39 research outputs found
On minimum sum representations for weighted voting games
A proposal in a weighted voting game is accepted if the sum of the
(non-negative) weights of the "yea" voters is at least as large as a given
quota. Several authors have considered representations of weighted voting games
with minimum sum, where the weights and the quota are restricted to be
integers. Freixas and Molinero have classified all weighted voting games
without a unique minimum sum representation for up to 8 voters. Here we
exhaustively classify all weighted voting games consisting of 9 voters which do
not admit a unique minimum sum integer weight representation.Comment: 7 pages, 6 tables; enumerations correcte
Ready for the design of voting rules?
The design of fair voting rules has been addressed quite often in the
literature. Still, the so-called inverse problem is not entirely resolved. We
summarize some achievements in this direction and formulate explicit open
questions and conjectures.Comment: 10 page
Minimal proper non-IRUP instances of the one-dimensional Cutting Stock Problem
We consider the well-known one dimensional cutting stock problem (1CSP).
Based on the pattern structure of the classical ILP formulation of Gilmore and
Gomory, we can decompose the infinite set of 1CSP instances, with a fixed
demand n, into a finite number of equivalence classes. We show up a strong
relation to weighted simple games. Studying the integer round-up property we
computationally show that all 1CSP instances with are proper IRUP,
while we give examples of a proper non-IRUP instances with . A gap larger
than 1 occurs for . The worst known gap is raised from 1.003 to 1.0625.
The used algorithmic approaches are based on exhaustive enumeration and integer
linear programming. Additionally we give some theoretical bounds showing that
all 1CSP instances with some specific parameters have the proper IRUP.Comment: 14 pages, 2 figures, 2 table
Average Weights and Power in Weighted Voting Games
We investigate a class of weighted voting games for which weights are
randomly distributed over the standard probability simplex. We provide
close-formed formulae for the expectation and density of the distribution of
weight of the -th largest player under the uniform distribution. We analyze
the average voting power of the -th largest player and its dependence on the
quota, obtaining analytical and numerical results for small values of and a
general theorem about the functional form of the relation between the average
Penrose--Banzhaf power index and the quota for the uniform measure on the
simplex. We also analyze the power of a collectivity to act (Coleman efficiency
index) of random weighted voting games, obtaining analytical upper bounds
therefor.Comment: 12 pages, 7 figure
Mostly sunny : a forecast of tomorrow's power index research
Power index research has been a very active field in the last decades. Will
this continue or are all the important questions solved? We argue that there
are still many opportunities to conduct useful research with and on power
indices. Positive and normative questions keep calling for theoretical and
empirical attention. Technical and technological improvements are likely to
boost applicability.Comment: 12 page
On the inverse power index problem
Weighted voting games are frequently used in decision making. Each voter has
a weight and a proposal is accepted if the weight sum of the supporting voters
exceeds a quota. One line of research is the efficient computation of so-called
power indices measuring the influence of a voter. We treat the inverse problem:
Given an influence vector and a power index, determine a weighted voting game
such that the distribution of influence among the voters is as close as
possible to the given target value. We present exact algorithms and
computational results for the Shapley-Shubik and the (normalized) Banzhaf power
index.Comment: 17 pages, 2 figures, 12 table
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The power index at infinity: Weighted voting in sequential infinite anonymous games
After we describe the waiting queue problem, we identify a partially observable 2n+1-player voting game with only one pivotal player; the player at the n-1 order. Given the simplest rule of heterogeneity presented in this paper, we show that for any infinite sequential voting game of size 2n+1, a power index of size n is a good approximation of the power index at infinity, and it is difficult to achieve. Moreover, we show that the collective utility value of a coalition for a partially observable anonymous game given an equal distribution of weights is n²+n. This formula is developed for infinite sequential anonymous games using a stochastic process that yields a utility function in terms of the probability of the sequence and voting outcome of the coalition. Evidence from Wikidata editing sequences is presented and the results are compared for 10 coalitions