Equilibrium or Simple Rule at Wimbledon? An Empirical Study

Transcription

1 Equlbrum or Smple Rule at Wmbledon? An Emprcal Study Shh-Hsun Hsu, Chen-Yng Huang and Cheng-Tao Tang Revson: March 2004 Abstract We follow Walker and Wooders (200) emprcal analyss to collect and study a broader data set n tenns, ncludng male, female and junor matches. We fnd that there s mxed evdence n support of the mnmax hypothess. Granted, the plays n our data pass all the tests n Walker and Wooders (200). However, we argue that not only the test on equal wnnng probabltes may lack power, but also the current serve choces may depend on past serve choces or the wnnng record of past serve choces. We therefore examne the role that smple rules may play n determnng the plays. For a sgnfcant number of top tenns players, some smple rules outperform the mnmax hypothess. By comparng junor players wth adult players, we fnd that the former tend to adopt smpler rules. The result of comparson between female and male players s nconclusve. Department of Economcs, Natonal Tawan Unversty

2 . Introducton The theory of mxed strategy equlbrum does not fare partcularly well n varous expermental settngs nvolvng human subjects n the past few decades. For nstance, though the experment by O Nell (987) s among the most celebrated desgns that support the mnmax hypothess, hs result was later challenged by Brown and Rosenthal (990) because of strong seral correlaton n players choces. Not untl recently does the theory of mnmax hypothess regan ts foothold. By analyzng feld data n professonal tenns matches n Grand Slam, Walker and Wooders (200) propose a study that lends emprcal support to the mnmax hypothess. They argue convncngly that, unlke subjects n labs, professonal players have adequate experence to play games well and are hghly motvated to wn the games. Therefore, f equlbrum theory can ever predct behavors, choces of the top players n tenns matches are the rght place to run emprcal tests on. Ther result ndcates that the wn rates across strateges are not dfferent and ths s consstent wth the equlbrum predcton. However, they farly note that even top players tend to swtch from one strategy to another too often, resultng n seral dependence. Palacos-Huerta (2003) goes a step further by examnng a data set from penalty kcks n professonal soccer games where both the kcker and the goale s choces are observable. He fnds stronger evdence n favor of the equlbrum predcton than Walker and Wooders (200). Not only the wn rates across strateges are not dfferent but also the seral ndependence of choces cannot be rejected. Chappor, Levtt and Groseclose (2002) further deal wth the heterogenety of players and they cannot reject that soccer players are behavng optmally n penalty kcks. Comparng these results wth those obtaned n labs, they all demonstrate that players experence and famlarty wth the For example, Erev and Roth (998) dscuss twelve such experments and ther equlbrum predctons.

3 strategc stuaton may play an mportant role n determnng whether the equlbrum theory fares well. We follow ths lne of dea and examne the mnmax hypothess by collectng and studyng a broader data set n tenns, ncludng male, female and junor matches. Snce a typcal tenns match lasts long (compared to penalty kcks n soccer), one may wonder whether players ndeed play ther equlbrum strateges or somehow learn throughout the game. We fnd that the support for the mnmax hypothess s mxed. Accordng to Walker and Wooders (200), f players are playng ther equlbrum strateges, the wnnng probablty for each of ther pure strateges must be equal. Moreover, snce players should maxmze the chance of wnnng at each pont, t s as f ther strateges are generated as a bnomal process whch s ndependently and dentcally dstrbuted (..d. henceforth). Our analyss shows that, the null hypothess of equal wnnng probablty across strateges cannot be rejected. Nether can the null hypothess of the serally ndependent choces be rejected. However, we fnd that the test on equal wnnng probablty may lack power. Moreover, when we test the..d. hypothess n a slghtly more general fashon, players current serve choces may depend on past serve choces or the performance of past serve choces. These fndngs together suggest that there mght be some lnks between current and past choces. Recently many studes have documented that learnng models can better descrbe and predct expermental results than statc Nash equlbrum, 2 we therefore propose several low-nformaton smple rules 3 and formulate a regresson framework. 4 Comparng the predctons of dfferent rules wth those of 2 See, for example, Erev and Roth (998), Camerer and Ho (999), and Feltovch (2000). 3 See Mookherjee and Sopher (994), Mtropoulos (200), and Ho, Camerer and Wang (2002). Ther studes examne some possble learnng rules under low nformaton. For nstance, Ho, Camerer and Wang (2002) study learnng n games where only the set of foregone payoffs from unchosen strateges are known. 4 A related and nterestng example can be found n the wrtng of the celebrated lngust Wttgensten. Wttgensten sometmes took tenns sport as an example when descrbng the learnng process of 2

4 the equlbrum theory on the bass of Akake nformaton and Schwarz crtera, we fnd that rules explan the data better than the equlbrum theory for a sgnfcant number of top tenns players. Moreover, for dfferent players, dfferent rules may best descrbe ther behavors. We further fnd that nterestngly, junor players tend to adopt smpler rules than adult players do. Meanwhle, there s no conclusve evdence that male and female players adopt rules n a dfferent way. The structure of the paper s as follows. In secton 2, we follow Walker and Wooders (200) to model each pont n a tenns match as a smple 2 2 normal form constant-sum game and brefly descrbe relevant aspects of Walker and Wooders (200) analyss. Secton 3 descrbes our data set. In secton 4, we frst perform the tests proposed by Walker and Wooders (200) on our data set. We then dscuss the power of the test on equal wnnng probablty and re-evaluate the result regardng the..d. hypothess. We test several rules and demonstrate the better performance of them n secton 5. Secton 6 contans some dscusson and secton 7 concludes. 2. Testng the Mnmax Hypothess We follow Walker and Wooders (200) to model each pont n a tenns match between the server and the recever as a smple 2 2 constant-sum normal form game. When the server serves, he can choose to serve to the left of the recever (L) or to the rght of the recever (R). Smultaneously, when the server serves, the recever s assumed to guess whether the serve wll reach to hs left (L) or rght (R). Each player s payoffs are the correspondng probabltes that he wll ultmately wn the pont, condtonal on both players have made ther left-or-rght choces for that pont. Let π sr denote the server s probablty of wnnng the pont, where languages and tred to make an analogy between tenns game and language game. (Wttgensten, Phlosophcal Investgatons, ss. 68, 7.) 3

