I want to characterize the probabilistic ordering of some (random) variables without going into a parametric from of the variables themselves. I couldn't easily find any theory for this and I am not sure what it is called, but I can try to explain through an example what I mean. Consider two random variables, $X$ and $Y$. There are only two possible orderings and I can characterize it as $P(X\le Y) = p_0$ and $P(X>Y) = 1-p_0$. What if I now have a third variable $Z$. Obviously, there now $3!=6$ orderings and we can give 6 parameters summing to 1, but I want to give more "separable" parameters, utilizing $p_0$ that we already have. Is it sufficient to also give $P(Y\le Z)$, $P(X\le Z)$? Once I give these parameters I should be able to compute the probability of observing any ordering. It seems to me that there must be a theory of such probabilistic orderings but I am unable to find it anywhere, or come up with a solution on first principles.
Probabilistic ordering
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probability
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order-theory
parametric
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1@Michael: If you don't have enough letters to add in an edit, you can always add `${}{}{}{}{}{}$` at the end of a paragraph. – 2011-08-02