and solve the following VI (F,X):

Find x∗ ∈ X such that 〈F (x∗), x − x∗〉 ≥ 0 ∀x ∈ X,

where F : �n → �n is defined by (4). The solution set of VI (F,X) is denoted by SOL (F,X) 1. Avector x∗ belongs to SOL (F,X) if and only if it is an optimal solution of the convex programmingproblem:

minimize 〈F (x∗), x〉subject to x ∈ X,

(8)

whose KKT condition is a special case of (7) with λ1 = · · · = λN . Specifically, the following resultis known [3, Theorem 3.6] (also see [26, Theorem 2] and [4, Theorem 2.1]).

Theorem 3.1. Every x∗ ∈ SOL (F,X) is a GNE. Furthermore, if x∗ together with some Lagrangemultiplier vector satisfies the KKT condition for (8), then x∗ and some λ∗ = (λ∗

ν)Nν=1 with λ∗

1 =· · · = λ∗

N satisfy (7).

1A solution of VI (F, X) is called a variational equilibrium in [5] and is shown to be a particular instance of a

normalized equilibrium introduced by Rosen [23]. In Section 3.3, we will show the relation between normalized

equilibrium and the solutions of VI formulations.

4

In many practical situations, since the players have different objective functions as well astheir own constraints, the multipliers of the shared constraints in GNEP may not be identical.Therefore, in general, there would be many GNEs that are not normalized equilibria. This isillustrated in the following example.

Example 1. Consider the two-person game, where the problems of player 1 and player 2 aredefined by

P1(x2) : minimize x21 − x1x2 − x1

subject to x1 ≥ 0x1 + x2 ≤ 1,

and

P2(x1) : minimize x22 − 1

2x1x2 − 2x2

subject to x2 ≥ 0x1 + x2 ≤ 1,

respectively. The set of GNEs consists of infinitely many vectors(x1

x2

)=(

t

1 − t

), 0 ≤ t ≤ 2

3 .

On the other hand, the corresponding F : �2 → �2 and X ⊆ �2 are given by

F (x) =(

2x1 − x2 − 1−1

2x1 + 2x2 − 2

), X = {x ∈ �2

+ | x1 + x2 ≤ 1},

and the solution of VI (F,X) is uniquely given by x =(

411 , 7

11

)T .

In this example, the mapping F is strongly monotone and hence SOL (F,X) is a singleton [6],but there are infinitely many GNEs. Uniqueness of GNE requires restrictive assumptions [8, 26]which cannot be expected to hold in most applications. In general, the above VI approach canfind only a part of the GNEs.

3.1 Price-Directed Parametrization

Now, we construct a family of VIs that contains VI (F,X) as a particular instance. Let Δ =(Δν)Nν=1 ∈ �m

+ with m = Nm and Δν ∈ �m+ , ν = 1, . . . , N , be a vector of parameters. Define the

function FΔ : �n → �n by

FΔ(x) := (∇xνθν(x) + ∇xνg(x)Δν)Nν=1 . (9)

Note that VI (FΔ,X) reduces to VI (F,X) if the parameter Δ is set to be zero. The follow-ing proposition shows that, under appropriate assumptions, the monotonicity of F implies themonotonicity of FΔ for any Δ ∈ �m

+ .

Proposition 3.1. Assume that F is monotone (strictly monotone, strongly monotone) on X.Assume further that the shared constraint function g is separable, that is, g(x) can be written as

g(x) =N∑

ν=1

gν(xν), (10)

where gν = (gν,i)mi=1 : �nν → �m, ν = 1, . . . , N , are differentiable convex functions. Then, forany Δ ∈ �m

+ , FΔ is also monotone (strictly monotone, strongly monotone) on X.

5

Proof. Since g is separable, we have

FΔ(x) = F (x) + (∇gν(xν)Δν)Nν=1 .

For each ν = 1, . . . , N and i = 1, . . . , m, ∇gν,i is monotone on Xν since gν,i is a differentiableconvex function. Therefore FΔ is monotone (strictly monotone, strongly monotone) on X for anyΔ ∈ �m

+ .

Under the assumptions of this proposition, we can apply, for example, Newton-type or projection-type methods to solve VI (FΔ,X). A sufficient condition for F to be strictly monotone in thesetting of spatial oligopolistic electricity models is given in [26, Theorems 5 and 6].

We now investigate the relationship between VI (FΔ,X) and GNEP. The KKT condition forVI (FΔ,X) can be written as

0 ∈ (∇xνθν(x) + ∇xνg(x)Δν) + ∇xνg(x)π + NXν (xν), ν = 1, . . . , N,

0 ≤ π ⊥ g(x) ≤ 0, xν ∈ Xν , ν = 1, . . . , N.(11)

For each GNE x, let

M(x) := {λ ∈ �m | (x, λ) satisfies the KKT condition (7)}.

By comparing this KKT condition (11) with (7), we have the following result.

Theorem 3.2. For any GNE x∗, if λ∗ ∈ M(x∗), then x∗ ∈ SOL (F λ∗,X).

Proof. Fix any GNE x∗, and assume that λ∗ ∈ M(x∗). Then (x∗, λ∗) satisfies the KKT condition(7). This in turn shows that (x∗, 0) satisfies the KKT condition for VI (F λ∗

,X) and hence x∗ isa solution of VI (F λ∗

,X).

Corollary 3.1. If M(x∗) = ∅ for every GNE x∗, then⋃Δ∈�m

+

SOL (FΔ,X) ⊃ SOL∗.

For an arbitrary GNE x∗, let λ∗ ∈ M(x∗) and define

δi := minν=1,...,N

λ∗ν,i i = 1, . . . , m

λν,i := λ∗ν,i − δi i = 1, . . . , m, ν = 1, . . . , N.

Then x∗ along with δ satisfies the KKT condition for VI (F λ,X), and hence x∗ also belongs toSOL (F λ,X). This implies that for any GNE x∗ satisfying M(x∗) = ∅, there always exists aΔ ∈ �m

+ such that x∗ ∈ SOL (FΔ,X) and, for each i, Δν,i = 0 for some ν. This observation yieldsthe following result that sharpens Corollary 3.1.