5 s { L,R} denotes the server s choce and { L,R} r denotes the recever s choce. Snce ether the server or the recever has to wn the pont, the recever s probablty of wnnng the pont s. If the Mxed Strategy Condton 5 holds, π sr the pont wll have a unque Nash equlbrum n whch both players use strctly mxed strateges. Followng Walker and Wooders (200), snce the server and the servng court could matter, there wll be four pont games n every tenns match, dependng on whch player s servng for that pont and whether the pont s a deuce-court or an ad-court pont. Fgure summarzes the bascs of a pont game. Accordng to Nash equlbrum, players should play ther mnmax strateges wthn a pont game. Ths mples the followng two testable predctons: () The wnnng probablty for each of the server s pure strategy should be the same. Ths predcton results because n a mxed strategy equlbrum, players should be ndfferent wth ther left-or-rght choces gven ther opponents are playng ther equlbrum strateges. Let q denote the recever s probablty of choosng L. From the server s perspectve, equlbrum mples that P = P s L s R where s L P q π + ( q ) π LR, s LL PR q π RL + ( q ) π RR. s Note that P L s the server s expected probablty of wnnng a pont by servng to s the left of the recever whle P R s that by servng to the rght of the recever. (2) The server s left-or-rght choces n a gven pont game must be serally ndependent because he should play the same Nash equlbrum strategy for every 5 It s reasonable to assume that the server s probablty of wnnng a pont s lower when the recever chooses the same strategy. That s to say, π LL < π RL and π RR < π LR, π LL < π LR and π RR < π RL. The above nequaltes are called the Mxed Strategy Condton. Ths reflects the dea that f the recever guesses correctly about the server s rght-or-left choce, he wll be better prepared and thus more lkely to wn that pont. 4

6 pont n a pont game. 6 Ths mples that the serve choces wll be random draws from a bnomal process whch s..d. across all serves n a gven pont game. By () and (2), we can therefore formulate and test these two fundamental hypotheses mpled by von Neumann s mnmax Theorem. 3. Data Our data set comprses three major groups (male, female and junor) and s colleted from vdeotapes or drectly recorded on the spot. We have 0 matches n male tenns, 9 n female and 8 n junor. Snce each match has 4 pont games (dependng on the server and the servng court), we therefore have 40 pont games n male tenns, 36 n female and 32 n junor. The data covers the top-level players n Grand Slam fnals (both male and female) over the past two decades. Therefore, t s far to say that they are all hghly motvated. In junor group, the matches nclude fnal, quarter fnal and second round n both Grand Slam and tournaments because t s hard to get data for junor players. 7 Accordng to tenns rules, male tenns matches n Grand Slam can last at most fve sets. Female and junor matches can last at most three sets. The frst three tables summarze the data. Each row of the tables corresponds to a pont game. We frst ndex each pont game. For each of them, we state the followng nformaton n order: the match and ts year, the server, the servng court, the number of tmes that the server chooses L, the number of tmes that the server chooses R, the total number of serves, the number of tmes that the server chooses L and wns, the number of tmes that the server chooses R and wns, the fracton of tmes that the server wns f he chooses L, the fracton of tmes that the server wns 6 See Walker and Wooders (999). 7 Few games are mssng at the begnnng of three junor matches. They are 2000 Wmbledon (Salern vs. Peredyns), 2003 Australan Open (Scherer vs. Cvetkovc) and 2003 Australan Open (Tsonga vs. Feeney). However, ths does not affect the contnuty of our data. 5

7 f he chooses R, Pearson statstc and ts p-value (the latter two wll be explaned n the next secton). 8 The wnner of each match s ndcated n boldface. 4. Testng the Equlbrum and a Reapprasal 4. Test of Equal Wnnng Probablty We frst run the test of equal wnnng probablty. Followng Walker and Wooders (200), we conduct both Pearson s ch-square goodness-of-ft and the Kolmogorov-Smrnov (KS henceforth) test. We ndex each pont game by. For each pont game, let when he uses strategy j { L, R} p j denote the probablty that the server wll wn the pont. Let n j denote the number of tmes that the server chooses j. Let N js denote the number of tmes that the server wns when he chooses j. Let N jf denote the number of tmes that the server loses when he chooses j. For each pont game, under the null hypothess, L R p = p = p. The maxmum lkelhood estmate of game s p s N LS nl + N + n RS R. The Pearson statstc for pont Q = j { L, R} 2 ( N js n j p ) ( N jf n j ( p ) + n p n ( p ) j j 2. If we substtute ts maxmum lkelhood estmate for p, the Pearson statstc s dstrbuted asymptotcally as ch-square wth degree of freedom f the null hypothess s true. The results n Tables to 3 show that the null hypothess s not rejected for most pont games n each group. In other words, under the conventonal 5% or 0% sgnfcance levels, the equal wnnng probablty hypothess n each group can hardly be rejected, although the number of rejectons n male group (2 for 5% 8 We consder only the frst serve drecton and whether the server ultmately wns that pont. 6

