May 31, 2012 - A over 2010-12 seasons, the wage dispersion has no effect on team performances. Keywords: Players ... group stage of the UCL tournament. As a whole there ... 2 The third team continues in the UEFA Europa League. 3 Obviously ... consists of more than 11 players), receive zero wages. Notice that, due to ...
M PRA Munich Personal RePEc Archive
A NLIP Model on Wage Dispersion and Team Performance Christos Papahristodoulou M¨alardalen University/Industrial Economics
30. May 2012
Online at http://mpra.ub.uni-muenchen.de/39149/ MPRA Paper No. 39149, posted 31. May 2012 10:04 UTC
A N LIP Mod el on Wage Dispersion and Team Perform ance
Christos Papahristod ou lou *
A bstract Using a N on-Linear Integer Program m ing (N LIP) m od el, I exam ine if w age d ifferences betw een Super talents and N orm al players im prove the perform ance of four team s w hich participate in a tournament, such as in the UEFA Cham pions League (UCL) group m atches. With ad -hoc w age d ifferences, the optimal solutions of the m od el show that higher w age equality seem s to im prove the perform ance of all team s, irrespectively if the elasticity of substitution betw een Super- and N orm al- players is high or low . In ad d ition to that, a U-type perform ance exists in tw o teams w ith the highest and the second high elasticity of substitution. With team d ata from the 2011-12 UCL group m atches and from the Italian Serie A over 2010-12 seasons, the w age d ispersion has no effect on team performances.
Keyw ord s: Players, Team s, Wages, Perform ance, Tournam ent *Departm ent of Ind ustrial Econom ics & Organization, Mälard alen University , Västerås, Sw ed en; christos.papahristod oulou@m d h.se Tel; +4621543176
1
1. Introduction The optim al w age structure of team s has been frequently d iscussed over the last d ecad es. The m ain hypothesis is w hether the com pressed or the d ispersed w ages am ong a team ’s players have a stronger im pact on the p erform ance of the team . As is very often in econom ics, there are at least tw o schools of thought. Lazear & Rosen (1981) argued that the team perform ance increases if the best talents are paid higher w ages than the norm al players. Milgrom (1988) and Lazear (1989) on the other hand , stressed the possibility of bad field cooperation betw een players, and consequently the perform ance could be inferior, if the un d er-paid players feel d iscrim inated . Levine (1991) took an extrem e position and favoured the egalitarian w ages. Fehr and Schm id t (1999) tried to balance these tw o effects and argued that, as a w hole, the team losses m ore if w ages are m ore unequal than equa l. Franck & N üesch (2007) review ed the em pirical stud ies from sport team s, in baseball, hockey, basket and football. Som e stud ies seem to support the com pressed w ages hypothesis. They argue that these find ings can partly be explained becau se the m ajority of em pirical stud ies assu m e a linear relationship betw een w age d ispersion and team perform ance. In their ow n stud y, based on 5281 ind ivid ual salary proxies from Germ an soccer players betw een 1995-07, they allow ed for squares of the Gini coefficient and the coefficient of variation of the w age d istribution a t the beginning of the season. They found a U-form ed sportive success, i.e. team s p erform better by either an egalitarian pay structure or a steep one 1. On the other hand , Pokorny (2004) found an inverted U-form ed success, i.e. the perform ance is higher w ith interm ed iate w age d ifferences. Avrutin & Som m ers (2007), using baseball d ata from 2001-05, found no effect. Torgler et al (2008), using also d ata from the Germ an Bund esliga (and the N BA), found that p layers care m ore about the salary d istribution w ithin the team and not ju st about their ow n salary. Generally, players prefer a red uced inequality and in that case their perform ance im proves. In ad dition, a d etailed investigation of the basketball d ata show s also that w hen a player m oves from a relative incom e ad vantage to a relative d isadvantage, his perform ance d ecreases in a statistically significant w ay. On the other hand , m oving from relative incom e d isad vantage to relative ad vantage has no effects. Wisem an & Chatterjee (2003), using baseball d ata from 1980-02, found a negative effect of w age d ispersion. In a sim ilar stud y recently, i.e. w ith baseball d ata from 1985-10, Breunig et al (2012) found also a negative effect of w age d ispersion. On the other hand , Sim m ons & Berri (2011), using basket statistics, supported the Lazear & Rosen (1981) hypothesis that higher w age d ispersion increases team perform ance.
