Quantiles and Medians

We provide a list of functional equations that characterize quantile functions for collections of bounded and measurable functions. Our central axiom is ordinal covariance. When a probability measure is exogeneously given, we characterize quantiles with respect to that measure through monotonicity with respect to stochastic dominance. When none is given, we characterize those functions which are simply ordinally covariant and monotonic as quantiles with respect to capacities; and we also find an additional condition for finite probability spaces that allows us to represent the capacity as a probability measure. Additionally requiring that a function be covariant under its negation results in a generalized notion of median. Finally, we show that all of our theorems continue to hold under the weaker notion of covariance under increasing, concave transformations. Applications to the theory of ranking infinite utility streams and to the theory of risk measurement are provided