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Adjustment process-based approach for computing a Nash-Cournot equilibrium
Compurers Ops Res. Vol. 21, No. 1, pp. 57-65, 1994 Printed in Great Britain. All rights rerrwd
ADJUSTMENT COMPUTING
0305-0548/94 $6.00 + 0.00
Copyright 0 1993 Pergamon Press Ltd
PROCESS-BASED APPROACH FOR A NASH-COURNOT EQUILIBRIUM JOANNA M. LELENO~
Department of Mathematical Sciences, Virginia Commonwealth University, Division of Operations Research and Statistics, 1015 West Main Street, Richmond, VA 23284-2014, U.S.A.
(Received November 1991; in revised form December 1992)
Scope and Purpose-The economic model is that of a noncooperative n-person game. We consider a homogeneous product market supplied by n producers (players). Their cost functions and the product demand function are assumed to be known. Each firm’s objective is to decide how much to produce (and sell) so as to maximize its own profit, and no firm is allowed to manipulate decisions of the remaining firms in order to increase its own profit. At a Nash-Cournot equilibrium no firm can increase its profit by a unilateral change of its output level. The main purpose of this paper is to examine properties of adjustment processes in an n-firm oligopolistic market. Adjustment processes are peculiar in that they emulate the entire sequence of steps each firm undertakes so as to maximize its profit. In this paper we consider two types of adjustment processes: sequential and simultaneous. Our analysis shows that they both lead to closed algorithms. Moreover, we show that if the demand function is linear, then the sequential adjustment process converges to an equilibrium point, and we explain why the simultaneous process fails to converge in this case. Abstract-This paper is focused on deriving properties of adjustment process-based algorithms for computing a Nash-Coumot equilibrium point. Two adjustment processes are considered: sequential and simultaneous. The corresponding numerical procedures are closely related to the Gauss-Seidel and Jacobi methods, respectively, for solving nonlinear systems. Our analysis shows that under quite general assumptions, both processes lead to closed algorithms. We also show that in case of a linear demand function (which implies the absence of the diagonal dominance of the Jacobian matrix of marginal profits), the sequential adjustment process-based algorithm converges to a Nash-Cournot equilibrium point. For this purpose we employ matrix splitting-based algorithms for solving a nonlinear complementarity problem, and an interesting equivalence between a Nash-Coumot equilibrium point and a certain n-dimensional optimization problem. This equivalence allows us also to gain more insight into why the simultaneous adjustment process fails to converge in this case.
1. INTRODUCTION
Consider a homogeneous product market supplied by n firms (players). Each firm is a Cournot firm, that is, it selects its decision so as to maximize its own profit given the decisions of all the remaining firms. In this paper we consider a static model in which ith firm’s decision variable is its output level, xi, i = 1, . ..,n.Letx_idenoteavector(x,x,...xi_,xi+,...x,),andletxi(xi;x_i) denote ith firm’s profit function, given the output levels of all its rivals. The function a&; x_~) is defined in the following way ni(Xi; X_i)=XiP
Xi+ (
C Xj -Ci(Xi), ) j#i
where p(Q) is the (inverse) demand function, Q 20, and ci(Xi)is the total cost function of the ith firm. Definition 1. A nonnegative vector x* = (XTxf . . . x.*) of output levels is said to be a Nash-Coumot (NC) equilibrium point if for each i= 1, . . ., n, xt is the profit maximizing output level of the ith firm, given the equilibrating output levels Xfi of all the remaining firms. That is, tJ. M. Leleno is Assistant Professor of Operations Research in the Department of Mathematical Sciences at Virginia Commonwealth University, Richmond, Va. She holds a Ph.D. in Industrial Engineering and Operations Research from Virginia Polytechnic Institute and State University, Blackburg, Va. Her current research interests include nonlinear complementarity problems, computing equilibrium points and bilevel optimization.
JOANNAM. LELENO
58
foreachi=l,...,n Ki(X:; X*_i)~Ki(q;
X*_i)
for all qa0.
In other words, for each i
1 X*,)-ci(x?)84p(4+zi xtP ( Xi * + jfi
for all qa0.
Xf)q(q)
Various sets of assumptions have been shown to guarantee the existence of an NC equilibrium point. For our purposes, the following conditions are assumed to hold: Assumption 1. The demand function p(Q) is nonincreasing for Q 20, strictly decreasing over its positive range, continuously differentiable and such that p(O)>O, p’(O)<0 and Qp(Q) is concave for Q > 0. Assumption 2. The cost functions ci(q) are convex, strictly increasing over q>O, continuously differentiable and such that c,(O)=O, i= 1, . . ., n. Assumption 3. For each i = 1, . . . , n there exists a positive number qi. such that p(q) - C!(q) 2 0 for all 4 > qi”. Assumptions 1 and 2 imply that each firm’s profit function is strictly concave in its decision variable and continuously differentiable. Assumption 3 is included to impose an upper bound on each firm’s output level. Let us point out that if there exists a positive number Q” such that p(Q) = 0 for all QaQ’, then assumption 3 is automatically implied by 1 and 2. Under assumptions l-3 a unique NC equilibrium point exists [l, 21. A number of papers [l-4] have dealt with computing an NC equilibrium point. The purpose of this paper is to examine an algorithm which emulates the way in which players make their decisions, and to show its convergence to a unique NC equilibrium point. This approach can be briefly described via the following generic scheme. Let x0 > 0 be any given initial vector of n output levels. At each step each firm chooses its output level given the most recent decisions of all its rivals. To be more specific, let xk = [XT]> 0 be an n-dimensional vector of output levels obtained at step k. Then the next iterate, xk+i = [x:’ ‘1, is a vector of optimal solutions to n one-dimensional optimization problems, that is, for each i= 1, . . ., n
xf+l solves
max q>O
qp p+C
xj+‘+
C xi” -ci(q) j>i
j
i(
>
. I
Thus players one after another make adjustments to the current total industry supply. This technique is referred to as a sequential adjustment process. Alternatively, a simultaneous adjustment process refers to an approach in which
xf+’ solves
qp
max q>O
i
q+ C x+ -c(q) (
j#i
I)
i
1.
