A Simple Proof "There is no consistent method by which a
democratic society can make a choice (when voting) that is always
fair when that choice must be made from among three or more
alternatives." Slide 2 Let A be a set of outcomes, N a number of
voters or decision criteria. The set of all full linear orderings
of A is then denoted by by L(A). Note: This set is equivalent to
the set S | A | of permutations on the elements of A). A social
welfare function is a function, F: L(A) N L(A), which aggregates
voters' preferences into a single preference order on A. The
N-tuple (R 1, , R N ) of voter's preferences is called a preference
profile. Slide 3 Arrow's Impossibility theorem states that whenever
the set A of possible alternatives has more than 2 elements, then
the following three conditions, called fairness criteria become
incompatible: Unanimity (Pareto efficiency): If alternative a is
ranked above b for all orderings R 1, , R N, then a is ranked
higher than b by F(R 1, , R N ). (Note that unanimity implies non-
imposition). Non-dictatorship: There is no individual i whose
preferences always prevail. That is, there is no i {1, ,N} such
that for every (R 1, , R N ) L(A) N, F(R 1, , R N ) = R i.
Independence of Irrelevant Alternatives: For two preference
profiles (R 1, , R N ) and (S 1, , S N ) such that for all
individuals i, alternatives a and b have the same order in R i as
in S i, alternatives a and b have the same order in F(R 1, , R N )
as in F(S 1, , S N ). Independence of Irrelevant Alternatives
http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem
Fairness CriteriaFairness Criteria Slide 4 Commonly restated as:
"No voting method is fair", "Every ranked voting method is flawed",
or "The only voting method that isn't flawed is a dictatorship".
But these are oversimplified, and thus do not hold universally
Actually says: A voting mechanism cant follow all the fairness
criteria for all possible preference orders Any social choice
system respecting unrestricted domain, unanimity, and independence
of irrelevant alternatives is a dictatorship
http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem Slide
5 Has been proven in numerous ways Graph Theory Proof, from a paper
by Nambiar, Varma and Saroch, submitted May 1992.
www.ece.rutgers.edu/~knambiar/science/ArrowProof.pdf Slide 6 Uses
two digraphs, D=(V,A), a preference and anonpreference.
Nonpreference: complete and transitive Preference graph is the
complement of the nonpreference graph Adjacency matrix of
preference graph is called the preference matrix Slide 7 Notation:
m is the total number of candidates C 1, .., C m n is the total
number of voters V 1, .., V n V k = [v i, j k ] is the preference
matrix of order m by m which gives the preference of the voter, V
k, where k is an element of {1, , n}. When v i, j k = 0, the voter
does not prefer candidate i over candidate j. When v i,j k = 1, the
voter prefers candidate i over candidate j. 0 (boldface) represents
a nonpreference set of voters, while 1 (also boldface) represents a
preference set of voters. v i, j k = * means that the voter has an
unspecified preference. The star also represents the unspecified
preference set of voters. Slide 8 S = [s i j ] is the preference
matrix of m by m order which gives the preference of society as a
whole, rather than individual voters. The voting function is F(V 1,
.., V n ) = S. The dictator function is also a projection function
and is as follows: D n k (x 1, .x n )= x k. Slide 9 (Previously
mentioned under Fairness criteria) Axiom of Independence S ij = f
ij (v ij 1,..v ij n ) for ij and s ij =0 States that s ij is a
function of the v ij k s only Axiom of Unanimity f ij (0,0,..0)=f
ij (1,1,..1)=1 States that if all the voters vote one way then the
voting system also votes the same way (definition of unanimous)
Slide 10 We want to prove: f ij (x 1, ..x n )= D n d (x 1, .x n )=
x d which means that S= V d Slide 11 Proof: Define: h= min ij {sum
from k=1,..n of the x k so that f ij (x 1,...,x n )=1} Note that
the m(m-1)2 n values of f ij are to be inspected before we can
obtain the value of h. We want to show that h = 1. f ij( 1; 0; 0) =
0 and f jk (1; 0; 0) = 0 f ik (*,*,0) = 0 since nonpreference
graphs are transitive f ik (1; 1; 0) = 0 Taking the contrapositive
of the above argument f ik (1; 1; 0) = 1 f ij (1; 0; 0) = 1 or f jk
(0; 1; 0) = 1 Slide 12 It immediately follows that h = 1. Note that
h cannot be zero because of the Unanimity axiom. Without loss of
generality we may assume f ab (1; 0) = 1, the position at which 1
occurs in f ab is of no concern to us. Here C a and C b are two
specific candidates. Now, f ia (1; 1) = 1 and f ab (1; 0) = 1 f ib
(1; *) = 1 since preference graphs are transitive, and f ib (1; *)
= 1 and f bj (1; 1) = 1 f ij (1;*) = 1 since preference graphs are
transitive f ji (0; *) = 0 since preference graphs are asymmetric f
ij (x 1 ; *) = x 1 Dictator Theorem immediately follows. Slide 13
Can be extended to symmetric tournaments Has been proven in several
other (more complicated) ways using graph theory Slide 14 Beigman,
Eyal. Extension of Arrows Theorem to Symmetric Sets of Tournaments.
Discrete Mathematics. Vol 301 pg 2074-2081. Namblar, K.K., Prarnod
K. Varma and Vandana Saroch. A Graph Theoretic Proof of Arrows
Dictator Theorem. May 1992. Powers, R.C. Arrows Theorem for Closed
Weak Hierarchies. Discrete Applied Mathematics. Vol 66 pg
271-278.
