Teaching competition in professional sports leagues

Transcription

1 Workg Paer Seres, Paer No Teachg cometto rofessoal sorts leagues Stefa Szymask May 006 Astract I recet years there has ee some dsute over the arorate ay to model decsomakg rofessoal sorts leagues. I artcular, Szymask ad Kesee (004), argue that formulatg the decso-makg rolem as a ocooeratve game leads to radcally dfferet coclusos aout the ature of cometto sorts leagues. Ths aer descres a smulato model that va e used a classroom to demostrate ho cometto orks a ocooeratve cotext. The suortg Excel sreadsheet used to coduct the game ca e doloaded from the author s ersoal eage htt://3.meral.ac.uk/eole/s.szymask JEL Classfcato Codes: A0, D43, L83 Keyords: sorts, teachg methods * Paer reseted at the Jot Aual Meetg 006 of the Iteratoal ad Germa-Seakg Assocatos of Sorts Ecoomsts (IASE ad AK), May 4-6, 006 Stefa Szymask, Taaka Busess School, Imeral College Lodo, South Kesgto camus, SW7 AZ, UK. Tel : (44) , Fax: (44) , e-mal: szy@meral.ac.uk. I ould lke to thak studets at the Uversty of Ater ad the Uversty of Zurch for ther cotruto to tryg out the smulato game descred ths aer.

2 . Itroducto Formal mathematcal modellg of sorts leagues ega th the ork of El-Hodr ad Qurk (97). Such models tycally assume that team oers make choces aout vestmet talet, that talet geerates s the league ad that tur s geerate reveues. Predctos are assocated th the equlrum dstruto of s ether a roft maxmsg model (e.g. Atkso et all (988), Fort ad Qurk (995), Vrooma (995), Marurger (997)) or a maxmsg model (e.g. Kesee (996, 000)). The urose of ths modellg has ee to make redctos aout the mact of olcy measures such as gate reveue sharg o quattes such as league-de rofts ad comettve alace. I recet years there has ee a techcal deate the lterature aout the ay hch the model s costructed. Essetally, models that follo the tradto of El- Hodr ad Qurk are ased o to sets of assumtos (a) each team oer chooses a quatty of talet to hre the market, ad ths quatty traslates oe-for-oe to a quatty of s the league () equlrum s detfed as the ot here margal eefts are equalsed across teams ( the roft maxmsg model ths meas the ot here the margal reveue of a s equalsed, the maxmsato model t meas the ot here the average reveue of a s equalsed). Szymask (003, 004a) ad Szymask ad Kesee (004) take ssue th the frst of these assumtos o oth theoretcal ad ractcal grouds. I terms of the theory, they argue that there s o coheret game theoretc terretato of the assumto that talet has a oe-to-oe corresodece th s, sce ths mles that each team s caale of choosg s deedetly of the other teams. Logcally, the s of oe team a league must deed o the talet choces of the other teams as ell. More ractcally, they argue that teams do ot fact choose s uless they are egaged match fxg. I ractce, teams allocate a udget to the hrg of talet the market, th the result that each team s share of talet the market s roughly roortoal to ts share of aggregate team udgets. They also sho that modellg the

3 choce of udgets as a o-cooeratve games geerates redctos that are dramatcally dfferet from the covetoal results the lterature. Some crtcs ve ths deate as a rather astract dsute over modellg assumtos th lttle or o ractcal cosequece. Moreover, the fact the models are usually framed usg very smle assumtos, most otorously the to-team league assumto, others have questoed the ractcal relevace of the ssue. Ths aer ams to llustrate the ractcal relevace of the deate y resetg a verso of the game theoretc model that ca e used for the uroses of classroom smulato of a sorts league. Egagg studets a smulato rgs to lfe the costrats faced y oers ad maagers the decso makg rocess. It ca also hel to rovde sghts to the effectveess of the tyes of mechasms that have ee desged to allevate some of the cosequeces of ecoomc cometto etee teams a league. The aer rovdes a full descrto of the model so that the reader ca ru the smulato model for themselves, ad to ad the use of the model a sreadsheet verso ca e doloaded from the author s ersoal e age. The detals of ths sreadsheet are also exlaed the aer. The aer roceeds as follos. I the ext secto the asc assumtos ehd the model are exlaed ad the ay the smulato ca e ru class s dscussed. The follog descres the smulato model ad assumtos ehd t. Secto 3 resets the mathematcal model o hch the smulato s ased ad the dervato of the Nash equlrum uder roft maxmsato ad the ot roft maxmsg soluto. Secto 4 resets the results of some smulatos ru th studets at the Uversty of Ater ad the Uversty of Zurch. Detals of the sreadsheet are rovded the aedces. htt://3.meral.ac.uk/eole/s.szymask 3

