Single crossing condition

In contract theory, the single crossing condition or single crossing property refers to how the probability distribution of outcomes changes as a function of an input and a parameter.

Given cumulative distribution functions "F" and "G", the functions obey the single crossing condition if there exists a "y" such that $forall\; x,\; x\; ge\; y\; implies\; F(x)\; ge\; G(y)$ and $forall\; x,\; x\; le\; y\; implies\; F(x)\; le\; G(y)$.

This property can be extended to two variables or more variables. Given x and t, for all x'>x, t'>t,

$F(x\text{'},t)\; ge\; F(x,t)\; implies\; F(x\text{'},t\text{'})\; ge\; F(x,t\text{'})$, and$F(x\text{'},t)\; >\; F(x,t)\; implies\; F(x\text{'},t\text{'})\; >\; F(x,t\text{'})$.

This condition could be interpreted as saying that for x'>x, the function g(t)=F(x',t)-F(x,t) crosses the horizontal axis at most once, and from below (although it may have zero value on an interval). The condition is not symmetric in the variables (i.e., we cannot switch x and t in the definition).