 In an important type of counterexamples to transitivity of strict preference, different properties of the alternatives dominate in different pairwise comparisons. Consider an agent choosing between three boxes of Christmas ornaments (Schumm 1987). Each box contains three balls, coloured green, red and blue, respectively; they are represented by the vectors \(\langle R_1,G_1,B_1\rangle,\) \(\langle R_2,G_2,B_2\rangle\), and \(\langle R_3,G_3,B_3\rangle\).  The agent strictly prefers box 1 to box 2, since they contain (to her) equally attractive blue and green balls, but the red ball of box 1 is more attractive than that of box 2. She prefers box 2 to box 3, since they are equal but for the green ball of box 2, which is more attractive than that of box 3. And finally, she prefers box 3 to box 1, since they are equal but for the blue ball of box 3, which is more attractive than that of box 1. Thus,