4 . The assumtos the smulato model The smulato model relates ecoomc uts to sortg ad ecoomc oututs. I the model teams hre talet hch the roduces s, ad s tur geerates roft, hch s the dfferece etee the cost of talet ad the reveue geerated y s. Thus there are to key ecoomc relatoshs that must e secfed: (a) the relatosh etee exedture ad success o the layg feld () the relatosh etee success o the layg feld ad reveues These relatoshs are defed y a set of arameters hch characterse the league cometto questo. The smulato model ths aer s ased o emrcal estmates of these relatoshs for the Amerca League, oe half of Maor League Baseall, for the year 003. Hoever, t s qute easy, oce the model s uderstood, to develo a model of ay league, ether usg arameters estmated from emrcal data or ased o guesstmates. (a) Exedture ad g Fe ould argue agast the roosto that teams hch sed more o average more o average. A gger udget for layer sedg meas etter layers ca e hred. Because there s a market for layg talet, exedture s correlated th exected success, ad actual success s correlated th exected success. Nether of these relatoshs s erfect: sometmes maagers make ad choces, ad sometmes layers do ot erform as exected (esecally he they are ured). The degree of sestvty of g to sedg s catured y the Tullock cotest success fucto () B B 4

5 Here reresets the ercetage of games layed y team that t s. If tes (dras) are ermtted, the each oe s treated as half of a. If there are teams the league the total ercetages sum to / (e.g. f there are 4 teams, as the Amerca League, the the total ercetages o sum to 7 or 700%). Gve ths total to e shared out, equato () says that they ll e allocated relato to team udgets B, ad each team s share of s s roortoal to ts share total team udgets. The degree of sestvty ths relatosh s measured y the arameter. If s very large, the small dffereces the B s traslate to large dffereces team erformace. If equalled zero the sedg ould make o dfferece to erformace. To llustrate the mact of sedg for dfferet values of, tale shos the exected ercetage of a team for a gve level of exedture y other teams a 4 team league. Tale : Exected ercetages for dfferet levels of exedture assumg a 4 team league hch the other teams each sed 00 udget Note that exected ercetage exceeds the maxmum feasle 00% f the udget exceeds a crtcal level. Therefore the smulato model e troduce a extra costrat- f team s udget s such that the value take y equato () ould e greater tha uty, the team s actual ercetage s costraed to equal uty (00% s). 5

6 () Wg ad reveues The relatosh etee g ad reveues ca e estmated for ay gve league usg hstorcal data. Wg creases reveues ecause the maorty of those ho atted league games suort the home team. Home team fas at to see ther team. There are a umer of factors that are ko to crease reveues. I the Amerca League t has ee sho that the costructo of a e allark creases attedace, hle other factors such as ast success cotrute to s. The assumed relatosh etee attedace ad g ths model thus takes the form () Attedace a + + c The dervato of these coeffcets for the Amerca League s exlaed the aedx. The estmated coeffcets, a, ad c for each team are gve Tale. Tale : Estmated arameters for the sestvty of attedace to s for the Amerca League Name a c Aahem Agels Baltmore Oroles Bosto Red Sox Chcago Whte Sox Clevelad Idas Detrot Tgers Kasas Cty Royals Mesota Ts Ne York Yakees Oaklad Athletcs Seattle Marers Tama Bay Devl Rays Texas Ragers Toroto Blue Jays

