Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
Chapitre 3. Calcul d’équilibre de Nash généralisé
3.1.4 Jointly convex case
In this subsection, we present reformulations for a subclass of GNEP called jointly convex
case. Firstly, the jointly convex setting requires that the constraint function is common to all
players g 1 = · · · = g N = g. Then, we assume, there exists a closed convex subset X ⊂ R n
such that for all player i,
{yi ∈ R ni , g(yi, x−i) ≤ 0} = {yi ∈ R ni , (yi, x−i) ∈ X}.
The convexity of X implies that the constraint function g is quasiconvex with respect to all
variables. However, we generally assume that g is convex with respect to all variables.
KKT conditions for the jointly convex case
In the jointly convex case, the KKT conditions (3.2) become
∇xi θi(x ⋆ ) + ∇xi g(x⋆ )λ i⋆ = 0, 0 ≤ λ i⋆ , −g(x ⋆ ) ≥ 0, g(x ⋆ ) T λ i⋆ = 0, (3.9)
since the constraint function is common to all players. But, there are still N Lagrange multipliers
λ i . Under the same condition as Subsection 3.1.1, a solution of this KKT system is also
a solution of the original GNEP.
VI formulation for the jointly convex case
In the jointly convex case, the QVI reformulation (3.5) simplifies to a variational inequality
problem VI(X, F )
∀y ∈ X, (y − x) T F (x) ≥ 0,
⎛
∇x1
⎜
with F (x) = ⎝
θ1(x)
.
∇xN θN(x)
⎞
⎟
⎠ , (3.10)
under certain conditions with X = {y ∈ R n , ∀i, g(yi, x−i) ≤ 0}. To understand that VI
problem solutions are a subclass of GNEs, we just compare the KKT conditions of the VIP
(3.10) and Equation 3.9. This is given in the following theorem, see, e.g., Theorem 3.9 of
Facchinei and Kanzow (2009), Theorem 3.1 of Facchinei et al. (2007).
Theorem. Assuming θi and g are C 1 functions and g is convex and θi player-convex. The
subset of variational equilibrium verifying Equation (3.10) are the solution of the KKT system
(3.9) with a common multiplier λ 1 = · · · = λN = λ ⋆ .
GNEs verifying the VI problem in Equation (3.10) are called variational or normalized
equilibrium, see also Kulkarni and Shanbhag (2010) for a detailed discussion of the VI representation
of the QVI reformulation of GNEPs.
NIF formulation for the jointly convex case
142
Recalling that for the Nikaido-Isoda function (3.7), the gap function is
ˆV (x) = sup
y∈X(x)
ψ(x, y).