A major objective of professional sport is to find out which player or team is the best. Unfortunately the structure of some sports means that this is often a 

difficult question to answer. For example, there may be too many competitors to run a round-robin league, whilst knock-out tournaments do not compare every player with every other player. 

The problem gets worse when one has to compare players whose performance varies over time.

Fortunately mathematical modelling can help and in this article, we use the Plackett-Luce model to estimate time-varying player strengths of golfers. 

We use the model to investigate how good golf's current biggest attraction, Tiger Woods, really is

Baker, RD

McHale, IG

English

A major objective of professional sport is to find out which player or team is the best. Unfortunately the structure of some sports means that this is often a difficult question to answer. For example, there may be too many competitors to run a round-robin league, whilst knock-out tournaments do not compare every player with every other player. The problem gets worse when one has to compare players whose performance varies over time.Fortunately mathematical modelling can help and in this article, we use the Plackett-Luce model to estimate time-varying player strengths of golfers. We use the model to investigate how good golf's current biggest attraction, Tiger Woods, really is

University of Salford Institutional Repository

How good is Tiger Woods?Rose D. Baker, C. Math, FIMA and Ian G. McHaleCentre for Operational Research and Applied Statistics, University of SalfordAbstractAmajor objective of professional sportis to find out which player or team isthe best. Unfortunately the structureof some sports means that this is of-ten a difficult question to answer. Forexample, there may be too many com-petitors to run a round-robin league,whilst knock-out tournaments do notcompare every player with every otherplayer. The problem gets worse whenone has to compare players whoseperformance varies over time. For-tunately mathematical modelling canhelp and in this article, we use thePlackett-Luce model to estimate time-varying player strengths of golfers. Weuse the model to investigate how goodgolf’s current biggest attraction, TigerWoods, really is.Introduction‘Who is the greatest golfer?’ is an of-ten discussed topic at the 19th holeof any golf course. In recent times, itseems likely that Tiger Woods is. Weknow Tiger is good, in fact we can bepretty certain he is the best amongsthis contemporaries, but just how goodis he? How much better than his com-petitors is he, and given his recentdrop in form, can we expect Tiger toget back to his best? It is difficult,if not impossible to give definitive an-swers to these questions. However, asis often the case in life, mathematicalmodelling and statistical analysis canshed light on the situation. We usea ratings or ranking model, and fit itto tournament data by the method ofmaximum likelihood.Ratings models have typically beenused to compare candidates at elec-tions, or compare the popularity ofa set of choices made by consumers.Given the main objective of profes-sional sport is to find the best player orteam, ratings models are naturally ap-plied to sport. Further, data on sportare often widely accessible making itan ideal playground for a statistician.Here we use a ratings model to com-pare the ability of golfers. The modeladopted is the Plackett-Luce modelwhich we use to estimate competitor‘strengths’.The strengths estimated from thestandard Plackett-Luce model arestatic in that they do not vary withtime. When used to estimate playerstrength in the context of golf, it seemsunreasonable to assume each and ev-ery player has a constant strengththroughout their playing career. Itis more likely that a player’s strengthwill vary during his or her career. Assuch, we need a model that can al-low a player’s strength to vary withtime. Our solution is to estimatethe strength parameters at regularly-spaced time points, with a smoothinterpolation between them. Splinesare very often used in statistical mod-elling (e.g. Harrell, 2001) but it iseven simpler and arguably better touse the newer but little-known methodof barycentric interpolation. This wasmentioned in a recent article in thismagazine (Trefethen, 2011), and webelieve we are the first workers to useit in a statistical analysis. Adoptingthis