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Generalized Nash games and quasi-variational inequalities
European Journal of Operational Research 54 (1991) 81-94 North-Holland
81
Theory and Methodology
Generalized N ash games and quasi-variational inequalities P a t r i c k T. H a r k e r D e p a r t m e n t o f Decision Sciences, The W h a r t o n School, University o f Pennsyh, ania, Philadelphia, P A 19104-6366, U S A Received October 1988: revised March 1989
Abstract: A generalized Nash game is an n-person noncooperative game with nondisjoint strategy sets; other names for this game form include social equilibria and pseudo-Nash games. This paper explores both the qualitative and quantitative properties of such games through the use of quasi-variational inequality theory. Several interesting relationships between the variational and quasi-variational inequality forms of this class of games are described and the practical implementation of generalized Nash games are explored at length. Keywords: Games, mathematical programming, computational analysis
1. Introduction In a standard normal form, noncooperative game it is usually assumed that the feasible set of the game is composed of the full Cartesian product of the individual strategy sets. In other words, it is assumed that players can only affect the utilities of the other players but not their feasible sets. However, the early development of the Nash equilibrium concept does allow for such feasibility interactions; e.g., Arrow and Debreu (1954), Rosen (1965) and Ichiishi (1983). This class of games with the feasible set being a proper subset of the full Cartesian product of the individual players' strategy sets has been given several names in the literature: social equilibria games, pseudo-Nash equilibria games, and generalized Nash equilibria (GNE) games. The G N E game can be defined as follows. Let N is the set of n players, where n is finite, X i C R m is the strategy set of player i; assume this set to be compact and convex in order to simplify the discussion, X = H i ~ N X I ~ R"m, and X N \ i = ~I] ~ N,I ~ t X J ' that is, X represents the full Cartesian product of the strategy sets and X N\~ equals this full set minus the i-th player's feasible region. Let K': X u \ ' ~ X ~ be a point-to-set mapping which represents the ability of players j 4: i to affect the feasible strategies of player i. Note that
K'(x)
X'
0377-2217/91/$03.50 ~) 1991
Vx
X.
Elsevier Science Publishers B.V. All rights reserved
(1)
82
P. T. Harker / GeneralizedNash games and quasi-variational in equalities
Finally, let the utility function for player i be represented by the function u': gr K ~---, R, where gr K i denotes the graph of the mapping K ~ and let K denote the mapping formed from the K ' Vi ~ N; i.e., for all x ~ X, K ( x ) = l-I i ~ NKi(x N\i) where x u\~ represents the vector x with the i-th subvector x ' removed. The generalized, pseudo- or social equilibria game is thus defined by the data { X', K ~, u ~},~ N, and an equilibrium of this game is defined as a point x* = (x .1, x *z . . . . . x *n) a X such that X*'Egi(x
*N\i )
(2)
Vi~U,
u i ( x * ) > ~ u i ( y i, x * N \ i )
Vyi~Ki(x*N\i)
i~N.
(3)
In words, x * is a G N E if it is feasible with respect to the mapping K ~ and if it is a maximizer of each player's utility over the constrained feasible set (3). Rosen (1965) studied a restricted form of this game in which the u ' ( x ) functions are assumed to be concave in x ~ and a special case of the mapping K ~ is assumed in order to establish the existence of a solution. Rosen also considered the questions of uniqueness and computation, although the results derived therein are overly restrictive in the sense of being difficult to verify and not explicitly dealing with the mapping K'. Ichiishi (1983) has provided the most general existence result for this class of games in his Theorem 4.3.1 which we now paraphrase: Theorem 1. Let ( X ~, K', u i ), ~ N be the data for a G N E game. Assume that X ~ is a nonempty, convex and compact subset of R m, K ~ is both upper and lower semicontinuous in X, K ' ( x N\~) is nonempty, closed and convex V x N\i ~ X N\i, u i is continuous in gr K ~, and that ui( ., x N\i) is quasi-concave in K i ( x N\i) given any x N\~ ~ X N\~. Then there exists a generalized Nash equilibrium of the game. In parallel with the results discussed above, several papers have also appeared in the literature dealing with the variational and quasi-variational inequality representations of the Nash and generalized Nash equilibria games which shall prove useful in the sequel. Briefly, a finite-dimensional variational inequality (VI) problem is to find an x* ~ ~2, a closed and convex subset of R t, such that r(x*)T(y-x*)>~O
Vy~n,
(4)
where F: ~2 --+ R t. The quasi-variational inequality (QVI) problem is defined by replacing ~2 in (4) with a point-to-set mapping /2: Rt--+ R/; i.e., to find an x* ~ I2(x*) such that F(x*)T(y--x*)>_.O
Vy~I2(x*).
(5)
The name quasi-variational inequality is attributed to Bensoussan, Goursat and Lions (1973), although the actual formulation appeared earlier in Bensoussan and Lions (1973). Recently, Chang and Pang (1982) have studied an extension of the basic QVI problem in which F is defined as a set-valued mapping. The recognition that Nash equilibria problems (full Cartesian products of strategy sets) could be cast as a VI dates back to the paper by Lions and Stampacchia (1967) which dealt with infinite-dimensional strategy sets. Recent papers dealing with the finite-dimensional case include G a b a y and Moulin (1980), Harker (1984) and Harker (1986). The formulation of the generalized Nash equilibrium game as a QVI is attributed to Bensoussan (1974); this work has recently been extended by Baiocchi and Capelo (1984). In both cases, the authors deal with infinite-dimensional strategy sets. To date, no work has been done on the special structure of finite-dimensional G N E games within the QVI framework. The purpose of this paper is to study the G N E game via finite-dimensional quasi-variational inequalities in order to achieve 'sharper' results than those which have been obtained in either the A r r o w - D e b r e u / R o s e n / I c h i i s h i literature or in the literature on infinite-dimensional QVIs. By 'sharper' results one means to derive (a) more easily understood and verifiable existence and uniqueness conditions, (b) results on efficient algorithms for the solution of such problems, and (c) sensitivity/stability results. In order to accomplish the above mentioned goals, a special case of a finite-dimensional QVI will be shown to simplify and extend the results found in Chang and Pang (1982), and these results will then be used to
P. T. Harker / Generalized Nash games and quasi-variational in equalities
83
analyze the G N E game. Thus, this paper presents results which are not only of interest to game theory, but to the general variational inequality literature as well. The outline of the remainder of this paper is as follows. The next section will state the Nash equilibrium and G N E problems in their respective VI and QVI representations, and will discuss the issues of existence, uniqueness, computation, and sensitivity/stability analysis. Section 3 presents the main results on finite-dimensional QVI problems which will then be applied to the G N E game. The paper concludes with a discussion of an economic interpretation of the K ~ mappings in Section 4, and the practical uses of the results of this paper in Section 5.
2. Variational and quasi-variational inequality formulations In order to begin the analysis of the generalized Nash equilibrium ( G N E ) game, let us reformulate (2)-(3) as a quasi-variational inequality (QVI) problem (5). However, let us first reformulate the traditional Nash equilibrium problem in its variational inequality (VI) form in order to compare and contrast the two problem structures. The standard Nash equilibrium game is defined by the data { X ~, u ~}, ~ N; i.e., no constraints across players represented by the mapping K i are permitted. Assuming that u'( ., x u\~) is concave and once continuously differentiable in X i and that X' is a closed and convex subset of R m for all i ~ N, the first-order conditions of player i's utility maximization problem max ui(x i, x ':\i)
(6)
x' ~ X'
can be stated as
-W,,u'(x'*,
X:~\')T(x'--X'*)>~O
Vx'~X',
(7)
where - W , , u ' ( x ~*, x N\~) is the gradient of u ~ with respect to player i's strategy vector and x *i is the solution to (6) for a given x N \ ( In words, condition (7) states that at the point x * ' , no feasible ascent direction exists. It is well-known that since there are no binding constraints across players (i.e., the feasible set of the game is equal to X, the full Cartesian product of individual player's strategy sets), the equivalent VI problem for the Nash game is simply the summation of the individual player's first-order conditions: find x* ~ X such that
E
-
' - x * ' ) >/0
Vx
x.
(8)
iEN
Defining F,(x*)=-W,u'(x
*i, x *NV)
and
F ( x * ) = ( F ~ ( x * ) T, F 2 ( x * ) T . . . . . F . ( x * ) ) T,
problem (8) can be rewritten in the standard VI form:
F(x*)T(x--x*)>_.O
VxeX.
(9)
The QVI formulation of the G N E game is derived in the exact same manner as the standard Nash game. Replacing X ~ with Ki(x N\~) in (6) and (7), the QVI formulation of the G N E game becomes: find x* ~ K ( x * ) such that
E -- ~7".'U'(X-i" x*N\i)T( "Xi-X*' ) ~0
~/xEK(x*),
(10)
or
Vx e K ( x * ) .
(11)
P.T. Harker / Generalized Nash games and quasi-variational in equalities
84
The existence of a solution to a finite-dimensional VI problem (4) can be proven by either assuming that F is a continuous function over ~2 and that /2 is compact and convex, or by assuming that I2 is closed and convex and that F is a continuous, coercive function; i.e. [Ix ]1 --' oo implies that
[ F ( x ) - F ( x ° ) ] T(x - x °) fl x -
(12)
x ° PI
for some vector x ° ~ ~2. Uniqueness is established by assuming that F is a strictly monotone function; i.e. [F(x)-F(y)]T(x-y)>O
Vx, y ~ ,
xey.
