TY - JOUR

T1 - The inverse Banzhaf problem

AU - Alon, Noga

AU - Edelman, Paul H.

N1 - Funding Information:
N. Alon’s research supported in part by an ERC Advanced grant, by a USA-Israel BSF grant, by NSF grant CCF 0832797 and by the Ambrose Monell Foundation.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

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U2 - 10.1007/s00355-009-0402-8

DO - 10.1007/s00355-009-0402-8

M3 - Article

AN - SCOPUS:77957280986

SN - 0176-1714

VL - 34

SP - 371

EP - 377

JO - Social Choice and Welfare

JF - Social Choice and Welfare

IS - 3

ER -