Chapter 10: Determining Stochastic Dominance Relations

Let $X={x1,…,xn}$ be a non-empty set of real numbers with x₁ < ⋯ < x_n. Let $F={Fi:1≤i≤N}$ be a set of cumulative probability distribution functions on $X$⁠, i.e. $Fi(xn)=1$ and if a random variable X follows $Fi$⁠, then $Pr[X≤xi]=F(xi)$⁠. For $1≤i≤N$⁠, let $fi$ be the probability mass function corresponding to the cumulative distribution function $Fi$⁠; let $μi$ be the mean for F_i, i.e. $μi=∑i=1nxif(xi)$⁠.

For $1≤i≤N$⁠, define

We consider the following well-known characterisations of stochastic dominance (SD) relations which are equivalent to the utility function-based definitions of SD.

First-order SD (FSD): F_i >₁ F_j if F_i(x) ≤ F_j(x) for all $x∈X$ and F_i(x) < F_j(x) for at least one $x∈X$⁠.