Download American Mathematical Monthly, volume 117, number 1, january by Daniel J. Velleman PDF

By Daniel J. Velleman

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Continuing to the right, one must eventually reach a left endpoint. Thus, there must be a sequence of endpoints of the form p x q y r z somewhere in the society. (Note that x = y, since each voter has only one right endpoint. ) Note that a S ( p) = a S (q) + 1, because supp S ( p) = supp S (q) ∪ {x} (and x ∈ supp S (q)); continuing this reasoning, we have a S ( p) = a S (q) + 1 = a S (r ) + 2. Let S be the society that results from swapping y and z , so that p x q y r z becomes p x q z r y . ) Since we are expanding A y and A z without altering any other agreement sets, this preserves (k, m)-agreeability.

D. in Georgia Tech’s interdisciplinary Algorithms, Combinatorics, and Optimization program in 2005. He spent a year as a post doc at the University of Waterloo, and he is currently a Humboldt Research Fellow at the University of Hamburg. Mathematisches Seminar der Universit¨at Hamburg, Bundesstr. de January 2010] VOTING IN AGREEABLE SOCIETIES 39 Agreement in Circular Societies Christopher S. Hardin 1. INTRODUCTION. In approval voting systems, each voter indicates approval or disapproval for each candidate or option (we use the term platform), and the platform with the most approval votes wins.

2. 1. For societies S and T , we say S validly transforms to T , and write S T , if |S| = |T |, a(S) ≥ a(T ), and (S is (k, m)-agreeable)⇒(T is (k, m)-agreeable) for all k, m. Observe that the relation on societies is reflexive and transitive (a preorder). An operation on societies is a valid transformation if applying the operation to a society S always yields a society S with S S. The transformation of an arbitrary society into a uniform society will occur in two phases. First, in Section 4, we transform to a society in which no voter’s approval set contains another’s.

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