Late in the season, three hockey teams are battling over the last remaining spot in the playoffs. How many remaining games will one of these teams have to win to have a high likelihood of securing the last playoff slot?

A team in the World Series is down three games to two. One loss will end the Series for them. If the team's top two pitchers are available for game six, but the best pitcher would be pitching with less rest than normal (and would probably be less effective than usual), should the team start the best pitcher in game six or hold him back for a potential seventh game?

Many instructors recognize the value of using sports examples such as these in the O.R. classroom; unfortunately, they generally have not documented their efforts as well as other quantitative disciplines. For example, statisticians have long acknowledged that sports provide a fruitful source of examples that many students understand; sports are a regular and popular topic in the statistics education literature and in education sessions at statistics conferences. Topics of these papers and presentations have included exploratory data analysis, descriptive statistics, basic probability theory, inference, correlation and linear regression, discrimination and classification, principal components, categorical data analysis, analysis of variance, logistic regression, ranking and forecasting. Several statisticians have even documented entire introductory statistics courses that they have taught using only sports examples, and Jim Albert of Bowling Green University has published a book titled "Teaching Statistics Using Baseball" [1].

Despite the lack of formal documentation, sports examples and activities unquestionably enjoy similar popularity in O.R. classrooms. I know this from informal conversations with several colleagues who effectively and creatively use sports examples to illustrate a wide variety of O.R. concepts. The INFORMS Education (INFORM-ED) forum and the INFORMS section on O.R. in Sports (SpORts) recently recognized that this lack of documentation represented a tremendous opportunity — "sports in the O.R. classroom" would provide the basis of a unique, practical and interesting jointly sponsored session. After preliminary discussions, the two subdivisions decided to cooperatively organize a session around this theme for the 2004 INFORMS conference.

The INFORM-ED/Section on O.R. in SpORts Sponsored Sessions


Within one hour, each of the nine initial invited speakers enthusiastically accepted an invitation, and almost immediately, the single planned session blossomed into three. Soon after, commitments from three more speakers were secured. The array of presentations was impressive; O.R. topics to be addressed included scheduling, mathematical programming, decision analysis, simulation, statistics and probability; cases, classroom examples, active learning exercises, projects, games and puzzles. As an added bonus, each of the five major North American team sports (baseball, hockey, basketball, soccer and American football) were represented.

The three sessions were extremely popular and enthusiastically received — each drew a standing-room only crowd (one at 8 a.m.) and turned away several who wanted to attend but arrived too late. As expected, the presentations were fascinating, and the presenters' creative uses of sports to illustrate O.R. concepts were very impressive.

The ITE Special Issue


A few weeks after INFORM-ED and SpORts began organizing these sessions, we realized that the presentations could provide the basis of very appealing and useful articles for INFORMS Transactions on Education (ITE). Erhan Erkut (ITE's editor in chief) agreed and asked me to act as guest editor for a special "Sports in the O.R. Classroom" issue of ITE. Publication of the special issue would (optimally) closely follow the INFORM-ED and SpORts-sponsored sessions in Denver, allowing individuals to attend a "Sports in the O.R. Classroom" presentation and subsequently read the associated ITE article for more details.

Although faced with a short turnaround time, seven of the "Sports in the O.R. Classroom" session presenters agreed to submit papers to ITE for this special issue. Two more papers, a note, a puzzle and a book review were eventually added to round out what would become ITE's largest issue (in terms of page count and number of papers) to date. Brief synopses of some of the articles published in this special issue follow under the loose classifications "Real Scheduling Using Spreadsheets," "Developing Modeling Skills" and "Thinking About Decision-Making."

Real Scheduling Using Spreadsheets


Undoubtedly the most common area of application of O.R. to sports problems is scheduling. The scheduling problems in sports are sufficiently rich to illustrate important concepts underlying successful use of integer programming. Furthermore, the problems themselves are important — North American sports are big business, generating several billion dollars in revenue annually.

In "Using Sports Scheduling to Teach Integer Programming," Mike Trick of Carnegie Mellon discusses how he uses sports scheduling examples to illustrate key integer programming concepts. Trick uses his examples in a Tepper School MBA elective course that follows an introductory "Optimization for Decision-Making" course; students have been exposed to linear programming taught through the standard version of Excel's Solver software. They have also been introduced to integer programming and have a basic understanding of the fundamentals of the branch-and-bound algorithm.

Trick begins with a six-team round robin in which the teams are divided into two equal-sized divisions, and each team plays one game per week. After demonstrating how to model this schedule when each team plays every other team once, he extends this example by incorporating other possible constraints — he prohibits the scheduling of certain games during certain weeks, requires certain games to be scheduled during certain weeks, imposes an order to some of the games played by a particular team, disallows intradivision games during certain weeks, and imposes restrictions so that particularly attractive games are not scheduled during the same week.