7 (c) Playg the Game Gve the arameters of the model the game ca o e ru o sreadsheet. I order to lay the game t s useful to rovde a structo sheet (see aedx ). The structo sheet should exla the relatoshs aove ad the artcats should e vted to choose or allocated a team. The game s essetally a oe-shot game. Each team chooses a udget hch, for examle, they ca rte do o a decso sl (see aedx ). Each team does so deedetly ad the hads the sl to the structor. Oce they have all ee collected the structor uts the decsos oto the sreadsheet, ad the comato of dvdual decsos the determes the ercetage, reveue ad roft of each team. The game s est layed several tmes over, order that studets ca lear from the exereces of earler rouds. Hoever, t s very mortat to emhasse to artcats that the game s oe-shot, so that choces made ast rouds do ot drectly affect curret decsos, ad curret decsos have o log term cosequeces (at least drectly). It s mortat to ote, hoever, that teams lear aout ehavour from revous rouds ad that therefore there are some drect effects; deed, a sese, ths s the hole ot of the smulato. 3 Gve that e have a relatosh etee sedg ad g (equato ) ad etee g ad attedace (equato ), e ca calculate the cosequeces of dfferet udget choces made y dvdual teams. Each team ca choose ts o udget, ut the umer of s for each team ll deed ot oly o ther o udget, ut o the udget of every other team. The g ercetage for each team the determes attedace. 4 The reveue deeds o oth attedace ad rces. I the Amerca League model e assume that each team geerates $60 of come er 3 It ould e ossle thout too much maulato to make the game dyamc, ut ths ould the make t dffcult or mossle to detfy a equlrum of the game, ad studets mght lear lttle more tha the trte oservato that aythg ca hae. 4 Although f sedg s lo eough, the egatve coeffcet a for some teams mles egatve attedace. Ths s ruled out the sreadsheet y requrg attedace to e o-egatve. 7

8 fa, ut fact the value ca e chaged the sreadsheets so that t ca e alloed to vary for each team. Thus the roft for team, gve equatos () ad (), s (3) π (a + + c ) B Where s the exected reveue er fa. It ll e oted that so far othg has ee set aout oectves. Tycally Amerca ecoomsts have assumed roft maxmsg ehavour ad Euroea ecoomsts have assumed maxmsg ehavour. These assumtos roduce dfferet theoretcal results. Hoever, the smulato t s u to the structor to decde ho to drect the artcats. It may ractce e easer to demostrate the ature of the model y askg artcats to act as roft maxmsers, sce there s the a clear echmark for success game. Ideed, a classroom stuato oe mght eve aard grades for the level of rofts geerated. The aalogy a maxmsg model s that a layer of the game s maxmally successful f they acheve a udget exactly equal to zero, ad the closer they are to zero, the greater ther success. Hoever, t mght e argued that there should e asymmetrc ealtes for over ad uder-sedg (sce the cosequeces of oversedg are lkely to e more severe). All ths s the gft of the structor- the smulato model s cosstet th ay comato of roft maxmsg ad maxmsg ehavour. Before dscussg some of the exereces of classes that have layed the game, t s useful to characterse the Nash equlrum of the game. 3. Nash equlrum ad otmalty Gve the assumed oectves of the layers, there ll geeral exst a teror Nash equlrum of the model. Gve the choces of every other team, each team ossesses a est resose - a choce of udgetary exedture that maxmses each team s oectve fucto. If every team s udget decso ere smultaeously a est 8

9 resose, the the udgets ould costtute a Nash equlrum. At a Nash equlrum o team ould sh to alter ts choce. The Nash equlrum s ot ecessary otmal from the ot of ve of the league. If e ere to take the ersectve of the league as a hole, magg t as a cartel hose oectve as to share out the s such a ay as to geerate the maxmum ossle attedace (ad therefore reveue), e ould eed to detfy the ot here the margal reveue (MR) of a as equal for every team. To see hy ths s so, mage that for a gve set of ercetages oe team had a hgher margal reveue tha aother. I such a case t ould e ossle to crease total reveue y takg oe aay from the lo MR team ad gvg t to the hgh MR team- the ga ould outegh the loss. If all MR s are equal, hoever, t s ot ossle to redstrute s to crease total reveues. Notce that ths otmalty codto s deedet of the udgets of the teams- these do ot lay a role determg the otmal dstruto of s. (a) Dervg the Nash equlrum Gve (), () ad (3) the frst order codto for roft to e a maxmum s thus π (4) ( + c ) 0 B B here B B B B (5) B B B so that e ca rerte (4) as 9

