Randomization and serial dependence in professional tennis matches: Do strategic considerations, player rankings and match characteristics matter?

Transcription

1 Judgment and Decson Makng, Vol. 13, No. 5, September 2018, pp Randomzaton and seral dependence n professonal tenns matches: Do strategc consderatons, player rankngs and match characterstcs matter? Leondas Splopoulos Abstract In many sports contests, the equlbrum requres players to randomze across repeated rounds,.e., exhbt no temporal predctablty. Such sports data present a wndow nto the neffcency of random sequence generaton n a natural compettve envronment, where the decson makers tenns players are both hghly experenced and ncentvzed compared to laboratory studes. I resolve a long-standng debate about whether professonal players tenns serve drectons are serally ndependent Hsu, Huang & Tang, 2007 or not Walker & Wooders, 2001 usng a new dataset that s two orders of magntude larger than those studes. I examne both between- and wthn-player determnants of the degree of seral ndependence. Evdence of the exstence of sgnfcant seral dependence across serves s presented, even among players ranked Number 1 n the world. Furthermore, sgnfcant heterogenety was found wth respect to the strength of seral dependence and also ts sgn. A novel fndng s that Number 1 and Number 2 ranked players tend to under-alternate on average, whereas n lne wth prevous fndngs, the lower-ranked the players, the greater ther tendency to over-alternate. Wthn-player analyses show that hgh-ranked players do not condton ther randomzaton behavor on ther opponent s rankng. However, the under-alternaton of top players would be consstent wth a best-response to belefs that the populaton of opponents over-alternates on average. Fnally, the degree of observed seral dependence s not systematcally related to other match varables proxyng for match dffculty, fatgue, and psychologcal pressure. Keywords: randomzaton, mxed strategy Nash equlbrum, mnmax, tenns, sports data analytcs 1 Introducton The producton and percepton of randomness has a long research hstory n cogntve psychology see Nckerson, 2002, for an overvew and rghtly so. The percepton or judgment of randomness s a core human competency see Oskarsson et al., 2009, for a revew. There s ample evdence that humans are capable of learnng patterns both mplctly and explctly n sequences of events Clegg et al., 1998; Remllard & Clark, Our ablty to dscover the correlatons e.g., Kareev, 1995; Kareev et al., 1997; Kareev, 2000 arsng from the causal relatonshps n our envronment allow us to adapt to and explot the envronmental structure. Wth respect to the producton or generaton of random behavor, subjects n laboratory tasks wthout strategc nteractons are typcally neffcent at creatng serally uncorrelated sequences. Subjects tend to produce over-alternatng sequences wth The author would lke to thank Andreas Ortmann and John Wooders for constructve feedback and gratefully acknowledges fnancal support for ths project from the Alexander von Humboldt Foundaton Humboldt Research Fellowshp for Experenced Researchers. Copyrght: The authors lcense ths artcle under the terms of the Creatve Commons Attrbuton 3.0 Lcense. Max Planck Insttute for Human Development, Center for Adaptve Ratonalty, Lentzeallee 94 Berln, 14195, Germany. Emal: splopoulos@mpb-berln.mpg.de too many runs and regress towards the representatve frequences of the dstrbuton they are emulatng Kahneman & Tversky, 1972; Bar-Hllel & Wagenaar, 1991; Rapoport & Budescu, Explanatons of these devatons n random generaton range from cogntve bounds such as short-term memory Kareev, 1992, 1995, 2000 and the complexty or dffculty of encodng of sequences Falk & Konold, 1997 to the statstcal propertes of small samples of random behavor Kareev et al., 1997; Sun & Wang, 2010, 2011, or the nteracton of both Hahn & Warren, 2009; Farmer et al., 2017; Warren et al., Although the judgment of randomness s typcally applcable to nteractons wth nature or ndvdual decson makng, the producton or generaton of random behavor s naturally most relevant to strategc nteractons wth other decson makers n our envronment,.e., n strategc games. In contrast to the above studes that nvestgate random sequence generaton n ndvdual decson-makng tasks, Rapoport & Budescu 1992 and Budescu & Rapoport 1994 used laboratory games where t s optmal to be unpredctable. Whle they found smlar qualtatve devatons n randomzaton behavor for both ndvdual and strategc decson-makng, Budescu & Rapoport 1994 show that people are more effcent randomzers n the latter. Random behavor s called for n strategc nteractons of conflct or competton, where one player s gan s another s loss and beng unpredctable 413