Both processes can be viewed as a sequence of repetitions of steps in which each player (firm) selects its output level so as to maximize its own profit given the total output of all the remaining firms. However, the simultaneous adjustment process requires that all players act simultaneously, whereas in the sequential adjustment process firms act in an organized way, so that one firm at a time is allowed to make an adjustment to the total industry supply. As a result of it each firm at the moment of announcing its decision is ‘at equilibrium’ with the remaining firms. This property is not shared by the simultaneous adjustment process because the moment the changes in the output levels are made, each firm is likely to face a totally different market situation than the one it used to make its own decision. In a more general framework, both processes can be represented as decomposition algorithms for solving variational inequality problems or nonlinear complementarity problems. Also, let us notice that the sequential and simultaneous adjustment processes are closely related to the Gauss-Seidel and Jacobi methods, respectively, for solving nonlinear systems. Both processes were analyzed in [l] in the context of stability of Nash equilibrium in noncooperative games. A similar approach was presented in [3], though more in the context of
Approach for computing a Nash-Cournot equilibrium
59
variational inequalities and convergence results derived in [S]. In general, no conditions are known to be both necessary and sufficient for the convergence of adjustment processes to a unique NC equilibrium point. In [l] it was demonstrated that strictly diagonal dominance of a certain matrix is a sufficient condition for the convergence of both adjustment processes. Economic interpretation of this condition is that for each firm, its own effect on marginal profit exceeds the aggregate cross-effects resulting from actions of all its rivals. In other words, each firm’s marginal profit is more sensitive to its own decision than to decisions of all the remaining firms. This condition also appeared in [4], where a sequential linear complementarity problem algorithm is examined for global convergence to a unique Nash-Cournot equilibrium point. Sufficient conditions derived in [S] require that a norm of certain matrices employed in computing an equilibrium point via variational inequality formulation be less than one. The above sufficient conditions are in fact difficult to verify a priori. Moreover, none of them holds if the demand function is linear. The main purpose of this paper is to examine properties of adjustment processes. Our interest in algorithms based on such processes is rooted in their structure which allows one to closely follow, step by step, the entire decision process in which firms (players) are involved. This paper is organized in the following way. First, we demonstrate that under assumptions l-3 both adjustment processes lead to a closed algorithm. Next we consider a linear demand function, and show that the sequential adjustment algorithm is convergent in this case. In [6] an illustrative numerical example is presented to demonstrate that the simultaneous adjustment process fails to converge if the demand function is linear and there are more than two players. Therefore, we also attempt to point out factors causing such instability. Finally, we show an interesting local property of the sequential adjustment process due to which at some point of the process, all firms that produce nothing at equilibrium are effectively eliminated from the market. We adopt the following notation. All vectors are column vectors and superscript T is used to denote the transpose. Symbol e denotes a vector of ones and I-an identity matrix. For any two n-dimensional vectors x= [Xi] and y= [yi], symbol (x, y) denotes its inner product, that is (x, y) =I:= 1 Xiyi. If A is an n x m matrix, and 9 and $ are subsets of {1,2, . . . , n} and {1,2, . . . . m}, respectively, then A, denotes a submatrix of A composed of those rows whose indices are in 9. Similarly, A,, denotes a submatrix of A composed of those rows and columns whose indices are in 9 and 3, respectively. R’!+denotes the nonnegative orthant in R”.
2. CHARACTERIZATION
OF ADJUSTMENT
PROCESSES
As already stated, under assumptions l-3 a unique NC equilibrium point exists. Moreover, due to differentiability and concavity of each firm’s profit function, the following two representations of an NC equilibrium point are valid [l, 31. 1. Variational inequality representation A nonnegative vector x* = (x7 x: x,*) of output levels is a Nash-Cournot if and only if it solves the following variational inequality in x (F(x), y-x)>0
for all y>O,
equilibrium point (1)
where F: R; + R”, with F(X) = Lfi(X)], and where fi(x) = - &ri(xi; x-,)/ax,.
2. Nonlinear complementarity problem representation x:) of output levels is a Nash-Cournot A nonnegative vector x* = (x: xt if and only if it solves the following nonlinear complementarity problem: find x such that
F(x)>O, x20 and (F(x), x)=0.
We find the two characterizations adjustment processes.
equilibrium point (2)
of an NC equilibrium point very useful in our analysis of
JOANNAM. LEL~NO
60
Definition 2. Sequential adjustment process is given by a mapping G: R; + R;
defined in the
following way : yEG(x) if and only if for each i=l, yi solves
max
@I q+ 1
q30
Yj+
C
xj
-Ci(q)
j>i
j
u
. . ,, n
)
,
I
whereas simultaneous adjustment process is given by a mapping G,: R’: + R: defined as y~G,(x)
if and only if for each i=l,
. . ., n
By assumptions 1 and 2 for each i = 1, . . . , n, optimization problems (3) and (3a) seek the maximum of a strictly concave function over the nonnegative real line. Moreover, by assumption A3, for each firm its feasible decision set is in fact reduced to a closed interval [O, q1J. Therefore, both problems (3) and (3a) have a unique solution, and thus mappings G(x) and G,(x) are well defined functions that map R: into a set X, where X is a Cartesian product [O, qlJx . . , x[O, qJ. As we show in Lemma 1 below, both functions are continuous. Lemma 1. The sequential adjustment process and the simultaneous adjustment process are closed
algorithm. Proof. We show that the sequential adjustment process is a closed algorithm. With minor changes
this proof can also be used to establish closedness of the simultaneous adjustment process. We have just shown that mapping G: R: + X which represents the sequential adjustment process is in fact a point-to-point algorithm G, so that x Ir+r = G(xk). To show that the algorithm G is closed we need to demonstrate that for any convergent sequence (x”}~ the following implication is true: ({X”>K3 % {YkIK+ 7, where yk = G(xk), k E K) =+-y = G(f). First note that the sequence {x”} is bounded (by assumptions), therefore it has a convergent subsequence. Suppose that {x”}~ + jz and {ykjK + y, where y” = G(xk), k E K. Then for each k E K we have YfP Yf+Cv”+C j
j,i”)-
c(yk)>qp i
i
’
q+ C (
yt+
j
1
x:
j>i
-ci(q)9
for all t?a&
)
i= 1, . . ., n.
By assumption, functions p( .) and ci( .) are continuous, therefore by taking the limit as k --) COwe obtain Jip
( j
j.+C ’
j>ixJ)
-. -
.(j)>qp
” i’
q+ (
1
jj+
j
C
-ci(q),
%j
j>i
forallqN
>
i= 1, . . ., n,
which shows that algorithm G is continuous at any accumulation point %.This completes the proof. 0 Having established closedness of the algorithmic mapping G, we need to show that the algorithm is descent in order to guarantee its convergence to a unique NC equilibrium point. This issue is addressed in the next section, where the inverse demand function p(Q) is assumed to be linear. 3. ADJUSTMENT
PROCESSES
IN CASE
OF A LINEAR
DEMAND
FUNCTION
Assume that the demand function p(Q) is linear, p(Q)=abQ, where a>O, b >O. Then by definition, x* is an NC equilibrium point if for each i= 1, . . ., n .
Approach for computing a Nash-Coumot
61
equilibrium
Next we state a result that allows one to identify an NC equilibrium point as an optimal solution to an n-dimensional optimization problem. For this purpose let M denote an n x n matrix whose all diagonal entries are identical and equal to 2b, and all remaining entries are identical and equal to b. Theorem 1. x* is an NC equilibrium point if and only if x* solves the following problem:
min
2 ci(xi) .
