Baseball has always been a delight for the mathematically

Is it ossible to loo at the inforation for each candidate and cobine it in such a way that we can redict how the voters will ran the candidates? For the Cy Young Award, the answer aears to be yes. A Matheatical Model to Predict Award Winners Rebecca L. Sars and David L. Abrahason Rhode Island College Baseball has always been a delight for the atheatically inclined. Nuerical records have been integral to following the sort, and oular easureents lie batting averages have enticed any youngsters to eerient with sile calculations. In the last three decades, a great any fans and even soe baseball eecutives have brought significant atheatical sohistication and the scientific ethod to the study of baseball. Evidence of this oveent aears in ublications ranging fro the daily sorts age to this agazine. We now that a layer s offensive contribution can be easured by things such as his slugging and on-base averages; we now that a tea s won-lost record can be redicted fro its totals of runs scored and runs allowed. There have also been any discussions on using atheatics to coare layers across eras. Organizations lie the Society for Aerican Baseball Research (SABR) have welcoed atheatical analyses; indeed, the nae SABR has given rise to the ter saberetrics to describe the atheatical study of the various facets of the gae. In this article, we wish to follow a soewhat uncoon direction. Rather than analyzing how soe easures of onfield erforance correlate with other occurrences within the gae, we want to see how they redict an off-field assessent. When a ballot is cast for an award the Rooie of the Year, the Most Valuable Player, and other such awards the voter resuably has soe criteria in ind. One voter ay ay articular attention to batting average, one ay ehasize runs batted in, one ay loo riarily at where a layer s tea finished in the standings, and so on. A voter ay or ay not ublish his criteria; indeed, he ay very well be unaware of eactly what those factors are as he fors his iressions of various layers. When the ballots are tallied, the voters will have raned the candidates for an award, fro first lace on down. Every year there are an untold nuber of sirited discussions of how voting will go, with redictions fro all corners about who will win. One cannot hel but wonder if a voting result is in fact redictable fro the data available to the voters. Is it ossible to loo at the inforation for each candidate and cobine it in such a way that we can redict how the voters will coare hi to the others? In at least one case, we believe the answer is in the affirative. In what follows, we will resent a way to redict the raning of candidates for one ajor award. The Cy Young Award In the 190s, baseball created the Cy Young Award to honor the ost outstanding itcher in ajor league baseball. (The award is naed after the legendary itcher Denton True Cy Young, fro the early era of baseball. His nicnae cae fro batters who felt his fastball blow by the lie a cyclone.) In the 1960s, the ractice changed to rovide each of the Aerican and National leagues with an honoree, and it reains that way today. The award is voted uon by ebers of the Baseball Writers Association of Aerica, two fro each city in the league. Each writer in the electorate casts a reference ballot, secifying his choices for first, second, and third lace, and the votes are tallied using a 3 1 oint count (that is, oints for each first lace vote, 3 oints for each second lace vote, and 1 oint for each third lace vote). The resulting oint totals deterine the final raning of to itchers within a league. The identities of the voters change frequently, but we will see that the voting results follow a redictable course. Voters ay use any criteria they wish. Perhas soe feel that a itcher s nuber of wins is araount, soe others loo first at earned run average, and others use still different statistics. There is roo for a great deal of honest disagreeent here. Let us loo at the case of hyothetical stars John and Kevin. John lays for a ennant winner and has a won-lost WWW.MAA.ORG/MATHHORIZONS