2 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 414 s benefcal. Such games have an equlbrum n mxedstrateges, where a player chooses to randomze over the actons at hs/her dsposal, rather than play one of them wth certanty. Stuatons where mxed strateges are relevant nclude bluffng n poker, penalty shootouts n soccer, and serve drectons n tenns, whch s the envronment that I wll study here. In repeated games, the normatve predcton s that the acton chosen n a round should be ndependent of chosen actons n the prevous round,.e., players should be randomzng perfectly. Otherwse, a player could learn the dependences or patterns n the opponent s behavor and explot them approprately Splopoulos, 2012, 2013a,b, 2018; Ioannou & Romero, Feld data from compettve sports are partcularly useful, combnng the benefts of a real-world doman where randomzaton s mportant guaranteeng hgh ecologcal valdty wth a hgh-level of ncentvzaton and the opportunty for sgnfcant learnng beyond what s feasble n the laboratory. The exstng lterature usng feld data has been prmarly conducted by game theorsts and economsts rather than cogntve psychologsts, despte ts obvous relatonshp to poneerng work by psychologsts on randomzaton. In ths paper, I analyse a large dataset of tenns serves wth the goal of resolvng an open debate on whether professonal players devate from effcent randomzaton n ther serve drecton Walker & Wooders, 2001 or not Hsu et al., Furthermore, I extend the exstng lterature by explotng the large number of wthn-player observatons to examne whether the degree of randomzaton depends on a player s own rank and the rank of the opponent, experence, the round of the match e.g., fnal, sem- or quarter-fnal, and the dffculty and length of the match. These analyses are related to exstng laboratory studes nvestgatng the mpact of learnng, feedback and other varables on the effcency of randomzaton. Specfcally, Lopes & Oden 1987 concluded that statstcally sophstcated subjects performed better than average subjects, although they exhbted the same qualtatve msperceptons of randomness. Regardng whether feedback nduces better randomzaton, the evdence thus far s mxed. Feedback has been found to mprove the dentfcaton of non-random sequences Zhao et al., 2014 and generaton of random sequences Neurnger, 1986; however, Budescu 1987 dd not fnd a sgnfcant effect of feedback. Of course, the degree of learnng that can occur n the laboratory s lmted by practcal celngs on the amount of exposure and the ncentves to perform well. The feld data from hghly-pad and compettve tenns tournaments addresses both of these lmtatons and permts the nvestgaton of other potental medators. Before proceedng, I summarse the state of the art n the game theory lterature. Recall that the normatve soluton to repeated games wth a stage unque mxed-strategy Nash equlbrum s perfect randomzaton,.e., actons must be ndependent of the pror hstory of play. One strand of expermental studes tests the equlbrum predctons n the laboratory, fndng sgnfcant devatons from the equlbrum predctons Bloomfeld, 1994; Brown & Rosenthal, 1990; Chappor et al., 2002; Ochs, 1995; Rapoport & Budescu, 1997; O Nell, 1987; Levtt et al., 2010; Wooders, 2010; Palacos-Huerta & Volj, 2008; Okano, 2013; Shachat, Experence can reduce the magntude of these devatons, however ths s condtonal on features of the game see Ochs 1995; Roth & Erev 1995; Erev & Roth 1998; Bnmore et al. 2001; Nyarko & Schotter Another fndng s that experence from the feld does not transfer well to the laboratory for new tasks. Despte ntal clams that professonals, to a large degree, transfer ther experence to new tasks n the laboratory Palacos-Huerta & Volj, 2008, later studes have not found evdence of ths effect Levtt et al., 2010; Wooders, 2010; Van Essen & Wooders, Fnally, subjects explot both devatons from the equlbrum margnal dstrbutons Shachat & Swarthout, 2004 and devatons from serally ndependent or random play, n ways that can be explaned by learnng models capable of detectng temporal patterns Splopoulos, 2012, 2013a,b, 2018; Ioannou & Romero, Another strand of research utlzes feld data from compettve sports. The frst paper to examne the optmalty of tenns serves n the feld s Walker & Wooders 2001 see also the comment by Hsu et al I refer to these two studes as WW and HHT respectvely. Both studes concluded that mxng proportons were not statstcally dfferent from the equlbrum; however, whle the former concluded that sgnfcant devatons exsted from the theoretcal predcton of seral ndependence, the latter concluded the opposte. The predctons of mnmax play n the feld have also been tested n other sports, such as soccer and the NFL.1 To summarze, the majorty of studes confrm equlbrum behavor n terms of mxng proportons, whereas the fndngs regardng seral ndependence are mxed. Of these dfferent sports, tenns allows for the most powerful tests of mnmax behavor for ndvdual players rather than a populaton of players. In soccer, snce players rarely make penalty shots, the data afford low statstcal power to reject the null hypothess of equlbrum behavor at the ndvdual level. Also, because there exst large ntervals between 1In soccer, Chappor et al conclude that the mxng proportons of penalty kcks are n accordance wth theoretcal predctons; a re-analyss by Coloma 2007 of ther data drectly testng mxng proportons and new data by Buzzacch & Pedrn 2014 confrm ths fndng. Smlarly, Palacos-Huerta 2003 found that mxng proportons are n lne wth the equlbrum predcton, and that seral ndependence across penaltes could not be rejected. Dohmen & Sonnabend 2016 conclude the same on both counts. Kovash & Levtt 2009 fnd sgnfcant devatons from the theory n baseball ptches and NFL plays for both mxng proportons and seral ndependence on average over-alternaton s more common. Smlarly, Emara et al conclude that there exsts a sgnfcant bas towards over-alternatng n NFL plays. On the other hand, McGarrty & Lnnen 2010 fnd that play n the NFL s not sgnfcantly dfferent from the equlbrum wth respect to mxng proportons and seral ndependence.