; xTMx-aeTx+
XERT
i=l
Proof. Matrix M is a symmetric and positive definite. This property together with assumption 2 implies that problem (4) involves minimization of a strictly convex and differentiable objective function over nonnegative orthant of R”. Therefore the Karush-Kuhn-Tucker system is then both necessary and sufficient for optimality. That is, x* is an optimal solution to (4) if and only if Mx* + C(x*) - ae > 0, (x*, Mx*+C(x*)-ae)=O,
(5)
x*20, where C(x) = [ci(Xi)]l is an n-dimensional vector of derivatives of total cost functions ci( .) evaluated at Xi. TO complete the proof let us notice that with p(Q)=a- bQ, the functions fi(x) employed in the nonlinear complementarity problem representation (2) of an NC equilibrium point become simply fi(X)= -(a-2bxi-b Cj+i Xj)+C:(XI), i=l, . . . , n, which further yields F(x) = Mx - ue + C(x). Therefore, the nonlinear complementarity characterization (2) of an NC equilibrium point and system (5) are equivalent, which completes the proof. 0 As evident from theorem 1, one can find an NC equilibrium point by simply solving problem (4), and various methods can be used for this purpose. What is perhaps more important is that an equilibrium point, which by definition ensures optimality for all n firms simultaneously, solves a single optimization problem for the whole industry. This is true when the demand function is linear. It would be desirable to drop the linearity assumption and still be able to claim such property. However, the shape of the demand function seems to play a crucial role in derivation of this result. The main reason for which problem (4) is presented in this paper is that it provides a useful tool in demonstrating that the sequential adjustment process is a convergent algorithm. Before we establish the main result let us interpret theorem 1 in economic terms. Simple algebraic operations allow to rewrite the negative of the objective function in problem (4) as (a - beTx)eTx -I;= 1 ci(xi) + 3b[(eTx)2 - xTx], which explicitly shows that an NC equilibrium point maximizes the total industry profit (the first two terms) and the net consumer surplus [7] adjusted according to the behavioral nature of Cournot firms. Sequential adjustment process can be now formulated in the following way: Initialization step. Choose any x0 > 0 as the initial vector. Step k, k = 1, . . . . Given xk - ‘, sequentially, find x:, xk,, . . ., xz, where xf is the unique solution to the following system in 4: b,Tix;+2bq+b
1 xi-‘-a+c;(q)aO j>i
q bC x;+2bq+b
=O
C x;-l-a+cgq)
j
j>i
q>o. It can be easily verified that for each i, xf is the unique solution to the equation: O,a-b
2bxf+c;(xf)=max (
c xjk+ c ( j
j>i
xf-’
. I
Given the above description, the entire process can be expressed as one that incorporates a matrix splitting approach for solving linear complementarity problem. Here, we employ the splitting (B, U)
JOANNA M. LELENO
62
of the matrix M (i.e. M=B+ U), where U is a strictly upper triangular matrix with all its nonzero entries equal to b, and B = D + UT, where D = 261. In this context, the sequential adjustment process translates into a procedure G(.) that generates a sequence ix’}, where xk, k= 1,2, . . . is the solution to the system: Bx+Uxk-l
+C(x)-ae>O
(x, Bx+Uxk-‘+C(x)--ue)=O
(6)
x20. That is x”=G(x’-‘) if and only if xk solves the system (6). A parallel description of the simultaneous adjustment process requires that in the equations above the terms Cji xj”-’ be replaced with Cjfi x!-‘. Consequently, xk=GJxk-i) if and only if xk solves the system Dx+(U+UT)xk-l+C(x)-ce~O (x, Dx+(U+UT)xk-l+C(x)-ue)=O
(6a)
x20. If all marginal cost functions are linear, then C(x)=Sx +r, where S is a nonnegative diagonal matrix, and r is a vector of constants. Consequently, algorithms G(.) and G,( .) represented by systems (6) and (6a) respectively, then become matrix splitting-based algorithms for solving a linear complementarity problem. This technique for solving linear complementarity problems has been intensively studied by many authors [S, 93, and several of their results can be cited here in order to assert the convergence of sequential adjustment process. Moreover, we have found some of the approaches presented therein useful in our analysis of algorithm G(. ) in the case of nonlinear marginal cost functions. We point out that forthcoming results do not employ a specific form of matrices D, B and M. They remain valid as long as matrix M is symmetric and positive definite, D is diagonal and positive definite, U is strictly upper triangular, and M = B+ U, B= D + UT. Lemma 2. Let D be an n x n positive definite diagonal matrix and let U be an n x n strictly upper triangular matrix. If B = D + UT, then the quadratic form xT(B - U)x is positive definite.
Proof. By definition of matrices B and U, we have xr(B- U)~=X~(D+U-U~)X=X~DX>O for all x#O, where the last inequality follows from positive definiteness of matrix D. Lemma 3. Let h(~)=)x~M~-c~e~x+C~= Moreover, h(x)= h(y) if and only if x = y.
1 Ci(Xi). For any ~20,
if y=G(x),
Cl
then h(x)ah(y).
Proof. First let us notice that by convexity of the cost functions we have
$‘i ci(xi)-i$l
ci(JJi)ai$l cl(yi)(xi-YJ=C(Y)T(x-Y)
for any x, y in R:+
Let x 20, and y = G(x). Then by the last inequality we obtain h(x)-h(y)8;
xrMx-aeTx-f
yTMy+aeTy+C(y)r(x-y)
=(x-y)‘(By+Ux-ae+C(y)+B(x-y))-; >(x-y)rB(x-y)-;
(x-y)‘M(x-y)=;
(x-y)rM(x-y) (x-y)r(B-U)(x-y)aO.
Here, the second equality results from rearrangement of terms, the third one from variational inequality representation (1) of the assumption y = G(x), x 20 and, the fourth one from the employed matrix splitting, and the last one from lemma 2.
