View source for 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|Principle of the largest sure result]]). If, for example, in a situation with efficiency criterion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o0684501.png" /> the undefined factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o0684502.png" /> takes values from a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o0684503.png" />, then the optimal guarantee strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o0684504.png" /> satisfies the equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o0684505.png" /></td> </tr></table>

If the last upper bound over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o0684506.png" /> is not attained, then the concept of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o0684508.png" />-optimal guarantee strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o0684509.png" /> arises, for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845011.png" />. Dependent on the set of strategies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845012.png" /> and the information on the undefined factor (the conditions under which the operation is carried out), the optimal guarantee strategy is concretely defined (see [[#References|[1]]]). So, if the set of strategies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845013.png" /> comprises all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845014.png" /> and the operation contains complete information on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845015.png" />, then the optimal guarantee strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845016.png" /> is called the absolutely optimal strategy and is defined by the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845017.png" /></td> </tr></table>

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

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



====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068450/o06845018.png" />-variable) are replaced by the worst possible. This idea is age old and universal in engineering and military analysis.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y.C. Ho,   G.J. Olsder,   "Differential games: concepts and applications"  M. Shubik (ed.) , ''Mathematics of Conflict'' , Elsevier &amp; North-Holland  (1983)  pp. 127–186</TD></TR></table>