3 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 415 a player s consecutve penaltes, ths could encourage equlbrum behavor by nducng memory-less behavor, whch may be conducve to the generaton of serally ndependence sequences. In the NFL, dfferent players are nvolved n each play and strateges are called by the coach; hence, tests of equlbrum behavor are essentally a test not of an ndvdual, but a jont test of the behavor of the coach and a group of players. Ths study s most smlar to WW and HHT, but uses a new tenns serve dataset that s two orders of magntude larger than those n exstng publshed studes. I wll specfcally address the conflctng fndngs regardng seral ndependence n WW and HHT, whch remans an mportant open ssue. Whle WW do not reject seral ndependence of serves, HHT fnd evdence of statstcally sgnfcant correlaton across serves. I extend the work n WW and HHT n three drectons. Frst, by ncludng analyses of behavor relatve to the player s own rankng,.e., examnng whether more hghly ranked players conform more closely to equlbrum predctons. A workng paper by Gaurot et al. 2016, henceforth GPW, uses another large dataset from another source to examne mnmax behavor n tenns and ts relatonshp to player rankng. Note, the latter manuscrpt also nvestgates the equlbrum predcton that wnnng rates for left and rght serves are equal. Whle there s some overlap between our manuscrpts n terms of the hypotheses tested, they are largely complementary. The followng hypotheses based on ndvdual player-level analyses rather than only populaton analyses dfferentate my work from WW, HHT and GPW. The frst hypothess regards whether players strategcally condton ther behavor on the rankng of ther opponent. Players capable of usng the equlbrum strategy but conscously choosng not to play accordngly may n fact be ratonal f they hold correct belefs that ther opponent wll not choose the normatve soluton Plott, Consder the case where low ranked players are mperfect randomzers. If hgh-ranked players are sophstcated n the sense of correctly predctng low-ranked players devatons from the equlbrum, then ratonalty dctates that they explot ths. Of course, ths would lead to non-equlbrum behavor by the hghly-ranked players, whch however, would ndcate ratonal behavor gven ther opponent s type. The second set of hypotheses regard whether players condton on, or are affected by, match characterstcs such as: a the tournament round of the match e.g., whether t s a fnal, sem-fnal etc., whch would factor n the effects of stress and the probablty of wnnng the tournament prze gven the tournament s progresson, b the dffculty of the match e.g., how close the score s, and the number of ponts played n a match a proxy for fatgue and dffculty. To the best of my knowledge ths s the frst feld study n tenns to address all of these addtonal questons. Table 1: The specfcaton of a pont game n terms of the server wnnng probabltes,π as,a r L Recever Server L π L,L π L,R R π R,L π R,R 2 Modelng tenns serves I brefly descrbe the tenns serve model ntroduced by WW and adopted by HHT see WW for more detals. Tenns serves alternate n terms of the area of the court box wthn whch each serve must land to be vald.e., not declared as a fault, referred to as deuce and ad courts. The collectons of ponts served by a player n each of these two courts are referred to as ether ad or deuce pont-games. For example, all ponts n a match where a specfc player s serve was drected to the ad box are referred to as that player s ad pont-game. Snce there are two players and two boxes, each match has four pont-games. Each pont-game n the match s modeled as a 2 2 normal form game wth acton spaces left L and rght R for both the server s and the recever r see Table 1. The payoffs of ths game are equvalent to the probabltesπ as,a r of wnnng each pont-game for the acton profle a s, a r consequently, ths game s constant-sum. The probabltes of wnnng dffer condtonal on whether serves are made to the ad or deuce court due to dfferences n servng and returnng abltes; ths s the reason why we must dstngush between pont-games. Walker et al show that tenns belongs to the class of Bnary Markov games, whch possess the property that the equlbrum play for every pont n the match can be solved ndependently of all other past ponts and outcomes n the match. That s, the equlbrum of the match corresponds to equlbrum play n each pont of a specfc pont-game. Every pont-game played has a unque mxed strategy Nash equlbrum under the assumptons that π L,L <π R,L,π R,R <π L,R,π L,L <π L,R andπ R,R <π R,L.2 3 Data The data orgnate from the crowd-sourced Match Chartng Project accessble at chartng/meta.html, whch comples tenns match statstcs.3 2These nequaltes follow from the reasonable assumpton that the server s more lkely to wn a pont f the drecton of the serve and the drecton antcpated by the recever are msmatched. 3The data at the Match Chartng Project are updated often wth new statstcs as volunteers upload nformaton from more tenns matches. The dataset used for the analyss was downloaded on 6/6/2016. Prelmnary analyses performed usng snapshots of ths dataset at varous ponts n tme n 2015.e., comprsed of subsets of the fnal dataset led to smlar conclusons. Further nformaton on the Match Chartng Project can be found at: and R

