Completeness and transitivity of preferences on mixture sets

This paper contributes to the interplay of the behavioral assumptions on a binary relation and the structure of the choice space on which it is defined: the (ES) research program of Eilenberg (1941) and Sonnenschein (1965) to which Schmeidler (1971) is an especially influential contribution. We show that the presence of the Archimedean and mixture-continuity properties, both empirically non-falsifiable in principle, foreclose the possibility of consistency (transitivity) without decisiveness (completeness), or decisiveness without consistency, or in the presence of a weak consistency condition, both indecisiveness and inconsistency altogether. Second, we delineate how semi-transitivity of a relation is already hidden in linearity assumptions; and third, offer sufficient conditions that yield an isomorphism theorem that reduces a general setting to the usual order on an interval, and thereby yields the classic theorem of Herstein–Milnor (1953). We remark on extensions to a generalized mixture set of Chipman–Fishburn.