# 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.

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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."

 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.

 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 є {1, …,N} such that for every (R 1, …, R N ) є 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

 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 can’t 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

 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

 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

 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.

 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.

 (Previously mentioned under Fairness criteria)  Axiom of Independence  S ij = f ij (v ij 1,……..v ij n ) for i≠j 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)

 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

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

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.

 Can be extended to symmetric tournaments  Has been proven in several other (more complicated) ways using graph theory

 Beigman, Eyal. “Extension of Arrow’s 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 Arrow’s Dictator Theorem.” May 1992.  Powers, R.C. “Arrow’s Theorem for Closed Weak Hierarchies.” Discrete Applied Mathematics. Vol 66 pg 271-278.

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