The least square nucleolus is a normalized Banzhaf value

Abstract

In this note we study a truncated additive normalization of the Banzhaf value. We are able to show that it corresponds to the Least Square nucleolus (LS-nucleolus), which was originally introduced as the solution of a constrained optimization problem (Ruiz et al., 1996). Thus, the main result provides an explicit expression that eases the computation and contributes to the understanding of the LS-nucleolus. Lastly, the result is extended to the broader family of Individually Rational Least Square values (Ruiz et al., 1998b).

Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record

“…Its computationally more convenient minimum sum integer representation is given by v 2 = [q; w 2 ] with q = 19 022 681 and the weights indicated in the w 2 -columns of Table 1. 1 The additional minimal blocking requirement (iii) can be described as v 3 = [25; 1, . .…”

“…So the dimension of v EU28 must be below 7.18 millions. 1 We remark that rounding populations to, say, thousands is common in applied work because this simplifies computations, e.g., of the voting power distribution in the EU Council. Rounding, however, leads to a different set of winning coalitions, i.e., is analyzing 'wrong' rules.…”

“…The mapping of given configurations of approving members to a collective "yes" (1) or "no" (0) defines a so-called simple game. It can often be described by a weighted voting rule: each member i gets a nonnegative weight w i ; a proposal is accepted iff the weight sum of its supporters meets a given quota q.…”

Dimension of the Lisbon Voting Rules in the EU Council: A Challenge and New World Record

“…Consider a group or committee whose members jointly decide whether to accept or reject a proposal (or, more generally, any system which outputs 1 if a minimal set of binary conditions are true and 0 otherwise). The mapping of given configurations of approving members to a collective "yes" (1) or "no" (0) defines a so-called simple game. It can often be described by a weighted voting rule: each member i gets a nonnegative weight w i ; a proposal is accepted iff the weight sum of its supporters meets a given quota q.…”