Corruption on the Court: The Causes and Social Consequences of Point-Shaving in NCAA Basketball

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1 Corruptio o the Court: The Causes ad Social Cosequeces of Poit-Shavig i NCAA Basketball by Yag-Mig Chag * ad Shae Saders ** October 8, 007 * Departmet of Ecoomics, Kasas State Uiversity, 39 Waters Hall, Mahatta, KS , Tel: (785) , Fax: (785) , ** Ecoomics Departmet, Aubur Uiversity Motgomery, 34C Clemet Hall, Motgomery, AL , Tel: (334) , Fax: (334) , correspodig author. We would like to thak Bhaveet Walia for commets ad suggestios. All errors are our ow.

8 7 opportuities i post-collegiate professioal basketball. To determie the optimal effort expediture by each team, we use () ad (3) to derive first-order coditios for the two teams: π e = V = 0 ; (4) e e + e ( ) π e = V σ = 0. e e + e ( ) (5) Solvig (4) ad (5) for the Nash equilibrium efforts by the two teams yields e = Vσ * ( + σ) ad e V * =. ( + σ) (6) I equilibrium, the expected probabilities of victory are: g * = σ + σ ad g * =. + σ (7) It follows from (6) ad (7) that Team, the favored team, exerts more effort tha its oppoet ad is thus more likely to wi the game. I what follows, we will use this meritorious (ucorrupted) game as a bechmark by which to evaluate the outcome of a corrupted cotest.. The Same Two Teams i a Corrupted Cotest I a corrupted cotest, or oe i which members of Team egage i poit-shavig activities, Team s uit cost of effort will rise. Assume each regular player o Team agrees to egage i poit-shavig. The uit cost of effort will rise because each additioal uit of effort icreases the likelihood that these players will ot receive the outcome cotiget poit-shavig bribe. Assume oly oe or two players o Team agree to egage i poit-shavig. The uit cost of effort for the team will still rise, as there are productivity spillovers i basketball productio (Kedall, 003). By puttig forth less effort (i.e., failig to set a scree for a drivig teammate, overlookig a pass to a teammate who has worked hard to become ope,

9 8 failig to double team a oppoet o defese oce a teammate has trapped him, or elicitig more playig time for a less taleted replacemet player), a corrupted player makes it more costly for his teammates to put forth productive efforts. This is true eve if his teammates are ot complicit i poit-shavig. I a corrupted game, the cotest success fuctio for each team is the same as equatio () i a meritorious game. However, Team s uit cost of effort rises i the presece of a poitshavig scheme. The expected payoff fuctios for the two teams become π = g V e ηe, (8) π = g V σ e, (9) where η ( > 0) represets the favored team s additioal uit cost of effort i the presece of poitshavig corruptio. follows: From each team s expected payoff fuctio, we calculate the first-order coditios as π e = V ( + η) = 0 ; (0) e e + e ( ) π e = V σ = 0. () e e + e ( ) Usig (0) ad (), we solve for the Nash equilibrium efforts ad expected probabilities of victory: e Vσ ** = ; ( + σ + η) e = V( + η) ( + σ + η) ** ; () g σ ** + η = g =. (3) + σ + η + σ + η ** ; Thus, the presece of poit-shavig activity reduces effort ad, to some extet, likelihood

10 9 of victory for Team. While icreasig likelihood of victory for Team, corruptio by Team has a ambiguous effect o Team s effort level..3 The Effects of Poit-Shavig Corruptio o Team Efforts ad Social Welfare Next, we calculate the effect of poit-shavig corruptio o the cotest s aggregate effort level. It is easy to verify from (6) ad () that * * * V E = e + e = ; (4a) + σ ** ** ** V E = e + e =. (4b) + σ + η We thus have * ** E > E. (4c) As i Amegashie ad Kutsoati (005), we take aggregate match effort as a partial determiat of match excitemet ad therefore of the demad fuctio. From iequality (4c), it is clear that poit-shavig corruptio reduces match excitemet ad, correspodigly, causes a leftward shift i the demad curve for subsequet NCAA basketball games. Further, before each match, we assume that some fas form a expectatio regardig relative efforts across competig teams. These educated fas value match itegrity, as it is ecessary i accurately assessig teams relative to oe aother. Thus, we take such fas as passively evaluatig the itegrity of a match as the differece betwee realized ad expected relative team efforts. For a corrupted match, the level of itegrity is measured accordig to e e ση d = = < 0. (5) e + η ** * ** * e As d approaches zero from the left, the level of match distortio (itegrity) falls (rises). From equatio (5), we fid that

