Sporting Contests and the Coase Theorem

Stefan Szymanski

Tanaka Business School, Imperial College London

October 2005

Paper presented at "Advances in the Theory of Contests and Tournaments", October 21-22, 2005, Hosted and organized by the Social Science Research Center Berlin (WZB) in collaboration with the University of Tromsø.

Stigler: “with zero transactions costs, private and social costs will be equal” Coase: “if private cost is equal to social cost, it follows that producers will only engage in an activity if the value of the product of the factors employed is greater than the value which they would yield in their best alternative use” Dixit and Olson (2000): “arguably the single largest influence on thinking about economic policy for the last three decades” Critics: Practicality: costless bargaining with full property rights not feasible, e.g. Canterbery and Marvasti (1992) Tautology: with zero transaction cost efficiency guaranteed among maximising agents, regardless of whether property rights, e.g. Usher (1998) Falsity: there exist initial allocations of property rights for which the core is empty, e.g. Aivazian and Callen (2003)

1

Coase Theorem in the sports literature: Sports (especially baseball) often cited as a natural test of the Coase Theorem Professional league sports: A form of multi-stage contest characterised by (a) asymmetry, (b) demand for competitive balance as well as demand for success Teams allocate annual budgets to player salaries plus player acquisition costs Strong correlation between (a) spending and success (e.g. percentage of matches won) and (b) between success and income Property rights over players in a sports league: Reserve Clause of baseball, Retain and Transfer System of English soccer

2

Rottenberg (1956) "the defense most commonly heard is that the reserve rule is necessary to assure an equal distribution of playing talent among opposing teams; that a more or less equal distribution of talent is necessary if there is to be uncertainty of outcome; and that uncertainty of outcome is necessary if the consumer is to be willing to pay admission to the game. This defense is founded on the premise that there are rich baseball clubs and poor ones and that, if the players' market were free, the rich clubs would outbid the poor for talent, taking all competent players for themselves and leaving only the incompetent for other teams." (p. 246) But he claims “a market in which freedom is limited by the reserve rule such as that which now governs the baseball labor market distributes players among teams about as a free market would” (p.255) = Invariance Principle Mathematical restatement (Quirk and El-Hodiri (1974)): market distribution of talent is equal to the joint-profit maximising distribution Conclusion: Invariance Principle + market efficiency = Coase Theorem

3

Natural experiments: 1. Free agency in baseball in 1976 Six year veterans free to move team Balance of evidence that competitive balance improved- distribution of players changed 2. The rookie draft (1965 in baseball) Weak teams given first choice of new players entering the league Balance of evidence that competitive balance improved- distribution of players changed Not very powerful tests

4

A simple model of talent allocation in a sports league A1. Each team generates attendance according to the number of wins, represented by a concave function Qi(wi) with Qi’ > 0 and Qi’’ ≤ 0; beyond some critical value it is possible that Qi’ < 0. A2. Win production: Each team purchases talent (t) in a competitive market. Talent is assumed to be measured in perfectly divisible units and sold at a constant marginal cost.

wi(0) = 0, wi(∞) = 1, 0>∂∂

i

i

tw

and 02

2

<∂∂

i

i

tw

.

A3. Teams maximize profits, and the league planner maximizes attendance. Suppose all teams constrained to charge the same price Proposition (a) If the marginal revenue functions are identical, then the noncooperative Nash equilibrium for the league will be perfectly balanced (planner’s allocation coincides with the Nash equilibrium). (b) With asymmetric marginal revenue functions, the planner’s equilibrium is less balanced than the noncooperative Nash equilibrium.

5

Proof (1) iiiii ctwQp −= )(π

(2) i

i

j

j

j

i

twtw

QQ

∂∂∂

∂

='

'

for all i and j. Thus at the Nash equilibrium if t i > tj then Q’i > Q’j.

At the planner’s optimum

(3) 1'

'

=j

i

QQ

for all i and j

Competitive externality (commonplace in IO contexts) Hirshleifer’s paradox of power

6

Empirical test: Estimate for each team the relationship between winning and attendance over some sample period: (4) Attendanceit = at + bi wpcit + ci wpcit

2 + εit Use the coefficients bi and ci to estimate the optimal distribution of wins

7

Actual and maximum attendance for the English second tier, 2002/03

Club

Actual winpercentage

Actual average attendance

Attendance maximising win percentage

constrained attendance maximising win percentage

maximum average attendance

Bradford City 0.413 12501 0.650 0.650 18944Brighton & Hove Albion 0.370 6651 0.781 0.781 23305Burnley 0.435 13977 0.597 0.597 17276Coventry City 0.413 14813 0.733 0.733 21695Crystal Palace 0.489 16867 0.749 0.749 22234Derby County 0.402 25470 0.393 0.393 18524Gillingham 0.500 8078 -0.104 0.043 2761Grimsby Town 0.326 5700 0.143 0.143 5049Ipswich Town 0.554 25455 0.360 0.360 19298Leicester City 0.717 29231 0.893 0.893 27293Millwall 0.511 8512 0.263 0.263 7927Norwich City 0.543 20353 0.843 0.843 25490Nottingham Forest 0.587 24437 0.495 0.495 21477Portsmouth 0.750 18906 0.660 0.660 19280Preston North End 0.489 13853 0.379 0.379 10942Reading 0.587 16011 0.426 0.426 12247Rotherham United 0.478 7522 0.167 0.167 5607Sheffield United 0.620 18073 0.403 0.403 16401Sheffield Wednesday 0.391 20327 1.362 1.000 37883Stoke City 0.413 14588 0.767 0.767 22843Walsall 0.424 6978 0.182 0.182 11136Watford 0.467 13405 0.528 0.528 15173Wimbledon 0.511 2787 -0.205 0.000 1983Wolverhampton Wanderers 0.609 25745 0.535 0.535 23285 Total 12 370240 12 11.991 408052Standard deviation 0.106 7327 0.340 0.277 8583