10 0 (6) B c + ) ( Hoever, e ca also defe the dvdual udget B of a team terms of the sum of udgets of all teams, usg (): (7) B B ad thus (6) ca e rertte as the equlrum codto (8) ) ( + B c Note that the RHS of (8) s commo to all teams. Aother ay to rerte the equlrum codto therefore s (9) + + c c ) ( ) ( for every team ad. It s straghtforard to solve ths system of equatos umercally f the arameters, c, ad are ko. ad c ca e recovered y regressg attedace o ercetage, the value of, hch s essetally reveue er fa, ca e derved drectly from come statemets of the clus, hle ca e derved from the relatosh etee ercetage ad team layer udgets (aedx 5 descres ho ths ca e doe o a Excel sreadsheet).

11 It s useful to cosder the values that ths codto takes for dfferet values of : (0) c c c c ) ( : 4 ) ( : 3 ) ( : ) ( : 3 () The otmum for the league cartel We o comare ths th the codto for league reveues as a hole to e maxmsed. Ths requres the margal reveue of a to e equalsed across teams, regardless of udgets. The codto s gve y the dervatve of () th resect to g ercetage ad hece: () ) ( ) ( c c + + We ca solve exlctly for as follos. Frst, from () () c c c + If e sum over all ot cludg, the (3) + c c c ut also (4) ( ) / -.

12 (ths s the addg u costrat troduced ()), ad hece (5) * c + c c The soluto (5) defes the dstruto of ercetages that maxmses total reveues. A laer terested maxmsg total reveue ould therefore dstrute layg talet such a ay as to roduce these ercetages. I theory there s o eed for the laer to use the market mechasm,.e. layg udgets. Hoever, t s ossle to detfy the udget dstruto that ould geerate the otmal ercetages y usg (7). For ay team the udget must e roortoal to * * (6) B K ( ) Where K s a costat that ca e adusted to defe the total layer udget. Of course, gve that oly the relatve sze of the udgets matter for allocatg talet, t ould e ossle for the laer to reduce total exedture to the level of the layers reservato age. 4. The classroom exerece The game has ee layed y studets takg a sorts ecoomcs class at the Uversty of Ater ad the Uversty of Zurch. Belo s descred the exerece of oe grou. Each team as rereseted y a sgle studet. Before makg ther frst decso, the teams ere told that the data rereseted roughly the stuato hch the Amerca League foud tself 003, ad that fact the teams had set aout $. llo o layer salares. After allog aout fve mutes to uderstad the rolem (ad aserg questos of clarfcato), the studets ere requred to sumt ther udget sls. Whe each team s rereseted y

13 a grou of studets somehat more dscusso tme s requred. The udget choces ere the etered to the sreadsheet, ad he ths as doe the outcome as the sho o a roector. Ths aeared as fgure. Fgure : Scree for roud of the game The aggregate sedg the frst roud as very hgh- aout 30% hgher tha the actual sedg. The rage vared from $5 mllo to $0 mllo. Gve that the studets ere suosed to e actg as roft maxmsers, those ho made losses ere asked to commet. Most quckly sa that reducg exedture ould crease rofts. Studets could see qute easly that some teams started th a stroger suorter ase, ut that hat really mattered terms of the udget choce as 3