Corollary 3.2. If M(x∗) = ∅ for every GNE x∗, then⋃Δ∈P

SOL (FΔ,X) ⊃ SOL∗,

where the set P is defined by

P :=m∏

i=1

(N⋃

ν=1

{Δ ∈ �N

+

∣∣ Δν = 0}) ⊂ �m

+ .

6

This corollary suggests that, for finding GNEs, it is enough to restrict ourselves to the parameterset P instead of �m

+ .In general, a solution of VI (FΔ,X) for some Δ ∈ �m

+ need not be a GNE. To see this, let usconsider the VI (FΔ,X) in Example 1 with Δ = (1, 1)T . Then

FΔ(x) =(

2x1 − x2

−12x1 + 2x2 − 1

),

and VI (FΔ,X) has the unique solution x =(

27 , 4

7

)T which is not a GNE.The next result gives a sufficient condition for a solution of VI (FΔ,X) to be a GNE.

Theorem 3.3. For any Δ ∈ �m+ and any (x∗, π∗) satisfying (11), a sufficient condition for x∗ to

be a GNE is that

〈g(x∗),Δν〉 = 0, ν = 1, . . . , N. (12)

If in addition the LICQ (6) holds at x∗, then (12) is also a necessary condition for x∗ to be aGNE.

Proof. Letλ∗

ν := π∗ + Δν ≥ 0, ν = 1, . . . , N.

By (11) and (12), we have

〈g(x∗), λ∗ν〉 = 〈g(x∗), π∗ + Δν〉 = 0.

Hence (x∗, λ∗) satisfies the KKT condition (7). This shows that x∗ is a GNE.Conversely, suppose that x∗ is a GNE and the LICQ (6) holds at x∗. By LICQ, the Lagrange

multiplier vector λ∗ satisfying (7) with x = x∗ and the Lagrange multiplier vector π∗ satisfying(11) with x = x∗ are both unique. Then we must have

λ∗ν = π∗ + Δν , ν = 1, . . . , N.

Moreover, 〈g(x∗), λ∗ν〉 = 0, ν = 1, . . . , N and 〈g(x∗), π∗〉 = 0, which together yield

〈g(x∗),Δν〉 = 〈g(x∗), λ∗ν − π∗〉 = 0, ν = 1, . . . , N.

Remark 3.1. When Δ = 0, we have FΔ ≡ F . Moreover, (12) clearly holds. Therefore, Theorem3.3 contains the result of Theorem 3.1.

Corollary 3.1 shows that SOL∗ is contained in the union of SOL (FΔ,X) over all Δ ∈ �m+ . We

show below that, under the following sequentially bounded CQ (SBCQ), the range of parameterΔ can be restricted to a bounded set Λ ⊆ �m

+ and the union of SOL (FΔ,X) still contains an“arbitrarily large” subset of SOL∗.

Definition 3.1 (SBCQ). For every bounded sequence {xk} ⊂ SOL∗, there exists a boundedsequence {λk} satisfying λk ∈ M(xk) for all k.

The SBCQ was introduced in the study of the mathematical program with equilibrium con-straints (MPEC) [15]. It is a unification of well-known CQs such as MFCQ and the constant rankCQ [12], and plays an important role not only in MPEC but also in GNEP [19]. It can be shownthat if the function g is affine and X1, . . . ,XN are polyhedral sets, then SBCQ holds [19].

7

Theorem 3.4. Assume that SBCQ holds. For any bounded set C ⊂ �n, there exists a boundedset Λ ⊂ �m

+ such that ⋃Δ∈Λ

SOL (F,X) ⊃ C ∩ SOL∗.

Proof. Fix any bounded set C ⊂ �n. We claim that there exists a bounded set Λ ⊂ �m+ such that

M(x) ∩ Λ = ∅ ∀x ∈ C ∩ SOL∗. (13)

If this were not true, then there would exist a sequence{xk} ⊂ C ∩ SOL∗ such that minλ∈M(xk) ‖λ‖ →

∞. (Here we use the convention that minλ∈M(xk) ‖λ‖ = ∞ whenever M(xk) = ∅.) However, since{xk} lies in a bounded set C, SBCQ would imply that minλ∈M(xk) ‖λ‖ is bounded, a contradiction.

Fix any x∗ ∈ C ∩ SOL∗. By (13), there exists λ∗ ∈ M(x∗) ∩ Λ. By Theorem 3.2, x∗ ∈SOL (F λ∗

,X). Thus every element of C ∩ SOL∗ belongs to SOL (F λ∗,X) for some λ∗ ∈ Λ. This

proves the desired inclusion.

If SOL∗ is bounded, then we can take C = SOL∗. Unfortunately, Theorem 3.4 does not sayhow large Λ should be. This is a question for further study.

3.2 Resource-Directed Parametrization

In the case where g is affine and X1, . . . ,XN are polyhedral, it is known that M(x) = ∅ forevery GNE x. Otherwise, M(x) could be empty for some GNE x, and the price-directed dualparametrization approach of Section 3.1 would not be able to find this GNE. In this sectionwe consider a resource-directed primal parametrization that does not rely on the existence of aLagrange multiplier vector. We motivate this with an example.

Example 2. Consider a modification of Example 1 where the shared constraint x1 + x2 ≤ 1 ischanged to x2

1 + x22 ≤ 1. It can be seen that the GNEs are the vectors(

x1

x2

)=(

t√1 − t2

), 0 ≤ t ≤ 4

5 .

The corresponding VI (F,X) still has only one solution since F is unchanged and remains stronglymonotone. At the GNE x = (0, 1)T , M(x) = ∅ (due to the constraint x2

1 ≤ 0 in the problem ofplayer 1). Thus the approach of Section 3.1 would not find this GNE.

The shared constraint function in Example 2 is separable, which we will exploit in developingour primal, or resource-directed, parametrization. Specifically, we make the following blanketassumption in this subsection.

Assumption C. The shared constraint function g has the form (10), where gν = (gν,i)mi=1 :�nν → �m, ν = 1, . . . , N , and each gν,i is a differentiable convex function.