After generalizing this example to allow for imbalanced schedules, Trick adds to the complexity and reality of the problem by requiring the late portion of the season to be comprised primarily of intradivision games. Finally, he eliminates the possibility that two teams would play each other on consecutive weeks, forces all intradivision games to be played during the first and last three weeks, and requires that no interdivision matchup can be repeated until all possible interdivision matchups have occurred once. At each step, Trick adds another layer of complexity and realism while demonstrating how to build logic rules, discrete decisions and scheduling constraints into his base model. He discusses how to explain the use of cuts to strengthen these formulations, and concludes his paper with a list of paths his course can follow after working through this example. One very interesting path involves constraint programming, and Trick outlines how to formulate these problems as both integer and constraint programs with ILOG's OPL studio product in an appendix.

Trick provides details on how all of the models he discusses can be solved using Solver. This is a key to the usefulness of his example; students gain an appreciation of the relevance, usefulness and power of integer programming while working on problems they can solve with software available on almost any desktop computer.

John Birge of the University of Michigan also contributes a paper ("Scheduling a Professional Sports League in Microsoft® Excel: Showing Students the Value of Good Modeling and Solution Techniques") on sports scheduling; he discusses a case he has written on scheduling Major League Soccer (MLS). Interestingly, Birge also helps his students gain an appreciation of the relevance, usefulness and power of integer programming by systematically adding layers of complexity and realism to a simple base formulation. However, Birge works with a five-team double round robin schedule (40 total games) and introduces an interesting wrinkle — balancing the home and away games on the schedule. He initially simplifies the problem by dividing the season into two halves, each consisting of one home game for each team against each opponent. He then asks students to find the number of possible schedules of this type; this generally convinces students that full enumeration is not desirable (or even possible). He utilizes the geometry implied by his problem to generate a graph of possible road trips for one team, and uses this graph to identify decision variables and constraints. An Excel spreadsheet containing distance matrices (included with the paper as a downloadable file) is introduced, and a corresponding objective function is developed. At this point the constraints are added to the Solver formulation, and the solution that minimizes travel distance for one team is found.

Birge then instructs his students to find the optimal road trips for each of the other teams and aggregate these optimal trips (traveling salesman problems) into a league schedule. Students are surprised to find that the resulting schedule is usually far superior to the schedule actually used by the league; Birge has quickly helped students understand the power of decomposing complex problems into smaller separable problems that are more manageable. Finally, he addresses the desirability of integrating all single team problems into one overall problem that addresses the league's objectives. The notions of column generation and set covering are introduced, and the resulting league-wide problem is solved. Again, powerful integer programming concepts are demonstrated through easily understood, realistic and meaningful examples that do not exceed the students' (or the standard version of Solver's) capability. He has used this case in sophomore level courses in the College of Engineering at the University of Michigan. His students generally were interested in industrial and operations engineering as a possible major, had some familiarity with the case methodology, and had no prior exposure to optimization models. The case is included with the article and is under review with INFORMS' Case and Teaching Materials initiative.

Developing Modeling Skills


Several articles in the ITE special issue are devoted to the use of sports to help students develop and refine their modeling skills. In "Bowie Kuhn's Worst Nightmare" (James Cochran), the 1981 Major League Baseball (MLB) season provides the backdrop for a case that I use to help students simultaneously develop a broad understanding of integer programming and Simpson's paradox. In 1981 MLB players went out on strike after the completion of approximately one-third of the season. After initially contemplating the use of minor league players to replace the striking major league players, the owners decided to "postpone" all scheduled games during the strike. After approximately one-third of the MLB season had been preempted, the players and owners were able to reconcile their differences and approve a new collective bargaining agreement. Both sides also agreed that the postponed games would not be rescheduled, and that the 1981 season would be divided into two "halves" (pre-strike and post-strike). In each division (East and West) of each league (National and American), the team with the best pre-strike record would play the team with the best post-strike record for the division championship. If the same team had the best pre- and post-strike records within a division, that team would play the team with the second best post-strike record for the division championship.

As the case indicates, the owners had to be concerned about whether either of two potentially embarrassing circumstances could arise: a) Could a team finish with the best overall (combined pre- and post-strike) record in its division and not qualify for its divisional playoffs? b) Could a team finish first in its division in both "halves" of the split season but not have the best combined (pre- and post-strike) record in its division?

MLB's most important asset — its perceived integrity among its fans — had already been severely damaged by the strike. Occurrence of either of these situations would further erode the fan base's confidence in MLB.

