Clemson Unversty From the SelectedWorks of Mchael T. Maloney March, 2008 Beatng a Lve Horse: Effort s Margnal Cost Revealed n a Tournament Mchael T. Maloney, Clemson Unversty Bentley Coffey, Clemson Unversty Avalable at: https://works.bepress.com/mchael_t_maloney/6/
Beatng a Lve Horse: Effort s Margnal Cost Revealed n a Tournament Bentley Coffey & M.T. Maloney * Abstract: There s ample evdence that ncentve pay structures such as tournaments result n ncreased performance, but whether ths s due to selecton or ncreased ndvdual effort s less clear. We show that emprcal specfcaton s the key. Msspecfcaton masks ndvdual effort and nterprets t as selecton. Lookng at data on horse racng, we compare a pure selecton model to the Lazear-Rosen tournament model. Whle both models organze the data, the tournament model does a better job, and t says that nearly two-thrds of the ncreased performance assocated wth hgher przes s due to ncreased ndvdual effort. Ths estmate s very smlar to estmates found n ndustral feld studes where performance pay s not structured as a tournament. We corroborate the horse-racng results by lookng at dog racng. Dogs are not expected to respond as elastcally as horses, and emprcally they do not. 1. Introducton Tournament theory s wdely held to apply n almost all labor/lesure settngs. The lterature s robust wth ndustral and manageral applcatons. 1 The model has cachet. 2 The one place that tournament theory s less well receved s n the actual tournament settng. Many people argue and wth some evdence that competng athletes do not vary ther effort levels: they clam that sportng partcpants go all out all the tme. The Lazear-Rosen tournament model, of course, says exactly the opposte. The tournament model says that players adjust ther effort n order to maxmze ther expected payout net of cost. So ndvduals should work harder when more s at stake. These opposng vews make a horse race. Whch of the two models can better organze the data? We propose to answer ths queston by lookng at data on horse and dog racng. Horse racng s an nterestng venue n whch to conduct our test. Frst, the data are rch. We know the feld for each race, odds that rank the horses, and the przes for whch they are competng. However, there s an even more compellng reason to look at horse racng. Horses clearly face a substantal loss * John E. Walker Department of Economcs, Clemson Unversty. Emal: bcoffey &/or maloney @clemson.edu. Thanks go to Skp Sauer for the data, encouragement, and vast nsttutonal knowledge. Helpful comments were receved from semnar partcpants at the Unversty of South Carolna and Clemson Unversty, and from Todd Kendall, Marty Smth, and Robert Tamura. 1 See O Keeffe, Vscusc, and Zeckhauser (1984), Malcomson (1984), Man, O Relly, and Wade (1993), Murphy (1999), Knoeber and Thurman (1984), Ferrall (1996), and Brckley, Lnck, and Coles (1999) to lst a few. 2 One measure s the tme lne of ctatons to Lazear & Rosen (1981). Cumulatve ctatons were around 10 over the frst ten years; they reached 200 by 2001; and are over 800 today. On the other hand, Demsetz (1997) argues that the L-R model has more cachet than t deserves. 1
functon assocated wth breakng down. Wtness the 2006 Kentucky Derby wnner, Barbaro, n hs run at the Preakness. Hence, there s every reason to beleve that ther rders and handlers would be margnally senstve to the cost of effort. On the other hand, horses are dumb anmals. Even more than human compettors, t may be mpossble to margnally vary effort n horses. Dog racng complements our nqury because dogs are not handled by humans durng the race. Thus, t s much less lkely that dogs adjust ther behavor n response to przes. The dog racng data that we have are not as rch as the horse racng data, but stll they provde corroboratve evdence. Ths research s mportant because t shows that only by careful theoretcal and emprcal dentfcaton can the two competng hypotheses be dstngushed. Yet, the polcy mplcatons of the two are dramatcally dfferent. If there s no scope for adjustment n ndvdual behavor then performance pay schemes for exstng workers are wasteful. On the other hand, f ndvduals can vary ther effort then these schemes have mportant effcency effects. We fnd that the tournament model, whch accounts for both selecton and ndvdual-effort effects, s the more powerful n organzng the data. Secton 2 dscusses pror research. Secton 3 dscusses and analyses some aspects of the horse racng data. Secton 4 dentfes the structure of the models and the estmaton strategy. Secton 5 shows the estmates and the comparson of the two models. Secton 6 ntroduces the dog racng data and shows results usng these. Fnally, secton 7 gves some concludng remarks. 2. Theory & Pror Research Everyone agrees that as pay goes up so does performance. However, there are two vews of why ths happens. The all-out model says that ndvduals adjust ther behavor n terms of where and how often they go to work, but when they are at work, effort s produced at a constant, fxed rate. The tournament model has the same feature n terms of selecton. If the expected net payout s not hgh enough, workers stay home. The addtonal feature of the tournament model s that workers respond margnally when they are at work. So the debate s whether observed performance dfferences across venues are attrbutable only to sortng or to margnal effort responses n addton to sortng. The trck for our research s to model the behavor underlyng the two propostons especally as t apples to horse racng data and devse an estmaton strategy. The all-out model s based on sortng horses across races wth horses runnng as fast as they can n each race they enter. The tournament model also recognzes the sortng of horses across races, but dentfes relatve effort levels based on the qualty of the other horses wthn the race and the structure of przes pad across fnshers. Ultmately the showdown between the tournament model 2
