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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
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<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>
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{{TEX|done}}
  
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
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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  $  f( x, y) $
 +
the undefined factor  $  y $
 +
takes values from a set  $  Y $,  
 +
then the optimal guarantee strategy $  {\widetilde{x}  } {}  ^ {*} $
 +
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/o06845010.png" /></td> </tr></table>
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$$
 +
\sup _ {\widetilde{x}  }  \inf _ {y \in Y }  f
 +
( \widetilde{x}  {}  ^ {*} , y )  = \
 +
\inf _ {y \in Y }  f ( \widetilde{x}  {}  ^ {*} , y ).
 +
$$
  
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
+
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
  
<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>
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$$
 +
\inf _ {y \in Y }  f( \widetilde{x}  {} _  \epsilon  ^ {*} , y )  \geq  \
 +
\sup _ {x
 +
tilde }  \inf _ {y \in Y }  f ( \widetilde{x}  , y ) - \epsilon ,
 +
$$
 +
 
 +
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 [[#References|[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, [[#References|[2]]] and [[#References|[4]]]).
 
Optimal strategies corresponding to other principles of optimality are also studied (see, for example, [[#References|[2]]] and [[#References|[4]]]).
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====References====
 
====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>
 
<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====
 
====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.
+
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====
 
====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>
 
<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>

Latest revision as of 08:04, 6 June 2020


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
How to Cite This Entry:
Optimal guarantee strategy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Optimal_guarantee_strategy&oldid=48053
This article was adapted from an original article by F.I. EreshkoV.V. Fedorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article