View source for Bell-shaped game

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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) $, 
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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> S. Karlin, "Mathematical methods and theory in games, programming and economics", Addison-Wesley (1959) {{ZBL|0139.12704}}</TD></TR>
</table>