Bell-shaped game

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)