en>Amkilpatrick: removing expert template, slightly expanding lead section

2014-01-06T11:04:48Z

removing expert template, slightly expanding lead section

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[[Image:Single Crossing Condition example.png|thumb|upright=1.25|alt=Example of two normal [[cumulative distribution function]]s F(x) and G(y) which satisfy the single-crossing condition.|Example of two [[cumulative distribution function]]s 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 function]]s ''F'' and ''G'' satisfy the single-crossing condition if there exists a ''y'' such that

<math>\forall x, x \ge y \implies F(x) \ge G(y)</math>

and

<math>\forall x, x \le y \implies F(x) \le G(y)</math>;

that is, function <math>h(x) = F(x)-G(y)</math> 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,

<math>F(x',t) \ge F(x,t) \implies F(x',t') \ge F(x,t')</math>

and

<math>F(x',t) > F(x,t) \implies F(x',t') > F(x,t')</math>.

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'.<ref>{{cite book |last=Karlin |first= Samuel |title=Total positivity |publisher=Stanford University Press |year=1968}}</ref> It was later used by [[Peter Diamond (economist)|Peter Diamond]], [[Joseph Stiglitz]], <ref>{{cite journal |author=Diamond, Peter A. & Stiglitz, Joseph E. |year=1974 |title=Increases in risk and in risk aversion |publisher= Journal of Economic Theory, Elsevier, vol. 8(3), pages 337-360, July}}</ref> and [[Susan Athey]], <ref>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.</ref> in studying the economics of uncertainty,<ref>{{Cite book |last=Gollier |first= Christian |title=The Economics of Risk and Time |publisher= The MIT Press |year=2001 |page= 103}}</ref> 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.

==See also==
* [[Brouwer fixed-point theorem]]

==References==
{{reflist}}

{{DEFAULTSORT:Single Crossing Condition}}
[[Category:Asymmetric information]]

{{Econ-stub}}

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en>Amkilpatrick: removing expert template, slightly expanding lead section