The behavioral meaning of the median

We extend the notion of statistical preference to the general framework of imprecise probabilities, by proposing a new notion of desirability of gambles called “sign-desirability”, different from the usual desirability notion in Walley’s framework. We axiomatically characterize coherent families of sign-desirable gambles. We furthermore prove that the pair of lower and upper previsions of a gamble, according to this new desirability notion, coincides with the pair of bounds (infimum and supremum) of the set of medians associated to a coherent family of linear previsions. Thus, a general notion of median is naturally derived, and provided with a behavioral meaning. As a consequence of these results, the connection between statistical preference in classical Probability Theory, and the sign of the median of the difference of two random variables is laid bare.