methodology means that, unlikeprevious work, we model the evolu-tion of player strength deterministi-cally rather than stochastically.Other authors, when modellingplayer or team strengths in othersports, have most commonly adopted1stochastic rather than deterministicevolution of player strength. This issometimes justifiable. In team sports,for example, there is a non-negliblestochastic component of strength evo-lution: as the team buys and sellstop players, its strength forms part ofa random walk. In individual sportshowever, there is a strong systematiccomponent in strength evolution. Forexample, a player may gain experiencein the early part of his career, reacha peak and then decline in strengthuntil he retires. Indeed, as we find,this would appear to describe the ca-reer path of a typical golf professional.A Plackett-Luce modelwith time-varying strengthsThe Bradley-Terry model (Bradleyand Terry, 1952) is a ratings modelused for contests between two play-ers. Such a model is appropriate forobtaining strength estimates of ten-nis players for example, where play-ers compete in head-to-head matches,but many players are observed play-ing many matches. In golf, the situ-ation is more complicated since manyplayers compete in the same ‘match’,and we observe many of these matches.Each match is of course a golf tour-nament, and at the end of each tour-nament we have a rankings list of theplayers who competed. But when wehave many such tournaments, withdifferent sets of competitors, how canwe combine the results? The answercomes from the generalisation of theB-T model, to be used when n playersare competing in each ‘match’, namedthe Plackett-Luce model, introducedby Luce (1959), and Plackett (1975).There are more complex models (e.g.Stern, 1990) which however requirecomputation of multivariate integrals.Here the probability that the ithplayer wins isPi =πi∑nj=1 πj,where the jth player has strength πj .After a player has been ranked first,another player ranks second by win-ning out of the remaining players, andso on down to the last, who ‘wins’ withprobability one.Time-varying player strengthsIn order to allow for time-dependentplayer strengths, π, we take as modelvariables for player i the strengths yiktabulated at mi equally-spaced epochstik from the first year of tournamentplay to the last. Next, we interpo-late these tabulated strengths, usingthe barycentric interpolation formulaπi(t) =∑mik=1 wikyik/(t− tik)∑mik=1 wik/(t− tik),where wik = (−1)k, with first and lastweights being halved. This methodis described in Berrut and Trefethen(2004), and is claimed to give smallererrors than spline interpolation. Itis trivial to obtain the differential∂πi(t)/∂yik which will be needed incomputing the likelihood function.To obtain a more precise estimateof a player’s maximum strength, af-ter a preliminary likelihood maximisa-tion, a new set of interpolation epochswas chosen so that the time of maxi-mum strength fell at a tabulated time.This necessitated the use of unequally-spaced points, when the weight for-mula iswik = (−1)k{1tik − ti,k−1+1ti,k+1 − tik},with terms with out-of-range epochsomitted.2The likelihood functionHunter (2004) describes the maximisa-tion of Bradley-Terry likelihood func-tions, including the Plackett-Luce ex-tension. He writes the log-likelihoodfunction (for constant strengths) asℓ(π) =N∑j=1mj−1∑i=1{lnπa(j,i)−lnmj∑s=iπa(j,s)},where there are N tournaments, withmj players competing in the jth,and a(j, i) denotes the identity of theplayer who came ith in the jth tour-nament.The sum is up to mj − 1 onlybecause the probability for the lastplayer is unity. With ties, however,the formula must be modified toℓ(π) =N∑j=1mj∑i=1{lnπa(j,i) − lnmj∑s=p(j,i)πa(j,s)},where p(j, i) is the position (place-ment) of the ith player in the jthmatch; several players can have thesame position. This approximationfor ties is that made by Cox andOakes (1984) in the context of theproportional hazards model; the anal-ogy with ranking models was pointedout in a short paper by Su and Zhou(2006).The meaning of the modelparametersA player’s strength π(t) is a functionof model parameters, but does not di-rectly have any observable meaning.However, it is easy to find such mean-ings. Thus, the player with greatermaximum strength would have beenable to beat the other player more of-ten than not. For lifetime achievementwe looked at the total strength, i.e.the integral of