DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

SECOND-ORDER DIFFERENTIAL GRADIENT
METHODS FOR SOLVING TWO-PERSON
GAMES WITH COUPLED VARIABLES
ANATOLY ANTIPIN
A two-person nonzero-sum game is considered both the classical statement and the form
of a coupled-variables game. Second-order gradient method for solving coupled-variables
game is offered and justified. The convergent analysis is given.
Copyright © 2006 Anatoly Antipin. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Statement of introduction
Let us consider the problem of computing a fixed point of a two-person nonzero-sum
game: find x1 ∗ ∈ X 1 , x2 ∗ ∈ X 2 such that
x1 ∗ ∈ Argmin{ ( ) ( ) }
f 1 z1 ,x2
∗ + ϕ1 z1 | z1 ∈ X 1 ,
x2 ∗ ∈ Argmin{ ( ) ( ) } (1.1)
f 2 x
∗
1 ,z 2 + ϕ2 z2 | z2 ∈ X 2 ,
where X 1 ∈ R n1
1 , X 2 ∈ R n2
2 are convex closed sets in finite-dimensional Euclidean spaces
with various dimensionality in general. Objective functions f 1 (x 1 ,x 2 )+ϕ 1 (x 1 ), f 2 (x 1 ,x 2 )+
ϕ 2 (x 2 ) are defined on the product spaces R n1
1 , R n2
2 . If the functions f 1 (z 1 ,x 2 )+ϕ(z 1 ),
f 2 (x 1 ,z 2 )+ϕ(z 2 ) are continuous and convex in their own variables, that is, the first function
is convex in z 1 and second one is convex in z 2 for any values of x 1 and x 2 ,whereX i ,
i = 1,2, are convex compact sets, then there exists a solution x ∗ = (x1 ∗ ,x2 ∗ )of(1.1)[10].
Any solution of (1.1) or Nash equilibrium describes some compromise with a summarized
gain f 1 (x1 ∗ ,x2 ∗ )+ϕ 1 (x1 ∗ )+ f 2 (x1 ∗ ,x2 ∗ )+ϕ 2 (x2 ∗ ). If the sum f 1 (x 1 ,x 2 )+ϕ 1 (x 1 )+
f 2 (x 1 ,x 2 )+ϕ 2 (x 2 ) ∀x 1 ∈ X 1 , x 2 ∈ X 2 is equal to zero, then a problem is called as zerosum
two-person game. Following conceptions accepted in decision making, we suppose
that functions ϕ i (x i ), i = 1,2, describe the preferences of every player in choosing their
alternatives over their own subsets X i , i = 1,2. In turn, the functions f i (x 1 ,x 2 ), i = 1,2,
describe a dependence or an effect of one player on the other player, if one of them is
taking his choice.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 67–79