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Using the Multilinear Extension to Study Some Probabilistic Power Indices

机译：使用多线性扩展来研究一些概率幂指数

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摘要

We consider binary voting systems modeled by a simple game, in which voters vote independently of each other, and the probability distribution over coalitions is known. The Owen's multilinear extension of the simple game is used to improve the use and the computation of three indices defined in this model: the decisiveness index, which is an extension of the Banzhaf index, the success index, which is an extension of the Rae index, and the luckiness index. This approach leads us to prove new properties and inter-relations between these indices. In particular it is proved that the ordinal equivalence between success and decisiveness indices is achieved in any game if and only if the probability distribution is anonymous. In the anonymous case, the egalitarianism of the three indices is compared, and it is also proved that, for these distributions, decisiveness and success indices respect the strength of the seats, whereas luckiness reverses this order.
机译：我们考虑由一个简单的游戏建模的二元投票系统，在该系统中，选民彼此独立地投票，并且知道联盟的概率分布。简单游戏的Owen多线性扩展用于改进此模型中定义的三个指标的使用和计算：果断指标（是Banzhaf指标的扩展），成功指标（是Rae指标的扩展） ，以及运气指数。这种方法使我们证明了这些指数之间的新性质和相互关系。特别是，证明了在且仅当概率分布是匿名的情况下，在任何博弈中都达到了成功与果断性指标之间的序数等效性。在匿名案例中，比较了三个指数的平均主义，并且还证明了，对于这些分布，决定性和成功指数尊重席位的强度，而运气则颠倒了这种顺序。

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