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Consider the possibility that the decision maker, at any point in time, rank orders the gambles from best to worst in a fashion consistent with. Mixture, aka, random preference models quantify this variability with a probability distribution over preference patterns such as, in this case, the four rankings ACD, ADC, DAC, DCA.

Figure 9 shows the binary choice probabilities if a person's preferences fluctuate or if the person is uncertain about their preference ranking, but permissible preference rankings are limited to the rankings ACD, ADC, DAC, DCA consistent with. The shaded region in Figure 9, that forms an irregular pyramid in 3D space, is called a convex polytope (see the Online Supplement for more details).