The generalized odds ratio is the product of the odds ratios of the
edges in a cyclic competition graph.In the two-by-two case, the maximum likelihood estimatore of the odds
ratio is just
$\frac{a_{11}a_{22}}{a_{12}a_{21}}$.
The generalization to larger boards with exactly one degree of freedom
is immediate: it is just the product of the entries on the diagonal,
divided by the product of the off-diagonal elements.
Ranking permissable tables in order of increasing odds ratio thus orders
the sample space and allows one to perform one-sided tests.