Equal probability for the best and the assignment of identical indivisible objects

We consider the problem of allocating several units of an indivisible object among agents with single-peaked utility functions. We introduce an axiom called equal probability for the best, and show that it is equivalent to both equal treatment of equals and symmetry in the presence of Pareto optimality. Moreover, we also show that the randomized uniform rule is the only randomized rule satisfying strategy-proofness, Pareto optimality, and equal probability for the best.