Ran Spiegler Inferring a Linear Ordering over a Power Set

An observer attempts to infer the unobserved ranking of two ideal objects, A and B, from observed rankings in which these objects are ‘accompanied’ by ‘noise’ components, C and D. In the first ranking, A is accompanied by C and B is accompanied by D, while in the second ranking, A is accompanied by D and B is accompanied by C. In both rankings, noisy-A is ranked above noisy-B. The observer infers that ideal-A is ranked above ideal-B. This commonly used inference rule is formalized for the case in which A,B,C,D are sets. Let X be a finite set and let be a linear ordering on 2. The following condition is imposed on . For every quadruple (A,B,C,D) ∈ Y , where Y is some domain in (2X)4, if A ∪ C B ∪ D and A ∪ D B ∪ C, then A B. The implications and interpretation of this condition for various domains Y are discussed.