Table 2: Actual attendance and maximum feasible attendance, 1994-2003 Season Sum of

average attendance per club (actual)

Sum of average attendance per club (constrained optimal)

difference

standarddeviation of win percentages (actual)

standard deviation of win percentages (constrained)

sum of win percentages (constrained)

1993/94 281938 296615 5.2% 0.086 0.274 11.5711994/95 261222 270633 3.6% 0.081 0.238 11.5031995/96 284512 296466 4.2% 0.072 0.266 11.5431996/97 300336 314789 4.8% 0.094 0.231 12.0001997/98 362128 397913 9.9% 0.115 0.289 12.1151998/99 327961 354169 8.0% 0.116 0.287 12.0601999/00 339712 368816 8.6% 0.111 0.256 12.0872000/01 344097 376491 9.4% 0.115 0.283 12.1242001/02 366164 402470 9.9% 0.112 0.306 11.9062002/03 370240 408052 10.2% 0.106 0.277 11.991

1

Second tier English football

050000

100000150000200000250000300000350000400000450000

1993

/94

1994

/95

1995

/96

1996

/97

1997

/98

1998

/99

1999

/00

2000

/01

2001

/02

2002

/03

Sum of averageattendance per club(actual)Sum of averageattendance per club(constrained optimal)

2

You Cannot Be Serious

1. Similar empirical results for Major League Baseball

2. Do fans demand competitive balance per se? 3. Do owners maximise profit?

4. Would redistributive measures help? – they are generally intended to create more

competitive balance, not less

5. Delegating control and the integrity of the sport

6. Other considerations- TV?

7. The Abramovitch effect 3

Pre-emption through bidding The Abramovitch effect: bookmakers have paid out to bettors on Chelsea to win the Premier League after only nine games of a 38 game season:

(A1) 1221

11 1, ww

tttw −=+

=

(A2) π1 = (σ - w1) w1 – ct1 , π2 = (1 - w2) w2 – ct2 , σ > 1

(A3) 1

1

t∂∂π

= (σ - 2w1) w2 - cTD = 0, 2

2

t∂∂π

= (1 - 2w2) w1 - cTD = 0

where TD = t1 + t2 , so that at the Nash equilibrium (A4) w1* = σ / (1 + σ)

4

(A5) cT

ct

ct D

2323

2

1 )1()1(

)1()1(,

)1()1(

+−

==>+−

=+−

=σ

σσσ

σσσ

σσ

TDTS

c*

(A6) STc 2)1(

)1(*+−

=σ

σσ

(A7) 2

221

1

1 *21)1(

1)1()1(*2

wRww

wR

∂∂

=−=+−

>+−

=−=∂∂

σσ

σσσσ

5

Joint profits (A8) π1 + π2 = (1 + σ)w1 - 2w1

2 – cT, (A9) w1

M = (1 + σ)/4 > w1* Inefficient result

6

The incentive to pre-empt At the interior equilibrium the profits of the dominant team is

(A10) 3

24

1 )1(*

σσσπ

++

=

If team 1 pre-empts by offering a wage rate c* + ε, it acquires all the talent (TS = t1* + t2*) has a win percentage w1 = 1 and obtains, for ε small enough, profits equal to

(A11) 2

3

)1(1*1

σσσ+−

=−− STc

Pre-emption can therefore be profitable if (A12) 84.1*123 ≈>⇒++> σσσσσ (Note w1

M = (1 + σ)/4 implies w1M = 1 for σ = 3)

7

Pre-emption and bidding for talent Under these conditions firm 1 will start to bid up the price of talent (A13) Pre-emption constraint: c = c** where π1(w1 = 1, c**) = π1(w1 = w1*(σ), c**) (A14) Participation constraint: π2(w2 = w2*(σ), c**) ≥ 0 To solve for c** when (13) binds

(A15) SS TcTc **

1)1(**1 2

3

σσ

σσσ

+−

+=−−

From this we derive

(A16) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−+

= 2

3

)1(11**

σσσσ

STc

8

Competition from firm 2 The participation constraint requires

(A17) π2(w2 = w2*(σ), c**) = σσσ

+−

+ 1**

)1( 2

STc≥ 0

From (A16) and (A17) we can see that both constraints bind when

(A18) 2101

1)1(

3

2 +==>=+

++−+

σσ

σσσσ

9

Contest wages and pre-emption

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 3 3.5 4

sigma (competitive imbalance)

wag

e ra

te

c*c**

no pre-emption no pre-emption but wages rise to deter it

pre-emption occurs but profit lower than under c* with no pre-emption

pre-emption occurs and is more profitable

10

Conclusion: 3 regimes 1. σ < σ* interior equilibrium, pre-emption never profitable 2. σ* < σ < 1+√2 pre-emption at c*, profitable, but wages bid up so that pre-emption

does not occur 3. σ > 1+√2 pre-emption occurs (NB pre-emption occurs at values of σ < 3)

11