Single-crossing condition
In monotone comparative statics, the single-crossing condition or single-crossing property refers to a condition where the relationship between two or more functions is such that they will only cross once. For example, a mean-preserving spread will result in an altered probability distribution whose cumulative distribution function will intersect with the original's only once.
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.
Example using cumulative distribution functions
Cumulative distribution functions F and G satisfy the single-crossing condition if there exists a such thatand
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.
Use in social choice and mechanism design
Social choice
In social choice theory, the single-crossing condition is a condition on preferences. It is especially useful because utility functions are generally increasing.Specifically, a set of agents with some unidimensional characteristic and preferences over different policies q satisfy the single crossing property when the following is true:
If and or if and, then
where W is the indirect utility function.
An important result extends the median voter theorem, which states that when voters have single peaked preferences, there is a majority-preferred candidate corresponding to the median voter's most preferred policy. With single-crossing preferences, the most preferred policy of the voter with the median value of is the Condorcet winner. In effect, this replaces the unidimensionality of policies with the unidimensionality of voter heterogeneity. In this context, the single-crossing condition is sometimes referred to as the Gans-Smart condition.