This example shows that the constants 1 / 3 and 2 / 3 in the conjecture are tight; if " q " is any fraction strictly between 1 / 3 and 2 / 3, then there would not exist a pair " x ", " y " in which " x " is earlier than " y " in a number of partial orderings that is between " q " and times the total number of partial orderings.
32.
This example shows that the constants 1 / 3 and 2 / 3 in the conjecture are tight; if " q " is any fraction strictly between 1 / 3 and 2 / 3, then there would not exist a pair " x ", " y " in which " x " is earlier than " y " in a number of partial orderings that is between " q " and times the total number of partial orderings.
33.
There is a sense in which one risk A can be said to be unambiguously greater than another risk B ( that is, greater for any reasonable risk function ) : namely, if A is a mean-preserving spread of B . This means that the probability density function of A can be formed, roughly speaking, by " spreading out " that of B . However, this is only a partial ordering : most pairs of risks cannot be unambiguously ranked in this way because different risk functions applied to the estimated distributions of these two risky variables will give different answers as to which is riskier.
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