306 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
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
Fast regocnition of planar non unit distance graphs
We study criteria attesting that a given graph can not be embedded in the
plane so that neighboring vertices are at unit distance apart and the straight
line edges do not cross.Comment: 9 pages, 1 table, 5 figure
Generalized roll-call model for the Shapley-Shubik index
In 1996 Dan Felsenthal and Mosh\'e Machover considered the following model.
An assembly consisting of voters exercises roll-call. All possible
orders in which the voters may be called are assumed to be equiprobable. The
votes of each voter are independent with expectation for an individual
vote {\lq\lq}yea{\rq\rq}. For a given decision rule the \emph{pivotal}
voter in a roll-call is the one whose vote finally decides the aggregated
outcome. It turned out that the probability to be pivotal is equivalent to the
Shapley-Shubik index. Here we give an easy combinatorial proof of this
coincidence, further weaken the assumptions of the underlying model, and study
generalizations to the case of more than two alternatives.Comment: 19 pages; we added a reference to an earlier proof of our main resul
Measuring voting power in convex policy spaces
Classical power index analysis considers the individual's ability to
influence the aggregated group decision by changing its own vote, where all
decisions and votes are assumed to be binary. In many practical applications we
have more options than either "yes" or "no". Here we generalize three important
power indices to continuous convex policy spaces. This allows the analysis of a
collection of economic problems like e.g. tax rates or spending that otherwise
would not be covered in binary models.Comment: 31 pages, 9 table
On the characteristic of integral point sets in
We generalise the definition of the characteristic of an integral triangle to
integral simplices and prove that each simplex in an integral point set has the
same characteristic. This theorem is used for an efficient construction
algorithm for integral point sets. Using this algorithm we are able to provide
new exact values for the minimum diameter of integral point sets.Comment: 9 pages, 1 figur
Enumeration of integral tetrahedra
We determine the numbers of integral tetrahedra with diameter up to
isomorphism for all via computer enumeration. Therefore we give an
algorithm that enumerates the integral tetrahedra with diameter at most in
time and an algorithm that can check the canonicity of a given
integral tetrahedron with at most 6 integer comparisons. For the number of
isomorphism classes of integral matrices with diameter
fulfilling the triangle inequalities we derive an exact formula.Comment: 10 pages, 1 figur
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