Approach for computing a Nash-Cournot
equilibrium
63
Theorem 2. For any x020, the sequence {x”) generated by the sequential adjustment process via system (6) converges to a unique NC equilibrium point x*. Proof. By lemma 1 the algorithm G( .) is closed, and by lemma 3 the sequence {h(xk)} is decreasing. Therefore, from Zangwill’s convergence theorem A, [lo], we can conclude that the sequence {x”} converges to a unique optimal solution to problem (4), which by theorem 1 completes the proof. 0
We note that by theorem 2 if the demand function is linear and the cost functions are convex and differentiable, then for any initial vector x0 the sequential adjustment process produces a sequence {x”} convergent to a unique equilibrium point. Thus in this case, convergence of the process is guaranteed under the same conditions that guarantee the existence of a unique equilibrium point. No additional requirements need to be imposed. This result stays in contrast with conclusions drawn in [l], where linearity of the demand function was perceived as the main obstacle in establishing convergence of adjustment processes. The analysis in [l] shows that the process is convergent if the Jacobian matrix of the function F(x) is diagonally dominant. In case of the linear demand function, we obtain F(x) = Mx + C(x)-ae, and therefore its Jacobian F’(x)= [fij(x)] is given by fli(X)=2b + C:i(Xi), i=l, . . ., n i, j=l 9 ‘. ., n, j#i, and flj(x)=b, where cy( .) denotes the second derivative of the ith firm cost function. It can be easily verified that for the matrix F’(x) = [f;j(X)] to be diagonally dominant for all x 3 0, we need c:((xi) - (n - 2)b > 0, for all x,20, i= 1, . . ., n. As also pointed out in [l], this condition fails when there are more than two firms, and all cost functions are linear. Yet, due to theorem 2, the process is convergent regardless of the number of firms and relationship between c:((Xi)and b. As shown by a simple numerical example in [6], the simultaneous adjustment process fails to converge to a unique equilibrium point. Analytically, it can be explained by the fact that, in contrast to the sequential adjustment process, the resulting sequence (h(xk)} is not decreasing. A closer look at (6a) and the relationship between h(x) and h(y), where y = G,(x), shows that h(x)-h(y)+
(x-y)T(D-U-UT)(x-y),
and matrix D-U-UT is not positive definite except when n = 2. In fact, this matrix may be indefinite after all, and this property makes the simultaneous adjustments diverge. Intuitively, this divergence can be viewed as a result of ‘instability’ that each firm experiences each time the decisions are announced. Recall that when the demand function is linear, for each firm, its own-effects on marginal profit are in fact dominated by the aggregate cross-effects. If all the firms are allowed to make their decisions simultaneously, they all will be willing to increase their output levels if the current market price is low, and decrease their output levels if the current market price is high. Consequently, the total industry supply will alternate between some small value and some large value. As demonstrated by theorem 2, this phenomenon does not occur when the adjustments to the total industry output are made sequentially, one at a time. Therefore, one is tempted to conclude that in the absence of dominance of each firm’s effects on its marginal profit over the cross-effects from its rivals, the manner in which firms make their decisions plays an important role as far as the convergence issue is considered, and hence market stability. Having established the convergence of the process, we next demonstrate its local property, due to which in some neighborhood of the equilibrium point, the algorithm essentially becomes the well-known Gauss-Seidel method for solving nonlinear systems. In case of linear complementarity problems, this property has been established by Mangasarian [S]. Our result is more general since it allows for nonlinear terms, however, in the proof we employ a very similar technique. Lemma 4. Let x* be a unique NC equilibrium point. Assume that the nonlinear complementarity problem (5) is nondegenerate, i.e. x* + Mx* - ae + C(x*) > 0, and that c;(O)= 0, i = 1, . . . , n. Let 9 and 9 denote sets defined as follows: B = {i: x: > 0} and 9 = (i: (Mx* - ae + C(x*)), > 0). There exists k* such that the sequential adjustment process results in {xk}, where for each k> k* B,,x~,+‘+C,(X~~)=
-Uu,,xk,+,e,P
and
xk,‘=O.
JOANNAM. LELENO
64
Proof. First let us notice that the sequential adjustment process as defined by system (6) can be, after rearrangement of terms, represented in the following way:
x’+D-‘C(xk)=(xk-r
+D-‘C(X~-~)-D-~[MX~-’
+UT(xk-xk-l)-ae+C(xk-l)]}+,
(7)
where for a vector y = [vi], y+ denotes a vector whose ith component is max(O, yi). For notational simplicity, let A(xk-r)=Mxkel +U’(~~-x~-~)-ae+C(x~-~). By definition of A(xk) and that of set 9, we have lim A(xk), = [Mx* - ae + C(x*)], > se9
for some E > 0.
k-m
By complementarity
requirement, continuity of C(x) and assumption C(0) = 0, we also have lim [x~-~+D-~C(X~-~)]~=O. k-m
Furthermore, since D is a positive definite diagonal matrix, we can conclude that there exists k, such that for all k > k, and some 6 > 0 while [x”- ’ +D-‘C(xk-‘)Jl<~eZ,
[D-‘A(xk-1)]52~e,
which by (7) implies [xk +D- ‘C(xk)lia = 0 for all k > k,. This conclusion together with C(x) > C(0) for all ~20, which follows from the concavity of the cost functions, results in Xk ia =o
for all k>k,.
In a similar manner it can be shown that there exists k, such that for all k> k, and some q>O [xk-1+D-1C(xk-1)]9>I]eg
while [D-lA(xk-l)]B<~e,,
which yields for all kak,.
[x~+D-~C(X~)]~=[X~-~+D-~C(X~-~)-D-~A(X~-~)],~ The two results and simple algebra finally lead to B,,xv’ xy ’ = 0 for all k > k*, which completes the proof.
+&(x2
‘)= -U,wxk,+aeg
and 0
It is worthwhile to make two brief comments on lemma 4. First let us point out that the assumption c:(O)=O, i= 1, . . . , n, used in lemma 4 is not restrictive at all. Its role is technical only. In order to show that lemma 4 is still true without this assumption all that one needs to do is rewrite system (6) so that vectors C(x) and -ae are replaced by vectors C(x) and r, respectively, where C(x)=C(x)C(0) and r = C(O)-ae, and next apply the proof to thus modified nonlinear complementarity system. The second remark deals with interpretation of lemma 4. What follows from it is that after finite number of iterations, the sequential adjustment process in fact reduces to Gauss-Seidel method for solving nonlinear system B,,9x, + C&x,)= - U,q,x,+ aeg. That is, at some point of the process, all firms that produce nothing at equilibrium will be eliminated from the game. We close this paper with a couple of final remarks:
(1) In this paper we showed that under general assumptions both adjustment processes produce a closed algorithm. We also derived a simple optimization problem that can be used to find an NC equilibrium solution when the demand function is linear. (2) One of the sufficient conditions for the global convergence of adjustment process requires that each firm’s marginal profit have the dominance property. If the demand function is linear this property cannot hold unless there are only two firms. In spite of the absence of this property we established convergence of the sequential adjustment process. We also showed reasons for which the simultaneous adjustment process does not necessarily converge. Results in Section 3 allow one to employ decomposition algorithm (6) to solve (3) a nonlinear optimization problem in which function f(x) = +xrMx +x1= 1 gi(Xi),
Approach for computing a Nash-Coumot
equilibrium
65
where matrix M is symmetric and positive definite, and Qi(Xi) is a convex, differentiable function, i= 1, . . ., n, needs to be minimized over R$. REFERENCES 1. D. Gabay and H. Moulin, On the uniqueness and stability of Nash-equilibria in noncooperative games. In Applied Stochastic Control in Econometrics and Management Science (Edited by A. Bensoussan, P. Kleindorfer and C. S. Tapiero). North-HoIiand, Amsterdam (1980). 2. F. H. Murphy, H. D. Sherali and A. L. Soys&r, A mathematical pro~amming approach for dete~ining oligopolistic market equilib~um. Mathf Pragr~. 24,92-106 (1982). 3. P. T. Harker, A variational inequality approach for the determination of ohgopohstic market equilibrium. Mathi Program. 30, 105-111 (1984). 4. C. D. Kolstad and L. Mathiesen, Computing Cournot-Nash equilibria. Ops Res. 39, 739-748 (1991). 5. S. Dafermos, An iterative scheme for variational inequalities. Math. Program. 26, 40-47 (1983). 6. R. D. Theocharis, On the stability of the Coumot solution in the oligopoly problem. Rev. Econ. Stud. 27,133-134 (1959). I. J. Tirole, The Theory of Industrial Organization. MIT Press, Cambridge, Mass. (1989). 8. 0. L. Mangasarian, Solution of symmetric linear complementarity problems by iterative methods. J. Optimiz. Theory and Applic. 22,465-485 (1987). 9. J. S. Pang, Necessary and sufficient conditions for the convergence of iterative methods for the linear complementarity problem. J. Optimiz. Theory Applic. 42, l-17 (1984). 10. W. I. Zangwill, Nonlinear Programming: A Unifier Approach. Prentice-Hall, Englewood Cliffs, N.J. (1969).