4 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 416 The dataset covers 391 male players whose career-hgh rankng ranged from Number 1 to Number 2076 mean = 123, medan = 71 n the world, and ncludes 1,093 matches from 1975 to In total, I analyze the data from 143,743 serves resultng from 4,372 pont-games. Ths s two orders of magntude larger than pror publshed studes of tenns serves: 3,026 serves from ten matches n WW and 2,490 serves from ten mens matches n HHT.5 The mean medan number of matches per player s 5.6 2; for top players who are more lkely to partcpate n these tournaments the dataset holds sgnfcantly more matches, e.g., the maxmum s for Federer 160 matches, followed by Nadal 142, Djokovc 126 and Murray 68. For these players there are 12413, 8708, 8984, and 4935 serve observatons respectvely. Ths amount of data permts much more powerful tests of the mxng behavor of athletes than prevous studes. Furthermore, hypothess testng targeted at the top players provdes the best chance of observng equlbrum behavor, as these players are the most capable and the most hghly ncentvzed to pursue optmal behavor. In the dataset, tenns serve drectons were encoded as ether 4, 5, or 6, correspondng to left, center and rght respectvely. I found 21,159 cases, where other symbols were used n the encodng. Ths may be ether due to dataentry error, or because the data-coders were uncertan how to categorze the serve drecton. These cases were not ncluded n the analyss, as s the case for serves n the drecton of the center the latter s standard practce n the lterature,.e., pror studes analyzed only the left and rght serve drectons. The complete dataset was compled by mergng pont-bypont data fles wth player-rankng data fles rangng from 22/12/1980 to 1/2/2016. I use both the career-hgh and current rankng at the tme of the match n the analyses; n some cases the former may be more approprate as players current rankngs may msleadngly fluctuate wldly due to njures.6 The followng notaton s used throughout. Let ndex the players, pg ndex the pont-games from all players matches, and pg ndex the pont games for a player. JeffSackmann/tenns_MatchChartngProject. The data are avalable under a Creatve Commons Attrbuton-NonCommercal-ShareAlke 4.0 Internatonal lcense CC BY-NC-SA by-nc-sa/4.0/. 4The data from one match were excluded as the column ttles clearly dd not match the data entres Match-ID= , Aptos Tournament, Sem-fnal between Klahn and Donskoy on 10/08/13. 5Magnus & Klaassen 1999; Klaassen & Magnus 2009 used a large dataset of 59,466 observatons from Wmbledon, but do not report nformaton about serve drectons. 6Seven mssng values for player career-hgh rankngs were replaced wth the rankngs as recorded at the ATP webste com/en/rankngs/sngles. 4 Results The seral ndependence of tenns serve drectons s tested at two dfferent levels of aggregaton: the ds-aggregated pont-game level and the player level aggregatng over the pont-games of each player. WW and HHT tested seral dependence usng the dstrbuton of pont-game statstcs snce they dd not have not enough observatons per player. Testng at the player-level s more desrable because t matches the expected structure of the data, partcularly the heterogenety that may exst between players based on ther ablty, experence etc.. Below, I summarze the statstcal procedures detals can be found n Appendx A. Seral dependence for each pont-game n the data s examned usng the two-sded exact runs test see Eq. 2. To test whether a set of these ponts-game statstcs ether for the whole populaton of players or for a specfc player are dstrbuted accordng to the null hypothess of no seral dependence requres the randomzaton of the test statstcs. These randomzed statstcs are generated accordng to Walker & Wooders 2001, p a set of these statstcs can then be tested usng the standard Kolmogorov-Smrnov test. The ndvdual pont-game level test s a KS-test on the dstrbuton of the randomzed exact run test statstcs at the pont-game level for all the players ths s the test n WW and HHT. For the player-level analyss t s the Kolmogorov- Smrnov test on the dstrbuton of pont-game statstcs for each player only. The latter permts the testng of seral ndependence for each player rather than the set of players. 4.1 Analyss at the pont-game level At the pont-game level, the null hypothess of serally ndependent serve drectons was rejected at the 5% level for 12% of the pont-games 9.5% for over-alternatng, 2.5% for under-alternatng. Controllng for multple comparsons usng the Bonferonn-Holm correcton, the hypothess of no seral correlaton s rejected for 172 ndvdual pontgames,.e., 3.9% of the cases 3.7% for over-alternatng, 0.2% for under-alternatng. Alternatvely, followng WW and HHT, the randomzed KS-test on the dstrbuton of the pont-games strongly rejects the null hypothess of seral ndependence K= 0.06, p= I conclude that the hypothess of seral ndependence s rejected, predomnantly due to over-alternaton of the serve drecton. Ths fndng corroborates the conclusons drawn by WW, but not HHT calculatons n Appendx C reveal that, gven the samples szes of these two studes, there would be roughly a 50% chance that two ndependent studes would arrve at the opposte conclusons. The GPW workng paper also rejects seral ndependence usng a large dataset wth suffcent power.

