The Geometry behind Paradoxes of Voting Power
The Geometry behind Paradoxes of Voting Power
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投票权悖论背后的几何学
DOI：
10.1080/0025570x.2009.11953603
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发表时间：
2009
影响因子：
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通讯作者：
Michael A. Jones
中科院分区：
文献类型：
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作者：
Michael A. Jones
安妮莉丝、布赖恩和卡洛斯拥有一家小公司全部1500股股票。在年度股东大会上，每个股东的投票数与他或她拥有的股份数相同。如果占全体股份三分之二的股东支持该措施，则会议通过该措施。由于即将举行的会议将对有关将制造业转移到亚洲和更换公司首席执行官的有争议的措施进行投票，布赖恩和卡洛斯每人从安妮莉丝那里购买100股股票，以获得更大的影响力。布赖恩发现，他有更多的影响力比以前，而卡洛斯发现，他的投票不能影响任何措施的结果，当它之前可以！怎么会这样？我将介绍简单的加权投票博弈来模拟股东情景和其他投票情景，并介绍权力指数来衡量选民对赞成/反对选举结果的影响。权力指数已被用于分析国际货币基金组织[9，19]、电子学院[21]、欧盟部长理事会[13，18]和以色列议会[17]的简单加权投票博弈模型。权力指数的计算也被用在关于制度设计的辩论中，如在关于改革对欧盟的影响和引入新成员的文章中[32，33]。像卡洛斯的困境这样违反直觉的结果通常被称为悖论。关于权力指数的文献充满了悖论，以及展示它们的现实生活机构。再分配悖论[9，25]、捐赠者和转移悖论[10]、成员争吵悖论[15]、新成员悖论[3，4]和大规模悖论[3，28]在简单的加权投票博弈中捕捉了反直觉博弈的不同方面。运用几何学，我将解释和分类投票权悖论的原因。令人惊讶的是，3个选民的例子足以理解几何形状。低维和3选民的例子的固有对称性往往足以证明，所有的权力指数都容易受到一个特殊的悖论。布拉德伯里[2]使用相同的几何方法来检查分配方法的悖论。为了将一个简单的加权投票博弈置于几何环境中，让投票者的权重（在股东的例子中是股票的数量）代表欧几里得空间中的一个点。投票规则（股东博弈的2/3）定义了将空间划分为不同部分或等价类的超平面，使得每个参与者的权力对于等价类中的所有博弈都是固定的;对于3投票者的例子，超平面是线。三种类型的几何现象描述了游戏中可能导致违反直觉结果的变化。的权重的变化，
Anneliese, Brian, and Carlos among them own all 1500 shares of a small company's stock. At the annual stockholders' meeting, each stockholder's vote counts the same as the number of shares that he or she owns. A measure at the meeting passes if stock holders accounting for 2/3 of all shares support the measure. Because controversial measures about moving manufacturing to Asia and replacing the CEO of the company are to be voted on at the upcoming meeting, Brian and Carlos each buy 100 shares of stock from Anneliese in an effort to gain more influence. Brian finds that he has more influence than before, while Carlos discovers that his vote cannot affect the outcome on any measure, when before it could! How can this happen? I will introduce simple weighted-voting games to model the stockholder scenario, and other voting situations, and power indices to measure the effect voters have on the outcome of yes/no elections. Power indices have been used to analyze the simple weighted-voting game models of the International Monetary Fund [9, 19], the Elec toral College [21], the European Union Council of Ministers [13,18], and the Israeli Knesset [17]. Power index calculations have also been used in the debate on the de sign of institutions, as in articles about the effects of reforms on, and the introduction of new members into, the European Union [32,33]. Counterintuitive results such as Carlos' predicament are often called paradoxes. The literature on power indices is full of paradoxes, as well as real-life institutions that exhibit them. The paradox of redistribution [9, 25], the donor and transfer paradoxes [10], the paradox of quarreling members [15], the paradox of a new member [3, 4], and the paradox of large size [3, 28] capture diverse aspects of counterintuitive be havior in simple weighted-voting games. Using geometry, I will explain and classify the causes of voting power paradoxes. Surprisingly, 3-voter examples are sufficient to understand the geometry. The low dimension and the inherent symmetry of the 3-voter examples are often enough to prove that all power indices are susceptible to a partic ular paradox. Bradberry [2] used the same geometric approach to examine paradoxes of apportionment methods. To place a simple weighted-voting game in a geometric setting, let the weights of the voters (the number of shares in the stockholder example) represent a point in Euclidean space. The voting rule (2/3 for the stockholder game) defines hyperplanes that partition the space into different parts or equivalence classes so that the power of each player is fixed for all games in an equivalence class; for the 3-voter examples, the hyperplanes are lines. Three types of geometric phenomena describe the changes in a game that may result in counterintuitive outcomes. A change in the weights of the