8 and 6 for 0%) s slghtly hgher than that n female (0 for 5% and for 0%) or junor groups ( for 5% and 3 for 0%). We then turn to the jont test to examne whether the data n each group s consstent wth the equlbrum theory. The statstc for the Pearson jont test s the sum of the ndvdual test statstc Q,.e. Nk = k Q, where N k denotes the number of pont games n group k and { male, female, junor} dstrbuted as ch-square wth allows the parameters. Under the jont null hypothess, ths statstc s p L and N k degrees of freedom. Note that ths jont test p R to vary across dfferent pont games wthn a group. The correspondng p-values are for male players, 0.76 for female, 0.55 for junor. Therefore, under the 0% sgnfcance level, the hypothess of equal wnnng probablty can be rejected for male players. In general, the equal-wnnng-probablty hypothess fares well for female and junor players. 9 Snce Pearson jont test would be problematc n detectng how the data s generated, 0 we therefore turn to compare the observed dstrbuton wth the predcted one by the KS test. As Walker and Wooders (200) suggest, the p-values assocated wth the realzed Q values should be N k draws from the unform dstrbuton U [ 0,] under the jont null hypothess for group k. We present a vsual comparson by drawng the cumulatve dstrbuton functon (CDF henceforth) of the p-values assocated wth the realzed Q values for each group and that of a unform dstrbuton n Fgure 2. As a result, the KS statstcs are 0.778, 0.578, and 0.64 for male, female and junor players respectvely, whch are all far from the crtcal value at 5% level. Thus, we cannot reject the null 9 One may be concerned by the results for the female and junor players because there are fewer observatons for these two groups. That s, fewer observatons may cause the problem that t s unlkely to reject the mnmax hypothess when t s false. One alternatve way to resolve ths s to merge ad-court and deuce-court data nto one for a gven server. The results are as follows. The p-values of Pearson statstc are and for female and junor players respectvely. The KS statstc ntroduced later yelds smlar results that the equal wnnng probablty hypothess cannot be rejected. 0 See p. 2 of Gbbons, J. and Chakrabort, S. (992). The crtcal values of the KS statstc at 5% and 0% level are.328 and.94, respectvely. 7

9 hypothess that jontly, players n each group are behavng accordng to equlbrum. Curously, the results of junor players contradct the orgnal conjecture we had before runnng the emprcal tests that junor players mght not play mnmax as well as adult players due to lmted ratonalty. Nevertheless, the results of both the Pearson jont test and the KS test so far ndcate the valdty of the equlbrum predcton for equal wnnng probablty. These are consstent wth Walker and Wooders (200). 4.2 Test of Seral Independence We next examne the seral ndependence of players serve choces. For each pont game, let = { s,s2,...,s } s n + n be the lst of drecton of serves n the order L R observed, where s n { L,R} s the drecton of the nth serve, L number of serves to the left (rght), and L R R n ( n ) s the n + n s the total number of serves. The examnaton of seral ndependence we conduct s runs test, where a run s the maxmal strng of dentcal serve drectons. 2 as Denote the number of runs n r. Under the null hypothess of seral ndependence, the probablty of havng r runs n a sequence wth n L serves to the left and s n R serves to the rght s r or denoted as f ( r ; n, n ). Let F ( r ; n, n ) be the probablty of havng L R L fewer runs. The null hypothess of seral ndependence n pont game s rejected at 5% sgnfcance level f ether F ( F ( r -; L R r ; n L, n ) < or n, n ) < , where the frst nequalty corresponds to the case that R the probablty of havng r or fewer runs s less than 2.5% (too few runs) and the second corresponds to the case that the probablty of havng R r or more runs s less than 2.5% (too many runs). Walker and Wooders (200) fnd that there are 2 For nstance, f s ={LRLLLLR}, then there are four runs n the lst. If we use commas to separate runs, we have s ={L,R,LLLL,R}. 8

10 too many runs n players choces and ths leads to the concluson that even the best tenns players tend to swtch from one drecton to another too often. In comparson wth ther fndng, our result s qute dfferent. Only fewer pont games n our data can the null hypothess be rejected. We report the results n Tables 4 to 6. For each pont game, n addton to the same basc nformaton as reported n Tables to 3, we report the number of runs, the CDF of havng r or fewer runs, and the CDF of havng r or fewer runs. At 5% sgnfcance level, there are 2 rejectons for male players because of too many runs, for female because of too many runs, and for junor because of too few runs. At 0% sgnfcance level, there are 4 rejectons for male players because of too many runs, for female because of too many runs, and 4 for junor (2 for too many runs and 2 for too few runs). It s nterestng to note that among all the rejectons, only junor players volate the null hypothess because of too few runs (they swtch drectons too nfrequently). As for the jont test, the KS statstcs of the jont null hypothess that the serves are serally ndependent wthn a certan group are for male, 0.83 for female and for junor. The p-values of these KS statstcs are all far from the rejecton regon under the conventonal sgnfcance level 3. Fgure 3 offers a vsual comparson of the emprcal CDF and the predcted CDF under the null hypothess. 4 In bref, we cannot reject the null hypothess that jontly, choces n each group are serally ndependent. 4.3 The Power of Tests So far we have bascally appled the tests n Walker and Wooders (200) to our 3 For the crtcal value of the KS statstc, please refer to footnote. 4 Followng Walker & Wooders (200), the jont KS statstc s constructed by pckng a random draw d from the unform dstrbuton U [ F ( r ; n L, n R ), F ( r ; n L, n R )] n each pont game. Under the null hypothess of seral ndependence n pont game, the statstc d s dstrbuted as U [ 0, ]. The average value of KS statstc we report here s obtaned by performng ten thousand trals wth such random draws for each pont game. 9

