The Copeland Method I; Relationships and the Dictionary

A central political and decision science issue is to understand how election outcomes can change with the choice of a procedure or the slate of candidates. These questions are answered for the important Copeland method (CM) where, with a geometric approach, we characterize all relationships among the rankings of positional voting methods and the CM. Then, we characterize all ways CM rankings can vary as candidates enter or leave the election. In this manner new CM strengths and flaws are detected. The Condorcet (or majority) winner [Cn] is the candidate who beats all others (by winning most votes) in pairwise contests. A glaring fault of this widely accepted concept is that it need not exist. Instead, for n ≥ 5 candidates, c1 could win all but one pairwise vote while all other candidates lose at least two. Although no one satisfies Condorcet’s criterion, c1 comes the closest, so it is arguable that she is who the voters want. She does win with Copeland’s method (CM) – an important, natural extension of the Condorcet winner [C]. More precisely, in a pairwise competition between cj and ck let (1.1) sj,k =  1 if cj beats ck 2 if cj and ck are tied 0 if ck beats cj The Copeland score for each cj , defined as (1.2) C(j) = ∑ k 6=j sj,k, is used to rank the candidates where more is better. Equivalent to these (1, 2 , 0) weights are the (3 , 1 6 , 0) and (1, 0, −1) choices that we use to simplify proofs. Notice that the CM is the method commonly used to rank hockey and other sport teams. Trivially, the CM ranking is transitive. (The CM score identifies each candidate with a point on the line, so the transitivity of the election ranking is inherited from the transitivity of points on the line.) Equally as trivial, when a Condorcet winner exists, she is CM topranked. (Only the Condorcet winner receives a point from each pairwise contest.) Other