The inverse Banzhaf problem

Noga Alon, Paul H. Edelman

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

LetF be a family of subsets of the ground set[n] = {1,2,n}.For each i ∈ [n] we let p(F,i) be the number of pairs of subsets that differ in the element i and exactly one of them is in F. We interpret p(F,i) as the influence of that element. The normalized Banzhaf vector of F,denoted B(F), isthevector(B(F,1),..., B(F,n)),where B(F,i) = p(F,i)/p(F) and p(F) is the sum of all p(F,i). The Banzhaf vector has been studied in the context of measuring voting power in voting games as well as in Boolean circuit theory. In this paper we investigate which non-negative vectors of sum 1 can be closely approximated by Banzhaf vectors of simple voting games. In particular, we show that if a vector has most of its weight concentrated in k < n coordinates, then it must be essentially the Banzhaf vector of some simple voting game with n - k dummy voters.

Original languageEnglish
Pages (from-to)371-377
Number of pages7
JournalSocial Choice and Welfare
Volume34
Issue number3
DOIs
StatePublished - 2010

Funding

FundersFunder number
USA-Israel BSF
National Science FoundationCCF 0832797
Ambrose Monell Foundation
Seventh Framework Programme226718
European Research Council

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