11 data and generally confrmed the valdty of the mnmax hypothess n these tests. We now reexamne these test results wth a crtcal eye. We frst turn to the ssue regardng the power of the tests. We concentrate on the Pearson jont test of equal wnnng probablty as Walker and Wooders (200) dd. Wthout loss of generalty, we take the male data as an example to see whether the test has the power to detect devatons from the mnmax play by runnng Monte Carlo smulatons. Consder frst a 2 2 pont game model where π LL = 0. 53, π LR = , π RL = 0.792, π RR = (see Fgure 4).5 If players follow ther mnmax strateges, ths model predcts that servers wll serve to the recevers left sde wth probablty Ths s to match that n our male tenns data, 56.8% of all serves are ndeed to the left of the recevers. The probablty that the servers wll wn a pont (.e., the game s value) s Ths agan s to match that n our male tenns data servers ndeed wn 64.3% of all ponts. Let θ be the proporton that recevers choose to play L, then under the null hypothess one can calculate that θ = 0.68 N. The Pearson statstc male = Q s dstrbuted as ch-square wth 40 degrees of freedom. To depct the power functon, we randomly generate ten thousand tmes for 40 pont games at any fxed value of θ, evaluate the servers probablty of wnnng across L and R by the payoff matrx, calculate the ndvdual test statstc Q, and then compute the frequency where the Pearson jont test rejects the null hypothess under 5% sgnfcance level (.e., an estmate of the power of the test under θ ). Ths process s performed for many values of θ. On the bass of the above payoff matrx, the Pearson jont test has good power aganst alternatve hypothess when the true value of θ dffers from Ths s consstent wth Walker and Wooders (200). 5 The payoff matrx satsfes the Mxed Strategy Condton so that there exsts a unque mxed strategy equlbrum n ths example. 0

12 However, to sort thngs out, we consder another 2 2 pont game wth the followng parameters: π = 0. LL 62, π LR = , π RL = , π RR = (see Fgure 5). Based on ths payoff matrx, the mnmax theory mples that servers serve to the left wth probablty and wn a pont wth probablty The null hypothess for recevers choosng L s stll Nevertheless, a quck glance at the power functon shows that t does poorly n detectng devatons from the mnmax play. Snce the payoff matrx of any pont games s not observable, we are led to queston that non-mnmax behavors may also lead to acceptance of equal wnnng probablty of server s left-or-rght choces. To make the pont clearer, note that as econometrcans, we only observe the value of the game and the proporton of servers serves to the left. The payoff matrx s not drectly observable. When applyng Monte Carlo smulatons, gven the observable value of the game and the proporton of servers serves to the left, we actually have lattude n choosng two addtonal parameters (θ, π LL ) due to lmted nformaton n the payoff matrx. 6 If we focus only on the parameter θ but gnore the fact that π LL can also vary, the power functon thus depcted can be msleadng. In other words, the surface that relates equal wnnng probabltes to the mnmax play could be qute flat around the equlbrum strategy for some payoff matrx. In statstcal termnology, tests based on equal wnnng probablty may lack power when some certan plausble alternatve hypothess s consdered. 7 Smlar remarks can be made about tests on female and junor groups. 4.4 Re-evaluaton of the I.I.D Hypothess 6 Let p, V denote the probablty of servers choosng L and the game s value respectvely. Gven these observable varables p and V, we can thus derve π LR, π RL and π RR gven any par (θ, π LL ) as V π LL θ V π LL p V θ ( V π LL p) follows: π LR =, π RL =, π RR =. θ p θ ( θ ) ( p) 7 Ths lack of power wll become severe when the test of equal wnnng probabltes s performed ndvdually on each pont game.

13 Although our results based on the runs tests are n favor of seral ndependence and thus dfferent from those n Walker and Wooders (200), yet we argue that the queston of seral ndependence should be further addressed. Specfcally, the..d. hypothess mples that the current serve choce should not only be ndependent of past serve choces, but also have nothng to do wth any other hstory of past plays, ncludng how successful past choces are. In other words, passng the runs test s necessary but not suffcent to pass the..d. hypothess. For ths reason, we further examne whether past serve choces, the performances of past serve choces and other possble varables concernng the hstory of past plays mght play a role n determnng the current serve choce. To address ths, we follow Brown and Rosenthal (990) and propose a probt equaton (Equaton No.) for each pont game. The dependent varable s a dchotomous ndcator of the server s choce of drecton. Denote ths by the dummy varable D (short for drecton ). D takes the value of f the server chooses R and 0 f the server chooses L. The ndependent varables we try are the followng: the frst and second lags of the server s choce of drecton; the frst and second lagged ndcators of whether the server wns that pont (denoted by W, short for wn, whch takes the value of f the server wns the pont and 0 f he loses); the frst and second lags for the product of D and W; and a tme trend denoted by T whch measures the number of serves that has occurred tll now. 8 The results are shown n the frst panel of Table 7. We perform fve tests here. In the frst test, the null hypothess s that all the explanatory varables are jontly nsgnfcant. It s rejected for four pont games at 5% level (eleven at 0%). Other tests help dentfy the source of the rejecton of 8 We have tred to nclude addtonal lags (the thrd or fourth lags) n the probt equaton. We fnd that these addtonal lags are nsgnfcant n explanng the current serve choces. Therefore, we only report the result wth two lags. 2

