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3 The Geometry of Binary Choice We now introduce a geometric framework within which we can simultaneously represent algebraic binary preference, binary choice probabilities, as well as empirically observed binary choice proportions all within one and the same geometric space 4. For the time being, we are interested in three-dimensional visualizations, but we will later move to high-dimensional abstract models. There are eight possible preference patterns among these gambles: the six rankings (each from best to worst): ABC, ACB, BAC, BCA, CAB, CBA, and two intransitive cycles that we label ABCA and ACBA.

For example, ranking ABC is the point (1,1,1) in the space with coordinate system (A,B), (A,C), and (B,C) 5.