Now, we construct a family of VIs that contains VI (F,X) as a particular instance. Let β =(βν)Nν=1 ∈ �m with βν ∈ �m, ν = 1, . . . , N , be a vector of parameters satisfying

∑Nν=1 βν = 0.

Define the set Xβ ⊂ X by

Xβ := Xβ11 × · · · × XβN

N with Xβνν := {xν ∈ Xν | gν(xν) ≤ βν} , ν = 1, . . . , N,

and consider VI (F,Xβ). By Assumption C, Xβ is closed and convex (possibly empty). In-tuitively, we parametrize the division of resources among the players, reminiscent of Bender’sdecomposition.

Now, we investigate the relationship between VI (F,Xβ) and GNEP. The following result iseasy to see.

8

Theorem 3.5. For any GNE x∗, we have x∗ ∈ SOL (F,Xβ) and∑N

ν=1 βν = 0, where we letβν = gν(x∗

ν) − ανg(x∗) with arbitrary real numbers αν such that∑N

ν=1 αν = 1 and αν > 0, ν =1, . . . , N .

Corollary 3.3. ⋃PN

ν=1 βν=0

SOL (F,Xβ) ⊃ SOL∗.

In Example 2, if we choose g1(x1) = x21 and g2(x2) = x2

2 − 1, then SOL (F,Xβ) ={(0, 1)T

}for

β = (0, 0)T . In general, VI (F,Xβ) need not have a solution or a solution need not be a GNE.The next result gives a sufficient condition for a solution of VI (F,Xβ) to be a GNE.

Theorem 3.6. For any β ∈ �m with∑N

ν=1 βν = 0 and any x∗ ∈ SOL (F,Xβ), a sufficientcondition for x∗ to be a GNE is that

for each i = 1, . . . , m,{

either gν,i(x∗ν) = βν,i ν = 1, . . . , N

or gν,i(x∗ν) < βν,i ν = 1, . . . , N.

}(14)

Proof. Since x∗ is a solution of VI (F,Xβ), for each ν = 1, . . . , N , we have that x∗ν is a solution

of VI (∇xνθν(·, x∗−ν),Xβνν ) or, equivalently, x∗

ν is an optimal solution of the convex programmingproblem:

minimize θν(xν , x∗−ν)

subject to xν ∈ Xν , gν(xν) ≤ βν .(15)

By (14) and Assumption C, for each i, we have either gν,i(x∗ν) = βν,i = −∑

ν′ �=ν βν′,i = −∑ν′ �=ν gν′,i(x∗

ν′),implying gi(x∗) = 0, or gν,i(x∗

ν) < βν,i = −∑ν′ �=ν βν′,i < −∑

ν′ �=ν gν′,i(x∗ν′), implying gi(x∗) < 0.

Thus, for each ν, x∗ν is also an optimal solution of the problem:

minimize θν(xν , x∗−ν)

subject to xν ∈ Xν , g(xν , x∗−ν) ≤ 0.(16)

In particular, the active inequality constraints at x∗ν in (15) coincide with those in (16). In view

of Assumption B and (3), the above problem is exactly Pν(x∗−ν). This shows that (2) holds andhence x∗ is a GNE.

Notice that, for any GNE x∗, the β given in Theorem 3.5 satisfies the sufficient condition (14).Thus, we can refine Corollary 3.3 to⋃

PNν=1 βν=0

(14) holds for some x∗∈SOL (F,Xβ)

SOL (F,Xβ) ⊃ SOL∗.

If there exist aν ∈ �m, ν = 1, . . . , N , such that

gν(xν) ≥ aν , ∀xν ∈ Xν , ν = 1, . . . , N,

then we can further restrict β to the bounded set{β ∈ �m

∣∣∣∣∣N∑

ν=1

βν = 0, βν ≥ aν , ν = 1, . . . , N

}.

In Example 2, if we choose g1(x1) = x21 and g2(x2) = x2

2 − 1, then we can take as lower boundsa1 = 0 and a2 = −1.

9

If a solution x∗ of VI (F,Xβ) has a Lagrange multiplier λ∗ν associated with each constraint

gν(xν) ≤ βν , i.e., 0 ≤ λ∗ν ⊥ gν(x∗

ν) − βν ≤ 0, then letting

π∗ = minν=1,...,N

λ∗ν , Δν = λ∗

ν − π∗, ν = 1, . . . , N,

where the “min” is taken componentwise, we see that (x∗, π∗) satisfies the KKT condition (11)for VI (FΔ,X). Thus, VI (FΔ,X) may be viewed as the dual of VI (F,Xβ), with the formerrequiring separability of shared constraints and the latter requiring a CQ to ensure existence ofLagrange multipliers.

3.3 Relation between some GNE concepts and VI approaches

In this subsection, we discuss the normalized equilibrium introduced by Rosen [23] in relation tothe solution sets obtained by the VI approaches discussed in the previous subsections. We makethe following assumption through this subsection.

Assumption D. A suitable CQ holds at every GNEs.

Assumption D implies that for every GNE x∗ there exists a Lagrange multiplier λ∗ = (λ∗ν)Nν=1 ∈

�m that satisfies the KKT system (5).The normalized equilibrium is defined as follows [23]:

Definition 3.2. Let a GNE x∗ together with a Lagrange multiplier λ∗ = (λ∗ν)Nν=1 ∈ �m satisfies

the KKT system (5). We call x∗ a normalized equilibrium if there exists vectors r ∈ �N++ and

λ ∈ �m such thatλ∗

ν = λ/rν ν = 1, . . . , N.

Rosen [23] proves that there exists a normalized equilibrium for any given r = (rν)Nν=1 ∈ �N++ if

the set X is compact. He defines the weighted Nikaido-Isoda-type function ρ : �n×�n×�N → �by

ρ(x, y, r) =N∑

ν=1

rνθν(x1, . . . , yν , . . . , xN ),

and shows that there exists a point x∗ ∈ X such that

ρ(x∗, x∗, r) = maxy∈X

{ρ(x∗, y, r)} (17)

to establish the existence of a normalized equilibrium. Note that it is easy to see that the KKTcondition for (17) is equivalent to the KKT condition for the VI (F r,X) where F r : �n → �n isgiven by

F r(x) = (rν∇xνθν(x))Nν=1 .