(13)
Proofs of these results can be found in Kinderlehrer and Stampacchia (1980). Applying these results to the Nash equilibrium game (8)-(9), one can either assume that X is compact or that the negative gradient vector is coercive along with the assumption of continuity of this gradient vector in order to establish existence. Uniqueness then follows from the assumption that the negative gradient vector is strictly monotone, a condition related to the notion of strict diagonal dominance found in the economics literature. Gabay and Moulin {1980) present these results in detail and Harker (1986) presents an application of these results in spatial economic theory. In recent years a substantial amount of work has been devoted to the efficient solution of finite-dimensional variational inequalities. The earliest method was transported from the infinite-dimensional literature: the projection algorithm. Later methods include the diagonalization (or relaxation or nonlinear Jacobi) algorithm and Newton's method. As shown by Pang and Chart (1982), both the projection and diagonalization algorithms have linear convergence rates, whereas Newton's method is quadratically convergent. However, the former two algorithms are somewhat easier to implement than Newton's method since the resulting subproblems are mathematical programming problems. Dafermos (1983) presents a general iterative scheme which subsumes the projection and diagonalization algorithms, and Harker (1988) presents an acceleration step which substantially enhances the performance of these algorithms. Other work in this area includes the dissertation by Hammond (1984), the extension of the ellipsoid algorithm by Liithi (1985), and a demonstration in Harker (1984) that the application of a diagonalization algorithm to (8)-(9) yields a convergent tatonnement process. In summary, several well-understood and efficient algorithms exist for finite-dimensional VI problems. The other work which is of interest in this paper is the recent research on the sensitivity/stability analysis of finite-dimensional V1 problems. Tobin (1986) has essentially specialized Robinson's (1980) generalized equation framework in order to derive useful analytic expression for the sensitivity of the VI solution, which is assumed to be unique, to perturbations in the problem data. Kyparisis (1987) has extended Tobin's results to allow for slightly more general second-order sufficiency conditions and has studied the directional differentiability of the VI solution when the assumption of strict complementary slackness is relaxed. In terms of the finite-dimensional QVI problem (5), far fewer results exist. In terms of existence theory, the best results appear in the paper by Chan and Pang (1982). In that paper, Chan and Pang consider a generalization of (5) in which F is assumed to be a general point-to-set map instead of a point-to-point map as in this paper. The results in this paper can be extended to the more general situation depicted in Chan and Pang (1982) but for ease of presentation, this extension shall be deferred to a subsequent paper. The most relevant theorem in the Chart and Pang (1982) paper for the analysis of G N E games is their Theorem 5.2 which is restated in the notion used in this paper. Theorem 2. Let F and $2 be respectively a point-to-point and point-to-set mapping from R / into itself. Suppose
that there exists a nonempty compact convex set 11 such that (i) $2(x) c F Vx~F, (ii) F is continuous on F, (iii) $2 is a nonempty, continuous, closed and convex valued mapping on F. Then there exists a least one solution to the Q I/1 problem (5).
P.T. Harker / Generalized Nash games and quasi-variational in equalities
85
By defining F to be X and F to be the vector defined in (9), the existence of a G N E equilibrium can be established with this result. The uniqueness of a QVI and thus a G N E solution is another matter. No general uniqueness results exist for the QVI problem. As we shall see in the sequel, one can establish conditions which insure uniqueness, but these conditions are often overly restrictive. Thus, uniqueness is a rarity in the QVI problem. In terms of algorithmic developments for the QVI problem, there exist only two methods of solution. The first is an adaptation of the projection algorithm for VI problems to the case where the feasible set and thus the set on which the projection is made depends upon the current vector. Convergence conditions for this algorithm can be found in Baiocchi and Capelo (1984) for the infinite-dimensional case and in Chan and Pang (1982) for finite-dimensional problems. The second class of algorithms deals with a special case of the QVI problem; namely the implicit complementary problem (ICP). The ICP defines the correspondence/2 to be the sum of a closed convex set ~ and a continuous point-to-point mapping m(x). Chan and Pang (1982) establish the convergence of an iterative scheme for this class of problems which possesses a linear rate of convergence. No other algorithms and no extensive empirical testing of the above mentioned algorithms have appeared in the literature. Finally, the lack of uniqueness a n d / o r regularity in the QVI framework has essentially blocked any work on sensitivity/stability analysis; no literature on the sensitivity of QVI solutions beyond what one can infer from Robinson's (1980) generalized equation framework currently exists.
3. Relationships between the variational and quasi-variational inequality formulations In this section, a special class of QVIs will be defined and its properties exploited in order to derive several interesting and useful results: a simplified existence proof, uniqueness conditions, and the recognition of several other algorithms for a special class of QVI problems. These results will also provide extensions of the existence, uniqueness and algorithmic literature concerning G N E games. To begin, a simple example of a G N E game will be presented which shall prove useful in the sequel. Consider the two-person game depicted in Figure 1. Each player picks a number x, between 0 and 10, and the sum of their numbers must be less than or equal to 15. The utility functions and mappings K s are defined by
UI(x1, X2) = 34.00x 1 - ( x ~ ) : - ( ~ ) x l x 2,
u2(x~, Xz) = 24.25x 2 - (x2) 2 - (~)x~x 2, K ' ( Y 2) = {x 110 ~< x, ~< 10, x, ~< 15 - x2}, K 2 ( y ' ) = {x210 ~
(~)x 2,
Fz(x ) = - 2 4 . 2 5 + 2x 2 + ( ~ ) x , .
(14) (15)
Consider the VI problem defined by the function F given in (14)-(15) over the set / ¢ = {x 10--,
K~(9)={x, lO<~x~<~6},
K2(5)={x210<~x2410}
P.T. Harker / Generalized Nash games and quasi-variational in equalities
86
X2
x2 ~ 10 10
\
\ X1 + X 2
8 -
~-~ 1 5
\
VI s o l u t i o n
\
\
\
\
-F 6 _
II 4
--
/ / /
F / / /
/ 2 -
QVI solutions
I / I / I / / I / / / / 1 1 1 / /
K(9.5,5.5)
/
/ / / I / / / / / / / / / / / / / / / / / / /
I
I 2
I
I
I
4
I
I
6
I
/ / /
xz<_ 10
r
I
8
X1
10
F i g u r e 1. E x a m p l e of G N E
no player has an incentive to alter this current strategy. The other G N E solutions lie between the points (9, 6) and (10, 5) on the line x 1 + x 2 = 15; Figure 1 illustrates these solutions. For example, the point (9.5, 5.5) yields functions values of F = ( - 3, 1 ~). Both players have an incentive to move from this point when considering the problem over/¢, but cannot move given the constraint defined by K(9.5, 5.5). Thus, the set of G N E solution is composed of the point (5, 9) and the interval [(9, 6), (10, 5)]. The behavior exhibited in the above example leads to the following theorem concerning a special class of QVI problems: 3. Let F and 12 be respectively point-to-point and point-to-set mappings from R I into itself. Suppose that there exists a nonempty compact convex set F such that (i) 12(x) ~ F Vx ~ F, and
Theorem
(ii) x ~ 12(x) V x ~ F.
Then any solution to the variational inequality defined with the function F over the set F is a quasi-variational inequality solution, but the converse is not true in general. Proof. Let x * ~ F be any VI solution, which implies that F(x*)~(x-x*)>O,
vx~r,
which implies by assumption (ii) that x * ~ g2(x*) and by (i) that F(x*)T(x--x*)>~O,
Vx ~ $2(x*);
i.e., x * is a QVI solution. The converse of this result is proven by the example just presented in which more QVI solutions exist than just the unique VI solution. [] The key assumption of this theorem is (ii), which states that the mapping 12 have the property of always containing its point of definition. This assumption rules out, for example, mappings which are projections
P. T. Harker / Generalized Nash games and quasi-variational in equalities
87
onto a proper subset of F. In terms of the applicability of this result of this result to G N E games, assumption (i) is always met by defining F =/~. Furthermore, one suspects that there exists a wide variety of applications of the G N E concept for which assumption (ii) is applicable. For example, any constraints of the type defining K ~ in the previous example satisfy assumption (ii). In order to prove this, let us consider the situation in which K ' 'qi ~ N and h e n c e / £ are polyhedral. Define /~ to be a nonempty set of the form
I~= ( x ~ R""'lAx < b },
(16)
where A is a p × nm matrix and b is a p × 1 vector. Letting A' denote the m columns of A associated with x' ~ R m, the mapping K'(x N\~ can be defined as
K'(x~~")=.{y'~R'~,A'y'
<~b- Y'~ A'xJ 1.
