A saddle point is a critical point of a function where the function attains neither a local maximum value nor a local minimum value. In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum. Saddle points mostly occur in multivariable functions, and they are defined as those points in the domain of the function where the tangent is parallel to the horizontal axis, but this point tends to be neither a local maximum nor a local minimum.
In game theory, a saddle point is the outcome that rational players would choose in a two-person constant-sum game. In a two-player zero-sum game defined on a continuous space, the equilibrium point is a saddle point.
To find a saddle point of a function, we can follow these steps:
- Find the critical points of the function.
- Calculate the second-order partial derivatives of the function.
- Evaluate the Hessian matrix of the function at each critical point.
- Identify the saddle points as the critical points where the Hessian matrix has both positive and negative eigenvalues.
In summary, a saddle point is a critical point of a function where the function attains neither a local maximum value nor a local minimum value. It is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero, but which is not a local extremum. In game theory, a saddle point is the outcome that rational players would choose in a two-person constant-sum game.
