# Single crossing condition

(Redirected from Single-crossing property)
Example of two cumulative distribution functions F(x) and G(y) which satisfy the single-crossing condition.

In economics, 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.

Cumulative distribution functions F and G satisfy the single-crossing condition if there exists a y such that

$\forall x, x \ge y \implies F(x) \ge G(y)$

and

$\forall x, x \le y \implies F(x) \le G(y)$;

that is, function $h(x) = F(x)-G(y)$ crosses the x-axis at most once, in which case it does so from below.

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

$F(x',t) \ge F(x,t) \implies F(x',t') \ge F(x,t')$

and

$F(x',t) > F(x,t) \implies F(x',t') > F(x,t')$.

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. The condition is not symmetric in the variables (i.e., we cannot switch x and t in the definition; the necessary inequality in the first argument is weak, while the inequality in the second argument is strict).

The single-crossing condition was posited in Samuel Karlin's 1968 monograph 'Total Positivity'.[1] It was later used by Peter Diamond, Joseph Stiglitz, [2] and Susan Athey, [3] in studying the economics of uncertainty,[4] The single-crossing condition is also used in applications where there are a few agents or types of agents that have preferences over an ordered set. Such situations appear often in information economics, contract theory, social choice and political economics, among other fields.