14 the jont null hypothess. In the second test we consder whether servers past choces of drecton affect ther current choce. The thrd test tackles whether prevous wnnng or losng experence may have an mpact on the current choce. The fourth test measures the nfluence of the nteracton terms. Fnally the ffth test deals wth the case whether servers rase the probablty of servng to a certan drecton as tme goes by. Consderng all the fve tests reported n the frst panel of Table 7, n twenty-one pont games the mpact of at least one lagged ndcator s sgnfcantly dfferent from zero at 0% sgnfcance level, suggestng a volaton of the..d. hypothess. At 5% sgnfcance level, there are eleven rejectons. Ths amounts to 9.4% at 0% sgnfcance level and.% at 5%. To ths pont, t casts doubt n the earler result n whch passng the runs test s nterpreted as passng seral ndependence. Recall that the runs test n secton 4.2 measures whether there are too many or too few runs n the server s ordered lst of the drecton of serves. Loosely speakng, the dea behnd the runs test corresponds well to that behnd the second test n Equaton No.. Ths s because the second test assesses whether the current serve choces depend on the prevous serve choces. Conceptually, f the prevous choce has a negatve mpact on the current serve drecton, then too many runs may result and vce versa. On the other hand, f the prevous choce has a postve effect on the current serve drecton, then too few runs may result and vce versa. Ths s generally confrmed here. There are only two rejectons at 5% sgnfcance level and three at 0% n the second test. 9 Recall that the runs test also fares qute well n secton These together suggest that we may drop the past serve 9 Note how dfferent the pcture s. If we examne the..d. hypothess by lookng at the second test (whch, as argued, s very smlar to the runs test) n Equaton No., snce there are only three rejectons at 0%, we may be led to accept the null hypothess. If we nstead look at the entre fve tests n Equaton No., the rejectons rse up to twenty-one. 20 Note that n pont games (pont games 22, 34 for male and 3 for male) where the null hypothess n the second test s rejected at 0%, the null hypothess n the runs test s ether rejected (pont game 34 for male) or at the margn of rejecton (pont games 22 for male and 3 for female). 3

15 choces from the regresson. Ths can help ncrease the degree of freedom and may be especally mportant for the analyss of the junor data because they are relatvely shorter. We therefore formulate another probt equaton (equaton No.2) by droppng the frst and second lags of serve choces. The second panel of Table 7 summarzes ths result. In comparson wth equaton No., the number of rejecton rses up n each test, especally for junor group. There are twenty-seven rejectons at 0% sgnfcance level and sxteen at 5%. Ths translates to 25% at 0% sgnfcance level and 4.8% at 5%. To summarze, secton 4.3 addresses the potental problem that the Pearson test may lack power and secton 4.4 demonstrates the dependence of the current serve drecton on the hstory of past plays. We thus tentatvely conclude that there mght be some alternatve hypothess that can better explan the data than the mnmax hypothess. Snce results n Table 7 suggest that past plays do affect current ones, we therefore turn to a more elaborate analyss n whch we propose varous rules to model ths nfluence. We am at characterzng the effect of varous past plays on the current serve choce by dfferent rules. Snce varous learnng models n the lterature provde good frameworks under whch past behavors affect current ones, we next turn to a bref dscusson about how to apply the concept of learnng models to our data (where payoff nformaton s very lmted). The dscusson gves us a gudelne to propose the rules that we wll later consder. Therefore, some of the rules we propose n the followng have the flavor of learnng embedded. Moreover, by proposng several rules and dscussng how smple or sophstcated they are, after we select the best rule n fttng players serve choces, we can categorze players level of smplcty. Ths wll ultmately lead us to address whether for dfferent group of players, the best rule may dffer n a certan nterestng way. 4

16 5. Varous Rules n Characterzng Players Choces 5. Learnng Models and Rules Learnng models can vary n the way that they focus on dfferent psychologcal effects. To a certan degree, they may generate dfferent results n predctng people s behavor. Despte the varety of learnng models, they can be classfed nto two broad branches: renforcement-based models and belef-based models. 2 Renforcement-based models assume that players strateges are renforced by the correspondng payoffs they earn. The hgher the payoffs are, the more strongly ther choces are renforced later. Therefore, players care about the payoffs from dfferent strateges and t s the relatve payoffs suffcent n gudng them to make choces. They do not explctly form belefs about what other people wll do once the nformaton about ther payoffs s realzed. Belef-learnng models, on the other hand, requre that players hold belefs by observng the prevous plays of other players. Gven these belefs, they then choose best responses to maxmze ther expected payoffs. Despte of the dfference between them, some authors argue that renforcement-based and belef-based models mght be dfferent only on the surface. The nature of the two learnng models, however, can be qute smlar. 22 In short, both learnng models provde consderable nsghts n explanng players behavors n strategc stuatons. It would seem straghtforward to apply ether model to our data. However, several obstacles stand n the way of such a foolhardy applcaton. When applyng belef-based models, not only every entry n the payoff matrx of a server has to be known n order to calculate hs best response, but also the recever s choces have 2 Several papers deal wth one knd of model only. Cheung and Fredman (997) estmate a fcttous play model on ndvdual-level data whle Roth and Erev (995) post a renforcement model n several games. The studes of belef or renforcement learnng have ther own explanatory power. Some studes compare the model of renforcement wth that of fcttous play, ncludng Ho and Wegelt (996), and Battalo, Samuelson and Van Huyck (997). Overall, comparsons appear to favor renforcement n constant-sum games and belef learnng n coordnaton games (see Camerer and Ho (999)). 22 See Camerer and Ho (999) and Hopkns (2002). 5

17 to be observed to us as econometrcans so that we can estmate how the server develops hs belef. However, the payoff matrx n a tenns match s not drectly observable. Moreover, unlke goales choces n penalty kcks, recevers guesses about servers choces can hardly be told at all. Overall, the nformaton avalable to us as econometrcans s qute low and partal. If we were to apply renforcement-based models to the data, these problems stll exst but may be less severe. Note that the sprt of renforcement-based models s that when a strategy does well, t wll be adopted more often later. Followng ths lne, we can model players behavors (wth the essence of renforcement learnng) as follows. Whenever the server wns a pont, ths choce of serve drecton wll be renforced. Ths corresponds well to the dea behnd renforcement learnng when nformaton s partal or low. 23 We now propose several rules followng the dscusson above. They can be dvded nto two general classes, based on whether the concept of renforcement learnng s nvolved. () Rules that renforcement learnng s not nvolved: We consder only one rule n ths class. That s, servers may smply keep track of past choces of drecton and these past choces may affect ther current choces. For ths smple rule, we regress the varable D on ts own lags. The sprt of ths rule resembles that underlyng the runs test n secton 4.2. Ths s the frst rule we consder. (2) Rules that renforcement learnng s nvolved: We consder three rules n ths class. Servers may be nfluenced by whether servng to a partcular drecton n the past wns. 24 To capture ths, we construct two dummy varables. The 23 A smlar dea has been used to descrbe subjects behavors n envronments wth partal nformaton by Ho, Camerer and Wang (2002). 24 For nstance, a server may care about the performance when servng to the recever s weaker sde. Recall that n secton 4.4, the nteracton terms n estmatng equatons No. and 2 take the value of f the serve s to the rght and wns. There are some rejectons n the null hypothess, ndcatng the nteracton terms may nfluence the current serve drecton. These rejectons also motvate our analyss here. 6

