On the Minimum Violations Ranking of a Tournament

This paper examines the problem of rank ordering a set of players or objects on the basis of a set of pairwise comparisons arising from a tournament. The criterion for deriving this ranking is to have as few cases as possible where player i is ranked above j while i was actually defeated by j in the tournament. Such a situation is referred to as a violation. The objective, therefore, is to determine the Minimum Violations Ranking (MVR). While there are situations where this ranking would be allowed to contain ties among subsets of objects, we will concern ourselves herein with linear ordering (no ties). A series of examples are given where this requirement would seem to be appropriate. In order to put the MVR problem into proper perspective we introduce the concept of a distance on the set of tournaments. A set of natural axioms is presented which any such distance measure should obey, and it is proven that in the presence of these axioms a unique such measure exists. It is then shown that the MVR problem is equivalent to the minimum distance problem, which can be represented in several forms---in particular as a problem of determining the minimum feedback edge set in a graph and as a mixed integer generalized network problem. This opens up a wide scope of possible solution procedures for the MVR problem. An optimal algorithm is presented along with computational results. In addition, various heuristics are discussed including an improved heuristic referred to as the Iterated Kendall method.group decisions, voting/committees