An assembly of $n$ voters needs to decide on $t$ independent binary issues. Each voter has opinions about the issues, given by a $t$-bit vector. Anscombe's paradox shows that a policy following the majority opinion in each issue may not survive a vote by the very same set of $n$ voters, i.e., more voters may feel unrepresented by such a majority-driven policy than represented. A natural resolution is to come up with a policy that deviates a bit from the majority policy but no longer gets more opposition than support from the electorate. We show that a Hamming distance to the majority policy of at most $\lfloor (t - 1) / 2 \rfloor$ can always be guaranteed, by giving a new probabilistic argument relying on structure-preserving symmetries of the space of potential policies. Unless the electorate is evenly divided between the two options on all issues, we in fact show that a policy strictly winning the vote exists within this distance bound. Our approach also leads to a deterministic polynomial-time algorithm for finding policies with the stated guarantees, answering an open problem of previous work. For odd $t$, unless we are in the pathological case described above, we also give a simpler and more efficient algorithm running in expected polynomial time with the same guarantees. We further show that checking whether distance strictly less than $\lfloor (t - 1) /2 \rfloor$ can be achieved is NP-hard, and that checking for distance at most some input $k$ is FPT with respect to several natural parameters.

翻译：一个由 $n$ 名投票者组成的议会需要对 $t$ 个独立的二元议题进行决策。每位投票者对各议题的观点由一个 $t$ 位向量表示。安斯康姆悖论表明：若遵循每个议题上的多数意见制定政策，该政策可能无法被同一组 $n$ 名投票者通过——即相比感到被代表的投票者，更多投票者可能认为该多数驱动政策未能代表其意愿。一种自然的解决方案是提出一项与多数政策略有偏离、但不再获得选民反对多于支持的政策。我们通过利用潜在政策空间的结构保持对称性，提出一种新的概率论证，证明总能保证与多数政策的汉明距离不超过 $\lfloor (t - 1) / 2 \rfloor$。除非全体选民在所有议题上对两个选项完全均分，我们实际上证明了在此距离界限内存在严格赢得投票的政策。我们的方法还导出了确定性多项式时间算法来寻找具有所述保证的政策，解决了前人工作的一个开放问题。对于奇数 $t$，除非处于上述病态情形，我们进一步给出更简单高效的随机多项式时间算法，其期望运行时间同样满足所述保证。我们进一步证明：检验能否实现严格小于 $\lfloor (t - 1) /2 \rfloor$ 的距离是 NP 困难的，而检验距离不超过给定输入 $k$ 的问题在若干自然参数下具有固定参数可解性（FPT）。