The problem to be addressed is: given N players who each compete and receive scores in different subsets of M tournaments that are not necessarily equally difficult, find a rating that represents the quality of each player's performance over the course of a period of time. By symmetry, ratings of the tournaments' difficulty are also produced. The research will develop the theory behind several rating procedures for this problem that have been used in the context of rating professional golfers and airlines' on-time performance, thereby laying the foundation for applications of greater substantive import. Future applications may include problems such as ranking students (i.e., improving on the use of simple GPA's) and performance evaluations (possibly overcoming the perennial problem of inflation in evaluation scores. There is no one 'correct' rating procedure. The goal of this research is to describe some reasonable candidates and develop criteria that help one decide which procedure is best suited to a particular application. This is analogous to examining properties such as unbiasedness, consistency, and efficiency when choosing from among various possible estimators in statistical problems.