and the all-out model s about predcton. We examne how well each can predct the observed speed of horses wthn and across races. Almost all the exstng research shows that tournament competton results n ncreased performance. Ths evdence spans almost every ndustral and commercal settng ncludng sports. Nonetheless, the ssue that we are addressng s the extent to whch tournament structures acheve ths result smply because they provde a settng where self-selecton of partcpants ensures that as pay ncreases, average ablty also ncreases, or whether there s an ndvdual ncentve effect. That s, when the gaps get bgger n tournament przes, ndvduals work harder. Nearly all of the theoretcal tournament lterature smply assumes that ndvduals can respond to ncentves by varyng ther effort. 3 However, t mght not be so. It mght be the case that tournament partcpants once engaged n the contest are unable to adjust margnally. Tournament przes may draw a selecton of partcpants nto a contest, but t may be that they go flat out once the gun sounds. In ths case the observed performance effects n tournament settngs are merely a result of sortng. The best partcpants mgrate to the tournaments wth the hghest payouts. A number of ntrgung questons are rased by comparng these competng assumptons on the behavor of tournament partcpants. One s, does the prze structure tself tell us whch model s better? Surprsngly, at least to us, the answer s no. It turns out that t s somewhat easer to ratonalze an exponentally declnng prze pattern, such as the one that we see n professonal golf and horse racng, n the context of the all-out model as opposed to the tournament model. 4 So we have to turn elsewhere for an answer. The emprcal work on the tournament model has addressed the ssue, but t has never staged a complete test of the competng hypotheses. Nearly all of the evdence (cted n note 1 above) shows the productvty effect of promoton n the context of tournament theory can be ratonalzed n the all-out model. The tournament settng selects people who wll work more hours and more days, but ths research does not show that people work more ntensvely when they are on the job. For nstance, there s some evdence dsputng the applcaton of the tournament model n one of the promnent examples from the commercal sector: 3 In addton to Lazear and Rosen (1981), see Green and Stokey (1983), Carmchael (1983), and Nalebuff and Stgltz (1983), and more recently Fullerton and McAfee (1999). 4 Galton (1902) s credted wth ntatng the queston of the optmal shape of the prze structure, whch he dd mplctly n the context of the all-out model. The tournament model s ambguous about the optmal prze schedule n feld tournaments and most research suggests that effort s maxmzed by payng all or nearly all to frst place, a feature rarely observed n practce. See Clark and Rs (1998a and b), Syzmansk and Vallett (2002), and Syzmansk (2003). 3
up-or-out decsons at large law frms. Whle the treatment of assocates at bg law frms may appear to be a tournament where some are kept and others let go (see Ferrall, 1996), Kordana (1995) argues that ths s merely selecton where assocates are montored drectly and the ones that leave after a gven perod of tme came wth the antcpaton of leavng. 5 Ther motvaton was to acqure human captal from on the job tranng. An early test of the tournament model by Ehrenberg and Bognanno (1990) nsghtfully attempted to show that there was an ncentve effect on top of the selecton effect. They dd ths by lookng at the performance of golfers n the fourth round of a tournament based on ther poston after the thrd round. Ther fndng s that the prze gaps affect performance. More recent work has been less conclusve. 6 Moreover, the selecton effect s not explctly accounted for. Knoeber and Thurman (1994) fnd that the level of pay holdng constant the ncremental pay does not affect performance n rasng chckens. They do fnd that movng from a tournament-style pay structure to a relatve performance pay schedule does ncrease performance. They clam that ths s evdence that workers can and do margnally adjust ther behavor. However, ther data do not allow them to control for the selecton effect. The more explct tests pont out the problem wth the methodology appled n the past. Maloney and McCormck (2000) report results that show ndvdual foot racers have better tmes when the przes are hgher, that s, Emly Wood had a faster tme n a $500/$200/$100 race than n a $200/$100/$50 event. Lynch and Zax (2000) dspute ths fndng also lookng at foot racers. We wll show that both of these studes are msspecfed. Indeed, there s no reason to necessarly expect a postve correlaton between effort and przes or purse. The relaton between effort and przes n the tournament model depends on ceters parbus condtons whch nclude the number of other contestants and ther abltes. So, for example, when Emly Wood runs n a race wth a 1 st place prze of $500 where she s the tenth best compettor, she may run slower than when she competes n a $200 race where she s the best. Hence, the postve relaton between performance and przes found by Maloney and McCormck and the lack of a relaton found by Lynch and Zax are both spurous. Our paper s about the specfc case of tournament compensaton and performance, but the more general ssue s that of the ncentve effects of performance-based compensaton. There has been a good bt of research on ths. 5 Ferrall (1996) attempts to compare a tournament model of promoton (the frm keeps k from a cohort of n) to promoton decsons based on a standard of performance. He models the behavor of entry-level lawyers who have unknown talent, even to themselves, so t s a model of symmetrc behavor. Moreover, on close nspecton hs estmatng form does not dfferentate behavor n the two regmes, whch makes the emprcal results unnformatve. 6 Bronnars and Oettnger (2001). 4
Prendergast (1999) surveys many papers. 7 All fnd performance responses from ncentve based compensaton. Of these, only a few (Lazear 1996, Paarsch- Shearer 1999, and Ferne-Metcalf 1999) have data that allow the selecton effect to be separated from ndvdual responsveness to ncentves. In partcular, Lazear fnds that n the auto-glass ndustry about a thrd of the overall ncrease n productvty of pece rate pay came from poorer workers beng replaced by superor ones. As Prendergast says, "even n stuatons were there s evdence consstent wth agency theory, the lterature has been plagued wth dentfcaton problems where outcomes are often equally consstent wth other plausble theores" (p. 11). Our results show that ths s ndeed true. Both models of pure selecton and ndvdual response organze the horse racng data. Nonetheless, the nterpretatons are substantally dfferent. In the results that we present below, the tournament model appled to horse racng shows that two-thrds of the performance ncrease observed when przes ncrease s due to ncreased ndvdual performance and only one-thrd to selecton. (Notce that ths s very smlar to Lazear s fndngs n the auto-glass ndustry). The alternatve hypothess mstakenly attrbutes all of the ncreased performance to selecton. 