Arrow's Theorem for Incomplete Relations

190 Pages·2012·1.32 MB·English

by  R. D. Maddux

Arrow’s Theorem for Incomplete Relations
R. D. Maddux
DepartmentofMathematics
IowaStateUniversity
Ames,Iowa,USA
Ames, Iowa
Tuesdays, October 9 and 16, 2012
R.D.Maddux Arrow’sTheoremforIncompleteRelations
R.D.Maddux Arrow’sTheoremforIncompleteRelations
Abstract
Let U be a set with three or more elements, let W be the set of
weak orderings of U, let T be the set of total orderings of U, and
let f be an n-ary function mapping Wn to W. Arrow’s
Impossibility Theorem asserts that if f satisfies
Arrow’s Condition P (“Pareto”) and Condition 3 (“independence
of irrelevant alternatives”) then f is a projection function on total
orderings, i. e., there is some k ∈ {1,...,n} such that
f(R ,...,R ) = R for all total orderings R ,...,R ∈ T.
1 n k 1 n
Th. If a transitive-valued multivariate relational operator f
satisfies versions of Arrow’s Conditions P and 3, and maps all
profiles from a diverse set R of binary relations on U to transitive
relations on U, then f must be the unanimous consent function for
some set of input variables, and if R is very diverse then f is a
projection function.
Cor. Characterizations of intersection and projection functions;
Arrow’s Theorem.
R.D.Maddux Arrow’sTheoremforIncompleteRelations
History
Kenneth J. Arrow (Nobel, 1973, Econonics)
learned the Calculus of Relations from Alfred Tarski in 1940
applied it to his problems in economics
obtained Arrow’s Impossibility Theorem (Arrow 1950, 1963)
created Social Choice Theory
R.D.Maddux Arrow’sTheoremforIncompleteRelations
History
Kenneth J. Arrow (Nobel, 1973, Econonics)
learned the Calculus of Relations from Alfred Tarski in 1940
applied it to his problems in economics
obtained Arrow’s Impossibility Theorem (Arrow 1950, 1963)
created Social Choice Theory
R.D.Maddux Arrow’sTheoremforIncompleteRelations
History
Kenneth J. Arrow (Nobel, 1973, Econonics)
learned the Calculus of Relations from Alfred Tarski in 1940
applied it to his problems in economics
obtained Arrow’s Impossibility Theorem (Arrow 1950, 1963)
created Social Choice Theory
R.D.Maddux Arrow’sTheoremforIncompleteRelations
History
Kenneth J. Arrow (Nobel, 1973, Econonics)
learned the Calculus of Relations from Alfred Tarski in 1940
applied it to his problems in economics
obtained Arrow’s Impossibility Theorem (Arrow 1950, 1963)
created Social Choice Theory
R.D.Maddux Arrow’sTheoremforIncompleteRelations
History
Kenneth J. Arrow (Nobel, 1973, Econonics)
learned the Calculus of Relations from Alfred Tarski in 1940
applied it to his problems in economics
obtained Arrow’s Impossibility Theorem (Arrow 1950, 1963)
created Social Choice Theory
R.D.Maddux Arrow’sTheoremforIncompleteRelations
History
Kenneth J. Arrow (Nobel, 1973, Econonics)
learned the Calculus of Relations from Alfred Tarski in 1940
applied it to his problems in economics
obtained Arrow’s Impossibility Theorem (Arrow 1950, 1963)
created Social Choice Theory
R.D.Maddux Arrow’sTheoremforIncompleteRelations
History, cont.
spring semester of 1940
Tarski was a Visiting Professor of Philosophy at the City
College of New York
class attended by Kenneth Arrow:
“It was a great course, Calculus of Relations. His organization
was beautiful — I could tell that immediately — and he was
thorough. In fact what I learned from him played a role in my
own later work — not so much the particular theorems but
the language of relations was immediately applicable to
economics. I could express my problems in those terms”
Arrow proofread Tarski’s Introduction to Logic for no pay at
Tarski’s request:
“I also owe many thanks to Mr. K. J. Arrow for his help in
reading proofs.”
R.D.Maddux Arrow’sTheoremforIncompleteRelations