Saddle point in game theory
A point of a function
defined on the Cartesian product
of two sets
and
such that
![]() | (*) |
For a function the presence of a saddle point is equivalent to the existence of optimal strategies (cf. Strategy (in game theory)) for the players in the two-person zero-sum game
.
Comments
A point satisfying the condition (*) is called a saddle point of
in general. If
is a differentiable function on
and
,
, while the Hessian matrix
is non-singular and neither positive definite nor negative definite, then locally near
,
is a saddle point. The corresponding splitting of
near
is determined by the negative and positive eigenspaces of the Hessian at
.
Indeed, by the Morse lemma there are coordinates near
such that
has the form
![]() |
where is the index of the quadratic form determined by the symmetric matrix
. (The index of a quadratic form is the dimension of the largest subspace on which it is negative definite; this is also called the negative index of inertia (cf. also Quadratic form and Morse index).)
Let be the spaces of strategies of two players in a zero-sum game and let
be (the first component of) the pay-off function (cf. Games, theory of). Then a saddle point is also called an equilibrium point. This notion generalizes to
-player non-cooperative games, cf. [a2], Chapt. 2; Games, theory of; Nash theorem (in game theory); Non-cooperative game.
References
[a1] | M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 6 |
[a2] | J. Szép, F. Forgó, "Introduction to the theory of games" , Reidel (1985) pp. 171; 199 |
Saddle point in game theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saddle_point_in_game_theory&oldid=48604