(17)
j~N i#=i
Choosing a vector 2 ~ / ~ implies Y~j~ NAJx I ~ b, or
A'Y' <~b -
~ AJY j, .]E N j#~i
that is, 2' ~ K'(Y N~'~) for all i ~ N. Thus, one has the following corollary: Corollary 3.1. Let the set 1( and the mapping K' in the QVI formulation of the generalized Nash equilibrium game (10)-(11) be defined by (16) and (17) respectively. Then every solution to the variational inequality: find x* ~ 1~ such that
-v,,u'(x*',
(18)
t ~/'v'
is a generalized Nash equilibrium, but the converse is not true in general. Similar results can also be derived for other classes of G N E games and QVI problems. One immediate implication of Theorem 3 and Corollary 3.1 for the QVI and G N E problems, respectively, concerns existence. Given the assumptions of the theorem, the proof of the existence of QVI solution follows immediately from the proof of the existence of a VI solution. Thus, if F is compact and convex and if F is continuous, or if F is closed and convex and F is a continuous, coercive function, then the existence of a QVI solution is established. Therefore, Theorem 3 greatly simplifies the proof of existence since one can use standard results in the finite-dimensional VI literature. The second major implication of this theorem is that it establishes a new class of algorithms for the QVI problem. In this special case of the QV! problem, any V1 algorithm such as Newton's method or diagonalization can be employed in order to calculate a solution. This result is of particular interest when one recognizes that the current state-of-the-art in QVI computation provides only linear rates of convergence and with Newton's method now being a possible solution technique, a quadratically convergent algorithm is available for the solution of a variety of QVI problems. The example depicted in Figure 1 also illustrates another property of QVI problems in general and G N E games in particular: the solution set is rarely connected let alone being a singleton. It is well-known in the VI literature that if ~2 is compact and convex and if F is monotone, then the VI solution set will be compact and convex. However, even strict monotonicity in the above example is not sufficient to insure connectedness of the solution set in a QVI, let alone uniqueness. This fact has immediate impact on the ability to perform classical sensitivity analysis on a QVI solution. Tobin (1986) and Kyparisis (1985) must assume local uniqueness in order to apply the implicit function theorem to the VI problem. Without local
P.T. Harker / Generalized Nash games and quasi-variational in equalities
88
uniqueness of the QVI solution set, this approach to deriving analytic expression for the perturbations of the solution vector must be abandoned. Further research is necessary in order to understand the sensitivity and stability of the QVI solution set. Although one lacks uniqueness in the Q V I / G N E setting, there is a question as to what makes the VI solution in the set of QVI solutions special. In order to analyze this question, let us define the mapping /2(x) in slightly more precise terms. Let C denote a nonempty, compact and convex subset of R t and let F be defined as
{xlg(x) >_.o},
r=cn
(19)
where g : R ~---, R p is assumed to be a concave and continuously differentiable function. Let us further assume that the function g ( x ) can be 'split' into 2 × n components g i ( x ) in order to define the mappings Ia(x):
n(x) =
+
gZ(x) >_.0}
n r,
(20)
where g'~ : R ~ - - , R rip, a 1, 2. In terms of the G N E game, this splitting of the binding constraints assumes that the function g ( x ) can be split according to the effect each constraint has on a particular player's strategies when the strategies of other players in the game are held fixed. A simple case of such constraints occurs with linear constraints as in (17). In this case, the constraints for player i, g'~i(x), can be written at the point Y as: ----
g " ( X ) = A'x',
g2'(X)
=
b-
(21)
~ AJ~ j.
(22)
jEN jv*i
Other splittings of g ( x ) could also be permitted in what follows, but we shall focus on (20) for ease of presentation. It is well known (e.g., Chan and Pang, 1982, Theorem 5.1) that the solution x * of a QVI can be characterized as the projection onto the set ~2(x)* of the vector x * - F ( x * ); i.e. x * = Arg m i n ½ [ I x - ( x * - F( x * ) ) rI.
(23)
x~Ja(x*)
Using the definition of ~2(x) in (20), this minimization problem can be rewritten as: rain s.t.
½(x-x*+F(x*))X(x-x*+F(x*)) x ~ F,
(24)
g ' ( x ) >_. -g:(x*). Assuming that the necessary constraint qualifications hold for g l ( x ) at the solution 2 to (24) (e.g., the rows of V'gl(~) corresponding to the constraints in (24) holding with equality are linearly independent), the first-order conditions are necessary and sufficient since the objective function is strictly convex and the constraint set is assumed convex. The first-order conditions of (24) at the solution ~ can be written as: [(2--x*+F(x*))--V'g'(~)Tx]T(x--2)>0 [gl(~)+g2(~)]T(x_~)>/0
V)~R/,
Vx~F,
(25a) (25b)
where ~ ~ R / is the vector of dual variables for the constraints comprising a2(x). This set of first-order conditions leads one to define the following equivalent VI formulation of the QVI problem when the constraints are of the form given in (20):
P.T. Harker / Generalized Nash games and quasi-variational in equalities
89
Theorem 4, Let F be defined by (19), [2(x) by (20), and let gl(x) and g2(x) be concave, once continuously
differentiable functions. Furthermore, assume that the necessary constraint qualifications hold for gl ( x ) at an)' solution to (24). Then x* E (2(x *) is a solution of the quasi-variational inequality F(x*)T(x--x*)>~O
Vx~2(x*)
(26)
if and only if there exists )~* ~ R"f such that (x*; X* ) solves the following equivalent variational inequality': [ r ( x * ) - V ' g t ( x * ) " x * ] T ( x - - x* ) + [ g ' ( x * ) + g Z ( x * ) l ' r ( x - - X*) >/0
forall (x; X ) ~ F X R y .
(27)
Proof. Under the assumed constraint qualifications and conditions on the feasible set, inequalities (25) are
both necessary and sufficient for a solution (2; ~) to (24). A QVI solution is defined as the point in (24) with 2 = x * . Substituting 2 = x * and ~ = X * in (25) yields the following necessary and sufficient conditions for a QVI solution x*" [F(x*)-%Tg'(x*)r)l*]l(x-x*)>/0 [g'(x*)+g2(x*)]T(x--X*)>/0
Vx~r,
(28a)
~'X~R~+p.
(28b)
To prove necessity, assume that x is a QVI solution. The point (x*;)~*) must satisfy (28) since these conditions are necessary for the solution of (26). Summing (28a) and (28b) yields (27), the desired VI formulation. To prove sufficiency, assume that (x*; 4*) satisfies (27) which implies that (x*:)~*) is a solution to the following linear program: min
[F(x*)--%Tg'(x*)T}~*]Tx+[gl(x * +g2(x*)]Tx.
s.t.
x ~ F,
(29)
)~ ~ R +p.
The first-order conditions of this linear program are precisely conditions (28), the sufficient conditions for a solution to (26). [] This equivalent VI formulation is similar to a saddle function in nonlinear programming and can be considered to be a generalized saddle function for QVIs. The equivalent VI formulation (27) is obviously applicable to the G N E game by defining 9 ( x ) = K ( x ) and F ( x ) to be the negative gradient of the utility functions. In the special case of linear constraints (16)-(17), the equivalent VI formulation of the GNE game simplifies to:
E
F,(x*)+(A') X*']
t~N
V.I:EXC//~,
E ,tJ *J (X'-X*' jGN
I
.I*i
j
)~ E R np,
(30) where ~' is a p-dimensional column vector for each i ~ N. The equivalent VI (27) can be used to establish conditions which insure the uniqueness of a QVI solution. Uniqueness of a VI solution is established by assuming that the function defining the VI is strictly monotone. If the function is also once continuously differentiable, strict monotonicity can be established by assuming that the Jacobian of the VI function is positive definite. Considering (27) and
P.T. Harker / Generalized Nash games and quasi-variational in equalities
90
assuming that F(x), V'gl(x), gl(x) and g2(x) are once continuously differentiable, the solution to (27) is unique if the following (l + np) × (l + np) matrix is positive definite for all 2 ~ F, ~ ~ R+P:
+ v,g2( )
(31)
"
Writing out the condition for positive definiteness over the set F × R+p one has
xT[v'F(~)--V'2gl(y)~]X+XTV'gZ(~)Tx>o
Vx~F,
X ~ R "p, (x; X ) • 0 .
(32)
Clearly, if no constraints of the form gl(x) + gZ(x) >10 existed, then (32) collapses to xTv'F(Y)x > 0; i.e., F being strictly monotone assures the uniqueness of a VI solution. However, when these constraints exist, condition (32) is very difficult to apply to the general QVI problem. For example, consider the linear constraints (16)-(17) in (30). The sufficient conditions for uniqueness to the G N E game simplify to:
E
[(x')TvF~(2) x ' ] -
i~N
E ()C) TAjxj
>0
Vx~XAI<,
X ~ R + p, (x; 2 t ) ¢ 0 .
(33)
jEN jq:i
The only general statement which can be made concerning (33) is that if ~ ( 2 ) is strictly monotone and if A ~ is the matrix of zeros for all i ~ N (i.e., there are no constraints across players), then uniqueness is assured. In specific applications it is possible to employ (32) or (33) to derive meaningful uniqueness criteria, but there exist no general conditions at this time. Other than providing a method for establishing the uniqueness of a Q V I / G N E solution, Theorem 4 can also be used to derive an interesting result concerning the the relationship between the solution sets of the QVI defined over (20) and the VI defined over (19): Theorem 5. Any solution x * to the quasi-variational inequality over J2( x ) as defined by (20) which is strictly
interior to the set
+
>1 o)
is a solution to the variational inequality over F as defined in (19) when the assumptions of Theorem 4 hold. Proof. If x* is strictly interior to the set defined above, then
g l ( x * ) + g 2 ( x * ) > 0,
(34)
which implies by complementary slackness of problem (24) that ~ * = 0, the vector of zeros. By the equivalent VI (27) one obtains
F(x*)T(x--x *)+[g'(x*)+g2(x*)]v()t-O)>~O Vx~l',
)t~R"f.
(35)
Since ~ = 0 is feasible, (35) implies
F(x*)V(x-x*)>~O that is, x* is a VI solution
Vx~F,
(36)
[]
The above theorem has a practical implication in that if F is strictly monotone and if the VI solution x* is interior to the binding constraints across players, then the only other place where the G N E could exist is on the boundary of these constraints. The example in the beginning of this section illustrates this situation. Thus, if the unique VI solution is interior, one need only search the boundaries for the remaining generalized Nash equilibria.
