page_title: Single crossing condition
hierarchical_section_title: Single crossing condition
context_page_description: 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 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=F-F crosses the horizontal axis at most once, and from below. The condition is not symmetric in the variables.
  The single-crossing condition was posited in Samuel Karlin's 1968 monograph 'Total Positivity'. It was later used by Peter Diamond, Joseph Stiglitz, and Susan Athey, 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. Such situations appear often in information economics, contract theory, social choice and political economics, among other fields.
caption: Example of two cumulative distribution functions F(x) and G(x) which satisfy the single-crossing condition.
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