18 varable RW, short for rght and wn, takes the value of f the serve s R and the server wns. It takes the value of zero otherwse. Symmetrcally, the varable LW, short for left and wn, takes the value of f and only f the serve s L and the server wns. The second rule we try s to regress D on lags of RW. Ths s to examne f the server s nfluenced by whether servng to R wns n the past. Symmetrcally, the thrd rule we try s to regress D on lags of LW. Lastly, we propose one more rule. If players are even more sophstcated, they may be motvated to play R ether because they wn a pont by servng to the rght or they lose that pont by servng to the left. 25 We construct another dummy varable WD, standng for wnnng dfference, to capture ths. WD takes the value of when a server serves to the rght and wns that pont, or he serves to the left and loses that pont. It takes the value of 0 otherwse. In the fourth rule, we regress D on lags of WD. For any X {D, RW, LW, WD}, let X t be that varable at the tth serve of pont game. Denote 5 X t ( s ) the sth lag of the varable X t. For nstance, D ( 3) s the thrd lag of D at the ffth serve n pont game. More specfcally, D 5 ( 3) = f the drecton of serve at the second serve n pont game s to the rght and 0 otherwse. In each pont game, a stochastc decson formula that allows for a catch-all ntercept s studed. Let G[ ] be the probt CDF. Recall that for rules to 4 above, we propose to regress D on lags of X {D, RW, LW, WD}. Therefore, for each rule, at the tth serve, R wll be chosen wth probablty Pr ( Dt = α, β,..., βm ) = G [ α + m s= βs X t ( s )], where X corresponds to the explanatory varable underlyng that partcular rule, m t reflects that the server s affected by what has happened up to m serves ago, 25 In Mookherjee and Sopher (994), a strategy s motvated (or vndcated) ether when t goes well or the other strategy does badly. 7

19 α s a catch-all ntercept, and β s measures how mportantly the current serve drecton tendency for servng to the rght. 5.2 Estmaton Procedure X ( s ) affects D t. Note that α descrbes a player s dosyncratc Our estmaton strategy s as follows. Frst, for each rule, we endogenously determne how many lags to nclude n the regressors. Second, we select whch rule, out of rules to 4, best ft the data. Ths rule wll be dubbed the best rule. Fnally, we compare the best rule to the equlbrum predcton. That s, we demonstrate whether the best rule can explan the data better than the equlbrum theory. Snce at all steps, some comparson and hence selecton has to be made, we therefore use Akake nformaton crteron (AIC henceforth) and Schwarz crteron (SC henceforth) as the selecton crtera. We frst explan these crtera and then the estmaton procedure step by step. Snce the log lkelhood cannot decrease when more regressors are ntroduced, to correct ths, the AIC and SC, based on the negatve of the log lkelhood, both nclude a penalty term dependng on the number of the regressors. The AIC s 2 LL / N + 2k / N and the SC s 2 LL / N + k ( log N ) / N where k s the number of estmated parameters, LL s the log lkelhood, and N s the number of observatons. Both crtera are to provde a measure that strkes a balance between the measure of goodness-of-ft and the parsmonous specfcaton of the model. The SC asks more penaltes relatve to the AIC. For each rule, to endogenously determne how many lags to nclude n the regressors, we search for the optmal number of m that mnmzes each crteron respectvely. We do so by ncreasng the value of m untl the AIC ncreases steadly and repeat the same procedure for the SC. In our data, we fnd that for most players, the optmal number of m s predomnantly less than 3, reflectng that servers are only affected by recent hstory. t 8

20 To sort out whch rule can better predct players choces, snce we have determned the optmal m for each rule, we smply compare the AIC (SC) values for rules to 4. The rule that mnmzes the AIC (SC) value s the best rule under the AIC (SC) respectvely. Fnally, to make a comparson to the equlbrum predcton, we run an addtonal regresson where the only regressor s the catch-all ntercept ( α ) for each pont game. Ths amounts to the equlbrum predcton where the probablty of choosng R s constant, therefore ndependent of any past hstory. We then compare the AIC (SC) value of the best rule wth the AIC (SC) value of the equlbrum. 5.3 Mnmax or Smple Rules? We report the results n Tables 8 to 0. For each pont game, we report the best rule under the AIC and that under the SC n the last column. The assocated AIC and SC values are reported n the second to last column. In the thrd to last column we report the AIC and SC values for the equlbrum. In summary, accordng to the AIC, the best rule fts better than the equlbrum predcton n a sgnfcant number of pont games. In twenty-seven out of forty pont games for male players, twenty-two out of thrty-sx for female players, and twenty-fve out of thrty-two for junor players, the best rules ft better than the equlbrum predcton. In total, n 68.5% of the 08 pont games, the best rules perform better than the equlbrum predcton. Snce the SC penalzes models wth more regressors more heavly than the AIC and n the regresson for testng the equlbrum, the only regressor s the constant ntercept term α, t s no wonder that the proporton where the best rules outperform the equlbrum wll be reduced f we use the SC nstead. Under the SC, n 36.% of the pont games, the best rules outperform the equlbrum predcton. In partcular, n seven out of forty for male players, ffteen out of thrty-sx for female players, and seventeen out of thrty-two for junor 9