P.T. Harker / Generalized Nash games and quasi-variational in equalities
91
In summary, a special case of the finite-dimensional quasi-variational inequality has been identified which has proven to be very useful in the analysis of generalized Nash equilibrium games. This class of VI problems has the property that its solution set contains the solution of a related VI problem. The implications of this results are that the existence theory and quadratically convergent algorithms for VI problems can be employed in the QVI case. Furthermore, one can define an equivalent VI formulation for this class of QVI problems which implies certain uniqueness criteria as well as providing a geometric characterization of the solution set. However, the delineation of the VI solution from the rest of the QVI solutions is still unclear; the purpose of the next section is to provide a further discussion of this issue with regards to the e c o n o m i c / g a m e theoretic interpretations of the VI and QVI solutions to the G N E game.
4. The meaning of nondisjoint strategy sets Ichiishi (1983), among others, has questioned the use of the G N E or social equilibria concept to model realistic situations due to the presence of the nondisjoint strategy sets in the game. For example, Ichiishi (1983, p. 60) states that the G N E / s o c i a l equilibria concept " . . . i s only useful as a mathematical tool to establish existence theorems in various applied contexts", There are, however, many situations in which the G N E concept can be used for more than just proving existence. Consider a situation in which a player is considered to be the 'dictator' of the rules of the game to its players (e.g., a leader in a Stackelberg game or the principal in a principal-agents game). In this case, the agents or followers respond to the constraints imposed on them by the leader. A good example of this type of situation is a governmental authority which imposes rules across all players in a game such as maximum pollution levels in a common body of water. We shall return to the practical implications of the G N E concept in the next section. It suffices for the moment to point out that in the modelling of real situations, the G N E concept can be used for more than proving existence. There are two basic ways by which the constraints across players arise: the imposition of these constraints by an external 'player' in the game, or by the joint imposition by the set of N players on their own actions. In either case, players must be given incentives to obey these constraints on their activities. These incentives take the form of either taxes or subsidies. For example, players in the example in the previous section could be taxed by a certain amount in order to insure that x 1 + x 2 ~< 15. The incentive levels necessary to obey these constraints are precisely the dual variables ~.' for each player i ~ N. In the example of the last section, the dual variable at the solution x* = (5, 9) are (?~1 ~2) = (0, 0) and in the interval x* = [(9, 6); 10, 5)], the dual variables take on the values in the interval [(0, 1); ( 2 47)]. Consider, for example, the point x* = (9.5, 5.5), ~,* = (~, ~ ) at which the utilities of each player are u l ( x ) = 93.41668, u2(x) = 37.8125. If either player 1 or player 2 were permitted to increase the value of their strategy x, by 0.01, their new utility values would increase by 0.00323 and 0.01365, respectively; i.e., the utility increases per unit increase in x, are approximately equal to the values of the dual variables, Thus, any tax structure with values greater than or equal to 2~* would provide sufficient incentives for the players not to violate the constraint xl + x 2 ~ 15. As the above example illustrates, it is the value of the dual variables at the VI versus the other QVI solutions which distinguishes this VI point; namely, that X~ = ~-~ only at the VI solution. This observation can be generalized by the following theorem:
Theorem 6. I f F is a continuous function in the variational inequality over F defined by (19), the conditions of Theorem 4 are assumed, and if it is assumed that the splitting of the constraints g( x ) >~0 is such that 17
~Tg(x) = Y'~ ~ T g " ( x ) /=1
Vx ~ F,
(37)
P. T. Harker / Generalized Nash games and quasi-variational in equalities
92
then the VI solutions are the only points in the set of solutions to the QVI over I2(x) as defined by (20) at which, in (27),)k* can be such that Vi, j = l , 2 ..... n.
~.i=~./
Proof. Consider the VI defined over the set F as in (19). The first-order conditions of this problem at a VI solution x* can be written as
VxEF,
[F(x*)-V'g(x*)~*]'r(x-x*)>~0
(38a)
X*>~O g(x*)>~O,
(?,*)Vg(x*)=0,
(38b)
where ~* is a p-dimensional vector of dual variables. By (37), condition (38a) can be rewritten as
F(x*)-
~Tg"(x*)~*
(x-x*)>~O
Vx~F;
(39)
i=1
that is, 2,*'= ?,* for all i = 1, 2 . . . . . n. Thus, there exists equal dual variables for this constraint splitting at any VI solution. Now consider any other QVI solution 2 not an element of the VI solution set, and assume that N = 7,-/= ~, Vi, j = 1, 2 . . . . . n, holds at this point. By (27) and (37), the equivalent VI can be written as:
F(2)-
E V g ' i ( 2 ) Tx
(x-2)+[gl(2)+g2(2)]v()
~_x)>~0
V(x; X) ~ F × RT:
i=1
(40) or
[ F ( 2 ) - V ' g ( 2 ) T ( ~ ) ] T ( X - - 2) + [g1(2) + g2()~)]T(Tt-- ~) >i 0
V(x;
2~)~FXRf.
(41)
Since
[g'(2).
X) 0
must hold at any QVI solution by (28b), the inequality (40) implies
[F(2)-~g(2)
T ^
T
Vx~F,
X] ( x - 2 ) > t 0
that is, 2 must be a VI solution since (38a) is a sufficient condition by the assumptions of the theorem. Thus, a contradiction is reached and the conclusion of the theorem obtains. [] The assumption made in this theorem concerning the method of constraint splitting (37) is not overly restrictive. Consider, for example, the linear constraints (17), (21) and (22). In this case g(x)
= b -
gl'(x)
O,
= -A'x'.
iGN
Calculating the gradient of these constraints, one has ~7(x)=[-A 1
vgl'(x)=[0
0
-A 2
....
--.
0
A"],
-A'
0
(42)
-..
0].
(43)
Summing (43) over all i E N clearly results in (42). Thus, restriction (37) on the constraint splitting is reasonable and can be expected to hold in a variety of applications.
P.T. Harker / Generalized Nash games and quasi-variational in equalities
93
Therefore, the VI solutions are distinguished from the remainder of the QVI solution set by having equal dual variables (taxes) across players. In the next section the practical implications of this result will be explored.
5. Practical uses of generalized Nash games One of the most difficult computational problems in operations research is the solution of a Stackelberg or principal agent game (Moulin, 1982). The difficulty in this problem is the fact that the constraint set consists of the equilibrium conditions for the followers or agents in the game. That is, the leader in the game possesses a set of strategies and the ability to foresee or predict the reactions of the followers to his strategy selection. The prediction of reactions to his strategy choice is typically modelled by placing conditions (8) in the constraint set of the leader's utility maximization problem. Thus, the mathematical programming formulation of this problem has an infinite number of constraints (8). Most applied game theorists have resorted to heuristic methods to solve the problem described above due to its extreme computational complexity for large-scale problems: e.g. Bard (1983) and Marcotte (1986). However, the results of the last few sections can be exploited in order to solve such a Stackelberg game in certain situations. Consider the case where the leader has as his strategies a vector of taxes or subsidies which he can impose on the followers in order to insure that constraints of the form (19) are satisfied. Assume that the utility of the leader is minus infinity if the constraints are violated and zero otherwise. Thus, the leader will make sure to impose the appropriate taxes k' on each player in order to insure compliance with the constraints. One need not, however, solve this problem in the standard manner but rather, one can simply employ the results of this paper. By solving the QVI formulation of the G N E game, the optimal strategy (taxes) for the leader can be inferred from the dual variables. Furthermore, if the leader is restricted to charge nondiscriminatory taxes (~.' = k ~ Vi, j), then the V1 solution of the G N E game is the appropriate solution to the Stackelberg game. Therefore, the results of this paper can be used for more than the proof of existence of G N E games. In fact, a wide variety of Stackelberg-like problems heretofore unsolvable by the traditional methods of solution can be addressed within the G N E / Q V I framework.
Acknowledgments This research has been supported by the National Science Foundation under grant CEE-840392 and by the Urban Mass Transit Administration under grant PA-11-0032. Several helpful discussions with James D. Laing and the careful comments of the referees are gratefully acknowledged.
References Arrow, K.J., and Debreu, G. (1954), "Existence of an equilibrium for a competitive economy", Econometrica 22, 265--290. Baiocchi, C., and Capelo, A. (1984). Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, New York. Bard, J.F. (1982), "An algorithm for solving the general bilevel programming problem", Mathematics of Operations Research 8, 260-272. Bensoussan, A. (1974), "Points de Nash dans le cas de fonctionnelles quadratiques et jeux diffdrentiels lindaires a N personnes", S I A M Journal Control 12, 460-499. Bensoussan, A., Goursat, M., and Lions, J.L. (1973), "Contr61e impulsionnel et inequations quasi-variationnelles stationnaires", Comptes Rendus des S~anees de l'Acad~mie des Sciences 276, 1279 1284. Bensoussan, A., and Lions, J.L. (1973), "Nouvelle formulation de problemes de controle impulsionnel et applications", Comptes Rendus des S~ances de rAcadOmie des Sciences 276, 1189-1192.
