# 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) Zbl 0139.12704
How to Cite This Entry:
Bell-shaped game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bell-shaped_game&oldid=54794
This article was adapted from an original article by V.K. Domanskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article