5 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 417 Table 2: Populaton averaged margnal and condtonal probabltes of serve drecton Margnal Condtonal Transton matrx Ad Deuce Ad Deuce L R L R L L L R R R Analyss at the player level Let the margnal probabltes of each player servng to the left and to the rght be denoted by q L and q R respectvely. Recall that these must be separately estmated for the ad and deuce pont-games; the subscrpt related to the pontgames s dropped for smplcty. Smlarly, the set of condtonal probabltes are denoted by { } q LL, q LR, q RL, q RR, where the frst letter of the superscrpt denotes the serve drecton at t 1 and the second to the serve drecton at t. The condtonal probabltes reveal whether players tend to over- or under-alternate. Snce these probabltes are condtoned only on the mmedately pror serve drecton for the same pont-game, they can be represented as the frstorder transton matrx of a two-state Markov-chan model: Drecton at t L R Drecton at t 1 L qll R q RL q LR q RR < q L and q RR < q R, Over-alternaton mples that q LL and under-alternaton mples the opposte sgns. The maxmum lkelhood estmates of the margnal and condtonal probabltes of serve drectons for both pont games are presented n Table 2 these are the averages of the estmates for each pont-game n the dataset. As expected, there are dfferences n the margnal and condtonal probabltes for the two pont games, arsng from dfferences n the ablty to serve and return for the ad and deuce courts. For both the ad and deuce pont-games and both serve drectons, the populaton of players exhbt over-alternaton on average, as q LL < q L and q RR < q R for both pont-games see Table 2. Asde from the condtonal probabltes, an alternatve measure of the degree of devaton from seral ndependence, whch can be used to compare across players, can be constructed based on the number of runs n a sequence. Let r dev be the % devaton for each player n the number of runs n all pont-games r pg compared to the expected number of runs Fgure 1: Hstogram of the ndvdual player percentage devaton n the number of runs, r dev Frequency r dev of a serally ndependent sequence Exp r pg : r dev = E r pg Exp r pg pg Exp r pg Fgure 1 presents a hstogram of the emprcal dstrbuton of r dev. Over-alternaton,.e., swtchng too often or negatve seral correlaton, occurs f r dev > 0 and underalternaton f r dev < 0. Negatve seral correlaton s found for 69% of the players and the mean of r dev s 5.55%, n lne wth the conclusons of over-alternaton on average drawn from the emprcal condtonal probabltes. Furthermore, the 5th and 95th percentles are large n magntude, 14.6% and 23.7% respectvely. The power of the statstcs condtonal on ths dataset s sgnfcantly hgher than pror nvestgatons, but vares accordng to the data avalable per player. Detaled smulatons verfyng the statstcal power can be found n Appendx B. Based on these calculatons, I refer to subjects wth at least ffty matches as the hgh-power group, between twenty and ffty matches as the moderate-power group, and less than twenty matches as the low-power group. For the hgh-power group, the statstcs have 80% power to detect an average effect sze, and for the moderate-power group, 80% power to detect a hgher yet stll plausble effect sze. Table 3 presents the statstcs of all players who are represented n the hgh-power and moderate-power groups. The hgh power group conssts of four players, three of whom were ranked Number 1 Federer, Nadal, Djokovc and the other Num-

6 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 418 ber 2 Murray n the world. The null hypothess of seral ndependence s strongly rejected p < for all four players. The mean percentage devaton n the number of runs, r dev, s 10.7%, 6.9%, 13.4%, and 3.4% respectvely. Also, the probablty of fndng over-alternaton n each player s pont-game runs statstcs s 0.25, 0.65, 0.2 and 0.42 respectvely. The moderate-power group of players conssts of fourteen players, all of whom are Top 10 career-hgh ranked players wth the excepton of two players. The null hypothess of seral ndependence s rejected for nne players. Notably, out of both power groups hgh and moderate, four out of the fve number 1 ranked players were found to exhbt seral dependence the excepton s Andre Agass. These results are not senstve to the groupng of players accordng to hgh- and moderate-power. Runnng the KS-tests on all players ncludng the low-power group leads to a rejecton of the null hypothess of no seral correlaton at the 5% level for 19.69% of the players. The B-H multple-comparsons correcton yelds a rejecton rate of 2.3% nne players ths correcton s overly conservatve due to the players n the low-power group. The rejectons stll nclude very hghly ranked players ncludng No. 1, e.g., Federer, Nadal, and Djokovc. A table reportng all the player-level statstcs can be found n the supplement. Returnng to the queston of economc sgnfcance of the observed devatons, note that the mean number of runs per pont-game s 16.5,.e., roughly 33 per match. Therefore, the devatons from seral ndependence of the top-ranked players n the hgh-power group Federer, Nadal, Djokovc correspond to approxmately 3.5 fewer runs, 2.3 more runs, and 4.4 fewer runs than expected respectvely. The moderatepower group also ncludes players whose statstcally sgnfcant devatons are approxmately of the same magntude see Berdych, Ferrer and Dmtrov. These ndvdual devatons may be dffcult to detect for the average player, unless two players are matched up often enough, whch s not unreasonable for the very best players. Furthermore, the average populaton tendency to over-alternate wll be more readly detectable by attentve players due to the large number of observatons. I return to ths ssue later n the manuscrpt Are wthn-subject devatons from seral ndependence a result of strategc best response to lower-ranked players or match characterstcs? Expermental and feld studes have shown that belefs about opponents ratonalty can affect the lkelhood of reachng the normatve soluton of a game. Palacos-Huerta & Volj 2009 fnd that the subgame-perfect equlbrum n the Centpede game s more lkely to result f both players are expert chess players, less