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
We consider a generalized game of N players defined by their objective function θi : R n ↦→ R
and their constraint function g i : R n ↦→ R mi . The generalized Nash equilibrium problem
(GNEP for short) extends standard Nash equilibrium, since ones’ player strategy depends
on the rival players’ strategies. Thus, when each player’s strategy set does not depend on
the other players’ strategies, the GNEP reduces to standard Nash equilibrium problem. The
GNEP consists in finding x⋆ such that for all i = 1, . . . , N, x⋆ i solves the subproblem
min
xi∈R n i
θi(xi, x ⋆ −i) s.t. g i (xi, x ⋆ −i) ≤ 0, (3.1)
where (xi, x−i) denotes the vector (x1, . . . , xi, . . . , xN) ∈ Rn with n =
i ni the total number
of variables and m =
i mi the total number of constraints.
GNEP arises from many practical problems, including telecommunications, engineering
and economics applications, see Facchinei and Kanzow (2009) and the references therein for
an overview of GNEPs. This paper aims to make a survey of computational methods to solve
general GNEPs defined in Equation (3.1).
The paper is organized as follows: Section 3.1 present the different reformulations of
GNEPs. Section 3.2 describes the numerous optimization methods that can solved a nonlinear
reformulation of the GNEP. Finally, Section 3.3 carries out a numerical comparison of all
algorithms presented in the previous section, before Section 3.4 concludes.
3.1 Problem reformulations
As presented in Equation (3.1), the generalized Nash equilibrium problem is not directly
solvable. This section aims to present the different reformulations of the GNEP. On Figure
3.1, we present a basic flow-chart of the relationship among the different reformulations that
we present below.
138
GNEP
KKT
Compl. reform. Constr. eq.
Figure 3.1: Map of GNE reformulations
QVI
NIF