Rosen’s results may be summarized with our notation as follows.

Theorem 3.7. [23, Theorem 3] For each r ∈ �N++, any solution of VI (F r,X) is a normalized

equilibrium.

Theorem 3.8. [23, Theorem 4] Assume that X is bounded and F r is strictly monotone for everyr ∈ Q, where Q is a subset of �N

++. Then there is a unique normalized equilibrium for each r ∈ Q.

10

Note that the monotonicity of F does not imply the monotonicity of F r. In fact, consider themapping F : �2 → �2 defined by

F (x1, x2) =(

2x1 + x2

x1 + 2x2

)

is monotone. However, a mapping F r : �2 → �2 with weight (r1, r2) = (15, 1) is given by

F r(x1, x2) =(

30x1 + 15x2

x1 + 2x2

),

which is not monotone because for vectors x = (1,−4) and y = (0, 0) we have

〈F r(x) − F r(y), x − y〉 = −2 < 0.

Now we investigate the difference between normalized equilibrium and GNE, and besides, wediscuss the relation with the solution set of the VI formulations. Below, the set of normalizedequilibria is denoted by SOL N .

From the definition of these concepts among with Theorems 3.1, 3.7 and Corollary 3.1, we havethe following relation:

SOL (F,X)

⊆

SOL N =⋃

r∈�N++

SOL (F r,X)⊆ ⊆

SOL∗ ⊆⋃

Δ∈�m+

SOL (FΔ,X).

For any (x, λ) satisfying the KKT system (5), we say strict complementarity holds at (x, λ) ifgi(x) = 0 implies λν,i > 0 for any ν = 1, . . . , N and i = 1, . . . ,m.

The next proposition states that, when there is only one shared constraint, all GNE are nor-malized equilibrium as long as the strict complementarity holds.

Proposition 3.2. Suppose that there is only one shared constraint and assume that M(x∗) = ∅for every GNE x∗. If strict complementarity holds at (x∗, λ∗) for every GNE x∗ and some λ∗ ∈M(x∗), then every GNE x∗ is a normalized equilibrium.

Proof. If the shared constraint is inactive, i.e., g(x∗) < 0, then we have λ∗ν = 0, ν = 1, . . . , N and

it is obvious that x∗ is a normalized equilibrium.Suppose that g(x∗) = 0. Then it implies λ∗

ν > 0 for all ν by the strict complementarityassumption. Let λ ∈ � and a weight vector r ∈ �N be given by

λ := 1

rν :=1λ∗

ν

, ν = 1, . . . , N.

Then we have λ∗ν = λ

rνfor ν = 1, . . . , N . This indicates that x∗ is a normalized equilibrium.

Corollary 3.4. Suppose that the assumptions of Proposition 3.2 hold. Then we have⋃r∈�N

++

SOL (F r,X) = SOL N = SOL∗,

and hence the following relation holds:

11

SOL (F,X)

⊆

SOL N =⋃

r∈�N++

SOL (F r,X)

= ⊆

SOL∗ ⊆⋃

Δ∈�m+

SOL (FΔ,X).

From the above results, in the single shared constraint case, we may expect to obtain all GNEby means of parametrization with weights rν on the objective functions θν .

In the case of multiple shared constraints, however, such parametrization may fail to yield theset of GNE, since a GNE is not necessarily a normalized equilibrium in general. This is illustratedin the next example.

Example 3. Consider the following two-person game, where player 1 and player 2 solve theminimization problems

P1(z) : minimizex,y

x2 + xy + y2 + (x + y)z − 25x − 38y

subject to x ≥ 0y ≥ 0x + 2y − z ≤ 143x + 2y + z ≤ 30,

P2(x, y) : minimizez

z2 + (x + y)z − 25z

subject to z ≥ 0x + 2y − z ≤ 143x + 2y + z ≤ 30,

respectively. This is a GNEP with two shared constraints.The solution set of this GNEP consists of vectors⎛

⎝x

y

z

⎞⎠ =

⎛⎝ t

11 − t

8 − t

⎞⎠ , t ∈ [0, 2].

Hence the shared constraints are both active at each GNE.Note that the VI (F,X) has a single solution⎛

⎝x

y

z

⎞⎠ =

⎛⎝ 0

118

⎞⎠ .

The Lagrange multipliers λ = (λν,i) ∈ �4 corresponding to the GNE (x(t), y(t), z(t)) = (t, 11 −t, 8 − t) is given by ⎛

⎜⎜⎜⎝λ1,1(t)λ1,2(t)λ2,1(t)λ2,2(t)

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎝

32t + 31 − t

2

s + 2 − 2ts

⎞⎟⎟⎟⎠ , s ≥ max{0, 2t − 2}.

Thus the tuple (λ1, λ2, r1, r2) ∈ �4+ satisfying

λ1

r1= λ1,1(t),

λ2

r1= λ1,2(t),

λ1

r2= λ2,1(t),

λ2

r2= λ2,2(t)

12

exists only if 0 ≤ t < 1. Hence, the following relation holds in this example:⎧⎨⎩⎛⎝ 0

118

⎞⎠⎫⎬⎭ = SOL(F,X)

�

⎧⎨⎩⎛⎝ t

11 − t

8 − t

⎞⎠∣∣∣∣∣∣ 0 ≤ t < 1

⎫⎬⎭ = SOL N

�

⎧⎨⎩⎛⎝ t

11 − t

8 − t

⎞⎠∣∣∣∣∣∣ 0 ≤ t ≤ 2

⎫⎬⎭ = SOL∗.

This example indicates that, when there are more than one shared constraints, some GNE maynot be obtained by the VI approach using parametrization with weights on the objective functions.

4 Numerical Experiments

In this section, we show some numerical experiments with our VI approaches. We give a briefdescription of the GNEPs taken from the literature in the next subsection and present numericalresults of our approaches in Subsection 4.2.