ADJUSTMENT COMPUTING
0305-0548/94 $6.00 + 0.00
Copyright 0 1993 Pergamon Press Ltd
PROCESS-BASED APPROACH FOR A NASH-COURNOT EQUILIBRIUM JOANNA M. LELENO~
Department of Mathematical Sciences, Virginia Commonwealth University, Division of Operations Research and Statistics, 1015 West Main Street, Richmond, VA 23284-2014, U.S.A.
(Received November 1991; in revised form December 1992)
Scope and Purpose-The economic model is that of a noncooperative n-person game. We consider a homogeneous product market supplied by n producers (players). Their cost functions and the product demand function are assumed to be known. Each firm’s objective is to decide how much to produce (and sell) so as to maximize its own profit, and no firm is allowed to manipulate decisions of the remaining firms in order to increase its own profit. At a Nash-Cournot equilibrium no firm can increase its profit by a unilateral change of its output level. The main purpose of this paper is to examine properties of adjustment processes in an n-firm oligopolistic market. Adjustment processes are peculiar in that they emulate the entire sequence of steps each firm undertakes so as to maximize its profit. In this paper we consider two types of adjustment processes: sequential and simultaneous. Our analysis shows that they both lead to closed algorithms. Moreover, we show that if the demand function is linear, then the sequential adjustment process converges to an equilibrium point, and we explain why the simultaneous process fails to converge in this case. Abstract-This paper is focused on deriving properties of adjustment process-based algorithms for computing a Nash-Coumot equilibrium point. Two adjustment processes are considered: sequential and simultaneous. The corresponding numerical procedures are closely related to the Gauss-Seidel and Jacobi methods, respectively, for solving nonlinear systems. Our analysis shows that under quite general assumptions, both processes lead to closed algorithms. We also show that in case of a linear demand function (which implies the absence of the diagonal dominance of the Jacobian matrix of marginal profits), the sequential adjustment process-based algorithm converges to a Nash-Cournot equilibrium point. For this purpose we employ matrix splitting-based algorithms for solving a nonlinear complementarity problem, and an interesting equivalence between a Nash-Coumot equilibrium point and a certain n-dimensional optimization problem. This equivalence allows us also to gain more insight into why the simultaneous adjustment process fails to converge in this case.
1. INTRODUCTION
Consider a homogeneous product market supplied by n firms (players). Each firm is a Cournot firm, that is, it selects its decision so as to maximize its own profit given the decisions of all the remaining firms. In this paper we consider a static model in which ith firm’s decision variable is its output level, xi, i = 1, . ..,n.Letx_idenoteavector(x,x,...xi_,xi+,...x,),andletxi(xi;x_i) denote ith firm’s profit function, given the output levels of all its rivals. The function a&; x_~) is defined in the following way ni(Xi; X_i)=XiP
Xi+ (
C Xj -Ci(Xi), ) j#i
where p(Q) is the (inverse) demand function, Q 20, and ci(Xi)is the total cost function of the ith firm. Definition 1. A nonnegative vector x* = (XTxf . . . x.*) of output levels is said to be a Nash-Coumot (NC) equilibrium point if for each i= 1, . . ., n, xt is the profit maximizing output level of the ith firm, given the equilibrating output levels Xfi of all the remaining firms. That is, tJ. M. Leleno is Assistant Professor of Operations Research in the Department of Mathematical Sciences at Virginia Commonwealth University, Richmond, Va. She holds a Ph.D. in Industrial Engineering and Operations Research from Virginia Polytechnic Institute and State University, Blackburg, Va. Her current research interests include nonlinear complementarity problems, computing equilibrium points and bilevel optimization.
JOANNAM. LELENO
58
foreachi=l,...,n Ki(X:; X*_i)~Ki(q;
X*_i)
for all qa0.
In other words, for each i
1 X*,)-ci(x?)84p(4+zi xtP ( Xi * + jfi
for all qa0.
Xf)q(q)
Various sets of assumptions have been shown to guarantee the existence of an NC equilibrium point. For our purposes, the following conditions are assumed to hold: Assumption 1. The demand function p(Q) is nonincreasing for Q 20, strictly decreasing over its positive range, continuously differentiable and such that p(O)>O, p’(O)<0 and Qp(Q) is concave for Q > 0. Assumption 2. The cost functions ci(q) are convex, strictly increasing over q>O, continuously differentiable and such that c,(O)=O, i= 1, . . ., n. Assumption 3. For each i = 1, . . . , n there exists a positive number qi. such that p(q) - C!(q) 2 0 for all 4 > qi”. Assumptions 1 and 2 imply that each firm’s profit function is strictly concave in its decision variable and continuously differentiable. Assumption 3 is included to impose an upper bound on each firm’s output level. Let us point out that if there exists a positive number Q” such that p(Q) = 0 for all QaQ’, then assumption 3 is automatically implied by 1 and 2. Under assumptions l-3 a unique NC equilibrium point exists [l, 21. A number of papers [l-4] have dealt with computing an NC equilibrium point. The purpose of this paper is to examine an algorithm which emulates the way in which players make their decisions, and to show its convergence to a unique NC equilibrium point. This approach can be briefly described via the following generic scheme. Let x0 > 0 be any given initial vector of n output levels. At each step each firm chooses its output level given the most recent decisions of all its rivals. To be more specific, let xk = [XT]> 0 be an n-dimensional vector of output levels obtained at step k. Then the next iterate, xk+i = [x:’ ‘1, is a vector of optimal solutions to n one-dimensional optimization problems, that is, for each i= 1, . . ., n
xf+l solves
max q>O
qp p+C
xj+‘+
C xi” -ci(q) j>i
j
i(
>
. I
Thus players one after another make adjustments to the current total industry supply. This technique is referred to as a sequential adjustment process. Alternatively, a simultaneous adjustment process refers to an approach in which
xf+’ solves
qp
max q>O
i
q+ C x+ -c(q) (
j#i
I)
i
1.
