Equilibrium or Simple Rule at Wimbledon? An Empirical Study

 Alexia Stevens
 7 months ago
 Views:
Transcription
1 Equlbrum or Smple Rule at Wmbledon? An Emprcal Study ShhHsun Hsu, ChenYng Huang and ChengTao 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. PalacosHuerta (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 lownformaton 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 constantsum 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 reevaluate 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 constantsum 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 leftorrght 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 deucecourt or an adcourt 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 leftorrght 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 leftorrght 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 rghtorleft 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 toplevel 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 pvalue (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 chsquare goodnessofft and the KolmogorovSmrnov (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 chsquare 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 chsquare 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 pvalues 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 equalwnnngprobablty 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 pvalues 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 pvalues 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 adcourt and deucecourt data nto one for a gven server. The results are as follows. The pvalues 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 pvalues 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 chsquare 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 nonmnmax behavors may also lead to acceptance of equal wnnng probablty of server s leftorrght 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 Reevaluaton 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 twentyone 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 twentyone. 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 twentyseven 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: renforcementbased models and belefbased models. 2 Renforcementbased 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. Beleflearnng 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 renforcementbased and belefbased 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 belefbased 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 ndvduallevel 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 constantsum 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 renforcementbased models to the data, these problems stll exst but may be less severe. Note that the sprt of renforcementbased 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 catchall 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 catchall 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 goodnessofft 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 catchall 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 twentyseven out of forty pont games for male players, twentytwo out of thrtysx for female players, and twentyfve out of thrtytwo 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 thrtysx for female players, and seventeen out of thrtytwo 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 goalefxed effects. The null hypothess s therefore the goalefxed 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 pvalue 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 pvalues of the Pearson jont statstc are for the frst half and 0.32 for the second half. The pvalues of the KS statstc are and for the frst half and the second half respectvely. As for the runs test, the pvalues 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 readdressed. 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, TH (999). ExperenceWeghted Attracton Learnng n Normal Form Games, Econometrca 67, Cheung, YW and Fredman, D. (997). Indvdual Learnng n Normal Form Games: Some Laboratory Results, Games and Economc Behavor 9, Chappor, PA; Levtt, S. and Groseclose, T. (2002). Testng MxedStrategy Equlbra When Players Are Heterogeneous: The Case of Penalty Kcks n Soccer, Amercan Economc Revew 92, 385 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). RenforcementBased vs. BelefBased Learnng Models n Expermental AsymmetrcInformaton Games, Econometrca 68, Gbbons, J. and Chakrabort, S. (992). Nonparametrc Statstcal Inference, New York: Marcel Dekker. Ho, TH and Wegelt, K. (996). Task Complexty, Equlbrum Selecton, and Learnng: An Expermental Study, Management Scence 42, Ho, TH; 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,
First digit of chosen number Frequency (f i ) Total 100
1 4. ANALYSING FREQUENCY TABLES Categorcal (nomnal) data are usually summarzed n requency tables. Contnuous numercal data may also be grouped nto ntervals and the requency o observatons n each nterval
More informationPERFORMANCE AND COMPENSATION ON THE EUROPEAN PGA TOUR: A STATISTICAL ANALYSIS
PERFORMANCE AND COMPENSATION ON THE EUROPEAN PGA TOUR: A STATISTICAL ANALYSIS C. Barry Pftzner and Chrs Spence, Department of Economcs/Busness, RandolphMacon College, Ashland, VA, bpftzne@rmc.edu, cspence@rmc.edu
More informationEvaluation of a Center Pivot Variable Rate Irrigation System
Evaluaton of a Center Pvot Varable Rate Irrgaton System Ruxu Su Danel K. Fsher USDAARS Crop Producton Systems Research Unt, Stonevlle, Msssspp Abstrat: Unformty of water dstrbuton of a varable rate center
More informationMajor League Duopolists: When Baseball Clubs Play in TwoTeam Cities. Phillip Miller. Department of Economics. Minnesota State University, Mankato
Major League Duopolsts: When Baseball Clubs Play n TwoTeam Ctes Phllp Mller Department of Economcs Mnnesota State Unversty, Mankato September 2006 Abstract: Ths paper focuses on examnng the attendance
More informationReferee Bias and Stoppage Time in Major League Soccer: A Partially Adaptive Approach
Econometrcs 2014, 2, 119; do:10.3390/econometrcs2010001 OPEN ACCESS econometrcs ISSN 22251146 www.mdp.com/journal/econometrcs Artcle Referee Bas and Stoppage Tme n Major League Soccer: A Partally Adaptve
More informationCS 2750 Machine Learning. Lecture 4. Density estimation. CS 2750 Machine Learning. Announcements
CS 75 Machne Learnng Lecture 4 ensty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square CS 75 Machne Learnng Announcements Homework ue on Wednesday before the class Reports: hand n before the
More informationCOMPENSATING FOR WAVE NONRESPONSE IN THE 1979 ISDP RESEARCH PANEL
COMPENSATING FOR WAVE NONRESPONSE IN THE 1979 ISDP RESEARCH PANEL 1. Introducton Graham Kalton, Unversty of Mchgan ames Lepkowsk, Unversty of Mchgan TngKwong Ln. Natonal Unversty of Sngapore The choce
More informationModeling the Performance of a Baseball Player's Offensive Production
Brgham Young Unversty BYU ScholarsArchve All Theses and Dssertatons 0060309 Modelng the Performance of a Baseball Player's Offensve Producton Mchael Ross Smth Brgham Young Unversty  Provo Follow ths
More informationWhat does it take to be a star?
