12,362 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 number of two-dimensional threshold functions
A two-dimensional threshold function of k-valued logic can be viewed as
coloring of the points of a k x k square lattice into two colors such that
there exists a straight line separating points of different colors. For the
number of such functions only asymptotic bounds are known. We give an exact
formula for the number of two-dimensional threshold functions and derive more
accurate asymptotics.Comment: 17 pages, 2 figure
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
On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
The problem of deciding whether CSP instances admit solutions has been deeply
studied in the literature, and several structural tractability results have
been derived so far. However, constraint satisfaction comes in practice as a
computation problem where the focus is either on finding one solution, or on
enumerating all solutions, possibly projected to some given set of output
variables. The paper investigates the structural tractability of the problem of
enumerating (possibly projected) solutions, where tractability means here
computable with polynomial delay (WPD), since in general exponentially many
solutions may be computed. A general framework based on the notion of tree
projection of hypergraphs is considered, which generalizes all known
decomposition methods. Tractability results have been obtained both for classes
of structures where output variables are part of their specification, and for
classes of structures where computability WPD must be ensured for any possible
set of output variables. These results are shown to be tight, by exhibiting
dichotomies for classes of structures having bounded arity and where the tree
decomposition method is considered
Approval-Based Shortlisting
Shortlisting is the task of reducing a long list of alternatives to a
(smaller) set of best or most suitable alternatives from which a final winner
will be chosen. Shortlisting is often used in the nomination process of awards
or in recommender systems to display featured objects. In this paper, we
analyze shortlisting methods that are based on approval data, a common type of
preferences. Furthermore, we assume that the size of the shortlist, i.e., the
number of best or most suitable alternatives, is not fixed but determined by
the shortlisting method. We axiomatically analyze established and new
shortlisting methods and complement this analysis with an experimental
evaluation based on biased voters and noisy quality estimates. Our results lead
to recommendations which shortlisting methods to use, depending on the desired
properties
Rooted Spiral Trees on Hyper-cubical lattices
We study rooted spiral trees in 2,3 and 4 dimensions on a hyper cubical
lattice using exact enumeration and Monte-Carlo techniques. On the square
lattice, we also obtain exact lower bound of 1.93565 on the growth constant
. Series expansions give and on a square lattice. With Monte-Carlo simulations we get the
estimates as , and . These results
are numerical evidence against earlier proposed dimensional reduction by four
in this problem. In dimensions higher than two, the spiral constraint can be
implemented in two ways. In either case, our series expansion results do not
support the proposed dimensional reduction.Comment: replaced with published versio
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
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