What constitutes a good solution to Formulation 9.7? Consider the game from the perspective of . It seems reasonable to apply worst-case analysis when trying to account for the action that will be taken by . This results in a choice that is equivalent to assuming that is nature acting under the nondeterministic model, as considered in Section 9.2.2. For a matrix game, this is computed by first determining the maximum cost over each row. Selecting the action that produces the minimum among these represents the lowest cost that can guarantee for itself. Let this selection be referred to as a security strategy for .
For the matrix game in (9.42), the security strategy is illustrated as
This can be formalized using the existing notation. A security strategy, , for is defined in general as
(9.45) |
Now swap roles, and consider the game from the perspective of , which would like to maximize . It can also use worst-case analysis, which means that it would like to select an action that guarantees a high cost, in spite of the action of to potentially reduce it. A security strategy, , for is defined as
(9.47) |
An interesting relationship between the upper and lower values is that for any game using Formulation 9.7. This is shown by observing that
(9.49) |