Unversty of Freburg Department of Internatonal Economc Polcy Dscusson Paper Seres Nr. 1 What does t take to be a star? The role of performance and the meda for German soccer players Erk E. Lehmann and
More informationWORKING PAPER SERIES Longterm Competitive Balance under UEFA Financial Fair Play Regulations Markus Sass Working Paper No. 5/2012
WORKING PAPER SERIES Impressum ( 5 TMG) Herausgeber: OttovonGuerckeUnverstät Magdeburg Fakultät für Wrtschaftswssenschaft Der Dekan Verantwortlch für dese Ausgabe: OttovonGuerckeUnverstät Magdeburg
More informationAn EnforcementCoalition Model: Fishermen and Authorities forming Coalitions. Lone Grønbæk Kronbak Marko Lindroos
An EnforcementCoalton Model: Fshermen and Authortes formng Coaltons Lone Grønbæ Kronba Maro Lndroos December 003 All rghts reserved. No part of ths WORKING PAPER may be used or reproduced n any manner
More informationCAREER DURATION IN THE NHL: PUSHING AND PULLING ON EUROPEANS?
UNC CHARLOTTE ECONOMICS WORKING PAPER SERIES CAREER DURATION IN THE NHL: PUSHING AND PULLING ON EUROPEANS? Crag A. Depken II Johnny Duckng Peter A. Groothus Workng Paper No. 2016005 THE UNIVERSITY OF
More informationEnglish Premier League (EPL) Soccer Matches Prediction using An Adaptive NeuroFuzzy Inference System (ANFIS) for
Englsh Premer League (EPL) Soccer Matches Predcton usng An Adaptve NeuroFuzzy Inference System (ANFIS) for Amadn, F. I 1 and Ob, J.C. 2 Department of Computer Scence, Unversty of Benn, Benn Cty. Ngera.
More informationADDITIONAL INSTRUCTIONS FOR ISU SYNCHRONIZED SKATING TECHNICAL CONTROLLERS AND TECHNICAL SPECIALISTS
A ADDITIONAL INSTRUCTIONS FOR ISU SYNCHRONIZED SKATING TECHNICAL CONTROLLERS AND TECHNICAL SPECIALISTS DIFFICULTY GROUPS OF FEATURES 1 DEFINITIONS: Change of Rotaton: Refers to TURNS or LINKING STEPS rotatng
More informationComparisons of Means for Estimating Sea States from an Advancing Large Container Ship
Downloaded from orbt.dtu.dk on: Jan 31, 18 Comparsons of Means for Estmatng Sea States from an Advancng Large Contaner Shp Nelsen, Ulrk Dam; Andersen, Ingrd Mare Vncent; Konng, Jos Publshed n: Proceedngs
More informationEXPLAINING INTERNATIONAL SOCCER RANKINGS. Peter Macmillan and Ian Smith
EXPLAINING INTERNATIONAL SOCCER RANKINGS Peter Macmllan and Ian Smth School of Economcs and Fnance Unversty of St Andrews St Andrews Ffe, KY16 9AL Unted Kngdom Tel: + 44 1334 46430 (Smth), + 44 1334 46433
More informationRisk analysis of natural gas pipeline
Rsk analyss of natural gas ppelne Y.D. Jo 1, K.S. Park 1, J. W. Ko, & B. J. Ahn 3 1 Insttute of Gas Safety Technology, Korea Gas Safety Corporaton, South Korea Department of Chemcal Engneerng, Kwangwoon
More informationPlanning of production and utility systems under unit performance degradation and alternative resourceconstrained cleaning policies
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Plannng of producton and utlty systems under unt performance degradaton
More informationSafety Impact of Gateway Monuments
*Manuscrpt Clck here to vew lnked References Ye, Venezano, and Lord 1 Safety Impact of Gateway Monuments Zhru Ye a,*, Davd Venezano a, Domnque Lord b a Western Transportaton Insttute, Montana State Unversty,
More informationHigh Speed 128bit BCD Adder Architecture Using CLA
Hgh Speed 128bt BCD Archtecture Usng CLA J.S.V.Sa Prasanth 1, Y.Yamn Dev 2 PG Student [VLSI&ES], Dept. of ECE, Swamy Vvekananda Engneerng College, Kalavara, Andhrapradesh, Inda 1 Assstant Professor, Dept.