The article details the use of this case in an undergraduate introductory O.R. course taught at Miami University and a core MBA management science course taught at Louisiana Tech. Various integer programming models that can be developed to answer these questions are discussed, as are methods for helping students develop these models independently. Although these models gradually increase in complexity as the student progresses through the case, they are all built on a few simple base models and can easily be solved using the basic version of Solver. The article also suggests several ways in which these formulations can be used to emphasize: 1) the value of alternative formulations of the same problem, 2) the importance of determining feasibility (as opposed to optimality) in some problems, and 3) the insights into a problem that can be gained by explicitly formulating the problem. Finally, various ways to use the final results to improve the students' appreciation of Simpson's paradox are provided. The case is included with the article and is under review with INFORMS' Case and Teaching Materials initiative.

In "Simulating NHL Games to Motivate Student Interest in OR/MS," Armann Ingolfsson discusses the use of Monte Carlo simulation to assess the likelihood, given the remaining games on the regular season National Hockey League (NHL) schedule, that a particular NHL team will qualify for the NHL playoffs. Of course, Ingolfsson (who teaches at the University of Alberta) utilizes a season in which the local Edmonton Oilers are on the cusp of qualifying for the NHL playoffs. Ingolfsson's simulations are all performed in Excel using the RAND() function and data tables, making his example even more meaningful and accessible to his students.

After explaining the NHL's point system (a team earns two points if it wins a game in either regulation time or overtime and one point if it loses a game in overtime), Ingolfsson uses the simulation to answer three interesting and relevant questions:

1. How many points will most likely be needed to qualify for the playoffs?
2. If the Oilers make the playoffs, which team will they most likely play?
3. How likely will the Oilers make the playoffs if they earn a given number of points?

Ingolfsson explains in detail the use of Excel to perform this simulation on the NHL Western Conference for games played on or after March 29, 2004, and provides a downloadable spreadsheet of this simulation. He also discusses a simulation of the 2004 NHL Western Conference playoffs and provides a downloadable spreadsheet of this simulation. Finally, Ingolfsson discusses ways he uses these simulations to motivate students in an introductory undergraduate O.R. course, an MBA operations management elective and an undergraduate elective course on service operations.

Thinking About Decision-Making


Eric Bickel demonstrates how he uses a half inning of a baseball game in his article with the self-explanatory title "Teaching Decision Making with Baseball Examples." Bickel begins by explaining how George Lindsey [2] estimated probabilities for winning a baseball game given the run differential and inning of play, then extends Lindsey's work to also account for the number of outs in the inning and bases that are occupied. After setting up his scenario (the Houston Astros are at bat against the Atlanta Braves in the bottom half of the ninth inning of a tied game), Bickel proceeds through the events of the inning and how these events affect the probability of winning. These probabilities form the basis of analysis of subsequent decisions faced by the manager of each team. In the example, Greg Maddox is pitching for Atlanta, there are no outs or base runners, and Houston's pitcher is due to bat. Under these conditions, the estimated probability the home team (Houston) will win is 0.607. Houston's manager, Terry Collins, elects to pinch hit (use a substitute hitter) for his weak-hitting pitcher. The Houston pinch hitter, Sean Berry, singles, and the estimated probability Houston will win improves to 0.677. Collins then replaces Berry with a much faster runner (James Mutton).

At this point, with a base runner on first base, nobody out and the Houston leadoff hitter (John Cangelosi) at bat, Collins has to make a decision. Should he instruct Cangelosi to attempt a sacrifice bunt (a strategy that will almost certainly result in an out but advance Berry to second base)? Should he allow Cangelosi to hit away and hope Cangelosi will get a hit, advancing Mutton to second base (or beyond) without giving up an out? Or should he instruct Mutton to attempt to steal second base, risking that he will be thrown out in the hope that he will be successful and advance to second base without assistance from Cangelosi?

Bickel graphically illustrates the decision faced by Collins, the various potential outcomes and resulting estimated probabilities that the Astros will win. He then uses this information to isolate assumptions that Collins must make in order to make his decision (What is the probability that Cangelosi would successfully execute a sacrifice bunt? If allowed to hit away, what is the probability that Cangelosi will get a hit? What is the probability that Mutton would successfully steal second base?).

After discussing these issues with the class, Bickel reveals Collins' actual decision (he instructed Cangelosi to attempt a sacrifice bunt, which Cangelosi successfully executed), outlines the new situation that resulted (Houston had a base runner on second base with one out, giving the Astros an estimated probability of winning of 0.698), and resumes analysis of the decisions now faced by each manager. This process is repeated until the inning has concluded (the Astros scored and won the game with two outs in the inning).

Although Bickel's example begins with perhaps the simplest scenario (a tied game, bottom of the ninth inning, no outs or base runners) that is easy for the students to understand, layers of complexity are quickly and naturally added. His students, new consultants at Strategic Decisions Group (most of whom have earned an MBA), gain a progressively deeper understanding of decision analysis by working through these increasingly complex situations.