lkely f chess players are matched wth students, and least lkely when students play other students. Smlarly, Bosch-Domenech et al fnd extensve terated belef-based reasonng about the sophstcaton of opponents n a guessng game many subjects who showed an understandng of the Nash equlbrum nevertheless chose to devate. For example, suppose that the recever s susceptble to the representatveness bas concernng random sequences or the law of small numbers Tversky & Kahneman, 1971,.e., beleves that a swtch n the serve drecton s more lkely after a sequence of the same serve drectons. Consequently, even f the server s randomzng effcently, the recever wll expect the sequence of the serves to overalternate. The latter could be exploted by a server choosng to under-alternate, leadng to an ncrease n the probablty of msmatch n the sender and recever drectons, thereby ncreasng the probablty of the server wnnng the pont. I examne whether such strategc devatons occur n tenns by usng a cross-sectonal regresson model wth fxed effects to absorb the between-subject varaton leavng the wthn-subject varaton to be modeled,.e., wthn-player strategc adaptaton to an opponent and/or match characterstcs. The followng ndependent varables are ncluded. The current match rankngs not career-hgh of both the server and the recever or opponent. The former captures wthn-player varaton n randomzaton, whch may occur as a result of the accumulaton of experence/expertse proxed by the player s own rankng. The latter represents the combned ablty and expertse of the opponent. If lowerranked players are more susceptble to the law of small numbers, then servers should devate more from seral ndependence n the drecton of under-alternatng, the lower-ranked ther opponent s. The current match rankngs of both the server and recever Rankt s, Rankr t are transformed nto Rt own = 8 log 2 Rankt s as suggested by Klaassen & Magnus 2009; the same transformaton s used for R opp t where the subscrpt t denotes the current not career hgh rankng. Other varables capture possbly mportant match characterstcs. The length of a match, specfcally the total number of ponts played, s ncluded as the varable N ponts. Ths varable could capture the effects of fatgue and dffculty of the match on the effcency of serve randomzaton. The varable L rally denotes the mean number of shots played per pont, or the length of a rally. Ths varable could nfluence serve randomzaton n two possble ways. Frst, the greater the rally length, the more tme that elapses between serves consequently, a player who ncorrectly condtons on pror behavor n a based attempt to randomze, may actually beneft from greater rally lengths. Second, although the rankng of the opponent would capture the expected dffculty of a match, a greater length rally mght ndcate that ths specfc match dffers n dffculty.7 Consequently, ths mght ncrease the ncentves for a player to exert more effort or greater care n randomzng effcently. The var- 7For example, the opponent rankng would not capture elements such as the effects of a recent njury, ncreased fatgue due to a busy schedule, the effects of dfferent court surfaces et cetera.

7 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 419 Table 3: Indvdual player statstcs Name Career-hgh rankng Matches Serves Runs K Runs p r dev % Hgh-power group Roger Federer , Rafael Nadal , Novak Djokovc , Andy Murray , Moderate-power group Stanslas Wawrnka , Tomas Berdych , Davd Ferrer , Mlos Raonc , Dego S. Schwartzman , Ke Nshkor , Juan Martn DelPotro , Pete Sampras , Bernard Tomc , Rchard Gasquet , Jo Wlfred Tsonga , Grgor Dmtrov , Andre Agass , Glles Smon , able W dff also captures the dffculty of a specfc match as t s the dfference between the rate at whch the player won and lost ponts n a gven match. If W dff s close to zero, then the match s relatvely even. Fnally, the varable ln Round denotes the round of the match and s a proxy for the expected tournament payoff factorng n the probablty of wnnng and the effects of stress or pressure n the later rounds. The varable Round s coded as follows: f the match s a fnal Round= 1, sem-fnal Round= 2, quarter-fnal Round= 3, pre-quarter-fnal or Top 16 Round= 4, or any lower qualfyng round Round = 5. Takng the logarthm of ths scale mposes a concave relatonshp,.e., that the effects of qualfyng for each round have an ncreasngly larger addtonal effect through the ncrease n player ncentves monetary or otherwse.8 Note, however, that the causalty of the varables Nponts, L rally, W dff may also run n the opposte drecton. That s, poor serve randomzaton could concevably have drect effects on these varables. For example, f a server exhbts seral correlaton and the recever explots ths, then the recever would be more lkely to return the serve leadng 8Results are smlar f nstead the regresson ncluded dummy varables of the round, whch however reduces the degrees of freedom. to longer ralles on average. Smlarly, ths could also affect the percentage of ponts won by the player or the number of ponts n the match. To remove the problem of endogenety, I calculate N ponts, L rally, W dff only usng data where the player was the recever, not the server. The complete model s shown below n Equaton 1, errors are normally dstrbuted.9 To allow for the possblty that players strategc adaptaton may depend on whether they are, on average, players who over- or under-alternate, the model estmates the set of coeffcents separately for these two groups the dstncton s made on the bass of the sgn of r dev. The coeffcents are denoted separately asβ + for > 0 andβ for players where r dev < 0. players where r dev r dev pg =α +β ± 1 Rown t +β ± 2 Ropp t +β ± 3 N ponts+ β ± 4 L rally+β ± 5 W dff+β ± 6 ln Round+ǫ pg 1 The results of the regresson are dsplayed n Table 4 the top half of the table presents the coeffcents for players that 9The conclusons are robust to the assumpton of normally dstrbuted errors, as bootstrapped standard errors dd not alter the results. Also, because current match rankngs were not avalable before 1980 n the database, ths analyss s based on 4,336 pont-games from 383 players; ths s a mnmal reducton compared to the total of 4,372 pont-games and 391 players.