3. Horse Racng Data The horse race data are from 712 races conducted at Churchll Downs n 1994. 8 We used 566 that pad to four places, had no multple entres by the same owner, tes n the prze wnnng places, and other anomales that caused the prze purse and the odds to dverge substantally, and had all horses fnsh the race. 9 The 566 races were comprsed of 27 stakes, 186 allowance, 2 starter allowance, 7 See Lazear (1996), Paarsch and Shearer (1996), Banker, Lee, and Potter (1996), Ferne and Metcalf (1996), McMllan, Whalley, and Zhu (1989), Groves, Hong, McMllan, and Naughton (1994), Kahn and Sherer (1990), and Foster and Rosenzweg (1994). More recent papers nclude Banker, Lee, Potter, and Srnvasan (2000), Brckley and Zmmerman (2001), and Lemmon, Schallhem, and Zender (2000). 8 These data were made avalable to us by Raymond Sauer who obtaned them when he was a Professor of Equne Studes at the Unversty of Lousvlle n the 1980s. 9 The bggest races, such as the Kentucky Derby, pad to fve places. However, to mantan homogenety n our sample for estmaton purposes, we excluded the few races payng 5 places. When there are multple horses by the same owner entered n a race, they are bet as a group so separate odds for each horse are not avalable. Tes happen and the przes are shared. From our data we only know the przes pad, not offered, so we are forced to delete races wth tes. When horses break down or pull up, ther fnshng tme s not avalable. Whle these outcomes are nterestng, we omt these races for fear that they wll bas the test between the two models. There s no way to treat these mssng observatons wthout favorng one model or the other. 5
289 clamng, and 62 maden races. 10 Table 1 shows the summary statstcs on the races. Table 2 shows the summary statstcs on the horses n the races. The most common dstance s 6 furlongs (4290 feet); the second most common s 1 1/16 mles. The average feld-sze s ten horses. The track purse pad to the horses vared from $7,320 to $233,950. On average, frst place pad 65 percent of the purse, second place 20 percent, thrd 10 percent, and fourth 5 percent. The par-mutuel bettng pool vared from $45,360 to $1,721,432 wth a mean of $155,004 and a medan of $132,792. 11 The sze of the bettng pool s postvely related to the purse of the race because the purse attracts faster horses. The sze of the bettng pool s also negatve related to the dsparty of horses n a race. There are several ways to characterze the dsparty of the feld. One s to look at the dstrbuton of the odds of wnnng across the horses. We can construct an entropy measure from the probablty of wnnng (prob ). It s prob ln( prob ) across the horses n each race. The entropy of the probabltes of wnnng ncreases as the odds become more equal across the horses. Alternatvely we can use the varance of the log of the probablty of wnnng 1 st place. Table 1 shows how these measures vary across the sample. Table 3 shows a regresson of the log of the bettng pool on the log of the purse and the log of the entropy measure. Both entropy and the varance of the log of the probablty of wnnng 1 st place are based on bettors' predctons about the performance of each horse. Both specfcatons show that dsparty of talent reduces spectator nterest. The fnal regresson n column (c) ncludes the actual race-outcome statstcs n the regresson. It shows that spectator nterest antcpates the outcome and spectators are most nterested n races that are fast, close, and have a lot of horses. By way of characterzng the magntudes of these coeffcents, a one standard devaton move n purse changes the bettng pool by slghtly more than 20 percent, a one standard devaton move n the varance of speed changes the bettng pool by 8 percent, and a one standard devaton move n the varance of the probabltes of each horse wnnng changes the bettng pool by 5 percent. 4. Models of Performance & Estmaton Strategy Horse beats horse j ff: 10 Maden races are for horses who have never won. In clamng races, horses entered n the race are up for sale at a posted prce. In allowance races, the horses are handcapped by carryng more or less weght based age, sex, and past performance. A starter allowance race s an allowance race for horses that have started for a gven clamng prce or less. In a stakes race the horse owners pay a fee to be n the feld. 11 Par-mutuel bettng means that the odds on each horse are determned by the amount of money bet on each after the bettng wndows close just before the begnnng of the race. 6
ln m + ε > ln m j + ε j where m s effort, unobserved by the econometrcan but known to the rders, and ε s luck, unobserved by the econometrcan but revealed to the rders. Table 1. Summary Statstcs on Races Varable Mean Std Dev. Mnmum Maxmum Dstance 4882 749 2970 9240 Feld-sze 10 2 4 12 Purse 28748 26122 7320 233950 Frst Place Prze 18663 16974 4758 152068 Second 5757 5273 1464 46790 Thrd 2889 2635 732 23395 Fourth 1440 1317 366 11698 Percent of Frst Place Prze to Purse 64.9 2.1 57.4 70.8 Second 20.0 1.6 15.3 25.2 Thrd 10.1 0.9 7.5 14.2 Fourth 5.0 0.4 3.7 6.5 Bettng Pool ($1000s) 153.41 110.14 45.36 1029.65 Varance log 1 st Place Probablty 1.01.38.09 2.28 Entropy 1.88.22 1.14 2.38 Notes: 566 races. Entropy s constructed from the probablty of each horse wnnng; as entropy ncreases the dsparty of talent n the race declnes. Purse, przes, and bettng pool n 1994 dollars. Table 2. Summary Statstcs on Horses across Races Varable Mean Std. Dev. Mnmum Maxmum Speed 54.35 2.44 48.89 60.69 Odds 20.45 24.91 0.30 260.50 Notes: 5504 Observatons. Speed n feet per second. Odds are the par-mutuel bettng odds of wnnng 1 st place. Let luck be d across the feld accordng to an extreme value dstrbuton. Ths produces the famlar logt form for the predcted probablty of horse wnnng the race: m Pr( places 1st) = (1) m j= 1, N We observe the probablty of each horse wnnng through the gamblng odds. j 7