14 hether the team had a large sestvty of reveues to s, catured the ad c arameters. We the moved to roud, the results of hch are sho tale 3. Tale 3: Roud of the game Roud Name udget $m(b) c(b) attedace Reveue $m roft $m Aahem Agels Baltmore Oroles Bosto Red Sox Chcago Whte Sox Clevelad Idas Detrot Tgers Kasas Cty Royals Mesota Ts Ne York Yakees Oaklad Athletcs Seattle Marers Tama Bay Devl Rays Texas Ragers Toroto Blue Jays Sum stadard devato 0.9 Total sedg as early halved. Reveues, hoever, fell oly slghtly, ad therefore rofts ere three tmes larger tha the frst roud. Note that the varace of ercetages also creased, sce the fe teams that cotued to sed at a hgh level acheved very hgh ercetages (ad relatvely lo rofts). The cocet of a est resose as o dscussed- could each team detfy a est resose? We the moved to roud 3, the results of hch are sho Tale 5. By roud 3 every team had recogsed the advatage to keeg sedg do ad aggregate sedg as o oe thrd of the level the frst roud, ad rofts ere aout four tmes larger. No studets foud that chagg ther decso made very lttle dfferece to total roft- most teams ere close to ther est resose. Roud 4 (Tale 6) llustrated that the grou as gettg closer to a equlrum. 4

15 Tale 5: Roud 3 of the game Roud 3 Name udget $m(b) c(b) attedace reveue $m roft $m Aahem Agels Baltmore Oroles Bosto Red Sox Chcago Whte Sox Clevelad Idas Detrot Tgers Kasas Cty Royals Mesota Ts Ne York Yakees Oaklad Athletcs Seattle Marers Tama Bay Devl Rays Texas Ragers Toroto Blue Jays Sum stadard devato 0.96 Tale 6: Roud 4 of the game Roud 4 ame udget $m(b) c(b) attedace reveue $m roft $m Aahem Agels Baltmore Oroles Bosto Red Sox Chcago Whte Sox Clevelad Idas Detrot Tgers Kasas Cty Royals Mesota Ts Ne York Yakees Oaklad Athletcs Seattle Marers Tama Bay Devl Rays Texas Ragers Toroto Blue Jays Sum stadard devato 0.6 I ths roud the aggregate result as qute smlar to the revous roud. As a result, although some grous chaged ther udget qute sgfcatly, t roved harder to 5

16 affect rofts sgfcatly. Ths led aturally to a dscusso of the dea that each team mght smultaeously e at a est resose. Some thought ths as ossle, others ot. At ths ot the cocet of the Nash equlrum as troduced ad the relevat values sho o the sreadsheet. As a fal stage the exercse the, dstruto of s hch maxmses total attedace as examed, ths eg dfferet from the comettve Nash equlrum. 6

17 Aedx : Istructos for artcats the Amerca League game. (NB the sreadsheet for the game ca e doloaded at htt://3.meral.ac.uk/eole/s.szymask) Image you are the oer of a team the Amerca League ad that your sole oectve s to maxmse rofts. Profts equal reveues mus costs. Costs equal the udget devoted to hrg layg talet. Reveue deeds o the ercetage of games o, hch ca rage etee zero ad 00%. The exact relatoshs deed o the follog to equatos: () B B (The ay-erformace relatosh) () Attedace a + + c (The attedace- relatosh) The ay-erformace relatosh Here reresets the ercetage of games layed y team that t s. For teams the league the total ercetages sum to /. I the Amerca League there are 4 teams ad so the total ercetages o sum to 7 (700%). B s the team udgets ad each team s share of s s roortoal to ts share total team udgets. The degree of sestvty ths relatosh s measured y the arameter. If s very large, the small dffereces traslate to large dffereces team erformace. If equalled zero the sedg ould make o dfferece to erformace. To llustrate the mact of sedg for dfferet values of, tale shos the exected ercetage of a team for a gve level of exedture y other teams a 4 team league. Tale : Exected ercetages for dfferet levels of exedture assumg a 4 team league hch the other teams each sed 00 udget I the game a value of 0.5 ll e assumed. 7