record of 24-8 with an earned run average of 2.9 while Kevin lays for a struggling tea and has a record of 17-11 with an earned run average (ERA) of 1.89. Ignoring other statistics for a oent, who sees to be the better itcher? Soe ight say that John is the an, as his 24 wins are a treendous accolishent. Others would vote for Kevin, citing his truly rearable 1.89 ERA. Those in the latter grou ight say that Kevin s oorer won-lost record is not his fault even if he itches well, his tea ay score very few runs on offense while those in the forer grou could say that wins are the botto line and that if John was lucy enough to get soe runs to wor with, he turned the into wins and ay not have even needed a icroscoic ERA. An arguent is at hand, even though this eale cites only three statistics. (By the way, our hyothetical en are in fact John Soltz and Kevin Brown of the 1996 National League. Soltz blitzed Brown and the rest of the league in the award voting, receiving 136 oints out of a ossible 140.) Our goal in this is not to argue how one should evaluate itchers; although we have our own strong oinions on what statistics are ost eaningful in describing a itcher s effectiveness, we don t have a vote for the award. The tas at hand is to try to redict how the actual award voting will go: can we tae the season statistics for each itcher in the league and coe u with a forula that will redict eactly how the voters will ran those itchers? Before we get to the atheatics, we need to coent on an issue regarding the roles itchers lay. Over the long history of baseball, starting itchers have been considered uch ore iortant than relief itchers. Starters itch any ore innings over the course of a season and, for a century, teas ade all their best itchers starters. Bullen dwellers were second-tier citizens on a itching staff. But in the second half of the twentieth century, relievers have coe to be considered uch ore iortant, and this ehasis continues to grow today. Fewer and fewer starting itchers colete nine-inning gaes, and ost teas have a few secialists in the bullen, including one whose job is to sla the door on the oonent when his tea has a sall lead in the last inning. This closer is often an ecellent itcher and is soething of a star to fans and tea officials alie. Most eole rate his effectiveness by how any saves he accuulates, and those nubers have ounted over the years as closers are used ore often. (There is, however, significant disagreeent about the actual relevance of the save as a eaningful statistic.) Relief itchers, articularly closers, are taen ore and ore seriously by Cy Young Award voters as the years go by, but they still do not often lace in the to sots in the voting. As a result, there is very little data regarding the aearance of relief itchers near the to of voting results, and so we will restrict our odel to starting itchers. (As luc would have it, in 2003 the National League Cy Young Award winner was bullen itcher Eric Gagne. We refer to this situation as a Cy of Relief.) A Matheatical Model For starting itchers, the ost coonly cited statistics are wins, losses, earned run average, and strieouts, so we will include the in our odel. With any award voting in baseball, etra attention sees to be aid to layers on the league s better teas there is a ercetion that if a layer s tea is a winner, he ust be esecially valuable and succeeding in ressureaced situations so we also include the winning ercentage of each itcher s tea. We want a sile odel, one whose araeters are failiar to all fans, so we ll use just those five statistics: wins (W), losses (L), earned run average (ERA), tea winning ercentage (TWP), and strieouts (K). For itcher l in year j, we wish to calculate a score S lj based on those five statistics. To ae the final results easier to understand, we will ut all those five on the sae scale. We choose to ut each on a scale of zero to ten by sile linear forulas, giving values lj1 through lj defined as follows: lj1 lj 2 lj 3 lj 4 lj These forulas are chosen with the idea in ind that a araeter value of zero should corresond to a erforance of no value to voters, while a value of ten should be historic in the odern era of baseball. If itcher l in year j wins no gaes, then he has lj1 0, while one who wins 30 gaes gets ten oints. A starter with just 0 strieouts earns a zero in the araeter lj, while one who fans 383 to tie Nolan Ryan s record gets ten oints, and so on. While negative values of soe araeters are an algebraic ossibility, it is not conceivable that a Cy Young Award contender will attain this dubious distinction. As an eale, John Soltz s 1996 season, entioned above, had his 24 wins, 8 losses, ERA of 2.9, tea winning roortion of 0.93, and 276 strieouts convert to ( lj1, lj2, lj3, lj4, lj ) (8.0000, 4.6667,.120, 6.8600, 6.7868). We will then calculate each score S lj as a weighted su of these araeters: S W 3 L 10 1 1 12. 2. ( ERA) 20( TWP 0. 2) K 0 10. 333 lj where the weights are to be found in such a way that, for lj, 6 APRIL 200