4.1 Examples

Example 4 (Harker’s example). This problem is taken from [8]. There are two players andthey solve the following problems:

P1(x2) : minimize x21 + 8

3x1x2 − 34x1

subject to 0 ≤ x1 ≤ 10x1 + x2 ≤ 15,

P2(x1) : minimize x22 + 5

4x1x2 − 24.25x2

subject to 0 ≤ x2 ≤ 10x1 + x2 ≤ 15.

This is a GNEP with one shared constraint and the solution set is given by

SOL∗ ={(

59

)}∪{(

t

15 − t

)∣∣∣∣ 9 ≤ t ≤ 10}

. (18)

The corresponding F : �2 → �2 and X ⊆ �2 are represented as

F (x) =(

2x1 + 83x2 − 34

54x1 + 2x2 − 24.25

), (19)

X ={x ∈ �2 | x1 + x2 ≤ 15, 0 ≤ xν ≤ 15, ν = 1, 2

}.

Since F is strongly monotone on �2, VI (F,X) has a unique solution, which is given by x = (5, 9)T .Note that this solution lies in the interior of the set X.

Example 5 (River basin pollution game). Consider the 3-person river basin pollution gamestudied in [13, Section 5.3], where the problem of player ν ∈ {1, 2, 3} is defined by

Pν(xν) : minimize (ανxν + β(x1 + x2 + x3) − χν)xν

subject to xν ≥ 0,3.25x1 + 1.25x2 + 4.125x3 ≤ 100,2.2915x1 + 1.5625x2 + 2.8125x3 ≤ 100,

13

with α1 = 0.01, α2 = 0.05, α3 = 0.01, β = 0.01, χ1 = 2.9, χ2 = 2.88, χ3 = 2.85. This GNEP hastwo shared constraints. The corresponding F : �3 → �3 and X ⊆ �3 are given by

F (x) =

⎛⎝0.04 0.01 0.01

0.01 0.12 0.010.01 0.01 0.04

⎞⎠x −

⎛⎝ 2.9

2.882.85

⎞⎠ , (20)

X ={

x ∈ �3+

∣∣∣∣(

3.25 1.25 4.1252.2915 1.5625 2.8125

)x ≤

(100100

)}.

Since F is strongly monotone on �3, VI (F,X) has a single solution, which given by x =(4673

221 , 5754359 , 567

208)T ≈ (21.14, 16.03, 2.73)T . Note that, at this solution, the first inequality defin-ing X is active, while the second inequality is inactive.

Example 6 (Electric power market model). We take a model in electric power marketswith endogenous arbitrage from [19, Section 5.3]. The model consists of several electricity firmscompeting on a spatial network of markets along with an arbitrager who attempts to make a profitby exploiting price differentials between regions. In the original model [11], each firm maximizesits profit with anticipating the arbitragers’ optimal response, resulting in a multi-leader-follower-game. In [19, Section 5.3], the arbitrager is removed from the model and the price differentialsare assumed to be less than the shipping costs. In this setting, the model can be formulated as aGNEP as described in the following.

The regions are represented by the nodes in a network and each firm has electric plants atthose nodes. Each firm determines how much it should produce at each plant and how much itshould sell at each node to maximize its profit.

We introduce the notations to formulate the problem.ParametersN : set of nodesF : set of firmscfi : cost per unit generation at node i by firm f

Pi : price intercept of sales function at node i

Qi : quantity intercept of sales function at node i

eij : unit cost of shipping from node i to j

CAPfi : production capacity at node i for firm f

Variablessfij : amount produced at node i and sold at node j by firm f

Sj : amount of total sales at node j

Sj :=∑t∈F

∑i∈N

stij, ∀j ∈ N

pj : market price at node j

pj(Sj) := Pj − Pj

QjSj, ∀j ∈ N

Each firm f ’s problem is to find sfij, (i, j) ∈ N × N that solve the following maximization

14

problem: Given sfij, t( = f) ∈ F , (i, j) ∈ N ×N ,

maximize∑j∈N

[pj(Sj)

∑i∈N

sfij

]−

∑(i,j)∈N×N

eijsfij −

∑i∈N

cfi

∑j∈N

sfij

subject to∑j∈N

sfij ≤ CAPf

i , ∀i ∈ N

pj(Sj) − pi(Si) − eij ≤ 0, ∀(i, j) ∈ N ×Nsfij ≥ 0, ∀(i, j) ∈ N ×N .

Note that these problems yield a GNEP with shared constraints, because the price differentialconstraints pj(Sj) − pi(Si) − eij ≤ 0 depend on the total sales at each node Sj =

∑t∈F

∑i∈N

stij.

4.2 Numerical Results

Now we show numerical results with our parametrized VI approaches for the following threeexamples: Harker’s example, the river basin pollution game and the electric power market model.All parametrized VI (FΔ,X) and VI (F,Xβ) are converted to equivalent linear complementarityproblems (LCPs) and solved by the MATLAB code pathlcp.m [7].

Example 4 (Harker’s example). In the price-directed parametrization, by (9) and (19), themapping FΔ is given by

FΔ(x) =(

2x1 + 83x2 + Δ1 − 34

54x1 + 2x2 + Δ2 − 24.25

),

with parameters Δ = (Δ1,Δ2)T ∈ �2+. The KKT condition of VI (FΔ,X) is then written as the

following LCP:

0 ≤ x1 ⊥ 2x1 + 83x2 + Δ1 + π + μ1 − 34 ≥ 0,

0 ≤ x2 ⊥ 54x1 + 2x2 + Δ2 + π + μ2 − 24.25 ≥ 0,

0 ≤ μ1 ⊥ 10 − x1 ≥ 0,

0 ≤ μ2 ⊥ 10 − x2 ≥ 0,

0 ≤ π ⊥ 15 − x1 − x2 ≥ 0.

In view of Corollary 3.2, we restrict the parameter space to the union of two line segments{Δ ∈ �2

+ | 0 ≤ Δ1 ≤ 2, Δ2 = 0} ∪ {Δ ∈ �2+ | Δ1 = 0, 0 ≤ Δ2 ≤ 2}, on which we pick grid

points. Specifically, we generate 512 = 2× 256 grid points on the segments and check whether anobtained solution x∗ of each parametrized VI satisfies the condition (12) approximately, i.e.,

|Δν(x∗1 + x∗

2 − 15)| < 10−6, ν = 1, 2.

