13 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
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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
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
The Inverse Shapley Value Problem
For a weighted voting scheme used by voters to choose between two
candidates, the \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of
provide a measure of how much control each voter can exert over the overall
outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley
and Martin Shubik in 1954 \cite{SS54} and are widely studied in social choice
theory as a measure of the "influence" of voters. The \emph{Inverse Shapley
Value Problem} is the problem of designing a weighted voting scheme which
(approximately) achieves a desired input vector of values for the
Shapley-Shubik indices. Despite much interest in this problem no provably
correct and efficient algorithm was known prior to our work.
We give the first efficient algorithm with provable performance guarantees
for the Inverse Shapley Value Problem. For any constant \eps > 0 our
algorithm runs in fixed poly time (the degree of the polynomial is
independent of \eps) and has the following performance guarantee: given as
input a vector of desired Shapley values, if any "reasonable" weighted voting
scheme (roughly, one in which the threshold is not too skewed) approximately
matches the desired vector of values to within some small error, then our
algorithm explicitly outputs a weighted voting scheme that achieves this vector
of Shapley values to within error \eps. If there is a "reasonable" voting
scheme in which all voting weights are integers at most \poly(n) that
approximately achieves the desired Shapley values, then our algorithm runs in
time \poly(n) and outputs a weighted voting scheme that achieves the target
vector of Shapley values to within error $\eps=n^{-1/8}.
On minimum integer representations of weighted games
We study minimum integer representations of weighted games, i.e.,
representations where the weights are integers and every other integer
representation is at least as large in each component. Those minimum integer
representations, if the exist at all, are linked with some solution concepts in
game theory. Closing existing gaps in the literature, we prove that each
weighted game with two types of voters admits a (unique) minimum integer
representation, and give new examples for more than two types of voters without
a minimum integer representation. We characterize the possible weights in
minimum integer representations and give examples for types of voters
without a minimum integer representation preserving types, i.e., where we
additionally require that the weights are equal within equivalence classes of
voters.Comment: 29 page
Complementary cooperation, minimal winning coalitions, and power indices
We introduce a new simple game, which is referred to as the complementary
weighted multiple majority game (C-WMMG for short). C-WMMG models a basic
cooperation rule, the complementary cooperation rule, and can be taken as a
sister model of the famous weighted majority game (WMG for short). In this
paper, we concentrate on the two dimensional C-WMMG. An interesting property of
this case is that there are at most minimal winning coalitions (MWC for
short), and they can be enumerated in time , where is the
number of players. This property guarantees that the two dimensional C-WMMG is
more handleable than WMG. In particular, we prove that the main power indices,
i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel
index, and the Deegan-Packel index, are all polynomially computable. To make a
comparison with WMG, we know that it may have exponentially many MWCs, and none
of the four power indices is polynomially computable (unless P=NP). Still for
the two dimensional case, we show that local monotonicity holds for all of the
four power indices. In WMG, this property is possessed by the Shapley-Shubik
index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the
Deegan-Packel index. Since our model fits very well the cooperation and
competition in team sports, we hope that it can be potentially applied in
measuring the values of players in team sports, say help people give more
objective ranking of NBA players and select MVPs, and consequently bring new
insights into contest theory and the more general field of sports economics. It
may also provide some interesting enlightenments into the design of
non-additive voting mechanisms. Last but not least, the threshold version of
C-WMMG is a generalization of WMG, and natural variants of it are closely
related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page