幂指数比较的复杂性
The Complexity of Power-Index Comparison
摘要：
我们研究以下问题的复杂性：给定两个加权投票游戏G'和G“，每个游戏都包含一个玩家p，在这些游戏中，p的权力指数值更高？我们研究了这个问题的Shapley-Shubik功率指数[SS 54]和Banzhaf功率指数[Ban 65，DS 79]。我们的主要结果是，对于这两个幂指数，问题对于概率多项式时间是完全的（即，PP是完整的）。我们应用我们的研究结果，部分解决了一些最近提出的问题，加权投票游戏的复杂性。我们还研究了原始Shapley-Shubik幂指数的复杂性。Deng和Papadimitriou [DP 94]证明了原始Shapley-Shubik幂指数是P度量完备的。我们通过证明原始Shapley-Shubik幂指数对于#P是多-一完全的来加强这一点。并且我们的加强不可能进一步改进到简约完备性，因为我们观察到，与原始Banzhaf幂指数相比，原始Shapley-Shubik幂指数不是#P-简约完备的。
英文摘要：
We study the complexity of the following problem: Given two weighted voting games G' and G'' that each contain a player p, in which of these games is p's power index value higher? We study this problem with respect to both the Shapley-Shubik power index [SS54] and the Banzhaf power index [Ban65,DS79]. Our main result is that for both of these power indices the problem is complete for probabilistic polynomial time (i.e., is PP-complete). We apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games. We also study the complexity of the raw Shapley-Shubik power index. Deng and Papadimitriou [DP94] showed that the raw Shapley-Shubik power index is #P-metric-complete. We strengthen this by showing that the raw Shapley-Shubik power index is many-one complete for #P. And our strengthening cannot possibly be further improved to parsimonious completeness, since we observe that, in contrast with the raw Banzhaf power index, the raw Shapley-Shubik power index is not #P-parsimonious-complete.
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