On the Minimum Violations Ranking of a Tournament

Abstract

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

Minimizing Game Score Violations in College Football Rankings

“…Another difference is that most previously offered models focused on developing the minimum-violations ranking for a so-called complete tournament (Ali et al 1986, Cook andKress 1990). From the perspective of sports, a complete (or round-robin) tournament is one in which all players (or teams) to be ranked have played each other at least once.…”

“…Goddard (1883Goddard ( , 1985 and Stob (1985) largely enforced similar restrictions. Although I could use Kendall's (1962) method of minimizing violations by ranking based on the number of wins or the iterated Kendall (IK) method of Ali et al (1986) that breaks ties by examining opponent wins, the accuracy would be suspect given the incomplete-tournament format in college football. None of these heuristics could ensure optimality for the current problem.…”

“…The largest complete-tournament, linear-ordering problems that authors reported solving with optimization models contain only 60 teams (or items) to be ranked (Grotschel et al 1984), roughly half the number of Division 1-A teams in 2003. The reason for the intractability of large-scale problems is the potential exponential proliferation of constraints and binary integer variables associated with the growth in problem size (Ali et al 1986, Grotschel et al 1984, Schrage 1991. Although Grotschel et al (1984) developed algorithms for solving large cases efficiently (and using today's technology, they would be capable of solving much larger problems than the 60-item cases reported in 1984), these algorithms require programming effort far beyond that required by my model, which can be solved by standard commercially available linear-programming software.…”

“…Although Grotschel et al (1984) developed algorithms for solving large cases efficiently (and using today's technology, they would be capable of solving much larger problems than the 60-item cases reported in 1984), these algorithms require programming effort far beyond that required by my model, which can be solved by standard commercially available linear-programming software. Even fairly efficient enumeration techniques, such as that Ali et al (1986) developed, can be far too time-consuming for a problem the size of a full college football season.…”

“…In the jargon of tournament theory, these reversals of game results are called violations in the ranking (Ali et al 1986). There are two primary reasons why such reversals are common among the math models (and why they are common among opinion polls).…”