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A [[Game on the unit square|game on the unit square]] whose pay-off function takes the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154501.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154502.png" /> is a positive analytic proper Pólya frequency function, i.e.:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154503.png" /> is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154504.png" />;
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2) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154505.png" /> and any sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154507.png" /> there is an inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154508.png" />;
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A [[Game on the unit square|game on the unit square]] whose pay-off function takes the form  $  \phi (x - y) $,
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where  $  \phi $
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is a positive analytic proper Pólya frequency function, i.e.:
  
3) for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b0154509.png" /> (correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b01545010.png" />) there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b01545011.png" /> (correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b01545012.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b01545013.png" />;
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1) $  \phi (u) $
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is defined for all  $  u \in (- \infty , \infty ) $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b01545014.png" />.
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2) for any  $  n $
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and any sets  $  - \infty < x _ {1} < \dots < x _ {n} < \infty $
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and  $  - \infty < y _ {1} < \dots < y _ {n} < \infty $
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there is an inequality  $  \mathop{\rm det}  \| \phi (x _ {i} - y _ {j} ) \| \geq  0 $;
  
An example of a bell-shaped game is a game with pay-off function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b01545015.png" />. The optimal strategies of players in a bell-shaped game are unique and are piecewise-constant distributions with a finite number of steps. The value of a game with pay-off function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b01545016.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015450/b01545017.png" />, moves towards zero, while the number of points in the supports of the optimal strategies grows unboundedly.
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3) for any set  $  \{ x _ {k} \} $(
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correspondingly,  $  \{ y _ {k} \} $)
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there is a set  $  \{ y _ {k} \} $(
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correspondingly,  $  \{ x _ {k} \} $)
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such that  $  \mathop{\rm det}  \| \phi (x _ {i} - y _ {j} ) \| > 0 $;
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4)  $  \int _ {- \infty }  ^  \infty  \phi (u)  du < \infty $.
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An example of a bell-shaped game is a game with pay-off function $  e ^ {- (x - y)  ^ {2} } $.  
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The optimal strategies of players in a bell-shaped game are unique and are piecewise-constant distributions with a finite number of steps. The value of a game with pay-off function $  \phi ( \lambda (x - y)) $,  
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as $  \lambda \rightarrow \infty $,  
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moves towards zero, while the number of points in the supports of the optimal strategies grows unboundedly.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Karlin,  "Mathematical methods and theory in games, programming and economics" , Addison-Wesley  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Karlin,  "Mathematical methods and theory in games, programming and economics" , Addison-Wesley  (1959)</TD></TR></table>

Revision as of 10:33, 29 May 2020


A game on the unit square whose pay-off function takes the form $ \phi (x - y) $, where $ \phi $ is a positive analytic proper Pólya frequency function, i.e.:

1) $ \phi (u) $ is defined for all $ u \in (- \infty , \infty ) $;

2) for any $ n $ and any sets $ - \infty < x _ {1} < \dots < x _ {n} < \infty $ and $ - \infty < y _ {1} < \dots < y _ {n} < \infty $ there is an inequality $ \mathop{\rm det} \| \phi (x _ {i} - y _ {j} ) \| \geq 0 $;

3) for any set $ \{ x _ {k} \} $( correspondingly, $ \{ y _ {k} \} $) there is a set $ \{ y _ {k} \} $( correspondingly, $ \{ x _ {k} \} $) such that $ \mathop{\rm det} \| \phi (x _ {i} - y _ {j} ) \| > 0 $;

4) $ \int _ {- \infty } ^ \infty \phi (u) du < \infty $.

An example of a bell-shaped game is a game with pay-off function $ e ^ {- (x - y) ^ {2} } $. The optimal strategies of players in a bell-shaped game are unique and are piecewise-constant distributions with a finite number of steps. The value of a game with pay-off function $ \phi ( \lambda (x - y)) $, as $ \lambda \rightarrow \infty $, moves towards zero, while the number of points in the supports of the optimal strategies grows unboundedly.

References

[1] S. Karlin, "Mathematical methods and theory in games, programming and economics" , Addison-Wesley (1959)
How to Cite This Entry:
Bell-shaped game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bell-shaped_game&oldid=46004
This article was adapted from an original article by V.K. Domanskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article