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.

翻译：在社会选择理论中，匿名性（所有代理人被平等对待）和中立性（所有备选方案被平等对待）被广泛视为公平与公正的“最低要求”和“无争议”公理。然而，ANR不可能性——即不存在同时满足匿名性、中立性和可解性（始终选出一个胜者）的投票规则——即使在仅有两个备选方案和两个代理人的简单场景下依然成立。如何设计能最优满足匿名性、中立性和可解性的投票规则仍是一个开放问题。我们针对包含排名列表和委员会在内的广泛偏好与决策场景，探讨了这一最优设计问题。我们的概念性贡献在于提出了一种新颖且强健的“最公平精炼”概念，该概念能在保留任何满足这两条公理的非决定性规则的前提下，最优地维持匿名性与中立性。我们的技术性贡献包含两点：首先，我们刻画了ANR不可能性在一般设置（尤其是代理人数量较大时）成立的条件；其次，我们提出了“最有利排列”（MFP）平局打破法以计算最公平精炼，并设计了在代理人偏好为完全排名时计算MFP的多项式时间算法。