every year j the scores S lj iic the order of finish in the award voting. We ll assign l 1 to the winner of the award, l 2 to the second-lace finisher, and l 3 to the itcher finishing third in the vote, and the inde j will vary fro 1 to if we are using seasons worth of data in our wor. Historically, there are usually no ore than three candidates who get serious consideration in the voting, so we will try to atch just that ortion of the voting results: the itcher who wins the vote should have the highest score (S 1j ) in the league that year, the second-lace finisher should have the net best score (S 2j ), and the third-lace finisher should have the net best score (S 3j ). The weights are to be nonnegative, of course, and we further require that their su be one. Requiring aes the score S lj a conve cobination of the araeters lj1 through lj, aing it easy to coare the relative iortance of their values by seeing how large a weight is assigned to each. At this oint, we have a forulation of the roble at hand. We wish to find nubers 1 through such that all of the following are true: It is worth writing out the eaning of the set of conditions in (3). They say that for each j we need > >, 1 j 2 j 1 () 1 0,,, ( 2) S > S > S, j 1,, ( 3) 1j 2j 3j or ( ) > 0 and ( ) > 0 j 1,,. ( 4) 1 j 2 j It is ossible to eliinate (1) by solving for one of the weights in ters of the others, but we will not need to do so. The other conditions, naely (2) and (4), are very nearly the sort of constraints which aear in a standard linear rograing roble. If we relace (4) by a set of conditions in which the inequality is not strict, say by relacing zero by a sall ositive nuber, then (2) and (4) together will secify eactly the sort of set that aears as the feasible region in a linear rograing roble. Since we will set u our roble as a linear rograing roble, it is necessary to choose an objective function that aes sense for this roble. Since we would lie our odel to redict the winner as it haens in real life, a natural one would be to aiize the scores of first-lace finishers. That is, we would lie to aiize S 1j for all years j in the data set. In order to tae into account all the winners at the sae tie, we chose our objective function to be 2j 3j 3 j No doubt the reader could coe u with other objective functions, but we suggest this one since it sees ost reasonable fro a baseball oint of view. Thus, the roble we will attet to solve is as follows: Proble (LP-Cy). Given a > 0, find ( 1,..., ) that solves the following linear rograing roble: Maiize: F ( ) S1 j Subject to: F ( 1,, ) S1j 1j. j 1 j 1 1j 2j 1 Finding a solution to (LP-Cy) The data used were fro both the Aerican and National leagues for the seasons 1993 through 2002 ( 20). In an attet to searate scores (thereby avoiding ties for first lace), we set a 0.001, and used the linear rograing acage available within Matheatica to find solutions to the roble. Using the data described above, no feasible solution eists. Is all our wor for naught? Not at all. After a closer loo at the 20 seasons worth of data, a ossible reason aears. In 199 in the Aerican League, the second-lace finisher in the award voting was David Cone of the New Yor Yanees, with an 18-8 record, 3.7 ERA, 191 strieouts, and a tea win roortion of 0.49, while Mie Mussina of the Baltiore Orioles finished third with 19 wins, 9 losses, a 3.29 ERA, 18 strieouts, and a tea win roortion of 0.493. Although Cone s statistics loo a little better than Mussina s in soe categories, his fewer wins and worse ERA led us to wonder if the snag in the roble ight be that Cone is suosed to have a lower score than Mussina in 199. We chose to delete that constraint, and then tried again. With that lone constraint deleted, the roble has a solution. The resulting values of the weights are as follows: 1 j. ( ) a, j,, ( ) a, j 1,, j 1 2j 3j 1 j 1 0,,,. 1 0. 78084, ( Wins) 2 0. 0099937, ( Losses) 3 0. 196700, ( ERA) 4 0. 078477, ( Tea Winning Proortion) 0. 136747, ( Strieouts) WWW.MAA.ORG/MATHHORIZONS 7