Both processes can be viewed as a sequence of repetitions of steps in which each player (firm) selects its output level so as to maximize its own profit given the total output of all the remaining firms. However, the simultaneous adjustment process requires that all players act simultaneously, whereas in the sequential adjustment process firms act in an organized way, so that one firm at a time is allowed to make an adjustment to the total industry supply. As a result of it each firm at the moment of announcing its decision is ‘at equilibrium’ with the remaining firms. This property is not shared by the simultaneous adjustment process because the moment the changes in the output levels are made, each firm is likely to face a totally different market situation than the one it used to make its own decision. In a more general framework, both processes can be represented as decomposition algorithms for solving variational inequality problems or nonlinear complementarity problems. Also, let us notice that the sequential and simultaneous adjustment processes are closely related to the Gauss-Seidel and Jacobi methods, respectively, for solving nonlinear systems. Both processes were analyzed in [l] in the context of stability of Nash equilibrium in noncooperative games. A similar approach was presented in [3], though more in the context of
Approach for computing a Nash-Cournot equilibrium
59
variational inequalities and convergence results derived in [S]. In general, no conditions are known to be both necessary and sufficient for the convergence of adjustment processes to a unique NC equilibrium point. In [l] it was demonstrated that strictly diagonal dominance of a certain matrix is a sufficient condition for the convergence of both adjustment processes. Economic interpretation of this condition is that for each firm, its own effect on marginal profit exceeds the aggregate cross-effects resulting from actions of all its rivals. In other words, each firm’s marginal profit is more sensitive to its own decision than to decisions of all the remaining firms. This condition also appeared in [4], where a sequential linear complementarity problem algorithm is examined for global convergence to a unique Nash-Cournot equilibrium point. Sufficient conditions derived in [S] require that a norm of certain matrices employed in computing an equilibrium point via variational inequality formulation be less than one. The above sufficient conditions are in fact difficult to verify a priori. Moreover, none of them holds if the demand function is linear. The main purpose of this paper is to examine properties of adjustment processes. Our interest in algorithms based on such processes is rooted in their structure which allows one to closely follow, step by step, the entire decision process in which firms (players) are involved. This paper is organized in the following way. First, we demonstrate that under assumptions l-3 both adjustment processes lead to a closed algorithm. Next we consider a linear demand function, and show that the sequential adjustment algorithm is convergent in this case. In [6] an illustrative numerical example is presented to demonstrate that the simultaneous adjustment process fails to converge if the demand function is linear and there are more than two players. Therefore, we also attempt to point out factors causing such instability. Finally, we show an interesting local property of the sequential adjustment process due to which at some point of the process, all firms that produce nothing at equilibrium are effectively eliminated from the market. We adopt the following notation. All vectors are column vectors and superscript T is used to denote the transpose. Symbol e denotes a vector of ones and I-an identity matrix. For any two n-dimensional vectors x= [Xi] and y= [yi], symbol (x, y) denotes its inner product, that is (x, y) =I:= 1 Xiyi. If A is an n x m matrix, and 9 and $ are subsets of {1,2, . . . , n} and {1,2, . . . . m}, respectively, then A, denotes a submatrix of A composed of those rows whose indices are in 9. Similarly, A,, denotes a submatrix of A composed of those rows and columns whose indices are in 9 and 3, respectively. R’!+denotes the nonnegative orthant in R”.
2. CHARACTERIZATION
OF ADJUSTMENT
PROCESSES
As already stated, under assumptions l-3 a unique NC equilibrium point exists. Moreover, due to differentiability and concavity of each firm’s profit function, the following two representations of an NC equilibrium point are valid [l, 31. 1. Variational inequality representation A nonnegative vector x* = (x7 x: x,*) of output levels is a Nash-Cournot if and only if it solves the following variational inequality in x (F(x), y-x)>0
for all y>O,
equilibrium point (1)
where F: R; + R”, with F(X) = Lfi(X)], and where fi(x) = - &ri(xi; x-,)/ax,.
2. Nonlinear complementarity problem representation x:) of output levels is a Nash-Cournot A nonnegative vector x* = (x: xt if and only if it solves the following nonlinear complementarity problem: find x such that
F(x)>O, x20 and (F(x), x)=0.
We find the two characterizations adjustment processes.
equilibrium point (2)
of an NC equilibrium point very useful in our analysis of
JOANNAM. LEL~NO
60
Definition 2. Sequential adjustment process is given by a mapping G: R; + R;
defined in the
following way : yEG(x) if and only if for each i=l, yi solves
max
@I q+ 1
q30
Yj+
C
xj
-Ci(q)
j>i
j
u
. . ,, n
)
,
I
whereas simultaneous adjustment process is given by a mapping G,: R’: + R: defined as y~G,(x)
if and only if for each i=l,
. . ., n
By assumptions 1 and 2 for each i = 1, . . . , n, optimization problems (3) and (3a) seek the maximum of a strictly concave function over the nonnegative real line. Moreover, by assumption A3, for each firm its feasible decision set is in fact reduced to a closed interval [O, q1J. Therefore, both problems (3) and (3a) have a unique solution, and thus mappings G(x) and G,(x) are well defined functions that map R: into a set X, where X is a Cartesian product [O, qlJx . . , x[O, qJ. As we show in Lemma 1 below, both functions are continuous. Lemma 1. The sequential adjustment process and the simultaneous adjustment process are closed
algorithm. Proof. We show that the sequential adjustment process is a closed algorithm. With minor changes
this proof can also be used to establish closedness of the simultaneous adjustment process. We have just shown that mapping G: R: + X which represents the sequential adjustment process is in fact a point-to-point algorithm G, so that x Ir+r = G(xk). To show that the algorithm G is closed we need to demonstrate that for any convergent sequence (x”}~ the following implication is true: ({X”>K3 % {YkIK+ 7, where yk = G(xk), k E K) =+-y = G(f). First note that the sequence {x”} is bounded (by assumptions), therefore it has a convergent subsequence. Suppose that {x”}~ + jz and {ykjK + y, where y” = G(xk), k E K. Then for each k E K we have YfP Yf+Cv”+C j
j,i”)-
c(yk)>qp i
i
’
q+ C (
yt+
j
1
x:
j>i
-ci(q)9
for all t?a&
)
i= 1, . . ., n.
By assumption, functions p( .) and ci( .) are continuous, therefore by taking the limit as k --) COwe obtain Jip
( j
j.+C ’
j>ixJ)
-. -
.(j)>qp
” i’
q+ (
1
jj+
j
C
-ci(q),
%j
j>i
forallqN
>
i= 1, . . ., n,
which shows that algorithm G is continuous at any accumulation point %.This completes the proof. 0 Having established closedness of the algorithmic mapping G, we need to show that the algorithm is descent in order to guarantee its convergence to a unique NC equilibrium point. This issue is addressed in the next section, where the inverse demand function p(Q) is assumed to be linear. 3. ADJUSTMENT
PROCESSES
IN CASE
OF A LINEAR
DEMAND
FUNCTION
Assume that the demand function p(Q) is linear, p(Q)=abQ, where a>O, b >O. Then by definition, x* is an NC equilibrium point if for each i= 1, . . ., n .
Approach for computing a Nash-Coumot
61
equilibrium
Next we state a result that allows one to identify an NC equilibrium point as an optimal solution to an n-dimensional optimization problem. For this purpose let M denote an n x n matrix whose all diagonal entries are identical and equal to 2b, and all remaining entries are identical and equal to b. Theorem 1. x* is an NC equilibrium point if and only if x* solves the following problem:
min
2 ci(xi) .