94
P.T. Harker / Generalized Nash games and quasi-uariational in equalities
Chan, D., and Pang, J.S. (1982), "The generalized quasi-variational inequality problem", Mathematics of Operations Research 7, 211-222. Dafermos, S. (1983), "An iterative scheme for variational inequalities", Mathematical Programming 26, 40-47. Gabay, D., and Moulin, H. (1980), "On the uniqueness and stability of Nashequilibria in noncooperative games", in: A. Bensoussan, P. Kleindorfer and C.S. Tapiero (eds.), Applied Stochastic Control in Econometrics and Management Science, North-Holland, Amsterdam, 271-293. Hammond, J.H. (1984), Solving Asymmetric Variational Inequality Problems ad Systems of Equations with Generalized Network Programming Algorithms, unpublished Ph.D. dissertation, Dept. of Mathematics, M.I.T., Cambridge, MA. Harker, P.T. (1984), "A variational inequality approach for the determination of oligopolistic market equilibrium", Mathematical Programming 30, 105-111. Harker, P.T. (1986), "Alternatives models of spatial competition", Operations Research 34, 410-425. Harker, P.T. (1988), "Accelerating the convergence of the diagonalization and projection algorithms for finite-dimensional inequalities", Mathematical Programming 41, 29-59. lchiishi, T. (1983), Game Theory for Economic Analysis, Academic Press, New York. Kinderlehrer, D., and Stampacchia, G. (1980), An Introduction to Variational Inequalities and Their Applications, Academic Press, New York. Kyparisis, J. (1987), "Sensitivity analysis framework for variational inequalities", Mathematical Programming 36, 105-113. Lions, J.L., and Stampacchia, G. (1967), "Variational inequalities", Communications on Pure and Applied Mathematics 20, 493-519. Liithi, H.J. (1985), "On the solution of variational inequalities by the ellipsoid method", Mathematics of Operations Research 10, 515-522. Marcotte, P. (1986), "Network design problem with congestion effects: A case of bilevel programming", Mathematical Programming 34, 142-162. Moulin, H. (1982), Game Theory for the Social Sciences, New York University Press, New York. Pang, J.S., and Chan, D. (1982), "Iterative methods for variational and complementary problems", Mathematical Programming 24, 284-313. Robinson, S.M. (1980), "Strongly regular generalized equations" Mathematics of Operations Research 5, 43-62. Rosen, J.B. (1965), "Existence and uniqueness of equilibrium points for concave n-person game", Econometrica 33, 520-534. Tobin, R.L. (1986), "Sensitivity analysis for variational inequalities", Journal of Optimization Theory and Application 48, 191-204.
81
Theory and Methodology
Generalized N ash games and quasi-variational inequalities P a t r i c k T. H a r k e r D e p a r t m e n t o f Decision Sciences, The W h a r t o n School, University o f Pennsyh, ania, Philadelphia, P A 19104-6366, U S A Received October 1988: revised March 1989
Abstract: A generalized Nash game is an n-person noncooperative game with nondisjoint strategy sets; other names for this game form include social equilibria and pseudo-Nash games. This paper explores both the qualitative and quantitative properties of such games through the use of quasi-variational inequality theory. Several interesting relationships between the variational and quasi-variational inequality forms of this class of games are described and the practical implementation of generalized Nash games are explored at length. Keywords: Games, mathematical programming, computational analysis
1. Introduction In a standard normal form, noncooperative game it is usually assumed that the feasible set of the game is composed of the full Cartesian product of the individual strategy sets. In other words, it is assumed that players can only affect the utilities of the other players but not their feasible sets. However, the early development of the Nash equilibrium concept does allow for such feasibility interactions; e.g., Arrow and Debreu (1954), Rosen (1965) and Ichiishi (1983). This class of games with the feasible set being a proper subset of the full Cartesian product of the individual players' strategy sets has been given several names in the literature: social equilibria games, pseudo-Nash equilibria games, and generalized Nash equilibria (GNE) games. The G N E game can be defined as follows. Let N is the set of n players, where n is finite, X i C R m is the strategy set of player i; assume this set to be compact and convex in order to simplify the discussion, X = H i ~ N X I ~ R"m, and X N \ i = ~I] ~ N,I ~ t X J ' that is, X represents the full Cartesian product of the strategy sets and X N\~ equals this full set minus the i-th player's feasible region. Let K': X u \ ' ~ X ~ be a point-to-set mapping which represents the ability of players j 4: i to affect the feasible strategies of player i. Note that
K'(x)
X'
0377-2217/91/$03.50 ~) 1991
Vx
X.
Elsevier Science Publishers B.V. All rights reserved
(1)
82
P. T. Harker / GeneralizedNash games and quasi-variational in equalities
Finally, let the utility function for player i be represented by the function u': gr K ~---, R, where gr K i denotes the graph of the mapping K ~ and let K denote the mapping formed from the K ' Vi ~ N; i.e., for all x ~ X, K ( x ) = l-I i ~ NKi(x N\i) where x u\~ represents the vector x with the i-th subvector x ' removed. The generalized, pseudo- or social equilibria game is thus defined by the data { X', K ~, u ~},~ N, and an equilibrium of this game is defined as a point x* = (x .1, x *z . . . . . x *n) a X such that X*'Egi(x
*N\i )
(2)
Vi~U,
u i ( x * ) > ~ u i ( y i, x * N \ i )
Vyi~Ki(x*N\i)
i~N.
(3)
In words, x * is a G N E if it is feasible with respect to the mapping K ~ and if it is a maximizer of each player's utility over the constrained feasible set (3). Rosen (1965) studied a restricted form of this game in which the u ' ( x ) functions are assumed to be concave in x ~ and a special case of the mapping K ~ is assumed in order to establish the existence of a solution. Rosen also considered the questions of uniqueness and computation, although the results derived therein are overly restrictive in the sense of being difficult to verify and not explicitly dealing with the mapping K'. Ichiishi (1983) has provided the most general existence result for this class of games in his Theorem 4.3.1 which we now paraphrase: Theorem 1. Let ( X ~, K', u i ), ~ N be the data for a G N E game. Assume that X ~ is a nonempty, convex and compact subset of R m, K ~ is both upper and lower semicontinuous in X, K ' ( x N\~) is nonempty, closed and convex V x N\i ~ X N\i, u i is continuous in gr K ~, and that ui( ., x N\i) is quasi-concave in K i ( x N\i) given any x N\~ ~ X N\~. Then there exists a generalized Nash equilibrium of the game. In parallel with the results discussed above, several papers have also appeared in the literature dealing with the variational and quasi-variational inequality representations of the Nash and generalized Nash equilibria games which shall prove useful in the sequel. Briefly, a finite-dimensional variational inequality (VI) problem is to find an x* ~ ~2, a closed and convex subset of R t, such that r(x*)T(y-x*)>~O
Vy~n,
(4)
where F: ~2 --+ R t. The quasi-variational inequality (QVI) problem is defined by replacing ~2 in (4) with a point-to-set mapping /2: Rt--+ R/; i.e., to find an x* ~ I2(x*) such that F(x*)T(y--x*)>_.O
Vy~I2(x*).
(5)
The name quasi-variational inequality is attributed to Bensoussan, Goursat and Lions (1973), although the actual formulation appeared earlier in Bensoussan and Lions (1973). Recently, Chang and Pang (1982) have studied an extension of the basic QVI problem in which F is defined as a set-valued mapping. The recognition that Nash equilibria problems (full Cartesian products of strategy sets) could be cast as a VI dates back to the paper by Lions and Stampacchia (1967) which dealt with infinite-dimensional strategy sets. Recent papers dealing with the finite-dimensional case include G a b a y and Moulin (1980), Harker (1984) and Harker (1986). The formulation of the generalized Nash equilibrium game as a QVI is attributed to Bensoussan (1974); this work has recently been extended by Baiocchi and Capelo (1984). In both cases, the authors deal with infinite-dimensional strategy sets. To date, no work has been done on the special structure of finite-dimensional G N E games within the QVI framework. The purpose of this paper is to study the G N E game via finite-dimensional quasi-variational inequalities in order to achieve 'sharper' results than those which have been obtained in either the A r r o w - D e b r e u / R o s e n / I c h i i s h i literature or in the literature on infinite-dimensional QVIs. By 'sharper' results one means to derive (a) more easily understood and verifiable existence and uniqueness conditions, (b) results on efficient algorithms for the solution of such problems, and (c) sensitivity/stability results. In order to accomplish the above mentioned goals, a special case of a finite-dimensional QVI will be shown to simplify and extend the results found in Chang and Pang (1982), and these results will then be used to
P. T. Harker / Generalized Nash games and quasi-variational in equalities
83
analyze the G N E game. Thus, this paper presents results which are not only of interest to game theory, but to the general variational inequality literature as well. The outline of the remainder of this paper is as follows. The next section will state the Nash equilibrium and G N E problems in their respective VI and QVI representations, and will discuss the issues of existence, uniqueness, computation, and sensitivity/stability analysis. Section 3 presents the main results on finite-dimensional QVI problems which will then be applied to the G N E game. The paper concludes with a discussion of an economic interpretation of the K ~ mappings in Section 4, and the practical uses of the results of this paper in Section 5.
2. Variational and quasi-variational inequality formulations In order to begin the analysis of the generalized Nash equilibrium ( G N E ) game, let us reformulate (2)-(3) as a quasi-variational inequality (QVI) problem (5). However, let us first reformulate the traditional Nash equilibrium problem in its variational inequality (VI) form in order to compare and contrast the two problem structures. The standard Nash equilibrium game is defined by the data { X ~, u ~}, ~ N; i.e., no constraints across players represented by the mapping K i are permitted. Assuming that u'( ., x u\~) is concave and once continuously differentiable in X i and that X' is a closed and convex subset of R m for all i ~ N, the first-order conditions of player i's utility maximization problem max ui(x i, x ':\i)
(6)
x' ~ X'
can be stated as
-W,,u'(x'*,
X:~\')T(x'--X'*)>~O
Vx'~X',
(7)
where - W , , u ' ( x ~*, x N\~) is the gradient of u ~ with respect to player i's strategy vector and x *i is the solution to (6) for a given x N \ ( In words, condition (7) states that at the point x * ' , no feasible ascent direction exists. It is well-known that since there are no binding constraints across players (i.e., the feasible set of the game is equal to X, the full Cartesian product of individual player's strategy sets), the equivalent VI problem for the Nash game is simply the summation of the individual player's first-order conditions: find x* ~ X such that
E
-
' - x * ' ) >/0
Vx
x.