strength over player’splaying lifetime. This can be givena simple meaning in terms of ob-servables. Imagine that a player ofstrength π1 regularly competes againsta player of very high ability π2. Theproportion of games won is π1/(π1 +π2) ≃ π1/π2. Over the player’s play-ing lifetime, the number of gameswon is (ρ/π2)∫π(t) dt, where ρ is therate at which these competitions oc-cur. Hence∫π(t) dt is proportionalto the total number of games that theplayer would win against a very strongplayer. This is a sensible measure oflifetime achievement.EstimationWhen maximising the likelihood func-tion for the tabulated ‘strengths’, wecannot use the slow but sure method oflikelihood maximisation recommendedby Hunter (2004).Solving for the tab-ulated ‘strengths’ yik rather than di-rectly for the π means that we can-not manipulate the likelihood maximi-sation condition so that yik is the sub-ject. This is the basis of the ‘fixedpoint’ method. A further complicat-ing factor here is that when fitting themodel to data from golf tournamentswe need to maximise a likelihood func-tion with respect to hundreds, or eventhousands, of parameters. This is noeasy feat, but the tractability of thelikelihood function means that firstand second derivatives can be foundanalytically, which helps. An ad hocmethod was developed which workedwell enough. The difficulty is thatone does not want to try to invert thehuge Hessian matrix. After some ex-perimentation, a Newton-type methodwas evolved, whereby only the diago-nal term of the Hessian was computed.This of course throws away the attrac-tive property of second-order conver-gence, but function maximisation stillproceeded rapidly until near the max-3imum.Backtracking was used if a step de-creased the likelihood function. In ad-dition, single-variable minimisationswere interspersed, because these canbe done without using the full Hessian,and the routine switched to ‘singlevariable mode’ if backtracking failed.It tried switching out of this modeagain if enough successful steps hadbeen made using the single-variable it-eration. The single variable to be iter-ated was chosen as the one promisingthe largest increase in log-likelihood,based on the first and second differen-tials. Finally, several random restartswere made.A small problem is that only theratio of the strengths π is determinablefrom tournament data. Hence thestrength of one player at one timepoint was fixed at unity, e.g. y11 = 1.The yik were rescaled after each itera-tion to preserve this property.The lack of a Hessian matrix againmakes it harder to do statistical in-ference, i.e. to find errors on modelparameter estimates, and to do signif-icance tests. Here the famous boot-strap method comes to the rescue, giv-ing us a complete methodology for pa-rameter estimation and inference.Implementing this methodologyrequires some heavy computing, evenwith the relatively simple Plackett-Luce model, and one of the authorswas obliged to upgrade her computer!Then using fortran95 from the Nu-merical Algorithms Group (NAG) anduse of their library of numerical sub-routines made rapid computation pos-sible. There was a wide variety ofcomplications to be sorted out, suchas the already mentioned existence ofties. Another obvious problem is thatif a player performs very well, theirstrength parameter may tend to in-finity, causing numerical problems andgreatly slowing down likelihood max-imisation. A small ‘prior’ term servesto constrain the parameters, removingthe problem and greatly speeding uplikelihood maximisation.ResultsWe obtained data from all four ofmen’s golf’s major tournaments (theMasters, the US Open, the BritishOpen and the USPGA) from 1996 to2011. For each tournament, we hadthe finishing position of all players.In all there were 63 tournaments and228 players. Table 1 shows the top10 players according to total ‘lifetime’strength.INSERT TABLE 1As expected Tiger Woods is thebest player of his generation. More in-teresting perhaps is the identity of theother players in the list. Jim Furyk israted as the fourth best player since1996 yet is not typically regarded sohighly in golfing circles. Furyk splitsthe big five of recent times, namely,Woods, Mickelson, Els, Goosen andSingh. The dominance of the US-based players is evident. Only LeeWestwood, and to some extent SergioGarcia are players based in Europe.The rise of European golf in the lastfew years (at the time of writing inearly 