Table 3. Predctng the Sze of the Bettng Pool Independent Varables (a) (b) (c) Purse 0.365 0.306 0.350 (0.042) (0.037) (0.041) Entropy 0.356 (0.087) Varance of log Probablty of Wnnng 1 st Place -0.123-0.135 (0.043) (0.047) Feld Sze 0.400 (0.089) Average Speed 0.800 (0.345) Varance of Speed (x 100) -12.239 (2.182) Intercept 7.486 8.88 4.468 (0.517) (0.359) (1.402) R 2 0.225 0.210 0.274 Notes: Sze of the par-mutuel bettng pool, track purse, feld sze, and speed n logs. Robust standard errors below coeffcent estmates. 566 observatons. To estmate total effort, we turn to the optmzng behavor of track owners. In horse racng, the track owner's profts ncrease wth the sze of the bettng pool. The volume of bets are ncreasng n total effort f gamblers prefer to bet on the more exctng races and f exctement s created by the effort of the horses n the race as s revealed n Table 3. Hence, we use the observed par-mutuel pool of wagered funds as a proxy for total effort n each race. Takng logs and rewrtng, we have: st ln m = ln Pr( places 1 ) + φ ln B + u (2) where B s the bettng pool of funds, φ s the nverse of the elastcty of the bettng pool wth respect to effort, and u s a whte nose term that vares by race. Equaton (2) s a reduced form relaton between effort and bettng pool. There are other characterstcs of the race that affect the bettng pool and the track owner's proft as shown n Table 3. Equaton (2) smply argues that the sum of effort n a race can be monotoncally proxed by the bettng pool. Log effort can also be estmated from nformaton extracted from a structural model of behavor. The d extreme value assumpton on luck allows us to compute the probablty of horse fnshng at any place n the money from the 8
probablty of each horse wnnng 1 st place. 12 Ths allows us to work wth the problem faced by horse, whch s to fnd the level of effort that maxmzes the expected prze net of the cost of effort: w w k C m st th 1 Pr( places 1 ) +... + k Pr( places ) (, α ) where w k s the prze pad to k th place and α s heterogeneous ablty causng dfferent rders to select dfferent levels of effort. 4.1. Case 1: The Tournament Model (Costly Effort) Assumng constant margnal cost, β, we can wrte: C( m ) = βm / α where agan α s ablty. The frst-order condton for the optmzng behavor of the rder s: th Pr( places k ) 1 β wk = ln m m α (3) k where the left-hand sde s the sum of the margnal effect of effort on the probablty of fnshng n the money weghted by the prze for that fnshng poston and the rght-hand sde s the margnal cost of effort. To smplfy notaton, we defne x: x th Pr( places k ) = w ln m k whch can be calculated from the observed probabltes gven our dstrbutonal assumpton on luck. By loggng the frst order condton, we obtan an addtonal expresson for log effort: 12 The logt probabltes generate the well-known Independence of Irrelevant Alternatves n the context of a random utlty model of ndvdual choce. In the context of a race, the logt probabltes generate an Independence of Irrelevant Compettors. Ths allows us to compute the probablty of a horse wnnng second place gven some other horse won frst place: the probablty of beatng the remanng horses. The remander of the computaton nvolves gong from the condtonal probablty to the jont probablty and then onto the margnal probablty. 9
ln m = ln α lnβ + ln x Addng and subtractng the wthn-race mean log effort, we obtan: ( ) ln m = lnβ + ln x + ln α + ln α ln α In order to capture mean log ablty wthn a race, we need to model how horses are selected nto races. 13 The nsttutonal detal of horse-racng suggests that horses are sorted nto races accordng to ther ablty and that ths stratfcaton s mrrored by the purse pad to the feld and horse characterstcs ncludng age and sex. In ths case, we can use a smple reduced form model of selecton: ( ) ln m = lnβ + ln x + γ Q + ln α ln α + v (4) where Q are the selecton crtera of purse, sex, and age, γ s the vector of elastctes of mean log ablty wth respect to the selecton crtera, and v s a whte nose term that vares at the race level. 14 Equatons (4) and (2) represent a system of equatons that dentfes the predcted level of effort by a horse based on the tournament model of behavor where effort s costly. 4.2. Case 2: The All-Out Model (Costless Effort) Now let the horse s cost functon be: (, ) C m 0 m bα α = m > b α 13 If horses were randomly selected nto races, then we would expect that the wthn-race mean log ablty would equal the populaton mean log ablty, whch could be set to zero. 14 The data are comprsed of 5504 observatons on ndvdual horse performance n 566 races. There 1349 horses that raced only once, and 1420 that raced multple tmes, a few as many as eght tmes. One mght argue that ablty s constant for these horses across races, but t can also be argued that ablty vares based on the vagares of lfe. We estmate the model both ways. The results are unaffected. See footnote 15, below. Moreover, by estmatng the model n an unrestrcted form, that s, treatng each horse n each race as an ndependent observaton, we then use the estmates of ablty across races n tests of model ft. 10
where b s a scalar and, agan, α s the ablty ndcator. Gven ths functon, the optmal soluton to the rder s optmzaton problem s smply to exert maxmum effort: m = bα (5) Note that ths would also be the soluton to the rders problem f the objectve was maxmzng the probablty of wnnng subject to the maxmum effort constrant. Substtuton of equaton (5) nto equaton (1) above yelds: Pr( places 1 st) = α j= 1, N Loggng equaton (5) and performng the analogous substtutons as case 1 yelds: α ( ) ln m = ln b + γ Q + ln α ln α + v (6) That s, we add and subtract the average log ablty level n each race. Subtractng t from the log ablty of each horse gves us a mean zero term of relatve ablty n a race. We then proxy for average log ablty n the race usng the race purse, sex, and age as the sortng mechansms mrrorng the tournament organzer. Equatons (6) and (2) represent a system of equatons that dentfes the predcted level of effort by a horse based on the all-out model where effort s costless. j 4.3. Completng the Models We have constructed two equatons for each case that predct log effort. Unfortunately, log effort s not drectly observable. However, we do observe speed and t s monotonc n performance, that s, log effort plus d luck: ( ln ) ln s = δ T + θ D + λ m + ε where s s speed, T s a set of dummes for track condtons wth δ effects, D s a set of race dstance dummes wth θ effects allowng speed to slow wth greater dstances holdng effort constant, and λ converts performance nto speed. Substtutng our two effort equatons nto the speed equaton yelds estmable forms for the two cases: 11
