  In social choice theory, anonymity (all agents being treated equally) and
neutrality (all alternatives being treated equally) are widely regarded as
``minimal demands'' and ``uncontroversial'' axioms of equity and fairness.
However, the ANR impossibility -- there is no voting rule that satisfies
anonymity, neutrality, and resolvability (always choosing one winner) -- holds
even in the simple setting of two alternatives and two agents. How to design
voting rules that optimally satisfy anonymity, neutrality, and resolvability
remains an open question.
  We address the optimal design question for a wide range of preferences and
decisions that include ranked lists and committees. Our conceptual contribution
is a novel and strong notion of most equitable refinements that optimally
preserves anonymity and neutrality for any irresolute rule that satisfies the
two axioms. Our technical contributions are twofold. First, we characterize the
conditions for the ANR impossibility to hold under general settings, especially
when the number of agents is large. Second, we propose the
most-favorable-permutation (MFP) tie-breaking to compute a most equitable
refinement and design a polynomial-time algorithm to compute MFP when agents'
preferences are full rankings.
