Comparison of probability measures: Dominance of the third degree

This note provides necessary and sufficient conditions for third-degree stochastic dominance between the probability measures on a finite grid of n points. Verifying these conditions involve n linear and n quadratic inequalities; the arithmetic operation of division is not required. The results are compared with those of Fishburn and Lavalle, and Milne and Neave. Finally, the solution to an open problem in Fishburn and Lavalle is presented.