Case 1: st ln s = δ T + θ D + λε + λ ln Pr( places 1 ) + λφ ln B + λ u + ( ε ε) ln s = δ T + θ D + λ lnβ + ε [ ] ( ) ( ) +λ ln x + λγ Q + λ ln α ln α + v + ε ε and Case 2: st ln s = δ T + θ D + λε + λ ln Pr( places 1 ) + λφ ln B + λ u + ( ε ε) [ ] ( ) ( ) ln s = δ T + θ D + λ ln b + ε + λγ Q + λ ln α ln α + v + ε ε (7) (8) The left-hand sde of equatons (7) and (8) s observed, the last term on the rght-hand sde of each equaton s a mean-zero d dsturbance term, and the remander of the rght-hand sde s observed regressors and ther parameters. Note that the only effectve dfference between the two models s the ncluson of ln x (the log margnal beneft of effort relatve to effort) n the second equaton of case 1, the tournament model. Its parameter, λ, also dentfed n the frst equaton, s restrcted to be the same across the equatons when estmated. Each structural parameter s fully recoverable. The parameter λ s dentfed off of the log probablty of wnnng n the frst equaton n the all-out model and also the log of the margnal beneft of effort relatve to effort n the tournament model; average luck s the ntercept of the frst equaton dvded by λ; φ s the quotent of λ dvded nto the coeffcent on the log of the bettng pool; u s estmated as the race-specfc mean resdual (dvded by λ) of the frst equaton, luck s devaton from the mean s the resdual (dvded by λ) less u; β (or b for case 2) s dentfed from the ntercept of the second equaton (removng mean luck as recovered from the frst equaton); γ are the quotent of λ dvded nto the coeffcents on the selecton crtera; v s estmated as the race-specfc mean resdual (dvded by λ) of moment 2; and log ablty s devaton from the mean s the resdual less demeaned luck and u (dvded by λ). The estmaton can be performed wth GMM by stackng the moments. To smplfy notaton of the stacked equatons, let y, Z, and ξ be defned as follows: Z ln s y = ln s st ( ) T D 1 0 ln Pr places 1 ln B 0 = T D 1 1 { ln x;0} 0 Q 12
δ θ λε ξ = λ β λ λφ λγ { ln ;ln b} where the bracketed expresson are the alternatve specfcatons for case 1 and case 2. The estmaton can be performed wth GMM under the approprate moment condtons on the dsturbance terms: ( ) u + ε ε E Z ' = 0 ( ln α ln α ) + v + ( ε ε) ( ) E Z ' y Zξ = 0 The frst stage of GMM s equvalent to OLS and the second s a mnmum dstance estmator where dstance s weghted by the nverse of the covarance of the moments. The estmator s gven by: (( ) ( )) ( ) ( ) 1 ˆ ˆ 1 ˆ 1 ξ = Z ' Z V Z ' Z Z ' Z V Z ' y Var( ξˆ ) where V s the covarance of the moments, estmated by: ˆ u + ( ε ε ) u + ( ε ε ) = ' ( ln α ln α ) + v + ( ε ε ) ( ln α ln α ) + v + ( ε ε ) 1 V Z E Z Under the assumpton of the model, the expectaton of the cross-product of the dsturbances works out to be the sum of the followng varance-covarance matrces sandwched by the covarates. The frst matrx captures dsturbances that are common across all horses n a gven race; the second matrx captures how a horse s dosyncratc luck appears n both equatons and heteroskedastcty n the varance of ablty across races. 13
2 2 σu σu σuv σuv 2 2 σu σu σuv σ uv 2 2 σu σu σuv σ uv 2 2 σu σu σuv σ uv + 2 2 σuv σuv σv σv 2 2 σuv σuv σv σv 2 2 σuv σuv σv σv 2 2 σ uv σuv σv σv 2 2 σε σε 2 2 σε σε 2 2 σε σε 2 2 σε σε 2 2 2 σε σ ε +σ1ln α 2 2 2 σε σ ε +σ1ln α 2 2 2 σε σ ε +σr lnα σ σ +σ 2 2 2 ε ε Rlnα 5. Results The parameters are shown n Table 4 wth the standard errors n parentheses. Every parameter assocated wth the two models has the expected sgn and all but one are statstcally sgnfcant. 15 The tournament model has a hgher R 2 than the all-out model. Ths ndcates that t predcts log speed better and wns the horse 15 As mentoned n note 14 above, we estmated the models allowng for ndvdual horse ablty to vary from race to race, as reported n Table 4, and holdng ndvdual horse ablty constant across races. The latter estmates are nearly dentcal to those reported n Table 4: λ s slghtly lower whch makes the elastctes on purse and pool slghtly hgher. 14
race between the two. Although the margn does not appear large, t s statstcally sgnfcant. We run a non-nested test of the two models n whch the costly model shows the all-out model to be msspecfed. The test begns wth a slght alteraton that combnes the two models nto one varyng by the parameter, τ. Ths parameter ndcates whch model appears: τ = 1 ndcates the tournament model and τ = 0 ndcates the all-out model: st ( ) ( ) lns = δ T + θ D + λε + λ ln Pr places 1 + λφ lnb + λ u + ε ε [ ] ( ) ( ) ln s = δ T + θ D + λε + + τλ ln x + λγ lq + λ ln α ln α + v + ε ε The parameter of nterest s recovered wth the same GMM estmaton, dvded by the estmated λ, and where the approprate adjustments are made to the standard errors. The estmated value of τ s 1.327 wth a standard error of 0.120; the null that the costless-effort model s correctly specfed gets rejected at the.01 level. 16 Although t s stll sgnfcantly postve, the mportance of the purse drops from the all-out model to the tournament model. Ths s exactly the result that we antcpated. The appearance of ln x n the second equaton of the tournament model dmnshes the nfluence of the purse. The tournament model attrbutes race-level varaton n speed to both a selecton affect (the purse drawng horses of a certan level of ablty) and the margnal beneft of effort (ncreasng n the purse as captured n ln x). The all-out model can only attrbute race-level varaton n speed to a selecton effect and hence wll enlarge the mportance of the selecton effect. The all-out model msnterprets ndvdual responses as selecton effects. The other selecton varables, age and sex, are smlar across the models and ther estmated effects on speed are also consstent wth our expectatons. Female horses and two year olds (the omtted age group) run slower on average. Also, our controls for dstance and track condtons follow the antcpated pattern. Horses run slower over longer dstances and faster on turf compared to drt. 17 16 A value of τ sgnfcantly dfferent from zero says that the all-out model s an ncorrect specfcaton and, strctly speakng, a value dfferent from one ndcates the same for the tournament model. Dependng on how conservatve the level of sgnfcance s, the tournament model s also msspecfed. Even so, we nterpret the value of 1.3 to mean that the tournament model s a better specfcaton than the all-out model. 17 We tred many varatons on the control for dstance. In the end, dummy varables for dfferent dstance groups were the most parsmonous specfcaton. We grouped races nto 6 furlongs or less, 6 ½ to 7 ½ furlongs, and 1 to 1 ¼ mle, allowng races longer than 1 ¼ mle to be the excluded group. Alternatve specfcatons for dstance have vrtually no effect on the estmated values of the coeffcents of nterest. 15
