Consider the GNEP from Section 1 with utility functions θνand a strategy set X satisfying the requirements from Assumption 1.1. In this section, our aim is to show that the GNEP can be reformulated as a nonsmooth equation and then we present several conditions guaranteeing the BD-regularity condition of the equation.
Suppose that x is a solution of the GNEP. Then if for player ν, a suitable constraint qualification (like the slater condition) holds, it follows that there exists a Lagrange multiplier such that the Karush-Kuhn-Tucker (KKT) conditions
(3.1)
are satisfied.
Let us consider the KKT conditions for the VI(F,X). Assuming that a suitable constraint qualification holds at a solution x, the KKT conditions can be expressed as
(3.2)
which is equivalent to
(3.3)
The next lemma from [13] relates the normalized Nash equilibria to the KKT conditions (3.3).
Lemma 3.1 (i) Let x be a solution of VI(F,X) at which the KKT conditions (3.3) hold. Then x is a solution of the GNEP (normalized Nash equilibria) at which the KKT conditions (3.1) hold with λ1 = λ2 = ⋯ = λN= λ.
(ii) Viceversa, let x be a solution of the GNEP at which KKT conditions (3.1) hold with λ1 = λ2 = ⋯ = λN. Then x is a solution of VI(F, X).
Using the minimum function , the KKT conditions (3.2) can equivalently be written as the nonlinear system of equations
(3.4)
where is defined by
and
From Assumption 1.1, we know that Φ is semismooth.
In the following, our aim is to present several conditions guaranteeing that all elements in the generalized Jacobian ∂Φ(ω) (and hence in the B-subdifferential ∂
B
Φ(ω)) are nonsingular. Our first result gives a description of the structure of the matrices in the generalized Jacobian ∂Φ(ω).
Lemma 3.2 Let. Then, each element H ∈ ∂Φ(ω)Tcan be represented as follows:
where
are diagonal matrices whose ith diagonal elements are given by
for any μ
i
∈ [0,1].
Proof. The first n components of the vector function Φ are continuously differentiable, so the expression for the first n columns of H readily follows. Then, consider the last m columns. Use the fact that
if i is such that -g
i
(x) ≠ λ
i
, then φ is continuously differentiable at (-g
i
(x), λ
i
) and the expression for the (n + i)th column of H follows. If instead -g
i
(x) = λ
i
, then, using the definition of the B-subdifferential, it follows that
Taking the convex hull, we get
This gives the representation of H ∈ ∂Φ(ω)T.
Our next aim is to establish conditions guaranteeing that all elements in the generalized Jacobian ∂Φ(ω) at a point ω = (x,λ) satisfying Φ(ω) = 0 are nonsingular.
Theorem 3.1 Letbe a solution of the system Φ(ω) = 0. Consider the following two statements:
-
(a)
The strong second-order sufficient condition and the linear independence constraint qualification (LICQ) for VI(F,X) holds at x*.
-
(b)
Any element in ∂Φ(ω*) is nonsingular.
It holds that (a) ⇒ (b).
Proof. For the sake of notational simplicity, let us define the following subsets of the index set I := {1,...,m},
Moreover, we need
The following relationships between these index sets can easily be seen to hold:
Using a suitable reordering of the constraints, every element H ∈ ∂Φ(ω*)Thas the following structure:
(3.5)
where D
a
(ω*)02 and D
b
(ω*)02 are positive definite diagonal matrices. Note that we abbreviated etc. by g+ etc. in (3.5). It is obvious that H is nonsingular if and only if the following matrix is nonsingular,
In turn, this matrix is nonsingular if and only if the following matrix is nonsingular:
(3.6)
Let be such that
(3.7)
we know that
(3.8)
By the first, second and third equations of (3.8), we obtain that
which, together with the last equation of (3.8), implies that
(3.9)
From the second equation of (3.8), we know that
where C(x*) denotes the critical cone of VI(F,X). Then, by (3.9) and the strong second-order sufficient condition that
Thus, the first equation of (3.8) reduces to
(3.10)
By the LICQ for VI(F,X), we have
This together with Δx1 = 0 shows that the matrix (3.6) is nonsingular, and then, H is nonsingular.
Now, we are able to apply Theorem 3.1 to some classes of GNEPs.
Proposition 3.1 Letsatisfying Φ(ω*) = 0, for all ν = 1,..., N the payoff functions θνare separable, that is
whereis stongly convex and. Assume that LICQ holds at x*. Then all elements H ∈ ∂Φ(ω*) are nonsingular.
Proof. We know that
then, by the definition of θν(·), we have
By the strong convexity of fν, we can conclude that ∇F(x*) is positive definite.
From and the convexity of g
i
, we obtain that
is positive semidefinite, which together with ∇F(x*) is positive definite implies that
is positive definite. Thus, the strong second-order sufficient condition for the VI(F, X) holds at x*. From Theorem 3.1, we obtain any element in ∂Φ(ω*) is nonsingular.
Proposition 3.2 Letbe such that Φ(ω*) = 0. Consider the case where the payoff functions are quadratic, i.e. for all ν = 1,...,N one has
where the matricesand A
νν
are symmetric. Suppose that LICQ holds at x*, and
is positive definite. Then all the elements in the generalized Jacobian ∂Φ(ω*) are nonsingular.
Proof. We show that ∇
x
L(ω*) is positive definite, which implies that the strong second order sufficient condition for the VI(F,X) holds at x*, and then apply Theorem 3.1. To this end, first note that
Moreover,
which together with and the convexity of g
i
implies that
is positive semidefinite. Hence, we obtain that ∇
x
L(ω*) is positive definite. The statement therefore follows from Theorem 3.1.