18 N.B. If team s udget s such that the value take y equato () ould e greater tha uty, the team s actual ercetage s costraed to equal uty (00% s). The attedace- relatosh The attedace relatosh s ased o hstorc data. For each team there s a quadratc relatosh hch reaches a maxmum at some ostve ercetage, ut that crtcal value ca e greater tha 00%. Tale : Estmated arameters for the sestvty of attedace to s for the Amerca League Name a c Aahem Agels Baltmore Oroles Bosto Red Sox Chcago Whte Sox Clevelad Idas Detrot Tgers Kasas Cty Royals Mesota Ts Ne York Yakees Oaklad Athletcs Seattle Marers Tama Bay Devl Rays Texas Ragers Toroto Blue Jays N.B. Gve the values of a Tale, t ould e ossle for a team to have egatve attedace f the team o fe games. Ths s ruled out y costrag attedace to e o-egatve. Gve these to relatoshs roft equals (3) π (a + + c ) B For ths smulato e assume (average reveue er fa) s equal to $60 for each team. Rules of the game Each team s requred to choose a ostve ad fte udget fgure. Colluso s ot ermtted. Based o these choces the ercetage, reveue ad roft of each team ll e determed. If more tha oe roud s layed, there s o coecto etee the decsos oe roud ad the decsos ay other. 8

19 Aedx : A examle of a udget sl used for the layg the game Aahem Agels Roud Budget $m: Aahem Agels Roud Budget $m: Aahem Agels Roud 3 Budget $m: Aahem Agels Roud 4 Budget $m: Aahem Agels Roud 5 Budget $m: 9

20 Aedx 3: Ho to solve for the Nash equlrum udgets o a Excel sreadsheet. The ut data requred for ths exercse cossts of the arameters ad c, reveue er fa, hch should e defed three searate colums, say colums A, B ad C.. Defe the ercetages colum D usg equato () here the B are umers utted colum E, hch e ca lael Rudgets ad the arameter, defed a free cell (for examle, a 4 team league, here the ames are defed the frst ro ad the ext 4 ros cota the team data, cell A7 could e used for the value of ). 3. I colum F ut the formula for LHS of the equlrum codto (9). Ths deeds o the arameters (colum A) ad c (colum B), reveue er fa (colum C), the umer of teams the league, ad the ercetages defed colum D. 4. Udereath the fgures colum F ut the average value of these fgures. 5. I colum G ut the dfferece etee value for the team colum F ad the average value for colum F. 6. From colum G detfy the team hose devato from the average s largest ad the adust ths team s Rudget fgure colum E utl the devato s zero (or close to zero). 7. Reeat ste 5 as ofte as s ecessary to reduce all of the devatos as close to zero as s requred. Note that as the devatos colum G aroach zero the values colum F aroach equalty, thus satsfyg equlrum codto (9). 8. To derve team udgets from the Rudget fgures colum E, ut colum H the LHS of equato (6), hch deeds o the ercetages colum D, the arameters (colum A), c (colum B), reveue er fa (colum C), the umer of teams ad. The fgures colum H are thus the Nash equlrum udgets (Rudgets are roortoal to the Nash Equlrum udgets, ut oly esure that the margal reveue of udget sedg s equalsed across all teams. For Nash equlrum e also requre that margal reveue equals margal cost). 0

21 Aedx 4: a lst of sheets from the Excel fle Base data: attedace ad ercet data for the Amerca League. Other varales clude dummes for date of a e allark oeed ad league hoours o. Exedture ad g: Ths sheet uses the mmum sum of squared devatos to estmate the value of hch est fts the data o ages ad ercet usg equato (). Reveue ad costs AL 03: Ths s actual reveue ad cost data for the Amerca League 003 Regresso Results: Ths sheet shos the results of the lear regressos of attedace o ercet. Quadratc estmates: Ths sheet shos ho the quadratc arameter estmates ere derved. The method s exlaed Szymask (004). Attedace ad g (chart): Ths shos the relatosh etee attedace ad ercetage for some of the teams. Model: Ths s sheet used to ut the udget choces made y artcats the smulato. Results: Ths sheet should e used to kee a record (y cuttg ad astg) of each roud. NE g 0.6: Ths sheet gves the Nash equlrum choces he 0.6, assumg roft maxmsato. NE g 0.5: Ths sheet gves the Nash equlrum choces he 0.5, assumg roft maxmsato. NE g 0.5: Ths sheet gves the Nash equlrum choces he 0.5, assumg roft maxmsato. NE g : Ths sheet gves the Nash equlrum choces he, assumg roft maxmsato. Plaer s equlrum: Ths sheet gves the dstruto of ercetages that maxmses total attedace Summary: Ths sheet summarses all the relevat varales for the dfferet Nash equlra ad the laer s equlrum. It also gves the udget choces he teams are maxmsers. Note that for some values of, there are some clus that caot avod losses he all teams are maxmsers.