Table 4. Parameter Estmates for the Speed Equatons from the Two Models Tournament Parm. Meanng Model: Costly Effort All-Out Model: Costless Effort δ 1 Turf Surface n Good Condton 0.029 0.029 (0.003) ** (0.003) ** δ 2 Turf Surface n Bad Condton 0.005 0.005 (0.007) (0.007) δ 3 Drt Surface n Good Condton 0.008 0.008 (0.003) * (0.003) * θ 1 Short Dstance 0.130 0.129 (0.007) ** (0.007) ** θ 2 Medum Dstance 0.056 0.055 (0.007) ** (0.007) ** θ 3 Long Dstance 0.031 0.030 (0.007) ** (0.007) ** λ Elastcty of Speed wrt Performance ( 10 3 ) 6.371 5.680 (.245) ** (.254) ** φ Inverse Elastcty of Pool wrt log Total Effort 1.850 2.080 (0.331) ** (0.381) ** ln β or ln Margnal Cost or ln Max Effort over Ablty -9.604 6.813 ln b (3.556) * (4.102) γ 1 Elastcty of Mean log Ablty wrt Purse 0.606 1.915 (0.195) ** (0.216) ** γ 2 Flles and Mares (females) -1.159-1.292 (0.263) ** (0.302) ** γ 3 Age = 3 years 5.418 6.096 (0.640) ** (0.761) ** γ 4 Age = 3 years and up 4.552 5.103 (0.500) ** (0.589) ** γ 5 Age = 4 years and up 4.341 4.883 (0.533) ** (0.638) ** R 2 0.809 0.800 Correlaton of Ablty for a Gven Horse across races 0.932 0.712 Kolmogorov-Smrnov test statstc for: log ablty ~ log normal 0.186 0.088 luck ~ extreme value 0.106 0.111 Skewness of Luck -0.527-0.573 Kurtoss of Luck 4.239 4.303 Notes: GMM estmates of two equaton model; absolute t-statstcs below coeffcents. Dependent varable s speed. All varables except the dummes for track are n logs. Observatons: 5504. Skewness : 1.1 for extreme value, 0 for normal. Kurtoss: 5.4 for extreme value, 3 for normal. Null for Kolmogorov-Smrnov test s that the dstrbutons are as predcted. Both tests reject the null. Sgnfcance levels shown by: one star ndcates sgnfcance at the 5% level, two stars at the 1% level. Short dstance s 6 furlongs or less, medum s 6 ½ to 7 ½ furlongs, and long s 1 to 1 ¼ mle. Dstance longer than 1 ¼ mle s the excluded group. 16
The value of margnal cost n the tournament model s statstcally sgnfcant at the 1 percent level. The magntude s also economcally reasonable. We can calculate the total varable cost of a horse enterng a raced by the antlog of the estmated coeffcent tmes effort dvded by ablty. The average over all horses s $835. 18 The favorte n the race wth the feld of average ablty had an expected payout of $5227 and expected varable cost of $314. 19 The all-out model says that all of the ncrease n effort assocated wth ncreased przes s due to selecton. The tournament model breaks the ncrease n effort assocated wth ncreased przes nto a selecton effect and ncrease n ndvdual performance. The rato of ndvdual performance to total can be wrtten as: ln x lnβ ln x lnβ + γq where the numerator s the ndvdual performance effect and the denomnator adds n the selecton effects of purse, age, and sex. Based on the estmated values for γ and lnβ, and averaged over the entre sample, we fnd a value of 63 percent. Ths says that nearly two-thrds of the ncrease n performance assocated wth payng hgher przes comes from ndvdual horses runnng harder. 5.1. Dagnostcs Table 4 shows the correlatons of estmated ablty for the same horse n dfferent races. There are 1420 horses that raced multple tmes. In our estmaton process, we treated these as ndependent observatons. By dong ths, we can emprcally assess how ablty changes race to race, but also we can compare the two models. As shown n Table 4, the tournament model says that the correlaton of estmated ablty from race to race s over 0.9; the correlaton of estmated ablty for the same horse from the all-out model s 0.7. Specfcaton tests ndcate a reasonable ft. Wthn a race, effort has a strctly postve monotonc relatonshp wth ablty n the tournament model. (Obvously, n the all-out model effort equals ablty.) The followng two fgures show the emprcal densty functon of log ablty alongsde the closest fttng log normal CDF. The tournament model s shown n Fgure 1. The all-out model s shown n Fgure 2. The test statstc shown n Table 4 rejects the null that log ablty s dstrbuted log normal, but the patterns shown n Fgures 1 and 2 ndcate that 18 The average for log effort s 19.12; the average for log ablty s 9.70. 19 We averaged the estmated ablty from the tournament model across horses n each race and then took the average of these. The favorte n the race wth the feld of average ablty fnshed second. 17
lognormal may appeal as a rough approxmaton, partcularly for the costless model. Although the model specfcaton assumes an extreme value dstrbuton for luck, the data do not appear to be extreme value. Just as n the non-nested model specfcaton test, the costless model s more msspecfed on ths dmenson than the costly model. The Kolmogorov-Smrnov test statstcs n Table 4 reject the null that luck s dstrbuted extreme value. When we nvestgate we see that the sample skewness s about half of the magntude of the assumed extreme value and a dfferent sgn; the luck dstrbuton appears to be skewed left (.e., a fat left tal) nstead of skewed rght. Moreover, the emprcal luck dstrbuton exhbts thnner tals (lower kurtoss) than the assumed dstrbuton; nstead, the kurtoss s somewhere n between the extreme value and normal dstrbutons. The extreme value assumpton s made n order to employ a closed form soluton n the calculaton of ln x and relatng effort to odds n (2). We could, wth some nontrval degree of dffculty, relax ths assumpton and use a more flexble dstrbuton of luck to derve ln x. Ths would probably mprove the goodness-offt for the tournament model relatve to the all-out model snce ln x s a regressor n equaton (7) but not n (8) and (2) s the same for both models. Snce we have already found that the tournament model s superor to the all-out model, we forgo ths task. Fgure 1. Emprcal and Hypotheszed CDF of log Ablty for the Tournament Model Fgure 2. Emprcal and Hypotheszed CDF of log Ablty for the All-Out Model Fnally we nvestgated the senstvty of both models to the frequency of favortes wnnng. Both models say that a dose of luck affects the outcome and stops the fnshng postons from beng perfectly correlated wth the odds. If the 18
