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：由n位选民组成的议会需对t个独立的二元议题进行决策。每位选民对议题的观点由t位二进制向量表示。安斯康悖论指出：若在每个议题上遵循多数意见形成政策，该政策可能无法通过同一组n位选民的投票验证，即多数驱动型政策导致的支持者数量可能少于反对者。自然的解决方案是提出一项与多数政策略有偏差、但不再获得多于支持票的反对票的政策。我们通过引入基于潜在政策空间结构保持对称性的新型概率论证，证明总能保证与多数政策的汉明距离不超过⌊(t-1)/2⌋。除非全体选民在所有议题上对两个选项势均力敌，我们进一步证明在此距离界限内存在严格赢得投票的政策。该方法还催生了具有保障性声明的确定性多项式时间算法，解决了前人工作中的开放问题。对于奇数t，除上述病态情形外，我们给出具有相同保障的期望多项式时间更简捷高效算法。此外，我们证明检验＜⌊(t-1)/2⌋距离可达性为NP困难问题，且检验距离不超过给定输入k的问题关于若干自然参数具有固定参数可解性。