4,811 research outputs found
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
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
Bounds for the Nakamura number
The Nakamura number is an appropriate invariant of a simple game to study the
existence of social equilibria and the possibility of cycles. For symmetric
quota games its number can be obtained by an easy formula. For some subclasses
of simple games the corresponding Nakamura number has also been characterized.
However, in general, not much is known about lower and upper bounds depending
of invariants on simple, complete or weighted games. Here, we survey such
results and highlight connections with other game theoretic concepts.Comment: 23 pages, 3 tables; a few more references adde
Bilateral symmetry and modified Pascal triangles in Parsimonious games
We discuss the prominent role played by bilateral symmetry and modified
Pascal triangles in self twin games, a subset of constant sum homogeneous
weighted majority games. We show that bilateral symmetry of the free
representations unequivocally identifies and characterizes this class of games
and that modified Pascal triangles describe their cardinality for combinations
of m and k, respectively linked through linear transforms to the key parameters
n, number of players and h, number of types in the game. Besides, we derive the
whole set of self twin games in the form of a genealogical tree obtained
through a simple constructive procedure in which each game of a generation,
corresponding to a given value of m, is able to give birth to one child or two
children (depending on the parity of m), self twin games of the next
generation. The breeding rules are, given the parity of m, invariant through
generations and quite simple.Comment: pp. 2
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
Twin relationships in Parsimonious Games: some results
In a vintage paper concerning Parsimonious games, a subset of constant sum
homogeneous weighted majority games, Isbell introduced a twin relationship
based on transposition properties of the incidence matrices upon minimal
winning coalitions of such games. A careful investigation of such properties
allowed the discovery of some results on twin games presented in this paper. In
detail we show that a) twin games have the same minimal winning quota and b)
each Parsimonious game admits a unique balanced lottery on minimal winning
coalitions, whose probabilities are given by the individual weights of its twin
game
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