models are systematcally mssng somethng assocated wth favortes t should show up as a correlaton between estmated luck and the average fnshng poston of the favortes. We defne favortes as the top three and the top fve horses and correlate estmated luck wth the average fnshng poston of these horses. The correlatons are unformly zero. 5.2. Reduced Form Estmates Further evdence comparng the ablty of the two models to organze the data can be found n a reduced form estmaton of the speed equaton. Essentally, the tournament model says that the performance of a gven horse wll be affected by the performance of the competton that t faces. The all-out model makes no such predcton. One way to nvestgate ths s to see f the revealed speed of a horse s affected by what the horses around t are dong. We regress speed on track, dstance, purse, age, and sex, and on the varance of speed n the race. We alternatvely use the varance of the log of the probablty of wnnng. We use the varance of speed of all the horses and also the varance n speed n the top fve favored horses. These results are shown n Table 5. We see that varance, however measured, causes average speed to declne. The horses appear to be reactng to the competton, and based on the results n Table 4, they are reactng n a way that s consstent wth the tournament model of behavor. Several specfcatons are shown ncludng the ex ante predcton about varance of speed gven by the varance n the bettng odds. The control varable effects are nearly the same as those shown n Table 4, though the magntudes are dfferent because the coeffcents shown n Table 4 are adjusted by the estmated value of λ. 19
Table 5. Reduced Form Estmates of Log Speed Independent Varables Coeffcent Std Error R 2 Intercept 3.720 (0.017) Turf Surface n Good Condton 0.027 (0.003) Turf Surface n Bad Condton 0.002 (0.006) + Drt Surface n Good Condton 0.010 (0.002) Short Dstance 0.136 (0.007) Medum Dstance 0.060 (0.007) Long Dstance 0.033 (0.007) Log of Purse 0.017 (0.001) Flles and Mares (female) -0.009 (0.001) Age = 3 years 0.040 (0.005) Age = 3 years and up 0.035 (0.005) Age = 4 years and up 0.043 (0.005) Varance of Speed Overall -59.816 (14.743) 0.887 Alternatve Specfcatons Independent Varable: Varance of Probablty of Wnnng ( 10 2 ) -0.516 (0.167) 0.882 Dependent Varable: Log Speed of Favortes Independent Varables: Varance of Speed Overall -95.153 (14.926) 0.883 Varance of Speed of Favortes -15.874 (7.474) # 0.868 Varance of Probablty of Wnnng ( 10 2 ) -0.915 (0.175) 0.872 Notes: Alternatve specfcatons nclude all of the track, dstance, and selecton varables. Coeffcent values on these varables change trvally across alternatve specfcatons. Robust standard errors n parentheses. All coeffcent estmates sgnfcant at the 1 percent level except (#) sgnfcant at the 5 percent level and (+) not sgnfcant. Favortes defned as top fve horses based on probablty of wnnng. Dstance dummes defned n Table 4. Another and possbly more powerful test of the tournament model s to look at horses that run multple tmes. We have already ponted out that our correlaton of estmated ablty for horses that run multple tmes s qute hgh, but we can go beyond ths and look at the relaton between estmated effort and the tme between races. If horses vary ther effort levels, then ths varance should be predctably lnked to the tme that t takes them to recover and race agan. That s, 20
f horse runs hard n one race, t wll take longer to recover and the tme untl ts next race should be longer. We measure the tme between races for 1285 horses that raced multple tmes wthn the same season (.e., fall and sprng). There are 2181 observatons. We control for drt versus turf because most races are drt and a turf racer wll have fewer opportuntes. We also control for age and sex because these determne whether a horse can compete n a gven race. 20 The results are shown n Table 6. The ndependent varable of nterest s our estmated value of effort expended by the horse n ts last race. As can be seen, effort expended n the last race sgnfcantly ncreases the tme before the horse races agan. Table 6. Tme Between Races Independent Varables: Coeffcent Std. Error Estmated Effort n Last Race 0.881 (0.168) * Drt v. Turf Race -1.595 (0.578) * Flles and Mares (female) 1.050 (0.389) * Age = 3 years -0.137 (1.419) Age = 3 years and up 0.817 (1.335) Age = 4 years and up -1.975 (1.323) Intercept 0.260 (3.967) R 2 0.046 Mean of Dependent Varable 18.643 Notes: 2181 observatons on 1285 horses. Robust standard errors n parentheses. Dependent varable measured n days. One star ndcates sgnfcance at the 1 percent level. 6. Dog Racng Dog racng offers an nterestng addtonal test of the comparson between the tournament model and the all-out model. Arguably, the effort level of horses can be controlled to some extent by the jockey. In dog racng, there are no jockeys. The dogs, when started, chase around a track after a smulated rabbt. The speed of the rabbt can be adjusted, but there s no way to make ntra-race adjustments for ndvdual dogs. The obvous predcton s that the all-out model should do a better job of predctng dog racng than horse racng. 20 It mght be argued that purse should be ncluded as an ndependent varable n ths regresson. The control varables that we use are physcal condtons of the race. That s, three year olds cannot races n a two year old race. However, three year olds can pck and choose among three-year-old races wth dfferent purses. Regardless, when purse s ncluded, t has the expected sgn (negatve), s statstcally sgnfcant, and does not affect the estmated coeffcent on lagged effort. 21
Unfortunately the nsttutonal characterstcs of dog racng data make the applcaton of our structural model problematc. In dog racng there s no ndependent determnaton of the purse and pool. The purse s a fxed percentage of the pool (slghtly less than 4 percent). Nonetheless we can estmate the reduced form model for dogs as we dd for horses as reported n Table 5. The predcton n the reduced form model s that f compettors are behavng accordng to the tournament model, average performance wll declne as the varance of performance ncreases. When facng more dsparate compettors, workers shrk. The all-out model makes no such predcton. By usng the dog racng data we can refne the predcton: dogs should be less affected by dsparty of the feld than horses. The dog racng data s taken from http://www.greyhound-data.com, a webste run by the nternatonal greyhound racng assocaton. It tracks races, ndvdual dog performance, and racetrack specfc races over tme. The data n queston s specfcally for the Jacksonvlle FL racetrack for