(8)
iEN
Defining F,(x*)=-W,u'(x
*i, x *NV)
and
F ( x * ) = ( F ~ ( x * ) T, F 2 ( x * ) T . . . . . F . ( x * ) ) T,
problem (8) can be rewritten in the standard VI form:
F(x*)T(x--x*)>_.O
VxeX.
(9)
The QVI formulation of the G N E game is derived in the exact same manner as the standard Nash game. Replacing X ~ with Ki(x N\~) in (6) and (7), the QVI formulation of the G N E game becomes: find x* ~ K ( x * ) such that
E -- ~7".'U'(X-i" x*N\i)T( "Xi-X*' ) ~0
~/xEK(x*),
(10)
or
Vx e K ( x * ) .
(11)
P.T. Harker / Generalized Nash games and quasi-variational in equalities
84
The existence of a solution to a finite-dimensional VI problem (4) can be proven by either assuming that F is a continuous function over ~2 and that /2 is compact and convex, or by assuming that I2 is closed and convex and that F is a continuous, coercive function; i.e. [Ix ]1 --' oo implies that
[ F ( x ) - F ( x ° ) ] T(x - x °) fl x -
(12)
x ° PI
for some vector x ° ~ ~2. Uniqueness is established by assuming that F is a strictly monotone function; i.e. [F(x)-F(y)]T(x-y)>O
Vx, y ~ ,
xey.
(13)
Proofs of these results can be found in Kinderlehrer and Stampacchia (1980). Applying these results to the Nash equilibrium game (8)-(9), one can either assume that X is compact or that the negative gradient vector is coercive along with the assumption of continuity of this gradient vector in order to establish existence. Uniqueness then follows from the assumption that the negative gradient vector is strictly monotone, a condition related to the notion of strict diagonal dominance found in the economics literature. Gabay and Moulin {1980) present these results in detail and Harker (1986) presents an application of these results in spatial economic theory. In recent years a substantial amount of work has been devoted to the efficient solution of finite-dimensional variational inequalities. The earliest method was transported from the infinite-dimensional literature: the projection algorithm. Later methods include the diagonalization (or relaxation or nonlinear Jacobi) algorithm and Newton's method. As shown by Pang and Chart (1982), both the projection and diagonalization algorithms have linear convergence rates, whereas Newton's method is quadratically convergent. However, the former two algorithms are somewhat easier to implement than Newton's method since the resulting subproblems are mathematical programming problems. Dafermos (1983) presents a general iterative scheme which subsumes the projection and diagonalization algorithms, and Harker (1988) presents an acceleration step which substantially enhances the performance of these algorithms. Other work in this area includes the dissertation by Hammond (1984), the extension of the ellipsoid algorithm by Liithi (1985), and a demonstration in Harker (1984) that the application of a diagonalization algorithm to (8)-(9) yields a convergent tatonnement process. In summary, several well-understood and efficient algorithms exist for finite-dimensional VI problems. The other work which is of interest in this paper is the recent research on the sensitivity/stability analysis of finite-dimensional V1 problems. Tobin (1986) has essentially specialized Robinson's (1980) generalized equation framework in order to derive useful analytic expression for the sensitivity of the VI solution, which is assumed to be unique, to perturbations in the problem data. Kyparisis (1987) has extended Tobin's results to allow for slightly more general second-order sufficiency conditions and has studied the directional differentiability of the VI solution when the assumption of strict complementary slackness is relaxed. In terms of the finite-dimensional QVI problem (5), far fewer results exist. In terms of existence theory, the best results appear in the paper by Chan and Pang (1982). In that paper, Chan and Pang consider a generalization of (5) in which F is assumed to be a general point-to-set map instead of a point-to-point map as in this paper. The results in this paper can be extended to the more general situation depicted in Chan and Pang (1982) but for ease of presentation, this extension shall be deferred to a subsequent paper. The most relevant theorem in the Chart and Pang (1982) paper for the analysis of G N E games is their Theorem 5.2 which is restated in the notion used in this paper. Theorem 2. Let F and $2 be respectively a point-to-point and point-to-set mapping from R / into itself. Suppose
that there exists a nonempty compact convex set 11 such that (i) $2(x) c F Vx~F, (ii) F is continuous on F, (iii) $2 is a nonempty, continuous, closed and convex valued mapping on F. Then there exists a least one solution to the Q I/1 problem (5).
P.T. Harker / Generalized Nash games and quasi-variational in equalities
85
By defining F to be X and F to be the vector defined in (9), the existence of a G N E equilibrium can be established with this result. The uniqueness of a QVI and thus a G N E solution is another matter. No general uniqueness results exist for the QVI problem. As we shall see in the sequel, one can establish conditions which insure uniqueness, but these conditions are often overly restrictive. Thus, uniqueness is a rarity in the QVI problem. In terms of algorithmic developments for the QVI problem, there exist only two methods of solution. The first is an adaptation of the projection algorithm for VI problems to the case where the feasible set and thus the set on which the projection is made depends upon the current vector. Convergence conditions for this algorithm can be found in Baiocchi and Capelo (1984) for the infinite-dimensional case and in Chan and Pang (1982) for finite-dimensional problems. The second class of algorithms deals with a special case of the QVI problem; namely the implicit complementary problem (ICP). The ICP defines the correspondence/2 to be the sum of a closed convex set ~ and a continuous point-to-point mapping m(x). Chan and Pang (1982) establish the convergence of an iterative scheme for this class of problems which possesses a linear rate of convergence. No other algorithms and no extensive empirical testing of the above mentioned algorithms have appeared in the literature. Finally, the lack of uniqueness a n d / o r regularity in the QVI framework has essentially blocked any work on sensitivity/stability analysis; no literature on the sensitivity of QVI solutions beyond what one can infer from Robinson's (1980) generalized equation framework currently exists.
3. Relationships between the variational and quasi-variational inequality formulations In this section, a special class of QVIs will be defined and its properties exploited in order to derive several interesting and useful results: a simplified existence proof, uniqueness conditions, and the recognition of several other algorithms for a special class of QVI problems. These results will also provide extensions of the existence, uniqueness and algorithmic literature concerning G N E games. To begin, a simple example of a G N E game will be presented which shall prove useful in the sequel. Consider the two-person game depicted in Figure 1. Each player picks a number x, between 0 and 10, and the sum of their numbers must be less than or equal to 15. The utility functions and mappings K s are defined by
UI(x1, X2) = 34.00x 1 - ( x ~ ) : - ( ~ ) x l x 2,
u2(x~, Xz) = 24.25x 2 - (x2) 2 - (~)x~x 2, K ' ( Y 2) = {x 110 ~< x, ~< 10, x, ~< 15 - x2}, K 2 ( y ' ) = {x210 ~
(~)x 2,
Fz(x ) = - 2 4 . 2 5 + 2x 2 + ( ~ ) x , .
(14) (15)
Consider the VI problem defined by the function F given in (14)-(15) over the set / ¢ = {x 10--,
K~(9)={x, lO<~x~<~6},
K2(5)={x210<~x2410}
P.T. Harker / Generalized Nash games and quasi-variational in equalities
86
X2
x2 ~ 10 10
\
\ X1 + X 2
8 -
~-~ 1 5
\
VI s o l u t i o n
\
\
\
\
-F 6 _
II 4
--
/ / /
F / / /
/ 2 -
QVI solutions
I / I / I / / I / / / / 1 1 1 / /
K(9.5,5.5)
/
/ / / I / / / / / / / / / / / / / / / / / / /
I
I 2
I
I
I
4
I
I
6
I
/ / /
xz<_ 10
r
I
8
X1
10
F i g u r e 1. E x a m p l e of G N E
no player has an incentive to alter this current strategy. The other G N E solutions lie between the points (9, 6) and (10, 5) on the line x 1 + x 2 = 15; Figure 1 illustrates these solutions. For example, the point (9.5, 5.5) yields functions values of F = ( - 3, 1 ~). Both players have an incentive to move from this point when considering the problem over/¢, but cannot move given the constraint defined by K(9.5, 5.5). Thus, the set of G N E solution is composed of the point (5, 9) and the interval [(9, 6), (10, 5)]. The behavior exhibited in the above example leads to the following theorem concerning a special class of QVI problems: 3. Let F and 12 be respectively point-to-point and point-to-set mappings from R I into itself. Suppose that there exists a nonempty compact convex set F such that (i) 12(x) ~ F Vx ~ F, and
Theorem
(ii) x ~ 12(x) V x ~ F.
Then any solution to the variational inequality defined with the function F over the set F is a quasi-variational inequality solution, but the converse is not true in general. Proof. Let x * ~ F be any VI solution, which implies that F(x*)~(x-x*)>O,
vx~r,
which implies by assumption (ii) that x * ~ g2(x*) and by (i) that F(x*)T(x--x*)>~O,
Vx ~ $2(x*);
i.e., x * is a QVI solution. The converse of this result is proven by the example just presented in which more QVI solutions exist than just the unique VI solution. [] The key assumption of this theorem is (ii), which states that the mapping 12 have the property of always containing its point of definition. This assumption rules out, for example, mappings which are projections
P. T. Harker / Generalized Nash games and quasi-variational in equalities
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onto a proper subset of F. In terms of the applicability of this result of this result to G N E games, assumption (i) is always met by defining F =/~. Furthermore, one suspects that there exists a wide variety of applications of the G N E concept for which assumption (ii) is applicable. For example, any constraints of the type defining K ~ in the previous example satisfy assumption (ii). In order to prove this, let us consider the situation in which K ' 'qi ~ N and h e n c e / £ are polyhedral. Define /~ to be a nonempty set of the form
I~= ( x ~ R""'lAx < b },
(16)
where A is a p × nm matrix and b is a p × 1 vector. Letting A' denote the m columns of A associated with x' ~ R m, the mapping K'(x N\~ can be defined as
K'(x~~")=.{y'~R'~,A'y'
<~b- Y'~ A'xJ 1.
