Optimal guarantee strategy

A strategy whose efficiency in a given situation is equal to the best guaranteed result (see Principle of the largest sure result). If, for example, in a situation with efficiency criterion $ f( x, y) $ the undefined factor $ y $ takes values from a set $ Y $, then the optimal guarantee strategy $ {\widetilde{x} } {} ^ {*} $ satisfies the equality

$$ \sup _ {\widetilde{x} } \inf _ {y \in Y } f ( \widetilde{x} {} ^ {*} , y ) = \ \inf _ {y \in Y } f ( \widetilde{x} {} ^ {*} , y ). $$

If the last upper bound over $ \widetilde{x} $ is not attained, then the concept of an $ \epsilon $- optimal guarantee strategy $ {\widetilde{x} } {} _ \epsilon ^ {*} $ arises, for which

$$ \inf _ {y \in Y } f( \widetilde{x} {} _ \epsilon ^ {*} , y ) \geq \ \sup _ {x tilde } \inf _ {y \in Y } f ( \widetilde{x} , y ) - \epsilon , $$

where $ \epsilon > 0 $. Dependent on the set of strategies $ \widetilde{x} = x( y) $ and the information on the undefined factor (the conditions under which the operation is carried out), the optimal guarantee strategy is concretely defined (see [1]). So, if the set of strategies $ \widetilde{x} $ comprises all functions $ x( y) $ and the operation contains complete information on $ y $, then the optimal guarantee strategy $ x ^ {*} ( y) $ is called the absolutely optimal strategy and is defined by the condition

$$ \sup _ { x } f( x, y) = \ f( x ^ {*} ( y), y) \textrm{ for } \textrm{ all } y \in Y. $$

Optimal strategies corresponding to other principles of optimality are also studied (see, for example, [2] and [4]).

References

[1]	Yu.B. Germeier, "Introduction to the theory of operations research" , Moscow (1971) (In Russian)
[2]	Yu.B. Germeier, "Non-antagonistic games" , Reidel (1986) (Translated from Russian)
[3]	J.-P. Aubin, "L'analyse non-linéaire et ses motivations économiques" , Masson (1984)
[4]	N.N. Borob'ev, "Game theory. Lectures for economists and cyberneticists" , Leningrad (1974) (In Russian)

Comments

The phrase "worst case strategyworst case strategy" is also used for "optimal guarantee strategy" . It provides a security level for the outcome of the efficiency criterion. Worst case designs naturally show up in two-person zero-sum games in which uncertainties of unknowns (the $ y $- variable) are replaced by the worst possible. This idea is age old and universal in engineering and military analysis.

References

[a1]	Y.C. Ho, G.J. Olsder, "Differential games: concepts and applications" M. Shubik (ed.) , Mathematics of Conflict , Elsevier & North-Holland (1983) pp. 127–186