More informationEvaluating the Effectiveness of Price and Yield Risk Management Products in Reducing. Revenue Risk for Southeastern Crop Producers * Todd D.
Evaluatng the Effectveness of Prce and Yeld Rsk Management Products n Reducng Revenue Rsk for Southeastern Crop Producers * Todd D. Davs ** Abstract A nonparametrc smulaton model ncorporatng prce and
More informationarxiv: v1 [cs.ne] 3 Jul 2017
Modelng preference tme n mddle dstance trathlons Iztok Fster, 1 Andres Iglesas, 2 Suash Deb, 3, 4 Dušan Fster, 5 and Iztok Fster Jr. 6 1 Unversty of Marbor, Faculty of Electrcal Engneerng and Computer
More informationComparative Deterministic and Probabilistic Analysis of Two Unsaturated Soil Slope Models after Rainfall Infiltration
Jordan Journal of Cvl Engneerng, Volume 11, No. 1, 2017 Comparatve Determnstc and Probablstc Analyss of Two Unsaturated Sol Slope Models after Ranfall Infltraton Manoj Kr. Sahs 1) and Partha Pratm Bswas
More informationALASKA DEPARTMENT OF FISH AND GAME DIVISION OF COMMERCIAL FISHERIES NEWS RELEASE
ALASKA DEPARTMENT OF FISH AND GAME DIVISION OF COMMERCIAL FISHERIES NEWS RELEASE Cora Campbell, Commssoner Jeff Regnart, Drector Contact: Cordova ADF&G Steve Mofftt, PWS Fnfsh Research Bologst 401 Ralroad
More informationInvestigating Reinforcement Learning in Multiagent Coalition Formation
Investgatng Renforcement Learnng n Multagent Coalton Formaton Xn L and LeenKat Soh Department of Computer Scence and Engneerng Unversty of NebrasaLncoln 115 Ferguson Hall, Lncoln, NE 665880115 {xnl,
More informationA nonparametric analysis of the efficiency of the top European football clubs
MPRA Munch Personal RePEc Archve A nonparametrc analyss of the effcency of the top European football clubs George Halkos and Nckolaos Tzeremes Unversty of Thessaly, Department of Economcs May 2011 Onlne
More informationCoalition Formation in a Global Warming Game: How the Design of Protocols Affects the Success of Environmental TreatyMaking
Coalton Formaton n a Global Warmng Game: How the Desgn of Protocols Affects the uccess of Envronmental TreatyMang Frst draft: March, 23 Ths verson: November, 23 Johan Eycmans K.U.Leuven, Centrum voor
More informationCOMPARATIVE ANALYSIS OF WAVE WEATHER WINDOWS IN OPERATION AND MAINTENANCE OF OFFSHORE WIND FARMS AT HSINCHU AND CHANGHUA, TAIWAN
Journal of Marne Scence and Technology, Vol. 5, No. 5, pp. 563570 (07) 563 DOI: 0.69/JMST070703 COMPARATIVE ANALYSIS OF WAVE WEATHER WINDOWS IN OPERATION AND MAINTENANCE OF OFFSHORE WIND FARMS AT HSINCHU
More informationResponse based sea state estimation for onboard DSS Safe and Efficient Marine Operations
Response based sea state estmaton for onboard DSS Safe and Effcent Marne Operatons Ulrk Dam Nelsen AMOS Workshop, NTNU, Trondhem, 13 th and 14 th November, 2014 Contents Introducton: Context Montorng and
More informationValuing Beach Quality with Hedonic Property Models
Valung Beach Qualty wth Hedonc Property Models Crag E. Landry* Department of Economcs Center for Natural Hazards Research East Carolna Unversty, Greenvlle, NC 27858 landryc@ecu.edu; 2523286383 and Paul
More informationPeace Economics, Peace Science and Public Policy
Peace Economcs, Peace Scence and Publc Polcy Volume 17, Issue 1 2011 Artcle 1 Lone Wolf Terrorsm Peter J. Phllps Unversty of Southern Queensland, phllpsp@usq.edu.au Copyrght c 2011 Berkeley Electronc Press.