(17)
j~N i#=i
Choosing a vector 2 ~ / ~ implies Y~j~ NAJx I ~ b, or
A'Y' <~b -
~ AJY j, .]E N j#~i
that is, 2' ~ K'(Y N~'~) for all i ~ N. Thus, one has the following corollary: Corollary 3.1. Let the set 1( and the mapping K' in the QVI formulation of the generalized Nash equilibrium game (10)-(11) be defined by (16) and (17) respectively. Then every solution to the variational inequality: find x* ~ 1~ such that
-v,,u'(x*',
(18)
t ~/'v'
is a generalized Nash equilibrium, but the converse is not true in general. Similar results can also be derived for other classes of G N E games and QVI problems. One immediate implication of Theorem 3 and Corollary 3.1 for the QVI and G N E problems, respectively, concerns existence. Given the assumptions of the theorem, the proof of the existence of QVI solution follows immediately from the proof of the existence of a VI solution. Thus, if F is compact and convex and if F is continuous, or if F is closed and convex and F is a continuous, coercive function, then the existence of a QVI solution is established. Therefore, Theorem 3 greatly simplifies the proof of existence since one can use standard results in the finite-dimensional VI literature. The second major implication of this theorem is that it establishes a new class of algorithms for the QVI problem. In this special case of the QV! problem, any V1 algorithm such as Newton's method or diagonalization can be employed in order to calculate a solution. This result is of particular interest when one recognizes that the current state-of-the-art in QVI computation provides only linear rates of convergence and with Newton's method now being a possible solution technique, a quadratically convergent algorithm is available for the solution of a variety of QVI problems. The example depicted in Figure 1 also illustrates another property of QVI problems in general and G N E games in particular: the solution set is rarely connected let alone being a singleton. It is well-known in the VI literature that if ~2 is compact and convex and if F is monotone, then the VI solution set will be compact and convex. However, even strict monotonicity in the above example is not sufficient to insure connectedness of the solution set in a QVI, let alone uniqueness. This fact has immediate impact on the ability to perform classical sensitivity analysis on a QVI solution. Tobin (1986) and Kyparisis (1985) must assume local uniqueness in order to apply the implicit function theorem to the VI problem. Without local
P.T. Harker / Generalized Nash games and quasi-variational in equalities
88
uniqueness of the QVI solution set, this approach to deriving analytic expression for the perturbations of the solution vector must be abandoned. Further research is necessary in order to understand the sensitivity and stability of the QVI solution set. Although one lacks uniqueness in the Q V I / G N E setting, there is a question as to what makes the VI solution in the set of QVI solutions special. In order to analyze this question, let us define the mapping /2(x) in slightly more precise terms. Let C denote a nonempty, compact and convex subset of R t and let F be defined as
{xlg(x) >_.o},
r=cn
(19)
where g : R ~---, R p is assumed to be a concave and continuously differentiable function. Let us further assume that the function g ( x ) can be 'split' into 2 × n components g i ( x ) in order to define the mappings Ia(x):
n(x) =
+
gZ(x) >_.0}
n r,
(20)
where g'~ : R ~ - - , R rip, a 1, 2. In terms of the G N E game, this splitting of the binding constraints assumes that the function g ( x ) can be split according to the effect each constraint has on a particular player's strategies when the strategies of other players in the game are held fixed. A simple case of such constraints occurs with linear constraints as in (17). In this case, the constraints for player i, g'~i(x), can be written at the point Y as: ----
g " ( X ) = A'x',
g2'(X)
=
b-
(21)
~ AJ~ j.
(22)
jEN jv*i
Other splittings of g ( x ) could also be permitted in what follows, but we shall focus on (20) for ease of presentation. It is well known (e.g., Chan and Pang, 1982, Theorem 5.1) that the solution x * of a QVI can be characterized as the projection onto the set ~2(x)* of the vector x * - F ( x * ); i.e. x * = Arg m i n ½ [ I x - ( x * - F( x * ) ) rI.
(23)
x~Ja(x*)
Using the definition of ~2(x) in (20), this minimization problem can be rewritten as: rain s.t.
½(x-x*+F(x*))X(x-x*+F(x*)) x ~ F,
(24)
g ' ( x ) >_. -g:(x*). Assuming that the necessary constraint qualifications hold for g l ( x ) at the solution 2 to (24) (e.g., the rows of V'gl(~) corresponding to the constraints in (24) holding with equality are linearly independent), the first-order conditions are necessary and sufficient since the objective function is strictly convex and the constraint set is assumed convex. The first-order conditions of (24) at the solution ~ can be written as: [(2--x*+F(x*))--V'g'(~)Tx]T(x--2)>0 [gl(~)+g2(~)]T(x_~)>/0
V)~R/,
Vx~F,
(25a) (25b)
where ~ ~ R / is the vector of dual variables for the constraints comprising a2(x). This set of first-order conditions leads one to define the following equivalent VI formulation of the QVI problem when the constraints are of the form given in (20):
P.T. Harker / Generalized Nash games and quasi-variational in equalities
89
Theorem 4, Let F be defined by (19), [2(x) by (20), and let gl(x) and g2(x) be concave, once continuously
differentiable functions. Furthermore, assume that the necessary constraint qualifications hold for gl ( x ) at an)' solution to (24). Then x* E (2(x *) is a solution of the quasi-variational inequality F(x*)T(x--x*)>~O
Vx~2(x*)
(26)
if and only if there exists )~* ~ R"f such that (x*; X* ) solves the following equivalent variational inequality': [ r ( x * ) - V ' g t ( x * ) " x * ] T ( x - - x* ) + [ g ' ( x * ) + g Z ( x * ) l ' r ( x - - X*) >/0
forall (x; X ) ~ F X R y .
(27)
Proof. Under the assumed constraint qualifications and conditions on the feasible set, inequalities (25) are
both necessary and sufficient for a solution (2; ~) to (24). A QVI solution is defined as the point in (24) with 2 = x * . Substituting 2 = x * and ~ = X * in (25) yields the following necessary and sufficient conditions for a QVI solution x*" [F(x*)-%Tg'(x*)r)l*]l(x-x*)>/0 [g'(x*)+g2(x*)]T(x--X*)>/0
Vx~r,
(28a)
~'X~R~+p.
(28b)
To prove necessity, assume that x is a QVI solution. The point (x*;)~*) must satisfy (28) since these conditions are necessary for the solution of (26). Summing (28a) and (28b) yields (27), the desired VI formulation. To prove sufficiency, assume that (x*; 4*) satisfies (27) which implies that (x*:)~*) is a solution to the following linear program: min
[F(x*)--%Tg'(x*)T}~*]Tx+[gl(x * +g2(x*)]Tx.
s.t.
x ~ F,
(29)
)~ ~ R +p.
The first-order conditions of this linear program are precisely conditions (28), the sufficient conditions for a solution to (26). [] This equivalent VI formulation is similar to a saddle function in nonlinear programming and can be considered to be a generalized saddle function for QVIs. The equivalent VI formulation (27) is obviously applicable to the G N E game by defining 9 ( x ) = K ( x ) and F ( x ) to be the negative gradient of the utility functions. In the special case of linear constraints (16)-(17), the equivalent VI formulation of the GNE game simplifies to:
E
F,(x*)+(A') X*']
t~N
V.I:EXC//~,
E ,tJ *J (X'-X*' jGN
I
.I*i
j
)~ E R np,
(30) where ~' is a p-dimensional column vector for each i ~ N. The equivalent VI (27) can be used to establish conditions which insure the uniqueness of a QVI solution. Uniqueness of a VI solution is established by assuming that the function defining the VI is strictly monotone. If the function is also once continuously differentiable, strict monotonicity can be established by assuming that the Jacobian of the VI function is positive definite. Considering (27) and
P.T. Harker / Generalized Nash games and quasi-variational in equalities
90
assuming that F(x), V'gl(x), gl(x) and g2(x) are once continuously differentiable, the solution to (27) is unique if the following (l + np) × (l + np) matrix is positive definite for all 2 ~ F, ~ ~ R+P:
+ v,g2( )
(31)
"
Writing out the condition for positive definiteness over the set F × R+p one has
xT[v'F(~)--V'2gl(y)~]X+XTV'gZ(~)Tx>o
Vx~F,
X ~ R "p, (x; X ) • 0 .
(32)
Clearly, if no constraints of the form gl(x) + gZ(x) >10 existed, then (32) collapses to xTv'F(Y)x > 0; i.e., F being strictly monotone assures the uniqueness of a VI solution. However, when these constraints exist, condition (32) is very difficult to apply to the general QVI problem. For example, consider the linear constraints (16)-(17) in (30). The sufficient conditions for uniqueness to the G N E game simplify to:
E
[(x')TvF~(2) x ' ] -
i~N
E ()C) TAjxj
>0
Vx~XAI<,
X ~ R + p, (x; 2 t ) ¢ 0 .