These weights tell us that the voters tae the itchers win totals uch ore seriously than any other category, as the weight of 0.78084 on that category indicates. Earned run average is the net ost iortant to the voters, followed by strieouts. Tea winning ercentage lays a saller but nontrivial role, and losses see nearly irrelevant to the voters. Now, if we wish to find the to three finishers (aong starting itchers) in the Cy Young balloting in any of the seasons considered, we just have to tae each itcher s vector of values on the ten-oint scales and find its dot roduct with the solution vector found above. The itcher with the highest result wins the award, the itcher with the second-highest result finishes second, and the itcher with the third-highest result finishes third. Going bac to our John Soltz eale fro 1996, his score for that year is: (0.78084, 0.0099937, 0.196700, 0.078477, 0.136747) (8.0000, 4.6667,.120, 6.8600, 6.7868) 7.1481. The botto line is that the score for each itcher is a weighted average of his (ten-oint converted) wins, losses, ERA, tea winning roortion, and strieouts. Fro scores calculated using this new ethod, our redicted to three finishers recisely atch the actual to three finishers every year in both leagues with the ecetion of the 199 Aerican League. In this case it is iortant to note, however, that our odel correctly redicts the winner of the Cy Young award that year; it only fails to agree on who should finish in second and third lace. To hel the reader get a feel for the ethod and the resulting scores, we suarize our results in Table 1. We can interret the scores in Table 1 further. Our susicions about the 199 Aerican League voting were correct. Considering all the data ecet the Cone/Mussina constraint, the voters erfor consistently and weigh wins the ost strongly and ERA second. That season, though, Cone finishes ahead of Mussina desite trailing in both of these ey categories. The voting attern that year has an inconsistency in it. (One ight seculate that the quir has to do with the vast ublicity garnered by New Yor teas.) Since a 10 in any category indicates a historic achieveent, it would be ost unliely to have any itcher attain a score near that value. The two highest scores which occur during the seasons of our study are Randy Johnson s 7.73261 in the 2002 N.L., and Pedro Martinez s 7.440 in the 1999 A.L.; readers ay recall the terrifying doinance of those two itchers. As noted above, the odel resented here used data fro 1993 to 2002. With two ore baseball seasons now colete, one ight as how the odel s forula fared in redicting further outcoes. Alying the forula derived above to the 2003 and 2004 Aerican League seasons correctly redicts the first-lace and second-lace finishers in the voting for both Winner Second lace Third lace A.L. 1993 Jac McDowell Randy Johnson Kevin Aier 6.039 6.0349.697 A.L. 1994* David Cone Jiy Key Randy Johnson 4.98132 4.97198 4.3823 A.L. 199* Randy Johnson David Cone Mie Mussina 6.2932.2666.36696 A.L. 1996 Pat Hentgen Andy Pettitte Charles Nagy.60894.60794.1776 A.L. 1997 Roger Cleens Randy Johnson Brad Rade 6.8880 6.73468.21891 A.L. 1998 Roger Cleens Pedro Martinez David Wells 6.43687 6.07660.46093 A.L. 1999 Pedro Martinez Mie Mussina Bartolo Colon 7.440.12298.00 A.L. 2000 Pedro Martinez Ti Hudson David Wells 6.2411.31983.22084 A.L. 2001 Roger Cleens Mar Mulder Freddy Garcia.8782.87482.68281 A.L. 2002 Barry Zito Pedro Martinez Dere Lowe 6.7293 6.923 6.10798 N.L. 1993 Greg Maddu Bill Swift To Glavine 6.4090 6.21047 6.08867 N.L. 1994* Greg Maddu Ken Hill Bret Saberhagen.81304 4.7484 4.6362 N.L. 199* Greg Maddu Pete Schoure To Glavine 6.313.38247 4.9834 N.L. 1996 John Soltz Kevin Brown Andy Benes 7.1481.66237 4.9887 N.L. 1997 Pedro Martinez Greg Maddu Denny Neagle 6.269 6.21342 6.0019 N.L. 1998 To Glavine Kevin Brown Greg Maddu 6.23146 6.21741 6.14203 N.L. 1999 Randy Johnson Mie Haton Kevin Millwood 6.413 6.4143.9000 N.L. 2000 Randy Johnson To Glavine Greg Maddu 6.2628.8198.78694 N.L. 2001 Randy Johnson Curt Schilling Matt Morris 7.1283 6.7801 6.2364 N.L. 2002 Randy Johnson Curt Schilling Roy Oswalt 7.73261 7.0018.7078 Table 1. To Cy Young Award Vote-Getters and Their Associated Scores *A labor disute shortened the 1994 and 199 seasons to significantly fewer than the usual 162 gaes. This lowers overall scores soewhat in those seasons by decreasing the oortunities for wins and strieouts. seasons. (In 2004, only two starting itchers received any oints in the voting, and in 2003, the odel s redictions invert the third-lace and fourth-lace finishers in the voting.) The 2003 National League award went to closer Eric Gagne for his astonishing relief wor, so the odel does not aly to that season. The 2004 National League season resented a snag for the odel, as it redicted that Houston s 20-gae winner Roy Oswalt would edge teaate Roger Cleens in the voting. The award actually went to the renowned Cleens and his 18 wins. It sees ossible that the intense edia coverage Cleens received in coing bac fro a brief retireent ay have layed a role with the voters. Beyond Baseball Can our constrained-otiization aroach be used on siilar robles? For the baseball fan, other awards are certainly a source of questions, but the issue of coaring itchers with Continued on age 13 8 APRIL 200