2012, European players have wonseven of the last eight major titles)may mean that this list will be domi-nated with European golfers in futureyears.Although of some interest, the re-sults shown in Table 1 do not displaythe time-varying strengths of the play-ers. Figure 1 shows these for the top4 players.INSERT FIGURE 1Ernie Els and Tiger Woods’ careersseem to be following a familiar pat-tern - an increasing strength in theearly part of their careers with a de-cline in later years. This is a verycommon shape of such strength tra-4jectories. Tiger’s peak is near 2001 -a year in which Tiger dominated theworld of golf. Jim Furyk and PhilMickelson are following a more pecu-liar career path. Jim Furyk has twoclear peaks in his ability, whilst PhilMickelson looks like he may be enter-ing another peak having peaked oncein 2003 (although recent form suggeststhis may not be long-lived). Exam-ining the trajectories of all players inour analysis shows that the invertedU-shape is very much the norm careerpath for modern golfers.Tiger Woods’ closest rival has beenPhil Mickelson. Our parameter es-timates suggest that, at their re-spective peak strengths (14.99 and10.08), a match between the twowould be won by Tiger with probabil-ity 14.99/(14.99 + 10.08) = 0.598.ConclusionsIn answer to the questions set at thestart of the paper, Tiger Woods doesindeed seem to be the best golferof his generation, but given the pat-terns observed in players’ strengths,it seems unlikely that he will onceagain reach the heights of his powersobserved around 2001. Golfers typ-ically increase in strength, and hav-ing reached a peak, face a slow de-cline in playing ability until they re-tire. Of course, part of Tiger’s recentdecline has been down to injury (thefew tournaments Tiger has played re-cently have seen him perform some-what below the standards he set inthe early part of his career), which,as Tiger gets older is more and morelikely to happen, suggesting his de-cline will continue. However, his de-cline in form was also partly due to hispersonal problems. As such, if theseare overcome, he may find himself en-tirely focussed on his golfing legacyonce more, and if anyone can have a‘twin peak career’, Tiger can.Future work in this area will be toanswer the ultimate question for anygolf fan: ‘who is the greatest golferever?’. To do so, one would needdata from a much longer time frame(golf’s major competitions began in1860 with the British Open), and evenmore heroic computing, but gettinganswers to questions such as ‘whowould win a match between TigerWoods and Jack Nicklaus’ would makethe effort worth it.5Bibliography[1] Berrut, J. P. and Trefethen, L.N. (2004). Barycentric Lagrangeinterpolation, SIAM Review, 46,501-517.[2] Bradley, R. A., and Terry, M.E. (1952). Rank analysis of in-complete block designs I: themethod of paired comparisons.Biometrika, 39, 324345.[3] Cox, D. R. and Oakes, D. (1984).Analysis of Survival Data, Chap-man and Hall, New York.[4] Harrell, Jr., F.E. (2001). Regres-sion modeling strategies. Springerseries in statistics, New York.[5] Hunter, D. R. (2004). MM Al-gorithms for generalized Bradley-Terry models, The Annals ofStatistics, 32, 384-406.[6] Luce, R. D. (1959). IndividualChoice Behavior: A TheoreticalAnalysis. Wiley, New York.[7] Plackett, R. L. (1975). The anal-ysis of permutations, AppliedStatistics, 24, 193-202.[8] Stern, H. (1990). Models for Dis-tributions on Permutations, Jour-nal of the American StatisticalAssociation, 85, 410, 558-564.[9] Su, Y. and Zhou, M. (2006). On aconnection between the Bradley-Terry model and the Cox propor-tional hazards model, Statisticsand Probability Letters, 76, 698-702.[10] Trefethen, L. N. (2011). Sixmyths of polynomial interpola-tion and quadrature, Mathemat-ics Today, August.7 0 2 4 6 8 10 12 14 16 1996  1998  2000  2002  2004  2006  2008  2010  2012StrengthYearTiger WoodsPhil Mickelson Ernie ElsJim FurykFigure 0.1: Lifetime strengths of the four players with highest lifetime strength.Figures and TablesPlayer total lifetime strength (cv)Tiger Woods 12.64 (0.17)Phil Mickelson 8.16 (0.19)Ernie Els 6.12 (0.18)Jim Furyk 4.56 (0.17)Vijay Singh 4.55 (0.16)Retief Goosen 4.11 (0.21)Lee Westwood 3.3 (0.15)Mike Weir 2.84 (0.16)Sergio Garcia 2.81 (0.15)Stewart Cink 2.66 (0.11)Table 0.1: Top 10 players by total ‘lifetime’ strength with coefficient of varia-tion.8

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