16 5 far greater degree tha i baseball, a basketball team must be ruig o all cyliders to be effective. Thus, we take each regular player o a basketball team as havig the same basic techology for team sabotage ad thus the same bribe opportuities. From the previous iequalities, the critical level of pay for a college player (i.e., the miimum amout to iduce hoest play) i the th period is equal to λ c% = a+ bt β( γ) m+ (9) λ Followig a backward iductive solutio path, we fid the critical level of compesatio for a player i period ( ) as follows c~ ~ ( a + βa + β γ m ) + ( λ)( c~ + b + βc~ β m ) + βc + β m+ = λ + t + +. (0) Should the player cheat i period ( ) ad ot be caught, his expected payoff i period is c ~ regardless of his actios i the latter period. This is due to the fact that his th period earigs are set such that he fids o advatage (i.e., o expected payoff greater tha c ~ ) by egagig i corruptio. Thus, the critical pay level of compesatio i period ( ) is Similarly, c~ ( λ)( β) ( γ) m + = a + bt β γ λ. () c~ ( λ)( β) ( γ) m + 3 = a + bt β γ c~ k λ ( λ)( β) ( γ) m + k+ k = a k + bt β γ λ for (,,3,..., ( ) ) () k (3) From equatios (9) to (3), we preset the followig propositio.

18 7 appreticeship). Thus, a college basketball player is less likely to egage i corruptio so as ot to jeopardize his professioal prospects. This disicetive effect becomes stroger as a college player approaches his th period. It is importat to ote that the value m + varies greatly amog regular collegiate players. This is true eve amog regular players o a give college team or team type. Lookig across professioal leagues, salary compesatio for professioal basketball players is quite o-liear with respect to skill level, meaig that small drop-offs i skill ca mea a disproportioately smaller paycheck. I a 007 article etitled Almost-NBA Players Take Home Paltry Salaries, Tom Goldma writes, With a average aual salary of more tha \$5 millio, NBA players are the highest-paid athletes i professioal sports. But for the may skilled professioals who have't quite made it ito the NBA, the fiacial gulf is huge Salaries i the developmet league (NBDL) rage from \$,000 to \$4,000 a seaso, paid i part by moey from the NBA. Based o the heterogeeity of a player s expected post-developmetal payoff, the, we expect cosiderable heterogeeity i the ature of corruptio decisios across the set of NCAA players. 3. Corruptible Player Types We ca ow determie which player-types are at a relatively high risk of egagig i corruptio accordig to the model. Give that compesatio, c, is similar across NCAA players, the model would describe a corruptible player as oe o a atioally strog team who, while playig regularly, does ot expect to ear a large professioal payday after his appreticeship. We might thik of a player who, while a key cotributor o a quality team, expects to fall somewhat short of the NBA level after his appreticeship. For istace, some effective college players do ot fall withi ay of the arrowly specified roles of a NBA team. A premier college player might be too short ad weak to play power forward i the NBA ad, at