By solving 512 LCPs, we obtained 98 GNEP solutions, as plotted in Figure 1. We observe thatthe solutions obtained by this approach are widely distributed in its solution set given by (18),while the VI approach of [26, 3, 4] can only find the unique equilibrium x = (5, 9)T .

Next we test the resource-directed parametrization approach. We let g1(x1) = x1 − 152 and

g2(x2) = x2 − 152 . Then the parametrized feasible set Xβ is defined by

Xβ ={(

x1

x2

) ∣∣∣∣ xν − 152 ≤ βν , 0 ≤ xν ≤ 10, ν = 1, 2

},

15

0 2 4 6 8 10 12

0

2

4

6

8

10

12

x1

x 2

(5, 9)

(9.05, 5.95)

(10, 5)

GNEP SolutionSOL (F,X)

Figure 1: GNEs obtained by the price-directed parametrization approach for Harker’s example

where β ∈ {(β1, β2)T ∈ �2 | β1 + β2 = 0}. The KKT condition of VI (F,Xβ) is written as thefollowing LCP:

0 ≤ x1 ⊥ 2x1 + 83x2 + λ1 + μ1 − 34 ≥ 0,

0 ≤ x2 ⊥ 54x1 + 2x2 + λ2 + μ2 − 24.25 ≥ 0,

0 ≤ μ1 ⊥ 10 − x1 ≥ 0,

0 ≤ μ2 ⊥ 10 − x2 ≥ 0,

0 ≤ λ1 ⊥ 152 − β1 − x1 ≥ 0,

0 ≤ λ2 ⊥ 152 − β2 − x2 ≥ 0.

Note that the function gν is bounded below on Xν := {xν | 0 ≤ xν ≤ 10}, that is,

gν(xν) ≥ −152

∀xν ∈ Xν .

Hence we can restrict β to the bounded set{(β1

β2

)∈ �2

∣∣∣∣ β1 + β2 = 0, βν ≥ −152

}={(

β

−β

)∈ �2

∣∣∣∣ − 152

≤ β ≤ 152

}.

We choose 256 points from the interval [−152 , 15

2 ] and for each β, check whether an obtainedsolution x∗ of the parametrized VI satisfies the condition (14) approximately, i.e.,

either |xν − 152 − βν | < 10−6, ν = 1, 2

or xν − 152 − βν < −10−6, ν = 1, 2.

As a result of solving 256 LCPs, we obtained 34 GNEP solutions, among which 16 GNEPsolutions correspond to the equilibrium x = (5, 9)T , and the shared constraint x1 + x2 ≤ 15 is

16

active at all the rest of computed GNEP solutions. Figure 2 shows those computed solutions. Wesee that the solutions obtained by this approach are also widely distributed in the solution set.

0 2 4 6 8 10 12

0

2

4

6

8

10

12

x1

x 2

(5, 9)

(9, 6)

(10, 5)

GNEP SolutionSOL (F,X)

Figure 2: GNEs obtained by the resource-directed parametrization approach for Harker’s example

Example 5 (River basin pollution game). In the price-directed parametrization approach,by (9) and (20), the mapping FΔ is given by

FΔ(x) =

⎛⎝0.04 0.01 0.01

0.01 0.12 0.010.01 0.01 0.04

⎞⎠x +

3∑ν=1

⎛⎝ 3.25

1.254.125

⎞⎠Δν,1 +

3∑ν=1

⎛⎝2.2915

1.56252.8125

⎞⎠Δν,2 −

⎛⎝ 2.9

2.882.85

⎞⎠

with parameters Δ =((Δν,1, Δν,2)T

)3ν=1

∈ �6+.

In view of Corollary 3.2, we restrict the parameter space to the set

P :=2∏

i=1

(3⋃

ν=1

Pi(ν)

),

where for i = 1, 2 and ν = 1, 2, 3,

Pi(ν) :={Δi ∈ �3

+ | Δν,i = 0, 0 ≤ Δν′,i ≤ 2, ν ′ ∈ {1, 2, 3}\{ν}} ,

on which we pick points randomly. Specifically, we generate

(a) 10,000 points from the set P1(ν1) × P2(ν2) for each (ν1, ν2) ∈ {1, 2, 3} × {1, 2, 3},(b) 1,000 points from the P1(ν) × {(0, 0, 0)T } for each ν ∈ {1, 2, 3},(c) 1,000 points from the {(0, 0, 0)T } × P2(ν) for each ν ∈ {1, 2, 3} and

17

(d) the origin (0, 0, 0, 0, 0, 0)T .

Thus we generate 96,001 points in total. Roughly speaking, these four cases aim to find GNEs suchthat (a) both of the shared constraints are active, (b) the constraint 3.25x1+1.25x2+4.125x3 ≤ 100is active, (c) the constraint 2.2915x1 + 1.5625x2 + 2.8125x3 ≤ 100 is active and (d) neither of theshared constraints is active, respectively. Then we check whether an obtained solution x∗ of eachparametrized VI satisfies the condition (12) approximately, i.e.,{

|Δν,1(3.25x∗1 + 1.25x∗

2 + 4.125x∗3 − 100)| < 10−6, ν = 1, 2, 3

|Δν,2(2.2915x∗1 + 1.5625x∗

2 + 2.8125x∗3 − 100)| < 10−6, ν = 1, 2, 3.

By solving 96,001 LCPs, we obtained 2,445 GNEP solutions as shown in Figure 3. Note thatthe polyhedral set in Figure 3 represents the feasible region X. We obtain the equilibrium x∗ =(21.14, 16.03, 2.73)T by setting Δ = (0, 0, 0, 0, 0, 0)T and the remaining 2,444 GNEP solutions areobtained when we pick the points from the P1(ν) × {(0, 0, 0)T } for ν = 1, 2, 3. Thus, the sharedconstraint 3.25x1 + 1.25x2 + 4.125x3 ≤ 100 is active at any of these GNEP solutions.