More informationA comparison study on the deck house shape of high speed planing crafts for air resistance reduction
csnak, 2014 Int. J. Nav. Archt. Ocean Eng. (2014) 6:867~875 http://dx.do.org/10.2478/ijnaoe20130218 pissn: 20926782, eissn: 20926790 A comparson study on the deck house shape of hgh speed planng crafts
More information2018 GIRLS DISTRICTSPECIFIC PLAYER DEVELOPMENT GUIDE
2018 GIRLS DISTRICTSPECIFIC PLAYER DEVELOPMENT GUIDE GENERAL OVERVIEW USA Hockey Grls Player Development DstrctSpecfc Gude The USA Hockey Grls Player Development DstrctSpecfc Gude outlnes the 2018 grls
More information2017 GIRLS DISTRICTSPECIFIC PLAYER DEVELOPMENT GUIDE
2017 GIRLS DISTRICTSPECIFIC PLAYER DEVELOPMENT GUIDE GENERAL OVERVIEW USA Hockey Grls Player Development DstrctSpecfc Gude The USA Hockey Grls Player Development DstrctSpecfc Gude outlnes the 2017 grls
More informationOCCASIONAL PAPER SERIES
OCCASIONAL PAPER SERIES NO. 4 / DECEMBER 5 WHAT DOES EUROPEAN INSTITUTIONAL INTEGRATION TELL US ABOUT TRADE INTEGRATION? by Francesco Paolo Mongell, Ettore Dorrucc and Ita Agur OCCASIONAL PAPER SERIES
More informationMass Spectrometry. Fundamental GCMS. GCMS Interfaces
Mass Spectrometry Fundamental GCMS GCMS Interfaces Wherever you see ths symbol, t s mportant to access the onlne course as there s nteractve materal that cannot be fully shown n ths reference manual.
More informationSCIENTIFIC COMMITTEE THIRTEENTH REGULAR SESSION. Rarotonga, Cook Islands 917 August, 2017
SCIENTIFIC COMMITTEE THIRTEENTH REGULAR SESSION Rarotonga, Cook Islands 917 August, 2017 Relatve abundance of yellowfn tuna for the purse sene and handlne fsheres operatng n the Phlppnes Moro Gulf (Regon
More informationInvestigating sailing styles and boat setup on the performance of a hydrofoiling Moth dinghy
Investgatng salng styles and boat setup on the performance of a hydrofolng Moth dnghy Abstract M W Fndlay, S R Turnock FludStructure Interactons Research Group, School of Engneerng Scences, Unversty
More informationChinese and foreign men s decathlon performance comparison and structural factor correlation test based on SPSS regression model
ISSN : 09747435 Volume 10 Issue 3 Chnese and foregn men s decathlon performance comparson and structural factor correlaton test based on SPSS regresson model Pengfe Zhang*, Jngjng Lu Insttute of Physcal
More informationArchimer
Please note that ths s an authorproduced PDF of an artcle accepted for publcaton followng peer revew. The defntve publsherauthentcated verson s avalable on the publsher Web ste Marne Polcy December 2014,
More information11. Contract or Grant No. Lubbock, Texas
Tee hnalr c eport D ocwnentaton Page 1. ReportNo. 2. Government Accesson No.. Recpent's Catalog No. TX 917951S. Ttle and Subttle 5. Report Date Proposed Geometrc Desgn for TwoLane, TwoWay Hghway ntermttent
More informationCrossshore Structure of Longshore Currents during Duck94
CHAPTER 283 Crossshore Structure of Longshore Currents durng Duck94 Falk Feddersen 1, R.T. Guza 2, Steve Elgar 3, and T.H.C Herbers 4 Abstract The crossshore structure of the mean longshore current on
More informationCENTRAL BLOOD PRESSURE AND HEART OUTPUT IN SURFACED AND SUBMERGED FROGS
J. Exp. Btol. (965), 42, 339357 339 Wth textfgura Prnted n Great Brtan CENTRAL BLOOD PRESSURE AND HEART OUTPUT IN SURFACED AND SUBMERGED FROGS BY G. SHELTON* AND D. R. JONES* Department of Zoology, The
More informationInvestigation on Rudder Hydrodynamics for 470 Class Yacht
Proceedngs Investgaton on Rudder Hydrodynamcs for 470 Class Yacht She Ln 1,, Yong Ma, *, Wetao Zheng, Song Zhang 1,, Xaoshan Le 1, and Yangyng He 1, 1 Graduate School of Wuhan Sports Unversty, Wuhan 430079,
More informationProceedings of the ASME nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 914, 2013, Nantes, France
Proceedngs of the ASME nd Internatonal Conference on Ocean, Offshore and Arctc Engneerng OMAE June 9,, Nantes, France OMAE6 ROLL DAMPING COEFFICIENTS ASSESSMENT AND COMPARISON FOR ROUND BILGE AND HARD
More informationAerator Performance in Reducing Phenomenon of Cavitation in Supercritical Flow in Steep Channel Bed
Engneerng Internatonal Conference UNNES Conservaton 13 Proceedng Aerator Performance n Reducng Phenomenon of Cavtaton n Supercrtcal Flow n Steep Channel Bed Yer Sutopo 1, Bud Wgnyosukarto, Istarto, dan
More informationIntroduction Match analysis is used to investigate technical and tactical behaviors within judo studies, and it can also be applied to improve
