In the single winner determination problem, we have n voters and m candidates and each voter j incurs a cost c(i, j) if candidate i is chosen. Our objective is to choose a candidate that minimizes the expected total cost incurred by the voters; however as we only have access to the agents' preference rankings over the outcomes, a loss of efficiency is inevitable. This loss of efficiency is quantified by distortion. We give an instance of the metric single winner determination problem for which any randomized social choice function has distortion at least 2.063164. This disproves the long-standing conjecture that there exists a randomized social choice function that has a worst-case distortion of at most 2.

翻译：在单一获胜者的确定问题中,我们有n个选民和m个候选人,如果选上候选人i,每个选民j就会产生c(i,j)成本。我们的目标是选择一个能够最大限度地减少选民预期总成本的候选人;然而,由于我们只能获得代理人优于结果的优先等级,效率的丧失是不可避免的。效率的丧失通过扭曲来量化。我们给出了一个标准单一获胜者确定问题的例子,任何随机的社会选择功能都会扭曲至少2.063164。这反驳了长期的推测,即存在一种随机的社会选择功能,其最坏的扭曲最多为2个。