Figure 3: GNEs obtained by the price-directed parametrization approach for Example 5

Next, we test the resource-directed parametrization approach. We let gν,i(xν) = γν,ixν − 1003

for ν = 1, 2, 3 and i = 1, 2, where γ1,1 = 3.25, γ2,1 = 1.25, γ3,1 = 4.125, γ1,2 = 2.2915, γ2,2 =1.5625, γ3,2 = 2.8125. Then the parametrized feasible set Xβ is given by

Xβ =

⎧⎨⎩⎛⎝x1

x2

x3

⎞⎠

∣∣∣∣∣∣ γν,ixν − 1003

≤ βν,i, xν ≥ 0, i = 1, 2, ν = 1, 2, 3

⎫⎬⎭ ,

whereβ ∈ {

(βν,i) ∈ �6 | β1,i + β2,i + β3,i = 0, i = 1, 2}

.

18

Note that each function gν,i is bounded below on Xν = �+, that is,

gν,i(xν) ≥ −1003

, ∀xν ∈ Xν .

So we can restrict β to the bounded set{(βν,i) ∈ �6

∣∣∣∣∣ β1,i + β2,i + β3,i = 0, βν,i ≥ −1003

,

ν = 1, 2, 3, i = 1, 2

}

=2∏

i=1

⎧⎨⎩⎛⎝β1,i

β2,i

β3,i

⎞⎠ ∈ �3

∣∣∣∣∣∣ β1,i + β2,i + β3,i = 0, βν,i ≥ −1003

⎫⎬⎭

=2∏

i=1

⎧⎨⎩⎛⎝ β1,i

β2,i

−β1,i − β2,i

⎞⎠ ∈ �3

∣∣∣∣∣∣ β1,i + β2,i ≤ 1003

, βν,i ≥ −1003

⎫⎬⎭ . (21)

To generate points that belong to this set, we pick 100, 000 points randomly from the rectangle{(β1,1, β2,1, β2,1, β2,2) ∈ �4

∣∣∣∣ −1003

≤ βν,i ≤ 2003

},

and then check whether each point β = (β1,1, β2,1,−β1,1 − β2,1, β1,2, β2,2,−β1,2 − β2,2)T satisfiesthe conditions in (21). In this way, we obtain 24,858 points that belong to the set given by (21).

Figure 4: GNEs obtained by the resource-directed parametrization approach for Example 5

As a result of solving 24,858 LCPs, we obtained 699 GNEP solutions as shown in Figure4. Thus, the resource-directed parametrization approach was also able to find GNEP solutionsdistributed widely on the face determined by the shared constraint 3.25x1+1.25x2+4.125x3 ≤ 100.

19

Table 1: Generation costs cfi and capacities CAPf

i

(f, i) (I, 1) (I, 2) (II, 2) (II, 3)cfi 15 15 15 15

CAPfi 100 50 100 50

Table 2: Price function data (Pi, Qi)node i 1 2 3

Pi 40 35 32Qi 500 400 600

Example 6 (Electric market model). We apply the price-directed parametrization approachto a simple example [19] consisting of two firms, named I and II, which compete in a 3-nodenetwork with six arcs {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}. Firm I owns electric plantsat nodes 1 and 2, and firm II owns electric plants at nodes 2 and 3. The other data for theexample are given in Table 1 and Table 2.

The unit cost of shipping eij are set to be

eij =

{1 (i = j)

0 (i = j).

The parameter space {Δ = (Δfij) ∈ �18

+ } is too large to enumerate fine grid points, so we just

pick up points from the restricted space P :=⋃

(i,j)∈N×NPij, where

Pij :={Δ ∈ �18

+ | 0 ≤ Δfij ≤ 20, Δf

kl = 0 ∀(k, l) = (i, j), f = 1, 2}

.

Specifically, for each (i, j) ∈ N × N , we generate 100 points (Δ1ij , Δ2

ij) randomly from therectangle [0, 20] × [0, 20] and hence 900 points Δ in total. Then we check whether a computedsolution of each parametrized VI satisfies the condition (12) approximately, i.e.,

|Δfi,j(pj(Sj) − pi(Si) − eij)| < 10−6 ∀(i, j), f = 1, 2. (22)

By solving 900 LCPs, we obtained 392 GNEP solutions. Note that, for (i, j) = (1, 1), (2, 2), (3, 3),the sufficient condition (22) is always satisfied for any Δf

ij. Hence, we got 300 GNEP solutionswhen we pick the points from Pii (i = 1, 2, 3) and all those solutions correspond to a single solu-tion, which is nothing but the equilibrium obtained by the approach of [26, 3, 4]. All the rest ofcomputed GNEP solutions are obtained when we pick the points from P31.

Here, instead of showing the whole picture of those GNEs, let us compare two particular GNEPsolutions, i.e., the equilibrium x obtained by the approach of [26, 3, 4] and the GNE x∗ obtained bythe price-directed parametrization approach for VI (FΔ,X) with (ΔI

3,1, ΔII3,1) = (14.1819, 8.6348)

and Δfij = 0 for all f and (i, j) = (3, 1). The sales and the nodal prices of GNEs x and x∗ are

summarized in Table 3 and Table 4, respectively. We observe that the nodal prices pi in Table 3are all identical for the two GNEs. However, comparing the firms’ profit, we have

θ1(x) = 1969.5 < 1975.1 = θ1(x∗)

θ2(x) = 1923.6 = 1923.6 = θ2(x∗),

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Table 3: Nodal pricesnode i 1 2 3

x 28.82 27.82 27.82x∗ 28.82 27.82 27.82

Table 4: Firms’ sales(f, i, j) (I, 1, 1) (I, 1, 3) (I, 2, 2) (I, 2, 3) (II, 2, 1) (II, 2, 2) (II, 3, 1) (II, 3, 3)

sfij 77.01 22.99 41.84 8.16 59.83 40.17 2.85 47.15

s∗fij 79.79 20.21 41.84 8.16 59.83 40.17 0.08 49.92

where θf (x) :=∑j∈N

[pj(Sj)

∑i∈N

sfij

]−

∑(i,j)∈N×N

eijsfij −

∑i∈N

cfi

∑j∈N

sfij. Thus, the GNE x obtained

by the approach of [26, 3, 4] is weakly dominated by the GNE x∗, in the sense that, at the GNEx, firm I has a motivation to move to the GNE x∗ by paying a small incentive ε > 0 to firmII. (Recall that each player is maximizing its profit.) Hence, the GNE x would not appear inthe actual situation. This result indicates that the approach of [26, 3, 4] may fail to find someimportant GNEs, while the proposed parameterization approach has a better chance to find thoseGNEs.

