Operations Research Letters 35 (2007) 159 – 164
Operations
Research
Letters
www.elsevier.com/locate/orl
On generalized Nash games and variational inequalities
Francisco Facchinei
a , ∗
, Andreas Fischer
b
, Veronica Piccialli
a
a
Department of Computer and System Sciences “A. Ruberti”, Università di Roma “La Sapienza”, Via Buonarroti 12, Rome 00185, Italy
b
Institute of Numerical Mathematics, Technische Universität Dresden, 01062 Dresden, Germany
Received 16 January 2006; accepted 19 March 2006
Available online 5 June 2006
Abstract
We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational
inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Generalized Nash equilibrium problem; Variational inequality; Multipliers
1. Introduction
The standard definition of a noncooperative game
in normal form usually requires that each player have
a feasible set that is independent of the rivals’ strate-
gies. However, it was well understood from the early
developments in the field, see e.g. [1,12,15], that in
many cases the interaction between the players can
take place (also) at the feasible set level. If one as-
sumes that each player’s feasible set can depend on
the rival players’ strategies we speak of generalized
Nash equilibrium problems (GNEPs in the sequel).
GNEPs have received an increasing amount of at-
tention in recent times because GNEPs naturally arise
in the modeling of complex and important economical
systems.
∗
Corresponding author.
E-mail address: facchinei@dis.uniroma1.it (F. Facchinei).
0167-6377/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.orl.2006.03.004
Let us give now a formal definition of the prob-
lem. There are N players, and each player con-
trols the variables x
∈ R
n
. We denote by x the
vector formed by all these decision variables: x ≡
((x
1
)
T
, (x
2
)
T
,...,(x
N
)
T
)
T
, and by x
-
the vector
formed by all the players’ decision variables except
those of player . To emphasize the -th player’s vari-
ables within x we sometimes write (x
, x
-
) instead
of x.
The strategy of player must belong to a set
X
(x
-
) ⊆ R
n
that depends on the rival players’
strategies. The aim of player , given the other play-
ers’ strategies x
-
, is to choose a strategy x
that
solves the minimization problem
minimize
x
(x
, x
-
)
subject to x
∈ X
(x
-
).
(1)
For any x
-
, the solution set of this problem is denoted
by S
(x
-
). The GNEP is the problem of finding a