; xTMx-aeTx+
XERT
i=l
Proof. Matrix M is a symmetric and positive definite. This property together with assumption 2 implies that problem (4) involves minimization of a strictly convex and differentiable objective function over nonnegative orthant of R”. Therefore the Karush-Kuhn-Tucker system is then both necessary and sufficient for optimality. That is, x* is an optimal solution to (4) if and only if Mx* + C(x*) - ae > 0, (x*, Mx*+C(x*)-ae)=O,
(5)
x*20, where C(x) = [ci(Xi)]l is an n-dimensional vector of derivatives of total cost functions ci( .) evaluated at Xi. TO complete the proof let us notice that with p(Q)=a- bQ, the functions fi(x) employed in the nonlinear complementarity problem representation (2) of an NC equilibrium point become simply fi(X)= -(a-2bxi-b Cj+i Xj)+C:(XI), i=l, . . . , n, which further yields F(x) = Mx - ue + C(x). Therefore, the nonlinear complementarity characterization (2) of an NC equilibrium point and system (5) are equivalent, which completes the proof. 0 As evident from theorem 1, one can find an NC equilibrium point by simply solving problem (4), and various methods can be used for this purpose. What is perhaps more important is that an equilibrium point, which by definition ensures optimality for all n firms simultaneously, solves a single optimization problem for the whole industry. This is true when the demand function is linear. It would be desirable to drop the linearity assumption and still be able to claim such property. However, the shape of the demand function seems to play a crucial role in derivation of this result. The main reason for which problem (4) is presented in this paper is that it provides a useful tool in demonstrating that the sequential adjustment process is a convergent algorithm. Before we establish the main result let us interpret theorem 1 in economic terms. Simple algebraic operations allow to rewrite the negative of the objective function in problem (4) as (a - beTx)eTx -I;= 1 ci(xi) + 3b[(eTx)2 - xTx], which explicitly shows that an NC equilibrium point maximizes the total industry profit (the first two terms) and the net consumer surplus [7] adjusted according to the behavioral nature of Cournot firms. Sequential adjustment process can be now formulated in the following way: Initialization step. Choose any x0 > 0 as the initial vector. Step k, k = 1, . . . . Given xk - ‘, sequentially, find x:, xk,, . . ., xz, where xf is the unique solution to the following system in 4: b,Tix;+2bq+b
1 xi-‘-a+c;(q)aO j>i
q bC x;+2bq+b
=O
C x;-l-a+cgq)
j
j>i
q>o. It can be easily verified that for each i, xf is the unique solution to the equation: O,a-b
2bxf+c;(xf)=max (
c xjk+ c ( j
j>i
xf-’
. I
Given the above description, the entire process can be expressed as one that incorporates a matrix splitting approach for solving linear complementarity problem. Here, we employ the splitting (B, U)
JOANNA M. LELENO
62
of the matrix M (i.e. M=B+ U), where U is a strictly upper triangular matrix with all its nonzero entries equal to b, and B = D + UT, where D = 261. In this context, the sequential adjustment process translates into a procedure G(.) that generates a sequence ix’}, where xk, k= 1,2, . . . is the solution to the system: Bx+Uxk-l
+C(x)-ae>O
(x, Bx+Uxk-‘+C(x)--ue)=O
(6)
x20. That is x”=G(x’-‘) if and only if xk solves the system (6). A parallel description of the simultaneous adjustment process requires that in the equations above the terms Cji xj”-’ be replaced with Cjfi x!-‘. Consequently, xk=GJxk-i) if and only if xk solves the system Dx+(U+UT)xk-l+C(x)-ce~O (x, Dx+(U+UT)xk-l+C(x)-ue)=O
(6a)
x20. If all marginal cost functions are linear, then C(x)=Sx +r, where S is a nonnegative diagonal matrix, and r is a vector of constants. Consequently, algorithms G(.) and G,( .) represented by systems (6) and (6a) respectively, then become matrix splitting-based algorithms for solving a linear complementarity problem. This technique for solving linear complementarity problems has been intensively studied by many authors [S, 93, and several of their results can be cited here in order to assert the convergence of sequential adjustment process. Moreover, we have found some of the approaches presented therein useful in our analysis of algorithm G(. ) in the case of nonlinear marginal cost functions. We point out that forthcoming results do not employ a specific form of matrices D, B and M. They remain valid as long as matrix M is symmetric and positive definite, D is diagonal and positive definite, U is strictly upper triangular, and M = B+ U, B= D + UT. Lemma 2. Let D be an n x n positive definite diagonal matrix and let U be an n x n strictly upper triangular matrix. If B = D + UT, then the quadratic form xT(B - U)x is positive definite.
Proof. By definition of matrices B and U, we have xr(B- U)~=X~(D+U-U~)X=X~DX>O for all x#O, where the last inequality follows from positive definiteness of matrix D. Lemma 3. Let h(~)=)x~M~-c~e~x+C~= Moreover, h(x)= h(y) if and only if x = y.
1 Ci(Xi). For any ~20,
if y=G(x),
Cl
then h(x)ah(y).
Proof. First let us notice that by convexity of the cost functions we have
$‘i ci(xi)-i$l
ci(JJi)ai$l cl(yi)(xi-YJ=C(Y)T(x-Y)
for any x, y in R:+
Let x 20, and y = G(x). Then by the last inequality we obtain h(x)-h(y)8;
xrMx-aeTx-f
yTMy+aeTy+C(y)r(x-y)
=(x-y)‘(By+Ux-ae+C(y)+B(x-y))-; >(x-y)rB(x-y)-;
(x-y)‘M(x-y)=;
(x-y)rM(x-y) (x-y)r(B-U)(x-y)aO.
Here, the second equality results from rearrangement of terms, the third one from variational inequality representation (1) of the assumption y = G(x), x 20 and, the fourth one from the employed matrix splitting, and the last one from lemma 2.