5 Concluding Remarks

In Section 4, we have applied the proposed VI approaches to three numerical examples and observethat the GNEs obtained by proposed methods are widely distributed in the GNEP solution setwhile the approach of [26, 3, 4] can only find some particular GNE.

We note that, in the price-directed parameterization, we have to restrict the parameter spaceheuristically in the price-directed parametrization because Theorem 3.4 does not say how largethe parameter space should be. This remains a question for further study. Moreover, in ournumerical experiments, we used fixed grid points or generated points randomly in the parameterspace. Although such a simple procedure works well for small examples, more efficient parametersearch procedures will be required for practical GNEPs.

Acknowledgment

First of all, I would like to express sincere appreciation to Professor Masao Fukushima. Al-though I sometimes troubled him, he always kindly looked after me and gave me plenty of preciseadvice. I am thankful to Professor Paul Tseng of University of Washington for his invaluablesuggestions to accomplish this thesis. Some results in this paper could not be established with-out his contribution. I would like to tender my acknowledgements to Associate Professor NobuoYamashita. He gave me a lot of invaluable comments for my research. I also wish to expressmy thanks to Assistant Professor Shunsuke Hayashi. He always gave me constructive commentsfor my study and research. Moreover, I am deeply impressed by his enduring passion for ramen.Finally, I would like to thank all members of Fukushima Laboratory, my friends and my family fortheir encouraging words. Especially, my special thanks are due to my parents Takashi Nabetaniand Ayako Nabetani for their understanding and financial support.

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References

[1] K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy,Econometrica, 22 (1954), pp. 265–290.

[2] D. P. Bertsekas, with A. Nedic, and A. E. Ozdaglar, Convex Analysis and Opti-mization, Athena Scientific, Belmont, 2003.

[3] F. Facchinei, A. Fischer, and V. Piccialli, Generalized Nash equilibrium problems andNewton methods, Mathematical Programming, Series B, to appear.

[4] F. Facchinei, A. Fischer, and V. Piccialli, On generalized Nash games and variationalinequalities, Operations Research Letters, 35(2007), pp. 159–164.

[5] F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR: A QuarterlyJournal of Operations Research, 5 (2007), pp. 173–210.

[6] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Comple-mentarity Problems, Springer-Verlag, New York, 2003.

[7] M. C. Ferris and T. S. Munson, Complementarity problems in GAMS and thePATH solver, Journal of Economic Dynamics & Control, 24 (2000), pp. 165–188,ftp://ftp.cs.wisc.edu/math-prog/solvers/path/matlab.

[8] P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journalof Operational Research, 54 (1991), pp. 81–94.

[9] P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinearcomplementarity problems: A survey of theory, algorithms and applications, MathematicalProgramming, Series B 48 (1990), pp. 161–220.

[10] A. von Heusinger and C. Kanzow, Optimization Reformulations of the GeneralizedNash Equilibrium problem using Nikaido-Isoda-type functions, Computational Optimizationand Applications, to appear.

[11] B. F. Hobbs, Linear complementarity models of Nash-Cournot competition in bilateral andPOOLCO power market, IEEE Transactions on Power Systems, 16 (2001), pp. 194–202.

[12] R. Janin, Directional derivative of the marginal function in nonlinear programming, Math-ematical Programming Study, 21 (1984), pp. 110–126.

[13] J. B. Krawczyk and S. Uryasev, Relaxation algorithms to find Nash equilibria witheconomic applications, Environmental Modeling and Assessment, 5 (2000), pp. 63–73.

[14] J. B. Krawczyk, Numerical solutions to coupled-constraint (or generalized Nash) equilib-rium problems, Computational Management Science, 4 (2007), pp. 183–204.

[15] Z. Q. Luo, J.-S. Pang, and D. Ralph, Mathematical Programs with Equilibrium Con-straints, Cambridge University Press, Cambridge, 1996.

[16] H. Nikaido and K. Isoda, Note on noncooperative convex games, Pacific Journal of Math-ematics, 5 (1955), pp. 807–815.

[17] J. V. Outrata and J. Zowe, A Newton method for a class of quasi-variational inequalities,Computational Optimization and Applications, 4 (1995), pp. 5–21.

22

[18] J.-S. Pang, Computing generalized Nash equilibria, Manuscript (2007).

[19] J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria,and multi-leader-follower games, Computational Management Science, 2 (2005), pp. 21–56.

[20] S. M. Robinson, Shadow prices for measures of effectiveness, I: Linear model, OperationsResearch, 41 (1993), pp. 518–535.

[21] S. M. Robinson, Shadow prices for measures of effectiveness, II: General model, OperationsResearch, 41 (1993), pp. 536–548.

[22] R. T. Rockafellar, Lagrange multipliers and optimality, SIAM Review, 35 (1993),pp. 183–238.

[23] J. B. Rosen, Existence and uniqueness of equilibrium points for concave N -person games,Econometrica, 33 (1965), pp. 520–534.

[24] S. Uryasev and R. Y. Rubinstein, On relaxation algorithms in computation of noncoop-erative equilibria, IEEE Transactions on Automatic Control, 39 (1994), pp. 1263–1267.

[25] H. R. Varian, Microeconomic Analysis, W. W. Norton Co., New York, 1992.

[26] J.-Y. Wei and Y. Smeers, Spatial oligopolistic electricity models with Cournot generatorsand regulated transmission prices, Operations Research, 47 (1999), pp. 102–112.

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