8 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 420 Table 4: The dependence of devatons n runs rpg dev player rankngs and match characterstcs # of obs. 4336, # of players 383 Independent var.: F 12, 3941= 0.38, p=0.97 Fxed-effects: F 382, 3941 = 1.19, p = 0.01 Coeff. s.e. t p α r dev > 0 R own t opp R t Nponts β 1 + β 2 + β 3 + β 4 + β 5 + β 6 + r dev Lrally Wdff ln Round < 0 R own t opp R t Nponts β1 β2 β3 β4 β5 β6 Lrally Wdff ln Round on Table 5: Indvdual regressons of rpg dev on player and match characterstcs Player Federer Nadal Djokovc Murray # of obs F p R Coeff. s.e. β 1 R own t opp β 2 R t β 3 Nponts β 4 Lrally β 5 Wdff β 6 ln Round α denotes sgnfcance at the 5% level. over-alternate on average, the bottom half those that underalternate. A jont F-test of the null hypothess that the set of ndependent varables are not dfferent from zero cannot be rejected F 12, 3941 = 0.38, p = Smlarly, tests for each ndependent varable cannot be rejected at the 5% level. I conclude that players do not systematcally strategcally manpulate ther serve randomzaton accordng to the rank of ther opponent, and furthermore that there s also no sgnfcant effect of match characterstcs on behavor. Ths fndng s robust to addng nteractons between Rt own and the other varables n the regresson model, whch would allow senstvty to match characterstcs to depend on a player s own rank see Table 8 Appendx D for the regresson results. Importantly, there s sgnfcant heterogenety n r dev between players as captured by the estmated fxed-effects F 382, 3941= 1.19, p=0.01. Table 5 presents ndvdual regressons usng the same set of regressors as above for the players n the hgh-power group. These regressons allow for the possblty of heterogenety not only n the fxed-effects but also n the estmated coeffcents of the ndependent varables. For example, t s possble that top-ranked players may adapt strategcally to ther opponent or the characterstcs of each match, but lower ranked players may lack ths ablty. The pror re- gresson on the whole set of players may thus have masked ths heterogenety. Note, that the bulk of the observatons of Rt own n these ndvdual regressons fall wthn the Top 10 rankng range. Therefore, conclusons regardng the wthnsubject varaton n randomzaton wth rank are vald only wthn ths range t s possble that learnng more effcent randomzaton may occur at much lower rankngs. By contrast, there s sgnfcant varaton n R opp t allowng for more general conclusons. From Table 5, none of the varables are statstcally sgnfcant for Federer and Murray; however, the β 1 R own t coeffcent for Nadal and β 3 Nponts and β5 Wdff coeffcents for Djokovc are sgnfcantly dfferent from zero p=0.023, and 0.036, respectvely. Of course, by ncreasng the sample sze enough t s possble to reject any hypothess for an arbtrarly small effect sze. Therefore, the economc sgnfcance, or effect sze, of the devatons s mportant f they are small, then we should be cautous n concludng that players are not servng optmally even f statstcal sgnfcance s found. Relatvely small devatons may be ether too dffcult or too costly to detect and/or explot. The economc sgnfcance, or effect sze of these coeffcents s more clearly llustrated byω 2

9 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 421 Fgure 2: Estmated weghted regresson of the relatonshp between r dev and career-hgh rank crcle sze s proportonal to the number of pont-games r dev Rank own max or convertng them to standardzed beta coeffcents.10 For Nadal,ω 2 forβ 1 R own t s equal to For Djokovc, ω 2 forβ 3 Nponts andβ5 Wdff s and 0.014, respectvely. Consequently, I conclude that, whle statstcally sgnfcant, these wthn-subject fndngs explan very lttle varaton, partcularly compared to the between-subject uncondtonal devatons from seral ndependence for these players found above. In conjuncton wth the nsgnfcant fndngs n the panel regresson Table 4, I conclude that there s no sgnfcant and systematc evdence of the exstence of strategc devatons condtonal ether on the opponent s rank or the characterstcs of a match. 10For Nadal, the standardzed coeffcent forβ 1 R own t s equal to For Djokovc, the beta coeffcents for β 3 Nponts and β5 Wdff are and , respectvely Are between-subject devatons from seral ndependence dependent on a player s own careerhgh rankng? observatons nto the player averages r dev. Snce the prevous results have ruled out any systematc wthn-subject varaton n serve randomzaton, n ths secton I focus solely on the between-player varaton after averagng the rpg dev Fgure 2 shows the estmated functon for all players relatng =δ 0 +δ 1 Rmax+ǫ own pg, where Rmax= own 8 log 2 Rankmax, own where the subscrpt max ndcates the career-hgh rank.11 The coeffcentδ 0 + 8δ 1 corresponds to the mean value of r dev for No. 1 ranked players. To account for the dfferent number of observatons determnng r dev for each player, a weghted regresson s employed wth weghts proportonal to the number of pont-games avalable for each player. Ro- r dev 11Usng nstead lnear and power law funcons of the rank n ths regresson led to hgher RMSE, confrmng the suggeston to use ths logarthmc transformaton by Klaassen & Magnus 2009.