19 8 the same time, ot sufficietly quick to play small forward i the NBA. 3. Applyig Becker ad Stigler s Notio of Hoesty Premium to a Heterogeeous Workforce We complete the aalysis by calculatig the preset value of salaries (at the oset of a college career) ecessary to keep the college player hoest throughout his college career. Usig equatios (9) to (3), the critical preset value of future salaries i decisio period oe is 7 PV c~ c~ c~... c~ = + β + β β ( λ) ( ) i β ai bt β γ m+. (4) i= λ = + Equatio (4) above states that the preset value of salaries ecessary to keep the collegiate basketball player hoest is equal to (i) the preset value of payoffs i a professioal appreticeship plus (ii) the preset value of payoffs from poit-shavig opportuities mius (iii) the preset value of post-developmetal professioal earigs lost i the evet that the player is caught for corruptio. This preset value formula ca be used to cosider a ati-corruptio policy for college basketball. Such a policy would require each eterig NCAA player to bod ito the orgaizatio by the amout of the secod ad third terms. Durig each period of hoest play, the player would receive a premium equal to the iterest icome geerated by the bod. Fially, the player would be retured the pricipal of the bod i the evet that he leaves college basketball with o evidece of poit-shavig ivolvemet. Give the NCAA s cocers, such a pay structure has the advatage of itroducig player icetives toward hoest play while simultaeously maitaiig a level of player pay cosistet with amateur status - whatever this term or ideal is iteded to mea. 7 See Appedix A- for a detailed derivatio of the preset value formula.

22 5. Appedices A-. Poit-shavig affects the supply of NCAA basketball games i three primary ways. First, paymets from bettors to players will allow NCAA teams to pay a lower wage to players. I this sese, poit-shavig shifts right the supply curve for NCAA basketball games. However, ay market surplus geerated by this shifted supply curve is merely a trasfer from outside bettors (i.e., those who do ot aticipate poit-shavig activity) to iside bettors, players, fas, ad the NCAA. As this shift creates o overall welfare gai, we abstract from it i our graphical welfare aalysis. Further, the presece of poit-shavig causes players to exert less aggregate effort toward what is take as the same aggregate prize (i.e., game experiece ad game exposure to professioal scouts). I this sese, players gai from poit-shavig. However, these gais are take as either egligible o a market scale or at least offset by expected costs, i the form of NCAA dismissal ad discouted professioal basketball earigs, bore by the corrupted player. 0 Give the origi of the first supply curve shift ad the offsettig ature of the latter two, we focus solely o the two demad curve effects i our welfare aalysis of Sectio. A-. To calculate the period- preset value of icome streams that discourages a player from participatig i poit-shavig corruptio, it is ecessary to make use of c% i derived i equatio (3). The term c% i measures the critical level of period-i compesatio such that the player fids o advatage by egagig i corruptio. It follows from (3) that 0 While we take these expected costs as sufficiet to overcome player gais from a reductio i aggregate effort, the cost of puishmet is obviously ot always sufficiet to overcome the value of the poit-shavig bribe itself.

23 ( λ)( β) ( γ) c% = a+ bt β γ m+ for k = ; λ ( λ)( β) ( γ) c% = a+ bt β γ m+ for k = ; λ c~ ( λ)( β) ( γ) m + = a + bt β γ λ for k = ; λ c% = a+ bt β( γ) m+ for k =. λ Pluggig this stream of critical earigs { c%,..., c% } ito the preset value formula: PV = c% + βc% β c% + β c%, we have ( λ)( β) PV = a + b β γ ( γ) m λ t + ( λ)( β) + β a + b β γ ( γ) m λ +... t + ( λ)( β) + β a + b β γ( γ) m λ λ β a b β( γ) m. λ t t + Simplifyig the above expressio yields ( λ)( β) ( λ) i i i = β i+ β t + β t β γ ( γ) + i= i= λ λ i= PV a b b m ( λ) ( ) i β ai bt β γ m+, i= λ = + i i sice β ( β) + β = ( β) + β = ad ( ) = ( ) i= β β i= γ γ γ γ = γ. γ

26 5 Figure. Welfare cosequece of poit-shavig corruptio p S F E p 0 p B C A Q=D (p, E * ) Q=D (p, E ** ) 0 Q Q 0 Q

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