论文标题
多数支持的价格
The Price of Majority Support
论文作者
论文摘要
我们考虑在许多相互独立的二元主题上找到一组个人之间的观点之间的妥协问题。在本文中，我们量化了代表性的损失，这是由于要求结果获得多数支持，换句话说，“多数支持的价格”。如果每个人至少在不同意的主题上同意结果，则假定每个人都支持结果。我们的结果也可以看作是量化ANSCOMBES悖论，该悖论指出，主题多数结果可能不受多数的支持。为了衡量结果的代表性，我们考虑两个指标。首先，我们寻找的结果与尽可能多的主题一致。我们证明，最大数字可以保证存在与大多数主题相符的结果，并且具有多数支持，等于$ \ ceil {（t+1）/2} $，其中$ t $是主题总数。其次，我们计算选民对一个主题意见的次数与该主题的结果相匹配。目的是通过最多的比赛获得多数支持，找到结果。我们考虑此数字和总体最佳结果的比赛数量之间的比率，这可能没有多数支持。我们试图找到最大比率，以便保证存在多数支持的结果和这种匹配比的比率。对于3个主题，我们显示此比率为$ 5/6 \约0.83 $。通常，我们证明了一个任意接近$ 2 \ sqrt {6} -4 \大约0.90 $的上限，因为$ t $倾向于无限。此外，我们在数值上计算了相关范围内的较高鞋面和非匹配下限，$ t $。
We consider the problem of finding a compromise between the opinions of a group of individuals on a number of mutually independent, binary topics. In this paper, we quantify the loss in representativeness that results from requiring the outcome to have majority support, in other words, the "price of majority support". Each individual is assumed to support an outcome if they agree with the outcome on at least as many topics as they disagree on. Our results can also be seen as quantifying Anscombes paradox which states that topic-wise majority outcome may not be supported by a majority. To measure the representativeness of an outcome, we consider two metrics. First, we look for an outcome that agrees with a majority on as many topics as possible. We prove that the maximum number such that there is guaranteed to exist an outcome that agrees with a majority on this number of topics and has majority support, equals $\ceil{(t+1)/2}$ where $t$ is the total number of topics. Second, we count the number of times a voter opinion on a topic matches the outcome on that topic. The goal is to find the outcome with majority support with the largest number of matches. We consider the ratio between this number and the number of matches of the overall best outcome which may not have majority support. We try to find the maximum ratio such that an outcome with majority support and this ratio of matches compared to the overall best is guaranteed to exist. For 3 topics, we show this ratio to be $5/6\approx 0.83$. In general, we prove an upper bound that comes arbitrarily close to $2\sqrt{6}-4\approx 0.90$ as $t$ tends to infinity. Furthermore, we numerically compute a better upper and a non-matching lower bound in the relevant range for $t$.
