# Single crossing condition

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

${\displaystyle \forall x,x\geq y\implies F(x)\geq G(y)}$

and

${\displaystyle \forall x,x\leq y\implies F(x)\leq G(y)}$;

that is, function ${\displaystyle h(x)=F(x)-G(y)}$ 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,

${\displaystyle F(x',t)\geq F(x,t)\implies F(x',t')\geq F(x,t')}$

and

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

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.

## References

1. ^ Karlin, Samuel (1968). Total positivity. Stanford University Press.
2. ^ Diamond, Peter A. & Stiglitz, Joseph E. (1974). "Increases in risk and in risk aversion". Journal of Economic Theory, Elsevier, vol. 8(3), pages 337-360, July.
3. ^ Athey, Susan, 2001. "Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information," Econometrica, Econometric Society, vol. 69(4), pages 861-89, July.
4. ^ Gollier, Christian (2001). The Economics of Risk and Time. The MIT Press. p. 103.