Average Weights and Power in Weighted Voting Games. Boratyn, D., Kirsch, W., Słomczyński, W., Stolicki, D., & Życzkowski, K. Technical Report 1905.04261 [cs.GT], May, 2019.
Average Weights and Power in Weighted Voting Games [link]Paper  abstract   bibtex   
We investigate a class of weighted voting games for which weights are randomly distributed over the unit simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform distribution. We analyze the average voting power of the $k$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of $n$ and a general theorem about the functional form of the relation between the average Banzhaf power index and the quota for the uniform measure on the simplex.
@techreport{BoratynEtAl19,
  type = {{{arXiv}}},
  title = {Average {{Weights}} and {{Power}} in {{Weighted Voting Games}}},
  author = {Boratyn, Daria and Kirsch, Werner and Słomczyński, Wojciech and Stolicki, Dariusz and Życzkowski, Karol},
  year = {2019},
  month = may,
  number = {1905.04261 [cs.GT]},
  eprint = {1905.04261},
  eprinttype = {arxiv},
  url = {http://arxiv.org/abs/1905.04261},
  urldate = {2019-09-05},
  abstract = {We investigate a class of weighted voting games for which weights are randomly distributed over the unit simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the \$k\$-th largest player under the uniform distribution. We analyze the average voting power of the \$k\$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of \$n\$ and a general theorem about the functional form of the relation between the average Banzhaf power index and the quota for the uniform measure on the simplex.},
  archiveprefix = {arXiv},
  keywords = {2010: Primary 91A12; Secondary 60E05; 60E10,Computer Science - Computer Science and Game Theory,Mathematics - Probability,Physics - Physics and Society}
}

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