10 Judgment and Decson Makng, Vol. 13, No. 5, September 2018 Randomzaton and seral dependence n professonal tenns 422 Table 6: Regressons of r dev on the rank of players. Regresson Player ranks ncluded: All Top 100 Top 20 Top 10 Obs F1, p RMSE δ s.e p δ s.e p δ 0 + 8δ s.e p bustness tests were performed by runnng the same regresson not only on the whole set of players, but also the Top 100, Top 20 and Top 10 players separately see regressons 1 4 n Table 6. In all regressons, the estmateδ 1 was negatve and statstcally dfferent from zero,.e., players were ncreasngly less prone to under-alternatng and more prone to over-alternatng [ the lower ranked they were. Indcatvely, the E r dev Rankmax] own condtonal on player rankngs [1, 10, 20, 50, 100, 500] are [ 2.8, 1.9, 3.3, 5.2, 6.6, 9.8] respectvely for the regresson ncludng all players see the plotted regresson [ ft n Fgure 2. In all the regressons, the value of E r dev Rankmax= own 1 ] s negatve and sgnfcantly dfferent from zero.12 GPW also fnd that more hghly ranked men players behavor s closer to equlbrum, but ther estmated logt regresson on serve drectons does not mply under-alternaton on average for No. 1 ranked players. The group of players ranked No. 1 and No. 2 therefore exhbt an average tendency to under-alternate, although at the ndvdual player level analyss above, we rejected seral ndependence for some Top 2 players both because of underand over-alternatng. Although under-alternatng serves s not an equlbrum strategy, f the majorty of players are over-alternatng as recevers, then ths would be consstent wth a best-response to the populaton of recevers. Recall however, that no evdence was found of condtonng server 12Ths s robust to the excluson of all other data, as the weghted mean of r dev for No. 1 ranked players only, s equal to -4.1% wth 95% confdence ntervals 8.1% and 0.0%. Smlarly, these statstcs for Top 2 ranked players are 3.75% wth 95% confdence ntervals 7.15% and 0.35%; from Top 3 players onwards, the 95% confdence nterval does not le solely n the negatve doman. randomzaton on the opponent s rank, so players would have to be learnng devatons at the populaton level. Unfortunately, ths cannot be drectly tested because the recevers actons are not easly observable. They would depend on the exact poston of the player n the court further to the left or rght, also the grp they are usng on the tenns racket,.e., whether t s more approprate for a backhand or forehand shot, and any other preparaton to receve the serve whether mental or physcal. 5 Concluson Usng a new dataset wth suffcent power to effcently nvestgate the seral dependence n serve drectons, I resolve the strkng dfference n the conclusons drawn by Walker & Wooders 2001 and Hsu et al wth respect to the seral ndependence of tenns serves. I corroborate the concluson of the former study that there exst statstcally sgnfcant devatons from seral ndependence n serves. Importantly, seral ndependence has been rejected even for players ranked Number 1 n the world at some pont n ther careers such as Federer, Nadal, and Djokovc. Over-alternaton, or swtchng too often negatve seral correlaton, was found to be more prevalent than under-alternaton n the whole group of players ths s n lne wth the earler results of the lterature both n the laboratory and the feld. Interestngly, Top 2 players were found to under-alternate on average ths would be a best response to a belef that the majorty of tenns players tend to over-alternate n ther drecton as recevers. Furthermore, the lower the rankng of a player, the hgher the degree of expected over-alternaton. Wthn-player analyses dd not fnd evdence of strategc devatons from seral ndependence by hgher-ranked players when competng aganst lower-ranked players. Consequently, the observed seral dependence cannot be explaned away as a ratonal response to non-equlbrum behavor of ndvdual lowerranked players wth less experence and/or ablty than the top players. These devatons mght be dffcult to detect and explot at the level of each ndvdual player, or wthn a sngle match, due to the small number of dataponts avalable for nference. However, learnng the populaton-level tendency outsde of the Top 2 players to over-alternate should be feasble and s one possble strategc explanaton for the Top 2 players under-alternatng on average. Ths s backed by extensve laboratory evdence that subjects playng repeated constant-sum games are capable of learnng and explotng the seral dependences n ther opponent s behavor gven enough rounds of play Splopoulos, 2012, 2013a,b, Future work could be drected at ascertanng whether the observed magntude of devatons from randomness are easly detectable gven the samples szes observed n tenns matches and whether dong so would lead to an economcally mportant advantage for a player. The latter would

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