Jack Brimberg and Bill Hurley also focus on decision-making, presenting an interesting Kahneman-Tversky-type probability example in a baseball strategy context. In their example, the manager of a team that is down three games to two in the World Series (MLB's end of season best of seven championship playoff) is trying to determine the optimal strategy for using his team's starting pitchers. Starting pitchers generally have four days of rest between games, and those who start a game with fewer days of rest are generally less effective. The manager is considering two starting pitchers — his top starting pitcher (or ace) and his second best starting pitcher — and is trying to decide which to use in the sixth game and which to reserve for the potential seventh game. To further complicate the issue, the ace would be pitching on only three days rest if he starts game 6, while the second pitcher will have enjoyed the customary four days rest if he starts game 6. Although the intuition of most students leads them to conclude the ace should start the sixth game (if you don't win game 6, there won't be a game 7), the optimal strategy (if the goal is to win the World Series) is to reserve the ace for the potential seventh game. After explaining this premise (and several similar premises), Brimberg and Hurley present several related multiple choice questions and encourage students to use their intuition by allocating a short time (10 minutes) for the students to answer. They then reveal and explain the correct answers to the students, providing insight into conditional probability. Finally, they add a level of complexity by facilitating a class discussion on why intuition consistently led students to incorrect answers. Brimberg and Hurley have successfully used this example to develop a deeper understanding and appreciation of decision analysis among undergraduates, master's students (MBA and master's in Defense Engineering and Management) at the Royal Military College of Canada, and attendees of military professional development seminars.

The Future of Sports in O.R. Classrooms


Is the sports context a panacea for applied quantitative methods courses? Of course not! Neither are cases, active learning, problem-based learning or any other pedagogical technique. Each instructor must find what is most effective in her or his classroom. However, an instructor who finds and uses contexts that are familiar and/or interesting to students will undeniably enjoy greater success (by any standard) in the classroom.

Some instructors may be concerned that the use of sports examples will be ineffective or counterproductive for students who are not knowledgeable about or interested in sports. Interestingly, each author of a paper in the ITE special issue and each presenter in the joint INFORM-ED and SpORts sponsored sessions addressed this legitimate issue in some way. They consistently reported:

• The success of sports examples depends heavily on the instructor's careful explanation of the context (e.g., what is a pinch-hitter?).
  
  
• Students who are not knowledgeable about or interested in sports frequently report that they have enjoyed the opportunity to learn something about sports.
  
  
• Sports examples can be used to bridge a "comprehension gap" to other applications — a student who develops command of a quantitative method through one context (such as sports) is relatively well-equipped to apply the technique in other contexts that become familiar later in her/his education and career.
  
  
• Students who are not knowledgeable about or interested in sports are not blinded by preconceived notions and frequently produce far superior analyses.

Of course, the examples discussed in this article represent a small portion of activity in this area — the special issue of ITE contains several additional articles (see the sidebar for a table of contents), and the joint INFORM-ED and SpORts sponsored sessions included several other presentation topics.

Inside ITE's Special Sports Issue INFORMS Transactions on Education special issue on "Sports in the O.R. Classroom": "A Baseball Decision Problem," Jack Brimberg and Bill Hurley "Bowie Kuhn's Worst Nightmare," James J. Cochran "Comparing the Impact of Star Rookies Carmelo Anthony and Lebron James: An Example on Simulating Team Performances in the NBA League," Salwa Ammar and Ronald Wright "An Intuitive Markov Chain Lesson from Baseball," Joel S. Sokol "A Metagame of Karate," Martin J. Chlond "Review of 'Teaching Statistics Using Baseball' by Jim Albert," Jerry Reiter  "Scheduling a Professional Sports League in Microsoft® Excel: Showing Students the Value of Good Modeling and Solution Techniques," John R. Birge "The Science of Sports: Combining Quantitative Analysis and Sports Applications in an Undergraduate Course," Keith A. Willoughby "Simulating NHL Games to Motivate Student Interest in OR/MS," Armann Ingolfsson "Teaching Decision-Making with Baseball Examples," J. Eric Bickel "Teaching Statistics with Sports Examples," Paul H. Kvam and Joel Sokol "Using Sports Scheduling to Teach Integer Programming," Michael A. Trick

Further Reading To learn more about... • INFORM-ED, visit http://education.forum.informs.org. • INFORMS' section on O.R. in SpORts, visit http://sports.section.informs.org. • INFORMS Transactions on Education, visit http://ite.pubs.informs.org. • INFORMS Case & Teaching Materials, visit www.informs.org/Pubs/Cases.

1. Jim Albert, 2003, "Teaching Statistics Using Baseball," Washington, D.C.: Mathematical Association of America.
2. George R. Albert, 1961, "The Progress of a Score During a Baseball Game," Journal of the American Statistical Association, Vol. 56, pp. 703-728.

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