21 players, best rules ft the data better than the equlbrum predcton. We take ths as strong evdence aganst the equlbrum. Moreover, even f the equlbrum predcton outperforms the best rule n a pont game, t s stll possble that the set of coeffcents β s for the best rule s jontly sgnfcantly dfferent from zero,.e. past hstory does affect the current serve drecton. To hghlght ths, we test, for the best rule n each pont game, whether the set of coeffcents β s s jontly dfferent from zero. Table summarzes ths result under both 5% and 0% sgnfcance levels applyng ether nformaton crteron. It suggests that the statstcal sgnfcance of the past hstory n explanng players behavor s the rule rather than an excepton. There are 53.7% of pont games where the correspondng β s are sgnfcant at 0% level under the AIC, 48.% under the SC. At 5% level, 37% of pont games exhbts ths sgnfcance under the AIC, 30.6% under the SC. Ths lends addtonal support to the dea that players are ndeed affected by the past plays and therefore descrbng ther choces by the varous smple rules we consder s approprate. 5.4 Junor vs. Adult Players As a frst cut, t s far to say that players whose choces are best characterzed by the equlbrum are sophstcated enough so that all the ratonalty assumptons requred by the mnmax hypothess are met. Then the four rules we consder can be thought of exhbtng dfferent degrees of sophstcaton. To begn wth, to mplement the frst rule where renforcement s not nvolved, players need to keep track of ther past serve choces only. On the other hand, to use the three rules where renforcement s nvolved (rules 2 to 4), they not only need to keep track of ther past serve choces but also the wnnng record or even the dfference n wnnng records of past choces. Ths s more demandng. If we nterpret the degrees of sophstcaton ths way, t brngs n an nterestng queston: Are junor players, n any sense, more or less smple than adult players? 20

22 We make an attempt to answer ths queston as follows. As dscussed above, we dvde all the rules and the equlbrum predcton nto three classes, dependng on the degree of smplcty or sophstcaton nvolved. We treat the frst rule as the Low class, presumably because t s smpler. Rules 2 to 4 are categorzed as the Medum class snce more sophstcaton s necessary n accordance wth these rules. Fnally, the equlbrum predcton s assgned to the Hgh class. Based on Tables 8 to 0, we can assgn each pont game nto ether the Low, Medum or Hgh class by comparng the AIC or SC values of the best rule wth those of the equlbrum. If the pont game s better characterzed by the equlbrum than the best rule, then that pont game s classfed nto the Hgh class. On the other hand, f the pont game s better characterzed by the best rule than the equlbrum, then the pont game s assgned to ether the Low or Medum class dependng on whch class the best rule belongs to. For nstance, for the frst pont game n male (Borg servng to McEnroe n the ad court n the 980 Wmbledon), the best rule, the 3rd rule, outperforms the equlbrum predcton under the AIC. Thus ths pont game s assgned to class Medum under the AIC snce the 3rd rule s categorzed nto ths class. We do ths for every pont game. Combnng all the pont games n male and female groups nto one adult group, we make bar charts to compare junor players choces wth adult players n Fgure 6. Under the AIC, we fnd the proporton of players usng Medum rules s hghest for both junor and adult players. Ths s not very surprsng snce there are three rules n ths class. More mportantly, the dstrbutons over the classes for adult players frst order stochastc domnate those for junor players under both AIC and SC. For nstance, by the AIC, the proporton of junor players classfed nto Low (rule ) s 0.29, hgher than that of adult players, whch s 0.7. The proporton of junor players classfed nto Medum (rules 2 to 4) s 0.562, hgher than that of adult players, whch s A smlar 2

23 result holds under the SC. We then turn to the comparson between the dstrbutons of classes of male and female players. Here the evdence s mxed. Under the AIC, the dstrbuton of female players frst order stochastc domnates that for male players. However, under the SC, the reverse holds,.e. the dstrbuton of male players frst order stochastc domnates that of female players. Note how ths s n sharp contrast to the Pearson test of equal wnnng probabltes where the number of rejectons arses mostly for male players. 6. Dscusson 6. Payoff Change wthn Pont Games? Our study s based on the assumpton that there s a fxed payoff matrx for every pont game. In partcular, the payoff matrx does not change wth tme n each pont game. We now verfy ths assumpton as below. The dea s as follows. A pror, we do not know when and whether the payoff matrx wll change n a pont game. Snce there are several sets n a pont game, we naturally focus on testng whether there s any payoff change from set to set nstead. Recall that for each pont, we only observe the server s choce of drecton and whether he wns that pont. If the payoff does not change, the observed choces and outcomes should not change from set to set ether. Followng Chappor, Levtt and Groseclose (2002), 26 we run two separate regressons for ths purpose. In one, we regress the server s choce of drecton on the constant and a collecton of dummes characterzng whch set the pont s n. In the other, we regress whether the server wns the pont on the constant and the same collecton of set dummes. The null hypothess s that the coeffcents of the set 26 Chappor, Levtt and Groseclose (2002) consder whether goales are homogeneous. They bascally regress four outcome varables (whether the kck s successful, whether the kcker shoots rght, whether the kcker shoots n the mddle, and whether the goale jumps rght) separately on goale-fxed effects. The null hypothess s therefore the goale-fxed effects are jontly nsgnfcant from zero. 22