Introducton Match analyss s used to nvestgate techncal and tactcal behavors wthn judo studes, and t can also be appled to mprove athletc performance. Ths type of data can be used n a varety of means to
More informationStability Analysis for the Design of 5000Tonnes Offshore Work Barge
Internatonal Journal of Engneerng and echnology Volume No. 9, September, Stablty Analyss for the Desgn of 5onnes Offshore Work arge Ntonye Samson a, Ezenwa Ogbonnaya b, uve Ejabefo a a Department of arne
More informationHydraulic DTH Fluid Hammer Drilling as a Seismic While Drilling (SWD) Source for Geothermal Exploration and Drilling Prediction
Proceedngs World Geothermal Congress 2015 Melbourne, Australa, 1925 Aprl 2015 Hydraulc DTH Flud Hammer Drllng as a Sesmc Whle Drllng (SWD) Source for Geothermal Exploraton and Drllng Predcton Poletto
More informationSEAKEEPING BEHAVIOUR OF A FRIGATETYPE TRIMARAN
Desgn & Operaton of Trmaran Shps, London, UK SEAKEEPNG BEHAVOUR OF A FRGATETYPE TRMARAN W Pastoor, Det Norske Vertas AS, Norway R van 't Veer, Martme Research nsttute Netherlands, The Netherlands E Harmsen,
More informationCost theory and the cost of substitution a clarification
Cost theory and the cost of substtuton a clarfcaton Walter J. ReMne The cost of substtuton has been wdely msnterpreted, whch has lmted ts utlty. Ths paper clarfes the cost concept and reestablshes ts
More informationNSPAA SLOW PITCH SOFTBALL POLICY AND PROCEDURES HANDBOOK
NSPAA SLOW PITCH SOFTBALL POLICY AND PROCEDURES HANDBOOK Revsed January 1, 2017 The Slow Ptch Softball Board (upon approval by the NSPAA Board) reserves the rght to edt, modfy, or change any polcy and/or
More informationPropagation of Big Island eddies
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. C1, PAGES 935944, JANUARY 15, 2001 Propagaton of Bg Island eddes Chrstna L. Holland and Gary T. Mtchum College of Marne Scence, Unversty of South Florda,
More informationMass Transfer of CO 2 into Water and Surfactant Solutions
Mass Transfer of CO 2 nto Water and Surfactant Solutons Rouhollah Farajzadeh, Al Barat, Harm A. Dell, Johannes Brunng, Pacell L.J. Ztha Delft Unversty of Technology, Department of Geotechnology, Mjnbouwstraat
More informationTeacher Resource for Unit 1 Lesson 1: Linear Models Refresher
Tme Frame: 34 Da Math 3 Unt Leon SID.6, 7, SIC. Math Practce 2, 3, 4 Teacher Reource for Unt Leon : Lnear Model Refreher Math Content Standard: SID.6, 7, SIC. S.ID.6 Repreent data on two quanttatve
More informationMinimax Play at Wimbledon
Minimax Play at Wimbledon By MARK WALKER AND JOHN WOODERS* In many strategic situations it is important that one s actions not be predictable by one s opponent, or by one s opponents. Indeed, the origins
More informationEvaluation of Wettability of Solid Surface with Oil/ Refrigerant Mixture
Purdue Unversty Purdue epubs Internatonal Compressor Engneerng Conference School of Mechancal Engneerng 16 Evaluaton of Wettablty of Sold Surface wth Ol/ Refrgerant Mxture Mtsuhro Fukuta Shzuoka Unversty,
More informationMultiCriteria Decision Tree Approach to Classify AllRounder in Indian Premier League
Internatonal Journal of Adanced Scence and Technology MultCrtera Decson Tree Approach to Classfy AllRounder n Indan Premer League Pabtra Kumar Dey 1, Dpendra Nath Ghosh 2, Abhoy Chand Mondal 3 1 Assstant
More informationIDENTIFICATION OF TRANSPORTATION IMPROVEMENT PROJECTS IN PHNOM PENH CONSIDERING TRAFFIC CONGESTION LEVEL
Proceedngs of the Eastern Asa Socety for Transportaton Studes, Vol. 5, pp. 12651280, 2005 IDENTIFICATION OF TRANSPORTATION IMPROVEMENT PROJECTS IN PHNOM PENH CONSIDERING TRAFFIC CONGESTION LEVEL Sambath
More informationSteelhead Broodstock Acclimation and Monitoring (BAM) Program in the Okanogan Basin, Annual Report 2012
2012 Annual Report Steelhead Broodstock Acclmaton and Montorng (BAM) Program n the Okanogan Basn, Annual Report covers performance under GPUD Contract No. 4303559 1/1/2012 12/31/2013 Tbbts, W.T., R.E.
More informationAlgorithms of Digital Processing and the Analysis of Underwater Sonar Images
9 Algorthms of Dgtal Processng and the Analyss of Underwater Sonar Images S.V. Sa, A.G. Shoberg and L.A. Naumov 2 Pacfc natonal unversty 2 Insttute of Marne Technology Problems FEB RUS Russa Open Access
More informationSTUDIES OF DIVERS' PERFORMANCE DURING THE SEALAB II PROJECT. Hugh M. Boweu Birger Andersen David Promisel. March Contract No.
