代议制民主中最优决策的数学模型
A Mathematical Model for Optimal Decisions in a Representative Democracy
摘要：
直接民主是分类器集合的一个特例，每个人（分类器）对每个问题进行投票。当平均投票者能力（分类器准确度）低于50%时，这会失败，这可能发生在投票者只有有限信息的嘈杂环境中，或者当有多个主题并且平均投票者能力对于某些主题可能不够高时。在这两种情况下，选民选择代表投票的代议制民主可以成为灵丹妙药。代议制民主是改进分类器集成的一种具体方法。我们引入一个数学模型来研究代议制民主，特别是了解代议制民主的参数，使最大的决策能力。我们的主要结果表明，在一般和自然条件下，1。代议制民主可以同时对多个嘈杂的问题做出正确的决定。2.当投票成本固定时，最优的代议制民主要求代表从固定规模的群体中选出：代表的数量应该与选民的数量成线性关系。3.当投票成本和投票收益均为多项式时，最优群体规模与投票人数近似线性相关。这项工作为研究代表性民主类型的合奏（更少的高素质代表与更多的低素质代表）的质量-数量权衡奠定了数学基础。
英文摘要：
Direct democracy is a special case of an ensemble of classifiers, where every person (classifier) votes on every issue. This fails when the average voter competence (classifier accuracy) falls below 50%, which can happen in noisy settings where voters have only limited information, or when there are multiple topics and the average voter competence may not be high enough for some topics. Representative democracy, where voters choose representatives to vote, can be an elixir in both these situations. Representative democracy is a specific way to improve the ensemble of classifiers. We introduce a mathematical model for studying representative democracy, in particular understanding the parameters of a representative democracy that gives maximum decision making capability. Our main result states that under general and natural conditions, 1. Representative democracy can make the correct decisions simultaneously for multiple noisy issues. 2. When the cost of voting is fixed, the optimal representative democracy requires that representatives are elected from constant sized groups: the number of representatives should be linear in the number of voters. 3. When the cost and benefit of voting are both polynomial, the optimal group size is close to linear in the number of voters. This work sets the mathematical foundation for studying the quality-quantity tradeoff in a representative democracy-type ensemble (fewer highly qualified representatives versus more less qualified representatives).
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