In economics

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, 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 function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

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



and

;

that is, function 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,



and

.

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

Samuel Karlin

Samuel Karlin was an American mathematician at Stanford University in the late 20th century.Karlin was born in Yanova, Poland and immigrated to Chicago as a child...

's 1968 monograph 'Total Positivity' It was later used by Peter Diamond, Joseph Stiglitz, and Susan Athey

Susan Athey

Susan Carleton Athey is an American economist. She is currently Professor of Economics at Harvard University and the first female winner of the John Bates Clark Medal.- Early life :...

 in studying the economics of uncertainty. 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

Ordered set

In order theory in mathematics, a set with a binary relation R on its elements that is reflexive , antisymmetric and transitive is described as a partially ordered set or poset...

. Such situations appear often in information economics

Information economics

Information economics or the economics of informationis a branch of microeconomic theory that studies how information affects an economy and economic decisions. Information has special characteristics. It is easy to create but hard to trust. It is easy to spread but hard to control. It...

, contract theory

Contract theory

In economics, contract theory studies how economic actors can and do construct contractual arrangements, generally in the presence of asymmetric information. Because of its connections with both agency and incentives, contract theory is often categorized within a field known as Law and economics...

, social choice and political economics, amongst other fields.