JL SSD66296 (571) ß CO 3 STUDES OF DVERS' PERFORMANCE DURNG THE SEALAB PROJECT Hugh M. Boweu Brger Andersen Davd Promsel March 1966 Contract No. Nonr 49^0 (00) Prepared for: Offce of Naval Research Washngton,
More informationWHOI MOORING OPERATIONS TECHNIQUES OF THE BUOY PROJECT AT THE WOODS HOLE OCEANOGRAPHIC INSTITUTION. Robert H. Heinmi11er, Jr.
WHO7669 )' MOORNG OPERATONS TECHNQUES OF THE BUOY PROJECT AT THE WOODS HOLE OCEANOGRAPHC NSTTUTON by Robert H. Henm11er, Jr. ~" co ft co.. Õ ru :~ cr :: 3: c: c: == ld,c,. ft c: C WOODS HOLE OCEANOGRAPHr'C
More informationContractor's Material and Test Certificate for Underground Piping
UNDERGROUND PPNG 1397 Contractors Materal and Test Certfcate for Underground Ppng PROCEDURE Upon completon of work, nspecton and tests shall be made by the contractors representatve and wtnessed by an
More informationIBIS: ATestbed for the Evolution of Intelligent Broadband Networks toward TINA
BS: ATestbed for the Evoluton of ntellgent Broadband Networks toward TNA Marc0 Lstant, Unversty of Roma "La Sapenza" Stefan0 Salsano, CoRTeL ABSTRACT Ths artcle analyzes a possble path for the evoluton
More informationInfluence of Gas Density on Hydrodynamics in a Bubble Column
Smulaton and Optmzaton Chna Petroleum Processng and Petrochemcal Technology 01, Vol. 16, No. 1, pp 6670 March 30, 01 Influence of Gas Densty on Hydrodynamcs n a Bubble Column Tang Xaojn; Hou Shuand; Zhang
More informationProduct Information. Gripper for small components MPG 80
Product Informaton MPG 80 MPG Precse. Compact. Relable. MPG grpper for small components 2fnger parallel grpper wth smooth roller gudes of the base jaws Feld of applcaton Grppng and movng of small to medumszed
More informationNumerical Analysis of Rapid Gas Decompression in Pure Nitrogen using 1D and 3D Transient Mathematical Models of Gas Flow in Pipes
Internatonal Journal of Chemcal and Bologcal Engneerng 6 01 Numercal Analyss of Rapd as Decompresson n Pure Ntrogen usng 1D and 3D Transent Mathematcal Models of as Flow n Ppes Evgeny Burlutsky Abstract
More informationApplications on openpdc platform at Washington State University
Applcatons on openpdc platform at Washngton State Unversty Chuanln Zhao Ebrahm Rezae Man V. Venkatasubramanan Washngton State Unversty Pullman WA WSU projects OMS  Oscllaton Montorng System Standalone
More informationAutumn Antics Open Play Day Horse Show
Slver Knolls Spurs 4H Club Presents The Twenty Frst Annual Autumn Antcs Open Play Day Horse Show Saturday, October 7, 2017 Start Tme: 9:00 AM UNR Equestran Center 1290 Valley Rd, Reno, NV 89512 Corner
More informationThe Effect of Pressure on MixedStrategy Play in Tennis: The Effect of Court Surface on Service Decisions
International Journal of Business and Social Science Vol. 3 No. 20 [Special Issue October 2012] The Effect of Pressure on MixedStrategy Play in Tennis: The Effect of Court Surface on Service Decisions
More informationMODEL : LDF7810WW/ LDF7810BB/ LDF7810ST LDF7811WW/LDF7811BB / LDF7811ST LDS5811WW/ LDS5811BB/ LDS5811ST LDF6810WW/LDF6810BB / LDF6810ST
_/ / _ III IRL III" Dshwasher MODEL : LDF7810WW/ LDF7810BB/ LDF7810ST LDF7811WW/LDF7811BB / LDF7811ST LDS5811WW/ LDS5811BB/ LDS5811ST LDF6810WW/LDF6810BB / LDF6810ST Please read ths manual carefully Retan
More informationThe Review of Economic Studies Ltd.