the year 2006. We have data on races between June 1 and September 4. There are generally 28 races per day broken nto afternoon and evenng sessons. We have complete data on 1037 races on 77 days. There are two race lengths: 89 percent of the races are 550 yards; 11 percent are 661 yards. These races all have eght dogs. Dogs are graded; dogs of smlar grade are raced aganst each other. There are 6 grades. Dogs vary by age, sex, and weght. The youngest dog raced when t was 16 months old; the oldest when t was 64 months old. Dogs vared between 23 and 40 kg wth the average at 30 kg. Males comprsed 51.2 percent of the dogs n the sample. The fastest dog n the short races ran 54.7 feet per second, and n the long races, 52.7 feet per second. Table 7 shows the reduced form regresson for dogs smlar to the one for horses gven n Table 5. The regresson shows that afternoon races are slghtly faster than evenng races, and that shorter races are 2 percent faster than longer ones. The grades order the qualty of the dogs wth the omtted class beng the fastest. The slowest class, M, whch s for dogs between one and two years old, s 2.6 percent slower than the fastest. Purse, age, weght, and sex do not predct speed, ceters parbus. The varable of nterest s the varance of speed n the race. We see that t does have a statstcally sgnfcant effect. As varance ncreases, average speed goes down. However, as predcted, the effect of varance on speed s smaller for dogs than t s for horses. Dogs do not vary ther effort as much based on the performance of the other compettors as do horses. The responsveness of dogs s approxmately one-thrd of the responsveness of horses. 22
Table 7. Reduced Form Estmate of Speed for Dogs Independent Varables Coeffcent Std. Error Afternoon 0.17 * 0.05 Grade A -0.47 * 0.18 B -1.17 * 0.18 C -1.71 * 0.17 D -2.06 * 0.17 M -2.64 * 0.20 Short Race 2.17 * 0.06 Purse -0.00 0.02 Age Average 0.28 0.19 Varance -0.08 0.60 Weght Average -0.39 0.82 Varance 2.00 3.80 Sex Average -0.24 0.17 Varance -0.03 0.36 Varance of Speed -21.09 * 2.38 R-squared 0.81 Notes: 1037 Races. Dependent varable s log speed. All coeffcents and standard errors except Varance of Speed are multpled by 100 and thus are percentage effects. Purse, age and weght are n logs. Sngle star ndcates 1 percent sgnfcance. Omtted grade s S, whch are the fastest. Fxed effects for date. 7. Conclusons We have nvestgated the performance effect of ncentve-based pay schemes. Everyone agrees that ncentve pay ncreases performance. However, there are two alternatve hypotheses that delver ths result. One model s that of all-out effort. Ths theory clams that compettors go all out, all of the tme, and the only thng that dstngushes one match from another s the sortng of partcpants. The other model s based on the Lazear-Rosen tournament theory. In ths model, players wegh the margnal payoff aganst the margnal cost of effort and vary ther level of exerton from one contest to the next. In the all-out model, ndvduals do not adjust ther effort. In the Lazear-Rosen model, they do. We compare these two theores to see whch better organzes the data. The polcy ssue s mportant because f the all-out model s correct, ncentve pay for exstng workers s neffcent. An mportant feature of our nvestgaton s the theoretcal structure of the emprcal nvestgaton. The tournament model has subtle features and the contrast to the all-out model requres attenton. Our results show that proper structural form s necessary to dstngush ndvdual 23
performance responses from feld selecton effects. We also explot mplcatons of the tournament model that dffer from the all-out model to perform reduced form tests. We examne performance n horse and dog racng. Usng structural estmaton n horse racng we fnd that the tournament model does organze the data more precsely. These estmates show that only around one-thrd of the performance ncrease resultng from hgher przes s due to the selecton effect. Ths result s very smlar to the result found n feld studes such as Lazear (1996) of ndvdual worker performance n ndustral settngs. The superorty of the tournament model and the pont estmate of the ndvdual versus selecton effect s corroborated both n both horse and dog racng by reduced form results. Importantly we fnd that compettors work less hard when talent s more heterogeneous a predcton of tournament theory and that ths effect s stronger for horses than for dogs, as we would expect because dogs are not under drect control by handlers durng the race. References Rajv D. Banker, S. Lee, and Gordon Potter "A Feld Study of the Impact of a Performance-Based Incentve Plan," Journal of Accountng and Economcs, 1996 Vol. 21, No.?, 195-226. Rajv D. Banker, S. Lee, Gordon Potter, and D. Srnvasan, "An Emprcal Analyss of Contnung Improvements Followng the Implementaton of a Performance-Based Compensaton Plan," Journal of Accountng and Economcs, Vol. 30, No.3, December 2000. Brckley, James A., James S. Lnck, and Jeffrey L. Coles, What happens to CEOs after they retre? New evdence on career concerns, horzon problems, and CEO ncentves Journal of Fnancal Economcs, June 1999, 52(3) 293-442. and Jerold L. Zmmerman, "Changng ncentves n a multtask envronment: evdence from a top-ter busness school," Journal of Corporate Fnance 7 2001 367 396. Bronars, Stephen G., and Gerald S. Oettnger, "Effort, Rsk-Takng, and Partcpaton n Tournaments: Evdence from Professonal Golf," manuscrpt, Unversty of Texas, October 2001. Clark D. and C. Rs (1998a) Competton over more than one prze, Amercan Economc Revew, 88, 1, 276-89. Clark D. and C. Rs (1998b) Influence and the Dscretonary Allocaton of Several Przes, European Journal of Poltcal Economy, 14, 605-25. Carmchael, H. Lorne, "The Agent-Agents Problem: Payment by Relatve Output," Journal of Labor Economcs, Vol. 1 (1), January 1983, 50-65. Demsetz, Harold, The Economcs of the Busness Frm: Seven Crtcal Commentares, Cambrdge: Cambrdge Unversty Press, 1997. Ehrenberg, Ronald G. and Bognanno, Mchael L., Do Tournaments Have Incentve Effects? Journal of Poltcal Economy, Vol. 98 (6), December 1990, 1307-1323. Ferne, Sue, and Davd Metcalf, "It's Not What You Pay t's the Way that You Pay t and that's What Gets Results: Jockeys' Pay and Performance," Labour, June 1999 13(2) 385-412. Ferrall, Chrstopher, "Promotons and Incentves n Partnershps: Evdence from Major US Law Frms," Canadan Journal of Economcs, November 1996, 29(4) 811-827. 24