In ractice, juers can control the robability of fouling by choosing an aiing line that is soe distance (on the order of a few centieters) before the official taeoff line. The closer the aiing line is to the taeoff line, the greater the chance of fouling. However, the length of the ju is easured fro the taeoff line, so choosing an aiing line that is too far fro the taeoff line decreases the net length of the ju. Other Ways of Scoring Using a best-of-three-attets scoring syste is not the only ossibility for events of this tye. One alternative would be to add (or average) the three attets. This would allow a juer to coensate for a oor ju with an ecetionally long ju, erhas by aiing closer to the official taeoff line. Such a scoring schee is used in any golf tournaents, where a golfer s score for the tournaent is the su of the scores in the four rounds. (The lowest total wins.) It is fairly coon that the golfer with the best round of the tournaent does not win. Consistency is ore iortant. Let s see what would haen if such a syste was used in the long ju. Let S X 1 + X 2 + X 3 be the su of the three jus. If the juer does not foul, then E[S] 3µ X. However, if the juer fouls once, then E[S] 2µ X, a substantial reduction. In contrast, under the current syste, a single foul reduces the eected score fro.846σ X + µ X to.64σ X + µ X,a uch less significant effect (deending on the juer s consistency). If θ is the robability of a successful ju, then conditioning on the nuber of successful jus, we have E[S] θ 3 (3µ X ) + 3 θ 2 (1 θ) (2µ X ) + 3θ(1 θ) 2 µ X 3µ X θ. So Allison s eected score reduces by 10% as θ decreases fro 1 to.9. Under the current syste, the decrease is 0.%. (Note: We could also derive this by observing that E[X i ] µ X θ for each i. Then E[S] E[X 1 ] + E[X 2 ] +E[X 3 ] 3µ X θ.) In essence, using the su of the three jus would eliinate any juer who fouls fro having any chance of winning against a juer who doesn t foul. We have, of course, assued that a foul counts as a ju of length 0. Alternatively, we could say that a foul results in soe distance (erhas 2 c) that is deducted fro the length of the ju. This would be siilar to what haens in soe coetitions, such as obstacle courses, where failure to colete soe asect of the course results in a tie enalty added to the final tie. In golf, certain offenses result in a one- or twostroe enalty, rather than having to relay the entire hole, or forfeiting the round. Now if we add the lengths of the jus with this enalty schee, a juer could ossibly coensate for a foul by eecuting two outstanding jus on the other attets. Further Reading Shaul P. Ladany and Georghios P. Shicas, Dynaic Policies in the Long Ju, in Manageent Science in Sorts,S. P. Ladany, R.E. Machol, and D. G. Morrison, eds., North- Holland Publishing, 1976. Continued fro age 8 osition layers has been a source of disagreeents aong voters and ight therefore give rise to soe unredictable voting. Studies of awards in basetball sound roising, for all layers accuulate statistics in essentially the sae categories. The sae ight be said for hocey if goalies are not under consideration. If not sorts, then what? One eale ay arise in business or even acadeia. Coanies are usually evaluated on a regular basis, and they ay be raned regarding earnings, size, seed of shiing roducts, affordability, and quality of their roducts. To Colleges are raned by coaring siilar categories. The oral is always the sae for the atheatical odeler: ore often than we ay now, there is a attern out there. We just have to ee thining creatively, and we have got a good chance of finding it. Further Reading D. L. Abrahason and H. Salzberg, A Matheatical Modeling Persective on Baseball Statistics, Matheatics in College, 1998. B. Jaes, The Bill Jaes Historical Baseball Abstract, New Yor: Villard Boos, 198. J. Thorn and P. Paler, The Hidden Gae of Baseball, Garden City, New Yor: 198. WWW.MAA.ORG/MATHHORIZONS 13