The Review of Economic Studies Ltd. Professionals Play Minimax Author(s): Ignacio PalaciosHuerta Source: The Review of Economic Studies, Vol. 70, No. 2 (Apr., 2003), pp. 395415 Published by: The Review
More informationAalborg Universitet. CLIMA proceedings of the 12th REHVA World Congress Heiselberg, Per Kvols. Publication date: 2016
Aalborg Unverstet CLIMA 2016  proceedngs of the 12th REHVA World Congress Heselberg, Per Kvols Publcaton date: 2016 Document Verson Publsher's PDF, also known as Verson of record Lnk to publcaton from
More informationASSEMBLY INSTRUCTIONS. TECHART roof spoiler
Part no. 070.100.650.009 Roof spoler Contents I Important safety nformaton...3 II Preparaton...5 III Installaton...6 1 Preparng the vehcle...6 2 Fttng the...6 3 Fnal nspecton...6 TECHART Verson 01 Page
More informationCIVL Paragliding and HangGliding Committees CIVL GAP. Centralised CrossCountry Competition Scoring for HangGliding and Paragliding
CIVL Paragldng and HangGldng Commttees CIVL GAP Centralsed CrossCountry Cometton Scorng for HangGldng and Paragldng 2014 Edton Revson 1.1 Publshed March 23, 2014 CIVL GAP Centralsed CrossCountry Cometton
More informationBRAIN INJURY CONFERENCE Tuesday 13 March 2018 Copthorne Hotel, Cardiff
BRAIN INJURY CONFERENCE 2018 Tuesday 13 March 2018 Copthorne Hotel, Cardff Bran Injury Conference 2018 The Hugh James Bran Injury Conference wll be of nterest to all healthcare professonals workng wth
More informationFK240 FLAT SOLAR COLLECTORS ROOFTOP INSTALLATION FOR WORCESTER SOLAR SYSTEMS
FK240 FLAT SOLAR COLLECTORS ROOFTOP INSTALLATION FOR WORCESTER SOLAR SYSTEMS G INSTALLATION INSTRUCTIONS About ths manual Ths nstallaton manual contans mportant nformaton for the safe and approprate nstallaton
More informationOffensive Strategy in the 2D Soccer Simulation League Using Multigroup Ant Colony Optimization
Internatona Journa of Advanced Robotc Systems ARTICLE Offensve Strategy n 2D Soccer Smuaton League Usng Mutgroup Ant Coony Optmzaton Reguar Paper Shengbng Chen *, Gang Lv Xaofeng Wang Key Lab of Network
More informationAMPOMATIC* SideFeed StripperCrimper III Machine No [ ] customer manual TOOLING ASSISTANCE CENTER
AMPOMATIC* SdeFeed StrpperCrmper III Machne No. 1320895[ ] Customer Manual 40910012 28 DEC 11 SAFETY A. PRECAUTIONS READ THIS FIRST!............................ 2 customer manual 1. INTRODUCTION.......................................................
More informationProduct Information. Gripper for small components MPGplus 40
Product Informaton MPGplus 40 MPGplus More powerful. Faster. Longer fngers. MPGplus grpper for small components 2fnger parallel grpper wth smooth roller gudes of the base jaws Feld of applcaton Grppng
More informationHUMAN POWER TECHNICAL JOURNAL OF THE IHPVA. Number 50 Spring 2000 $5.50 NUMBER 50, SPRING 2000
HUMAN POWER TECHNICAL JOURNAL OF THE IHPVA NUMBER 50, SPRING 000 Number 50 Sprng 000 $5.50 Summares of artcles n ths ssue; mast................ Contrbutons to Human Power....................... Artcles
More informationCoastal Cutthroat Trout Maturity Studies in Southeast Alaska,
Fshery Data Seres No. 1332 Coastal Cutthroat Trout aturty Studes n Southeast Alaska, 1997 1998 by Roger D. Hardng Alaska Department of Fsh and Game July 2013 Dvson of Sport Fsh and Commercal Fsheres Symbols
More informationChampionship Draw Template team Premier Leagues
hamponshp raw Template 0 team Premer Leagues MINTON, SKETLL, FENING, FOOTLL, FUTSL, HOKEY, LROSSE, NETLL, RUGY LEGUE, VOLLEYLL, WTER POLO Last (round ) Quarter s Sem s st Prem N th Prem S th Prem N nd
More informationPower Generation Scheduling of Thermal Units Considering Gas Pipelines Constraints
World Academy of Scence, Engneerng and Technology Internatonal Journal of Electrcal and Computer Engneerng Vol:3, No:1, 29 ower Generaton Schedulng of Thermal Unts Consderng Gas pelnes Constrants Sara
More informationBy BETTY DEBNAM. All About Japan. Many tourists visit Mount Fuji, Japan's highest mountain.
41 (98) Especally and for famles ther e 1998 by Unversal Press Syndcate By BETTY DEBNAM from The Mn Page by Betty Debnam @1998 Unversal Press Syndcate The Host Country of the Wnter Olympcs All About Japan
More informationDOMINIQUE MALTAIS DETERMINATION OLYMPIAN STORIES CANADIAN OLYMPIC SCHOOL PROGRAM 2014/2015. SILVER LEVEL Grades 45.
