14,785 research outputs found
Arrovian juntas
This article explicitly constructs and classifies all arrovian voting systems
on three or more alternatives. If we demand orderings to be complete, we have,
of course, Arrow's classical dictator theorem, and a closer look reveals the
classification of all such voting systems as dictatorial hierarchies. If we
leave the traditional realm of complete orderings, the picture changes. Here we
consider the more general setting where alternatives may be incomparable, that
is, we allow orderings that are reflexive and transitive but not necessarily
complete. Instead of a dictator we exhibit a junta whose internal hierarchy or
coalition structure can be surprisingly rich. We give an explicit description
of all such voting systems, generalizing and unifying various previous results.Comment: 22 pages, 1 figur
A defense of an entropy based index of multigroup segregation
This paper defends the use of the entropy based Mutual Information index of multigroup
segregation for the following five reasons. (1) It satisfies 14 basic axioms discussed in the
literature when segregation takes place along a single dimension. (2) It is additively
decomposable into between- and within-group terms for any partition of the set of
occupations (or schools) and the set of demographic groups in the multigroup case. (3) The
underlying segregation ordering has been recently characterized in terms of 8 properties. (4) It
is a monotonic transformation of log-likelihood tests for the existence of segregation in a
general model. (5) It can be decomposed so that a term independent of changes in either of the
two marginal distributions can be isolated in pair wise segregation comparisons. Other
existing measures of segregation have not been characterized, fail to satisfy one or more of the
basic axioms, do not admit a between- within-group decomposition, have not been motivated
from a statistical approach, or are based on more restricted econometric models
An approach to basic set theory and logic
The purpose of this paper is to outline a simple set of axioms for basic set
theory from which most fundamental facts can be derived. The key to the whole
project is a new axiom of set theory which I dubbed "The Law of Extremes". It
allows for quick proofs of basic set-theoretic identities and logical
tautologies, so it is also a good tool to aid one's memory.
I do not assume any exposure to euclidean geometry via axioms. Only an
experience with transforming algebraic identities is required.
The idea is to get students to do proofs right from the get-go. In
particular, I avoid entangling students in nuances of logic early on. Basic
facts of logic are derived from set theory, not the other way around.Comment: 22 page
Set mapping reflection
In this note we will discuss a new reflection principle which follows from
the Proper Forcing Axiom. The immediate purpose will be to prove that the
bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2
and that L(P(omega_1)) satisfies the Axiom of Choice. It will also be
demonstrated that this reflection principle implies that combinatorial
principle Square(kappa) fails for all regular kappa > omega_1.Comment: 11 page
Relations between some cardinals in the absence of the Axiom of Choice
If we assume the axiom of choice, then every two cardinal numbers are
comparable. In the absence of the axiom of choice, this is no longer so. For a
few cardinalities related to an arbitrary infinite set, we will give all the
possible relationships between them, where possible means that the relationship
is consistent with the axioms of set theory. Further we investigate the
relationships between some other cardinal numbers in specific permutation
models and give some results provable without using the axiom of choice
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