(33)
jEN jq:i
The only general statement which can be made concerning (33) is that if ~ ( 2 ) is strictly monotone and if A ~ is the matrix of zeros for all i ~ N (i.e., there are no constraints across players), then uniqueness is assured. In specific applications it is possible to employ (32) or (33) to derive meaningful uniqueness criteria, but there exist no general conditions at this time. Other than providing a method for establishing the uniqueness of a Q V I / G N E solution, Theorem 4 can also be used to derive an interesting result concerning the the relationship between the solution sets of the QVI defined over (20) and the VI defined over (19): Theorem 5. Any solution x * to the quasi-variational inequality over J2( x ) as defined by (20) which is strictly
interior to the set
+
>1 o)
is a solution to the variational inequality over F as defined in (19) when the assumptions of Theorem 4 hold. Proof. If x* is strictly interior to the set defined above, then
g l ( x * ) + g 2 ( x * ) > 0,
(34)
which implies by complementary slackness of problem (24) that ~ * = 0, the vector of zeros. By the equivalent VI (27) one obtains
F(x*)T(x--x *)+[g'(x*)+g2(x*)]v()t-O)>~O Vx~l',
)t~R"f.
(35)
Since ~ = 0 is feasible, (35) implies
F(x*)V(x-x*)>~O that is, x* is a VI solution
Vx~F,
(36)
[]
The above theorem has a practical implication in that if F is strictly monotone and if the VI solution x* is interior to the binding constraints across players, then the only other place where the G N E could exist is on the boundary of these constraints. The example in the beginning of this section illustrates this situation. Thus, if the unique VI solution is interior, one need only search the boundaries for the remaining generalized Nash equilibria.
P.T. Harker / Generalized Nash games and quasi-variational in equalities
91
In summary, a special case of the finite-dimensional quasi-variational inequality has been identified which has proven to be very useful in the analysis of generalized Nash equilibrium games. This class of VI problems has the property that its solution set contains the solution of a related VI problem. The implications of this results are that the existence theory and quadratically convergent algorithms for VI problems can be employed in the QVI case. Furthermore, one can define an equivalent VI formulation for this class of QVI problems which implies certain uniqueness criteria as well as providing a geometric characterization of the solution set. However, the delineation of the VI solution from the rest of the QVI solutions is still unclear; the purpose of the next section is to provide a further discussion of this issue with regards to the e c o n o m i c / g a m e theoretic interpretations of the VI and QVI solutions to the G N E game.
4. The meaning of nondisjoint strategy sets Ichiishi (1983), among others, has questioned the use of the G N E or social equilibria concept to model realistic situations due to the presence of the nondisjoint strategy sets in the game. For example, Ichiishi (1983, p. 60) states that the G N E / s o c i a l equilibria concept " . . . i s only useful as a mathematical tool to establish existence theorems in various applied contexts", There are, however, many situations in which the G N E concept can be used for more than just proving existence. Consider a situation in which a player is considered to be the 'dictator' of the rules of the game to its players (e.g., a leader in a Stackelberg game or the principal in a principal-agents game). In this case, the agents or followers respond to the constraints imposed on them by the leader. A good example of this type of situation is a governmental authority which imposes rules across all players in a game such as maximum pollution levels in a common body of water. We shall return to the practical implications of the G N E concept in the next section. It suffices for the moment to point out that in the modelling of real situations, the G N E concept can be used for more than proving existence. There are two basic ways by which the constraints across players arise: the imposition of these constraints by an external 'player' in the game, or by the joint imposition by the set of N players on their own actions. In either case, players must be given incentives to obey these constraints on their activities. These incentives take the form of either taxes or subsidies. For example, players in the example in the previous section could be taxed by a certain amount in order to insure that x 1 + x 2 ~< 15. The incentive levels necessary to obey these constraints are precisely the dual variables ~.' for each player i ~ N. In the example of the last section, the dual variable at the solution x* = (5, 9) are (?~1 ~2) = (0, 0) and in the interval x* = [(9, 6); 10, 5)], the dual variables take on the values in the interval [(0, 1); ( 2 47)]. Consider, for example, the point x* = (9.5, 5.5), ~,* = (~, ~ ) at which the utilities of each player are u l ( x ) = 93.41668, u2(x) = 37.8125. If either player 1 or player 2 were permitted to increase the value of their strategy x, by 0.01, their new utility values would increase by 0.00323 and 0.01365, respectively; i.e., the utility increases per unit increase in x, are approximately equal to the values of the dual variables, Thus, any tax structure with values greater than or equal to 2~* would provide sufficient incentives for the players not to violate the constraint xl + x 2 ~ 15. As the above example illustrates, it is the value of the dual variables at the VI versus the other QVI solutions which distinguishes this VI point; namely, that X~ = ~-~ only at the VI solution. This observation can be generalized by the following theorem:
Theorem 6. I f F is a continuous function in the variational inequality over F defined by (19), the conditions of Theorem 4 are assumed, and if it is assumed that the splitting of the constraints g( x ) >~0 is such that 17
~Tg(x) = Y'~ ~ T g " ( x ) /=1
Vx ~ F,
(37)
P. T. Harker / Generalized Nash games and quasi-variational in equalities
92
then the VI solutions are the only points in the set of solutions to the QVI over I2(x) as defined by (20) at which, in (27),)k* can be such that Vi, j = l , 2 ..... n.
~.i=~./
Proof. Consider the VI defined over the set F as in (19). The first-order conditions of this problem at a VI solution x* can be written as
VxEF,
[F(x*)-V'g(x*)~*]'r(x-x*)>~0
(38a)
X*>~O g(x*)>~O,
(?,*)Vg(x*)=0,
(38b)
where ~* is a p-dimensional vector of dual variables. By (37), condition (38a) can be rewritten as
F(x*)-
~Tg"(x*)~*
(x-x*)>~O
Vx~F;
(39)
i=1
that is, 2,*'= ?,* for all i = 1, 2 . . . . . n. Thus, there exists equal dual variables for this constraint splitting at any VI solution. Now consider any other QVI solution 2 not an element of the VI solution set, and assume that N = 7,-/= ~, Vi, j = 1, 2 . . . . . n, holds at this point. By (27) and (37), the equivalent VI can be written as:
F(2)-
E V g ' i ( 2 ) Tx
(x-2)+[gl(2)+g2(2)]v()
~_x)>~0
V(x; X) ~ F × RT:
i=1
(40) or
[ F ( 2 ) - V ' g ( 2 ) T ( ~ ) ] T ( X - - 2) + [g1(2) + g2()~)]T(Tt-- ~) >i 0
V(x;
2~)~FXRf.
(41)
Since
[g'(2).
X) 0
must hold at any QVI solution by (28b), the inequality (40) implies
[F(2)-~g(2)
T ^
T
Vx~F,
X] ( x - 2 ) > t 0
that is, 2 must be a VI solution since (38a) is a sufficient condition by the assumptions of the theorem. Thus, a contradiction is reached and the conclusion of the theorem obtains. [] The assumption made in this theorem concerning the method of constraint splitting (37) is not overly restrictive. Consider, for example, the linear constraints (17), (21) and (22). In this case g(x)
= b -
gl'(x)
O,
= -A'x'.
iGN
Calculating the gradient of these constraints, one has ~7(x)=[-A 1
vgl'(x)=[0
0
-A 2
....
--.
0
A"],
-A'
0
(42)
-..
0].
(43)
Summing (43) over all i E N clearly results in (42). Thus, restriction (37) on the constraint splitting is reasonable and can be expected to hold in a variety of applications.
P.T. Harker / Generalized Nash games and quasi-variational in equalities
93
Therefore, the VI solutions are distinguished from the remainder of the QVI solution set by having equal dual variables (taxes) across players. In the next section the practical implications of this result will be explored.
5. Practical uses of generalized Nash games One of the most difficult computational problems in operations research is the solution of a Stackelberg or principal agent game (Moulin, 1982). The difficulty in this problem is the fact that the constraint set consists of the equilibrium conditions for the followers or agents in the game. That is, the leader in the game possesses a set of strategies and the ability to foresee or predict the reactions of the followers to his strategy selection. The prediction of reactions to his strategy choice is typically modelled by placing conditions (8) in the constraint set of the leader's utility maximization problem. Thus, the mathematical programming formulation of this problem has an infinite number of constraints (8). Most applied game theorists have resorted to heuristic methods to solve the problem described above due to its extreme computational complexity for large-scale problems: e.g. Bard (1983) and Marcotte (1986). However, the results of the last few sections can be exploited in order to solve such a Stackelberg game in certain situations. Consider the case where the leader has as his strategies a vector of taxes or subsidies which he can impose on the followers in order to insure that constraints of the form (19) are satisfied. Assume that the utility of the leader is minus infinity if the constraints are violated and zero otherwise. Thus, the leader will make sure to impose the appropriate taxes k' on each player in order to insure compliance with the constraints. One need not, however, solve this problem in the standard manner but rather, one can simply employ the results of this paper. By solving the QVI formulation of the G N E game, the optimal strategy (taxes) for the leader can be inferred from the dual variables. Furthermore, if the leader is restricted to charge nondiscriminatory taxes (~.' = k ~ Vi, j), then the V1 solution of the G N E game is the appropriate solution to the Stackelberg game. Therefore, the results of this paper can be used for more than the proof of existence of G N E games. In fact, a wide variety of Stackelberg-like problems heretofore unsolvable by the traditional methods of solution can be addressed within the G N E / Q V I framework.
Acknowledgments This research has been supported by the National Science Foundation under grant CEE-840392 and by the Urban Mass Transit Administration under grant PA-11-0032. Several helpful discussions with James D. Laing and the careful comments of the referees are gratefully acknowledged.
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