Copeland Method Ii: Manipulation, Monotonicity, and Paradoxes

An important issue for economics and the decision sciences is to understand why allocation and decision procedures are plagued by manipulative and paradoxical behavior once there are n ≥ 3 alternatives. Valuable insight is obtained by exploiting the relative simplicity of the widely used Copeland method (CM). By use of a geometric approach, we characterize all CM manipulation, monotonicity, consistency, and involvement properties while identifying which profiles are susceptible to these difficulties. For instance, we show for n = 3 candidates that the CM reduces the negative aspects of the Gibbard-Satterthwaite theorem. This paper continues our investigation into the properties and flaws of the widely used Copeland’s method (CM). Interest in CM derives from its relationship to the Condorcet winner [Cn]; this is a candidate who always is preferred by a majority of the voters when compared with any other alternative. The CM winner [C] is a natural choice when a Condorcet winner does not exist. Namely, award the winning and losing alternative from each pairwise contest, respectively, one and zero points; with a tie, each receives half a point. A candidate’s CM score is the sum of assigned points and the scores determine the CM ranking of the candidates. (To simplify proofs, we use the equivalent weights (1, 0, −1).) In spite of its wide use (e.g., the CM is commonly used to rank sports teams), surprisingly little is known about it. Our first paper [SM] examines single profile CM properties by contrasting the CM rankings with those of positional procedures, by describing how the CM rankings vary as candidates enter or drop out of the election, and by comparing the “natural” and CM rankings associated with certain profiles. (Recall; a profile is a listing of the voters’ preferences.) This paper addresses multiple profile issues. As examples, if profile p1 represents the current, sincere preferences of the voters, another profile p2 might be where some voters now vote for the p1-CM winner, or they vote strategically to try to alter the outcome, or they forget to vote. The goal is to understand how the p1, p2 CM outcomes are related. Thus, multiple profile issues include not only strategic voting, but also those traditional normative themes which