22 Aedx 5: A ote o arameter values the model (a) the value of For the smulato a value of equal to ½ s coveet. It s ossle, hoever, to derve a value of. The sheet laelled Exedture ad g estmates the arameter usg data from the Amerca League for the erod Ths s doe y defg the exected ercetage for each team each seaso ased o the ayrolls of all team secfed equato (), ad the varyg the arameter so the sum of squared devatos of exected from actual ercetage s mmsed. The value of that does ths for the Amerca League s 0.6, suggestg a relatvely lo sestvty. Estmates for other leagues at other tmes could dffer sgfcatly. () the value of the a, ad c arameters Usg the Base data sheet, the regresso results shos the ecoometrc results for the relatosh etee attedace at the allark ad these factors. For each clu, a crease g ercetage creases attedace, ut at each clu the sestvty vares. The estmated relatosh s lear, suggestg that, hatever, the level of ercetage, a addto to ercetage roduces the same crease attedace. More realstcally, t mght e exected that creases ercetages roduce a smaller ad smaller addto to attedace (dmshg returs). It mght eve e the case that attedace decreased f ercetage rose too hgh, sce fas ould lose the elemet of uredctalty that makes sortg cotests attractve. Ths o-learty s hard to estmate for the Amerca League, sce teams seldom acheve extreme ercetages. Out of the 363 team seasos the dataase here, there ere oly three cases of ercetages elo 33% ad oly three aove 66% (the hghest as 7% ad the loest 6%). Hoever, t s lkely that f a team o more tha 80% of ts games t ould at least start to face caacty costrats. The sheet quadratc estmates shos a o-lear estmate ca e roduced from the regresso results ad the assumto of a caacty costrat, ad the chart Attedace ad g llustrates the relatosh for several clus.

23 Refereces Atkso, Scott, Lda Staley ad Joh Tschrhart Reveue Sharg as a cetve a agecy rolem: a examle from the Natoal Footall League Rad Joural of Ecoomcs, 9,, 7-43 El-Hodr, Mohamed ad James Qurk. 97. A Ecoomc Model of a Professoal Sorts League Joural of Poltcal Ecoomy, 79, 30-9 Fort, Rodey ad James Qurk Cross Susdzato, Icetves ad Outcomes Professoal Team Sorts Leagues Joural of Ecoomc Lterature, XXXIII, 3, Késee, Stefa. 996 League Maagemet Professoal Team Sorts th W Maxmzg Clus, Euroea Joural for Sort Maagemet, vol. /,. 4- Késee, Stefa Reveue Sharg ad Comettve Balace Professoal Team Sorts Joural of Sorts Ecoomcs, Vol, No, Marurger, Dael "Gate reveue sharg ad luxury taxes rofessoal sorts" Cotemorary Ecoomc Polcy, XV, Arl, 4-3. Stefa Szymask, 003. The Ecoomc Desg of Sortg Cotests Joural of Ecoomc Lterature, XLI, Stefa Szymask, 004a, Professoal team sorts are oly a game: the Walrasa fxed suly coecture model, Cotest-Nash equlrum ad the Ivarace Prcle Joural of Sorts Ecoomcs, 5,, -6. Stefa Szymask, 004, Tltg the Playg Feld: Why a sorts league laer ould choose less, ot more, comettve alace, mmeo. Stefa Szymask ad Stefa Késee, 004. Comettve alace ad gate reveue sharg team sorts Joural of Idustral Ecoomcs, LII,, Vrooma, Joh A Geeral Theory of Professoal Sorts Leagues Souther Ecoomc Joural, 6, 4,