Chapter Eighteen

Epilogue

In writing this book we had to make some decisions. One decision we faced often was: should Method X appear in the book or not? Ultimately, we had to stop writing at some point, which meant omitting some interesting methods. As a partial remedy, we have written this epilogue. So while the following methods did not appear as chapters in the book, we recommend them for those readers who are eager to learn more and wish the book hadn’t ended just yet. However, there is always the hope of a second edition, so we welcome reader feedback and suggestions.

Analytic Hierarchy Process (AHP)

In the 1970s, Thomas Saaty invented his Analytic Hierarchy Process (AHP) to help decision makers make complex, multi-criteria decisions [68, 69]. For the method’s widespread use and impact, which includes governments and militaries worldwide, INFORMS (the Institute For Operations Research and Management Science) awarded Dr. Saaty and his AHP method its prestigious Impact Prize in 2008.

The heart of the AHP method is its reciprocal pair-wise comparison matrix, from which a rating vector is produced by computing the dominant eigenvector of this matrix. In this sense, AHP has a strong connection to the Keener method of Chapter 4. In another sense, AHP has a strong connection to the Massey method of Chapter 2. In a very clever analysis, David Gleich has shown that the geometric AHP method, which replaces the standard AHP’s arithmetic mean with a geometric mean, is mathematically equivalent to the Massey method [31]. The AHP method was applied to college football in [11] and Israeli soccer in [71].

The Redmond Method

In a 2003 Mathematics Magazine article [61], Charles Redmond introduced a rating method that is a natural generalization of the win-loss rating system. The Redmond method begins with the idea of a team’s average dominance that is computed by summing a team’s point differentials, both positive and negative, and dividing by the number of games that team played.

It was a tough choice to omit Redmond’s Method because it involves some interesting linear algebra. However it falls into the YAMM category (yet another matrix method). Its results are often in the same ball park as other YAMMs, but Redmond’s method is limited because it requires all teams (or competitors) to play the same number of games.

The Park-Newman Method

In [59], Juyong Park and M. E. J. Newman take a network approach to ranking U.S. college football teams. Their method considers both direct wins and indirect wins to compute both a win score and a loss score for each team. An indirect win of team i over team j occurs when a team i beats team k who beats team j. Thus, even though teams i and j did not play in a direct matchup, some information is still inferred from the indirect relationship of length 2. Length 3, 4, and higher relationships can also be considered, though each with successively discounted weight. The Park-Newman method uses some very elegant mathematics to consider relationships of all lengths. The user sets the discounting parameter that controls how much each length distance is downgraded. This method draws interesting connections to both the Markov method of Chapter 6 and the OD method of Chapter 7.

Logistic Regression/Markov Chain Method (LRMC)

The LRMC rating method developed by Sokol and Kvam [48] was designed to use point score information plus home court advantage to rank teams in college basketball. Their method has been successful at predicting games in the March Madness tournament and enabled many fans to win their office pools.

The Markov chain part of the LRMC method is similar in some respects to the Markov method of Chapter 6. The ultimate goal is the same—to calculate the stationary, or dominant, eigenvector of the Markov transition matrix. One difference is that the LRMC method uses logistic regression to cleverly estimate the elements in the Markov transition matrix, accounting for home court advantage. The authors of LRMC also show a nice connection between the LRMC and the Colley and Massey methods, which are built around the strength of schedule philosophy.

Hochbaum Methods

Dorit Hochbaum, an expert in the theory of optimization, has built several ranking methods using network optimization methods [2, 38, 39, 37]. Hochbaum has analyzed her methods with respect to their computational effort, complexity, and susceptibility to manipulation. These methods are adaptable given that the objective functions can be tailored as needed. When certain properties are satisfied, some of these optimization methods for ranking can compete, in terms of computation time, with linear-algebra based methods for ranking.

Monte Carlo Simulations

Simulation is a popular technique favored by many technicians, particularly those interested in analyzing baseball. Commercial sports forecasting companies such as Accuscore.com often use simulations as their primary tool. By using statistics compiled from past performances, a game between two teams can be simulated in the computer by running a Markov chain whose states are various aspects of the game (e.g., a hit against a given pitcher, a fly ball, a runner on first base being thrown out at second base given a hit to left field, etc.), and whose transition probabilities are constructed from past statistics. Simulating thousands of games between two teams and averaging the results is one way to produce ratings and make predictions. Simulation works pretty well when applied to baseball, but it is more or less on par with many of the less involved techniques covered in this book when applied to other sports, especially NFL football. Simulation is an interesting and somewhat deep subject that can fill a book by itself. The interested and more advanced reader will find many rich and varied discussions simply by doing a simple Google search.

Hard Core Statistical Analysis

We decided to forgo purely statistical methodology, which is probably a disappointment to hard core statisticians. Statistical analysis is a viable approach, particularly when ample statistics are available, and a tremendous array of statistical techniques can be brought to bear. But like simulation, statistical analysis is an area unto itself that can fill volumes, so we decided not to open Pandora’s box in this regard. It would nevertheless make for interesting comparisons between some of the algebraic methods contained in this book and those predicated on fitting distributions to observed data for the purpose of formulating ratings and rankings. Massey hints on his Web site that he now relies more on statistical techniques than on the algebraic methods described in Chapter 2.

And So Many Others

It would require many books to completely survey all of the rating and ranking models that have been proposed. The number of models for football alone is staggering. Listed below is a sample of the vast number of sources compiled by David Wilson. Many of these are available from the following Web site that was active at the time this was written.
         homepages.cae.wisc.edu/^~dwilson/rsfc/rate/
biblio.html.

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