Three Approaches to the Comparison of Random Variables

Abstract

Decision making inevitably involves the comparison of real variables. In the presence of uncertainty, this entails the comparison of real-valued random variables. In this short contribution, we briefly review three approaches to such comparison: (i) stochastic dominance: an approach based on a pointwise comparison of cumulative distribution functions; (ii) statistical preference: an approach based on a pairwise comparison in terms of winning probabilities; (iii) probabilistic preference: an approach based on multivariate winning probabilities. Whereas the first and third approaches are intrinsically transitive, the second approach requires considerable mathematical effort to unveil the underlying transitivity properties. Moreover, the first approach ignores the possible dependence between the random variables and is based on univariate distribution functions, the second approach is by definition built on bivariate joint distribution functions, while the third approach is based on the overall joint distribution function.