24 dummes are jontly zero, reflectng no effect of the sets on servers choces and the outcomes of each pont. At the 0. sgnfcance level, regardng whether the server wns the pont, n seven out of 00 pont games are the coeffcents of the set dummes sgnfcant. For the server s choce of drecton, there are eleven out of 00 pont games n whch the null hypothess s rejected. 27 Admttedly, one may queston whether the sets do affect the server s choce of drecton snce there are eleven rejectons whereas by chance, one would expect only ten. We therefore turn to conduct a jont lkelhood rato test where the null hypothess s that the set dummes do not affect the server s choce of drecton n the 00 pont games. The p-value s 0.84, whch s not sgnfcant at any usual sgnfcance level. 28 These results are thus consstent wth the vew that there s no statstcally sgnfcant payoff change, at least from set to set. We further note that f we adopt a more cautous vew to drop the pont games where the coeffcents of the set dummes are sgnfcant at 0% level n ether regresson, among the remanng pont games, the relaton that the dstrbuton of rules characterzng the adult players behavors frst order stochastc domnates those of the junor players stll holds Learnng Mnmax? Our results so far generally pont to the drecton that the mnmax hypothess should be more carefully consdered. Whle dong so, we adopt the concept of renforcement learnng and propose several smple rules to account for the seral dependence n the data. A logcal queston s: even f the mnmax hypothess does 27 The total number of pont games where we run these regressons s 00, because two junor matches are too short to do so. 28 N N The null hypothess mples that the statstc 2( = L = l ) has the asymptotc χ 2 (d) dstrbuton, where N denotes the total number of pont games, L s the maxmum log lkelhood of the regresson wthout the set dummes, l s the maxmum log lkelhood of the regresson wth the set dummes, and d s the number of the estmated parameters n the regresson wth the set dummes. 29 Under the AIC, the proportons of Low, Medan and Hgh for adult players are 0.206, 0.43 and 0.38, whle those of junor players are 0.3, 0.4 and 0.3, respectvely. Under the SC, the proportons of Low, Medan and Hgh n adult players are 0.095, 0.9 and 0.74, whle those of junor players are 0.2, 0.4 and 0.4, respectvely. 23

25 not ft the data well, can t descrbe players behavors better n the later stage of a game when they have more experences? In other words, players may learn to play mnmax. To address ths possblty, we take male tenns matches, whch have longer observatons, as an example and repeat some of our earler analyses. As a frst and prelmnary attempt, we dvde the data n each pont game nto the frst half and the second half of the game, and perform the Pearson test of equal wnnng probablty and the runs test on them separately. If the null hypotheses fare better n the second half of the game than n the frst half, then t may be suggestve of some learnng to play mnmax. We fnd that the p-values of the Pearson jont statstc are for the frst half and 0.32 for the second half. The p-values of the KS statstc are and for the frst half and the second half respectvely. As for the runs test, the p-values of the KS statstc are for the frst half and 0.35 for the second half. Therefore, among the three tests, there does not seem to be strong evdence n support of the dfference n the frst and second part of the game. We do not want to make too much out of ths prelmnary fndng as dvdng the data nto two halves s qute arbtrary. 7. Concludng Remarks Whle the theory of mxed strategy equlbrum has not been entrely consstent wth the flurry of the emprcal evdence from expermental settngs nvolvng human subjects n the past few decades, the emprcal fndng n Walker and Wooders (200) undoubtedly makes a mark n testng the mnmax hypothess. However, our fndng, from studyng a broader natural data set n tenns, suggests that the mnmax hypothess should be further questoned. Not only should one examne the robustness of equal wnnng probablty, but also the phenomenon of seral dependence needs to be re-addressed. 24

26 Snce we do fnd evdence regardng mpacts of past plays on current choces, we therefore turn to model ths dependence by dscussng how learnng models can apply to our data and proposng several rules. When estmatng varous rules to account for players choces, we fnd that for a sgnfcant number of pont games, the best rule outperform the equlbrum predcton. More nterestngly, we demonstrate that junor players tend to adopt smpler rules than adult players do. Overall, we see our work as a prmtve attempt n applyng rules (or learnng models) to feld data. More elaborate emprcal analyses seem mmnent to mprove our understandng of choces under strategc stuatons. 25

27 References Brown, J. and Rosenthal, R. (990). Testng the Mnmax Hypothess: A Reexamnaton of O Nell s Experment, Econometrca 5, Battalo, R.; Samuelson, L. and Van Huyck, J. (2003). Rsk Domnance, Payoff Domnance and Probablstc Choce Learnng, Wsconsn Madson Socal Systems Workng Paper No. 2. Camerer, C. and Ho, T-H (999). Experence-Weghted Attracton Learnng n Normal Form Games, Econometrca 67, Cheung, Y-W and Fredman, D. (997). Indvdual Learnng n Normal Form Games: Some Laboratory Results, Games and Economc Behavor 9, Chappor, P-A; Levtt, S. and Groseclose, T. (2002). Testng Mxed-Strategy Equlbra When Players Are Heterogeneous: The Case of Penalty Kcks n Soccer, Amercan Economc Revew 92, 38-5 Erev, I. and Roth, A. E. (998). Predctng How People Play Games: Renforcement Learnng n Expermental Games wth Unque, Mxed Strategy Equlbra, Amercan Economc Revew 88, Feltovch, N. (2000). Renforcement-Based vs. Belef-Based Learnng Models n Expermental Asymmetrc-Informaton Games, Econometrca 68, Gbbons, J. and Chakrabort, S. (992). Nonparametrc Statstcal Inference, New York: Marcel Dekker. Ho, T-H and Wegelt, K. (996). Task Complexty, Equlbrum Selecton, and Learnng: An Expermental Study, Management Scence 42, Ho, T-H; Camerer C. and Wang, X. (2002). Indvdual Dfferences n EWA Learnng wth Partal Payoff Informaton, mmeo. Hopkns, E. (2002). Two Competng Models of How People Learn n Games, Econometrca 70,