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Open in a separate window Figure 2 Two different views of the same geometric representation of eight algebraic and one probabilistic model(s) for Gambles A, B, C.
Each of the eight possible preference patterns forms a vertex of the unit cube. Modal choice consistent with preference ranking ABC (choose A over B at least 50, A over C at least 50, and B over C at least 50 of the time) forms the smaller shaded cube. For example, Figure 2 also shows a probability model, namely the modal choice consistent with the ranking ABC: If a person chooses A over B at least 50, B over C at least 50, and A over C at least 50 of the time, then their binary choice probabilities must lie somewhere in the smaller shaded cube attached to the vertex ABC.
In particular, if a person acts deterministically and chooses A over B 100, B over C 100, and A over C 100 of the time, then this person's (degenerate) choice probabilities coincide with the vertex ABC that has coordinates (1,1,1) and that also represents the deterministic preference ABC.