Randomized collective choices based on a fractional tournament

An extension rule assigns to each fractional tournament x (specifying, for every pair of social alternatives a and b, the proportion x ab of voters who prefer a to b) a random choice function y (specifying a collective choice probability distribution for each subset of alternatives), which chooses a from { a, b} with probability x ab . There exist multiple neutral and stochastically rationalizable extension rules. Both linearity (requiring that y be an affine function of x) and independence of irrelevant comparisons (asking that the probability distribution on a subset of alternatives depend only on the restriction of the fractional tournament to that subset) are incompatible with very weak properties implied by stochastic rationalizability. We identify a class of maximal domains, which we call sequentially binary, guaranteeing that every fractional tournament arising from a population of voters with preferences in such a domain has a unique admissible stochastically rationalizable extension.