In social choice there often arises a conflict between the majority principle (the search for a candidate that is as good as possible for as many voters as possible), and the protection of minority rights (choosing a candidate that is not overly bad for particular individuals or groups). In a context where the latter is our main concern, veto-based rules -- giving individuals or groups the ability to strike off certain candidates from the list -- are a natural and effective way of ensuring that no minority is left with an outcome they find untenable. However, such rules often fail to be anonymous, or impose specific restrictions on the number of voters and candidates. These issues can be addressed by considering the proportional veto core -- the solution to a cooperative game where every coalition is given the power to veto a number of candidates proportional to its size. However, the na\"ive algorithm for the veto core is exponential, and the only known rule for selecting from the core, with an arbitrary number of voters, fails anonymity. In this paper we present a polynomial time algorithm for computing the core, study its expected size, and present an anonymous rule for selecting a candidate from it. We study the properties of core-consistent voting rules. Finally, we show that a pessimist can manipulate the core in polynomial time, while an optimist cannot manipulate it at all.

翻译：在社会选择中，多数原则（寻求尽可能多选民尽可能满意的候选人）与保护少数群体权利（选择对特定个人或群体不会产生过度不利结果的候选人）之间常常存在冲突。在后者成为主要关注点的情境下，基于否决的规则——赋予个人或群体从候选名单中剔除某些候选人的能力——是确保少数群体不会面临无法接受结果的自然且有效方式。然而，这类规则往往缺乏匿名性，或对选民和候选人数量施加特定限制。这些问题可以通过考虑比例否决核来解决，即一种合作博弈的解，其中每个联盟被赋予与其规模成比例地否决一定数量候选人的权力。然而，否决核的朴素算法是指数级的，且已知的针对任意选民数量从核中选择候选人的规则缺乏匿名性。本文提出了一种计算核的多项式时间算法，研究了其期望大小，并给出了一种从核中选择候选人的匿名规则。我们研究了与核一致的投票规则的性质。最后，我们证明悲观主义者可以在多项式时间内操纵核，而乐观主义者则完全无法操纵它。