#### Saddle points in a sequential game

A saddle point will be obtained once again by defining security strategies for each player. Each player treats the other as nature, and if the same worst-case value is obtained, then the result is a saddle point for the game. If the values are different, then a randomized plan is needed to close the gap between the upper and lower values.

Upper and lower values now depend on the initial state, . There was no equivalent for this in Section 10.5.1 because the root of the game tree is the only possible starting point.

If sequences, and , of actions are applied from , then the state history, , can be derived by repeatedly using the state transition function, . The upper value from is defined as

 (10.107)

which is identical to (10.33) if is replaced by nature. Also, (10.108) generalizes (9.44) to multiple stages. The lower value from , which generalizes (9.46), is

 (10.108)

If , then a deterministic saddle point exists from . This implies that the order of and can be swapped inside of every stage.

Steven M LaValle 2020-08-14