# Saddle point in game theory

A point $( x ^ {*} , y ^ {*} ) \in X \times Y$ of a function $F$ defined on the Cartesian product $X \times Y$ of two sets $X$ and $Y$ such that

$$\tag{* } F ( x ^ {*} , y ^ {*} ) = \ \max _ {x \in X } \ F ( x, y ^ {*} ) = \ \min _ {y \in Y } \ F ( x ^ {*} , y).$$

For a function $F$ 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 $\Gamma = ( X, Y, F )$.

A point $( x ^ {*} , y ^ {*} ) \in X \times Y$ satisfying the condition (*) is called a saddle point of $F$ in general. If $F$ is a differentiable function on $\mathbf R ^ {n}$ and $( \partial F / \partial x _ {i} ) ( x ^ {*} ) = 0$, $i = 1 \dots n$, while the Hessian matrix $( \partial ^ {2} F / \partial x _ {i} \partial x _ {j} ) ( x ^ {*} )$ is non-singular and neither positive definite nor negative definite, then locally near $x ^ {*}$, $x ^ {*}$ is a saddle point. The corresponding splitting of $\mathbf R ^ {n}$ near $x ^ {*}$ is determined by the negative and positive eigenspaces of the Hessian at $x ^ {*}$.

Indeed, by the Morse lemma there are coordinates $y _ {1} \dots y _ {n}$ near $x ^ {*}$ such that $F$ has the form

$$F( y) = F( x ^ {*} ) - y _ {1} ^ {2} - \dots - y _ {r} ^ {2} + y _ {r+} 1 ^ {2} + \dots + y _ {n} ^ {2} ,$$

where $r$ is the index of the quadratic form determined by the symmetric matrix $( \partial ^ {2} F / \partial x _ {i} \partial x _ {j} ) ( x ^ {*} )$. (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 $X, Y$ be the spaces of strategies of two players in a zero-sum game and let $F: X \times Y \rightarrow \mathbf R$ 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 $n$- 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
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