OLYMPIAN STORIES DOMINIQUE MALTAIS DETERMINATION CANADIAN OLYMPIC SCHOOL PROGRAM 2014/2015 SILVER LEVEL Grades 45 www.olympc.ca/educaton JUSTIN WADSWORTH FAIR PLAY TEACHERS GUIDE CREATING A CANADIAN OLYMPIC
More informationLETTER Size and functional diversity of microbe populations control plant persistence and longterm soil carbon accumulation
LETTER Ecology Letters, (2005) 8: xxx xxx do: 10.1111/j.14610248.2005.00813.x Sze and functonal dversty of mcrobe populatons control plant persstence and longterm sol carbon accumulaton Sébasten Fontane
More information20 fascinating facts about the Paralympics
20 fascnatng facts about the Paralympcs By James Moore Fled under: Holdays, Stayng n After the drama of the Olympcs come the equally mpressve Paralympcs. Ths year s set to be the bggest ever staged. But
More informationBy BETTY DEBNAM. InLine Skating
49 (94) Especally for and famles ther e I 994 by Unversal Press Syndcate By BETTY DEBNAM from The Mn Page by Betty Debnam 994 Unversal Press Syndcate In. It'S In, In,., InLne Skatng Inlne skatng s the
More informationMANUAL ROAD BIKE PURE CYCLING. Your bicycle and this manual comply with the safety requirements of the EN ISO standard.
PURE CYCLING MANUAL ROAD BIKE 1 13 14 2 3 4 a 15 16 5 6 7 8 9 10 11 12 e d c b 17 18 19 20 21 22 23 24 25! Your bcycle and ths manual comply wth the safety requrements of the EN ISO 42102 standard.! Attenton!
More informationi Rear foot and midfoot OPCT City Man OPCT City Lady 6 OPCT S OPCT O OPCT CITY
OPCT Cty Man OPCT Cty Lady DRESS SHOES DRESS SHOES Ths orthotc offers the best combnaton of comfort and thnness. Wth ts narrow pattern and leathereffect top layer, t s perfectly suted to street shoes.
More informationTaekwondo roundhouse kick leg technique biomechanical feature research and application
ISSN : 09747435 Volume 0 Issue 3 Taekwono rounhouse kck leg technque bomechancal feature research an applcaton Yunpeng Meng Insttute of Physcal Eucaton, Fuyang Normal College, Fuyang, Anhu, (CHINA) BTAIJ,
More informationCAUTION When using any equipment, always follow the manufacturer s instructions for safe operation.
Cleanng 3M Controltac Decorated Graphc Vnyl Awnng Flmand Sgn Facngs Instructon Bulletn 6.1 Release I, Effectve March 2017 (replaces H, Oct 03) These cleanng nstructons are furnshed as a gude. The enduser
More informationEvaluation of Sustainable Development in the Member States of the European Union. Ocena poziomu zrównoważenia rozwoju krajów Unii Europejskiej
PROBLEMY EKOROZWOJU PROBLEMS OF SUSTAINABLE DEVELOPMENT 2015, vol. 10, no 2, 7178 Evaluaton of Sustanable Development n the Member States of the European Unon Ocena poomu równoważena rowoju krajów Un
More informationChapter 3 Reserve Estimation. Lecture notes for PET 370 Spring 2012 Prepared by: Thomas W. Engler, Ph.D., P.E.
Chapter 3 Reserve Estmaton Lecture notes for PET 370 prng 2012 Prepared by: Thomas W. Engler, Ph.D., P.E. Volumetrc N ol n 7758A B o place, n h f (1 1 ) here, N = ol n place, stb A = dranage area, acres
More informationBICYCLE MANUAL ROAD BIKE
PURE CYCLING BICYCLE MANUAL ROAD BIKE 1 13 14 2 3 4 5 c a 15 16 17 6 7 8 9 10 11 12 e d b 18 19 20 21 22 23 24 25 Attenton! Assembly nstructons page 10. Before your frst rde please read pages 59.! Your
More informationu.s. til_rala Seni.ce Al_1taa o.t.r CoIItlM11tal Shelf legion U.S. u.,c. of Iat.dor 603,,., EMt 36tla A.... A8eb0ra, Aluke or luis SIA PI.
OCSt.., "'1~2 Ecologc.1 Allocue. SY'tUsS 0' ~c. (( 1'B PnCfS OP MOSE Al) DsmuA1K2 a, Uc. ADLOft m.::da1'ons or lus SA P.Pms fr LGL Muke rda Aaeoc:at_, nc 505 "t ortbera Lllbta.1,"Sat. 201 ABdaonp, AlMke
More informationThe final set in a tennis match: four years at Wimbledon 1
Published as: Journal of Applied Statistics, Vol. 26, No. 4, 1999, 461468. The final set in a tennis match: four years at Wimbledon 1 Jan R. Magnus, CentER, Tilburg University, the Netherlands and Franc
More information#16  Profiles & HMMs 9/28/07
#  Profles & HMMs 9/28/7 BCB 444/544 Lecture Profles & Hdden Markov Models (HMMs) #_Sept28 Requred Readng (before lecture) Mon & Wed Sept 24 & 2 Lecture 4 & 5 Revew: Nucleus, Chromosomes, Genes, RNAs,
More information