Approach for computing a Nash-Cournot
equilibrium
63
Theorem 2. For any x020, the sequence {x”) generated by the sequential adjustment process via system (6) converges to a unique NC equilibrium point x*. Proof. By lemma 1 the algorithm G( .) is closed, and by lemma 3 the sequence {h(xk)} is decreasing. Therefore, from Zangwill’s convergence theorem A, [lo], we can conclude that the sequence {x”} converges to a unique optimal solution to problem (4), which by theorem 1 completes the proof. 0
We note that by theorem 2 if the demand function is linear and the cost functions are convex and differentiable, then for any initial vector x0 the sequential adjustment process produces a sequence {x”} convergent to a unique equilibrium point. Thus in this case, convergence of the process is guaranteed under the same conditions that guarantee the existence of a unique equilibrium point. No additional requirements need to be imposed. This result stays in contrast with conclusions drawn in [l], where linearity of the demand function was perceived as the main obstacle in establishing convergence of adjustment processes. The analysis in [l] shows that the process is convergent if the Jacobian matrix of the function F(x) is diagonally dominant. In case of the linear demand function, we obtain F(x) = Mx + C(x)-ae, and therefore its Jacobian F’(x)= [fij(x)] is given by fli(X)=2b + C:i(Xi), i=l, . . ., n i, j=l 9 ‘. ., n, j#i, and flj(x)=b, where cy( .) denotes the second derivative of the ith firm cost function. It can be easily verified that for the matrix F’(x) = [f;j(X)] to be diagonally dominant for all x 3 0, we need c:((xi) - (n - 2)b > 0, for all x,20, i= 1, . . ., n. As also pointed out in [l], this condition fails when there are more than two firms, and all cost functions are linear. Yet, due to theorem 2, the process is convergent regardless of the number of firms and relationship between c:((Xi)and b. As shown by a simple numerical example in [6], the simultaneous adjustment process fails to converge to a unique equilibrium point. Analytically, it can be explained by the fact that, in contrast to the sequential adjustment process, the resulting sequence (h(xk)} is not decreasing. A closer look at (6a) and the relationship between h(x) and h(y), where y = G,(x), shows that h(x)-h(y)+
(x-y)T(D-U-UT)(x-y),
and matrix D-U-UT is not positive definite except when n = 2. In fact, this matrix may be indefinite after all, and this property makes the simultaneous adjustments diverge. Intuitively, this divergence can be viewed as a result of ‘instability’ that each firm experiences each time the decisions are announced. Recall that when the demand function is linear, for each firm, its own-effects on marginal profit are in fact dominated by the aggregate cross-effects. If all the firms are allowed to make their decisions simultaneously, they all will be willing to increase their output levels if the current market price is low, and decrease their output levels if the current market price is high. Consequently, the total industry supply will alternate between some small value and some large value. As demonstrated by theorem 2, this phenomenon does not occur when the adjustments to the total industry output are made sequentially, one at a time. Therefore, one is tempted to conclude that in the absence of dominance of each firm’s effects on its marginal profit over the cross-effects from its rivals, the manner in which firms make their decisions plays an important role as far as the convergence issue is considered, and hence market stability. Having established the convergence of the process, we next demonstrate its local property, due to which in some neighborhood of the equilibrium point, the algorithm essentially becomes the well-known Gauss-Seidel method for solving nonlinear systems. In case of linear complementarity problems, this property has been established by Mangasarian [S]. Our result is more general since it allows for nonlinear terms, however, in the proof we employ a very similar technique. Lemma 4. Let x* be a unique NC equilibrium point. Assume that the nonlinear complementarity problem (5) is nondegenerate, i.e. x* + Mx* - ae + C(x*) > 0, and that c;(O)= 0, i = 1, . . . , n. Let 9 and 9 denote sets defined as follows: B = {i: x: > 0} and 9 = (i: (Mx* - ae + C(x*)), > 0). There exists k* such that the sequential adjustment process results in {xk}, where for each k> k* B,,x~,+‘+C,(X~~)=
-Uu,,xk,+,e,P
and
xk,‘=O.
JOANNAM. LELENO
64
Proof. First let us notice that the sequential adjustment process as defined by system (6) can be, after rearrangement of terms, represented in the following way:
x’+D-‘C(xk)=(xk-r
+D-‘C(X~-~)-D-~[MX~-’
+UT(xk-xk-l)-ae+C(xk-l)]}+,
(7)
where for a vector y = [vi], y+ denotes a vector whose ith component is max(O, yi). For notational simplicity, let A(xk-r)=Mxkel +U’(~~-x~-~)-ae+C(x~-~). By definition of A(xk) and that of set 9, we have lim A(xk), = [Mx* - ae + C(x*)], > se9
for some E > 0.
k-m
By complementarity
requirement, continuity of C(x) and assumption C(0) = 0, we also have lim [x~-~+D-~C(X~-~)]~=O. k-m
Furthermore, since D is a positive definite diagonal matrix, we can conclude that there exists k, such that for all k > k, and some 6 > 0 while [x”- ’ +D-‘C(xk-‘)Jl<~eZ,
[D-‘A(xk-1)]52~e,
which by (7) implies [xk +D- ‘C(xk)lia = 0 for all k > k,. This conclusion together with C(x) > C(0) for all ~20, which follows from the concavity of the cost functions, results in Xk ia =o
for all k>k,.
In a similar manner it can be shown that there exists k, such that for all k> k, and some q>O [xk-1+D-1C(xk-1)]9>I]eg
while [D-lA(xk-l)]B<~e,,
which yields for all kak,.
[x~+D-~C(X~)]~=[X~-~+D-~C(X~-~)-D-~A(X~-~)],~ The two results and simple algebra finally lead to B,,xv’ xy ’ = 0 for all k > k*, which completes the proof.
+&(x2
‘)= -U,wxk,+aeg
and 0
It is worthwhile to make two brief comments on lemma 4. First let us point out that the assumption c:(O)=O, i= 1, . . . , n, used in lemma 4 is not restrictive at all. Its role is technical only. In order to show that lemma 4 is still true without this assumption all that one needs to do is rewrite system (6) so that vectors C(x) and -ae are replaced by vectors C(x) and r, respectively, where C(x)=C(x)C(0) and r = C(O)-ae, and next apply the proof to thus modified nonlinear complementarity system. The second remark deals with interpretation of lemma 4. What follows from it is that after finite number of iterations, the sequential adjustment process in fact reduces to Gauss-Seidel method for solving nonlinear system B,,9x, + C&x,)= - U,q,x,+ aeg. That is, at some point of the process, all firms that produce nothing at equilibrium will be eliminated from the game. We close this paper with a couple of final remarks:
(1) In this paper we showed that under general assumptions both adjustment processes produce a closed algorithm. We also derived a simple optimization problem that can be used to find an NC equilibrium solution when the demand function is linear. (2) One of the sufficient conditions for the global convergence of adjustment process requires that each firm’s marginal profit have the dominance property. If the demand function is linear this property cannot hold unless there are only two firms. In spite of the absence of this property we established convergence of the sequential adjustment process. We also showed reasons for which the simultaneous adjustment process does not necessarily converge. Results in Section 3 allow one to employ decomposition algorithm (6) to solve (3) a nonlinear optimization problem in which function f(x) = +xrMx +x1= 1 gi(Xi),
Approach for computing a Nash-Coumot
equilibrium
65
where matrix M is symmetric and positive definite, and Qi(Xi) is a convex, differentiable function, i= 1, . . ., n, needs to be minimized over R$. REFERENCES 1. D. Gabay and H. Moulin, On the uniqueness and stability of Nash-equilibria in noncooperative games. In Applied Stochastic Control in Econometrics and Management Science (Edited by A. Bensoussan, P. Kleindorfer and C. S. Tapiero). North-HoIiand, Amsterdam (1980). 2. F. H. Murphy, H. D. Sherali and A. L. Soys&r, A mathematical pro~amming approach for dete~ining oligopolistic market equilib~um. Mathf Pragr~. 24,92-106 (1982). 3. P. T. Harker, A variational inequality approach for the determination of ohgopohstic market equilibrium. Mathi Program. 30, 105-111 (1984). 4. C. D. Kolstad and L. Mathiesen, Computing Cournot-Nash equilibria. Ops Res. 39, 739-748 (1991). 5. S. Dafermos, An iterative scheme for variational inequalities. Math. Program. 26, 40-47 (1983). 6. R. D. Theocharis, On the stability of the Coumot solution in the oligopoly problem. Rev. Econ. Stud. 27,133-134 (1959). I. J. Tirole, The Theory of Industrial Organization. MIT Press, Cambridge, Mass. (1989). 8. 0. L. Mangasarian, Solution of symmetric linear complementarity problems by iterative methods. J. Optimiz. Theory and Applic. 22,465-485 (1987). 9. J. S. Pang, Necessary and sufficient conditions for the convergence of iterative methods for the linear complementarity problem. J. Optimiz. Theory Applic. 42, l-17 (1984). 10. W. I. Zangwill, Nonlinear Programming: A Unifier Approach. Prentice-Hall, Englewood Cliffs, N.J. (1969).