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, Math. Operat. Res. 20 (1995) 513-525 and Milne and Neave, Management Sci. 40 (1994) 1343-1352. Finally, the solution to an open problem in Fishburn and Lavalle, Math. Operat. Res. 20 (1995) 513-525 is presented.