Also they found evidence that teams with dispersed wages entertain the public better since the number of seasonal dribbling and runs increases significantly! 1
2
In this paper I form ulate and solve a N LIP m od el to exam ine m ainly the effects of w age d ifferences betw een Super (S) - and N orm al (N ) - players in the perform ance of four team s, w hich participate in a tournam ent like the UEFA Cham pions League (UCL) group m atches. Using ad -hoc w age d ifferences and various “team prod uction functions”, it is of interest to see if the optim al solutions favour the com pressed or the d ispersed w ages hypotheses for all, or som e team s. In ad d ition, tw o sm all d ata sets (UCL group m atches from 2011-12 and Italian Serie A from 2010-12 seasons) are used to test the above hypotheses. 2.1 The model There are four football team s w hich p lay three hom e and three away m atches in the group stage of the UCL tournam ent. As a w hole there are 12 m atches and the m axim um num ber of points is 36. The tw o team s w hich collect m ost points are qualified for the next round and the other two are elim inated 2. The four participating team s have d ifferent qualities and consequently d ifferent ranking. To d ifferentiate the team s, four d ifferent “team prod uction functions” are assum ed . I follow Kesenne (2007) and assum e that the form ation of the team s consists of a certain num ber of S- and N -players. The S- and N -players are consid ered as being from alm ost com plem ents, to alm ost substitu tes. All S-players are equally “Super” and all N -players are equally “N orm al” 3. Team s use their S- and N - players in ord er to “prod uce” points. The supply of S- and N -players is unlim ited . Since the value of the m arginal prod uct of players can’t be observed or m easured , team s have a certain w age structure, from very d ispersed t o very com pressed . The w ages is the policy variable of the m od el. It is assum ed that team s have no other fixed costs (like m anagers or other facilities); their only variable costs consist of the w ages they pay to their players. It is also assum ed that all team s receive sim ilar revenues, either d irectly from UEFA, and / or from their public, TV-rights and sponsors and the w ages they pay can’t be higher than their revenues. All team s p lay a “Cournot” type game and m axim ize sim ultaneously their points. The m od el is rather general and can explain, not only the ow n perform ance, but the effect to the other team s as w ell, even if they keep their w ages unchanged . It can also
2
The third team continu es in the UEFA Eu rop a Leagu e. Obviou sly, there are p layers w ho, objectively, belong to one category or the other. Perhap s, m ost p layers are neither that excellent to be classified in the first category, n or that “N orm al” to be classified in the second category. In the football w orld , it is very com m on that the su p p orters of a team tend to overvalu e their ow n p layers and u nd ervalu e the com p etitor’s p layers. This is an em p irical issu e, left to su p p orters, to m anagers, to jou rnalists or even to those w ho d o research in efficiency analysis. 3
3
show w hether the tournam ent rem ains balanced and w hether team s w ho have m ore S-players can collect m ore points. Let P1, P2, P3, P4, be the points collected by the team s; let S 1, S 2, S 3, S 4, be the num ber of S-players, and N 1, N 2, N 3, N 4 the num ber of N-players the team s use in these m atches. Let w 1, w 2, w 3, w 4, be the victories of the team s, d1, d2, d3, d4, the d raw m atches and l1, l2, l3, l4, the losses (d efeats) of the team s. All these 24 variables are positive integers. Since all firm s aim at m axim izing points (sim ultaneously), their objective function is given by (1), und er the follow ing constraints. The key constraints that d ifferentiate team s are the team s’ “prod uction” functions , (2a – 2d ). All fu nctions are of Constant Elasticity of Substitutio n (CES) type of d egree one, w ith d ifferent elasticity. The function of team 1 is closed to Leontief, i.e. very low elasticity of substitution betw een its S- and N -players 4. Team 4 on the other hand , has a closed to Cobb-Douglas “prod uction” function, i.e. alm ost excellent elasticity of substitution, team 2 is close to team 1, w hile team 3 is close to team 4. The use of S- and N -players for each team w ill be end ogenously d eterm ined from the optim al solution. When the team form ation has been d eterm ined , it is assum ed that the sam e team w ill be used for all six m atches, unless the w age structure ha s changed . Team s of course change the com position of their players for tactical reasons, such as if they play aw ay against a stronger team or if they play at hom e against a w eaker team , or because som e of their players m ight be injured or punished and are not available for a particular m atch. The m od el neglects such possibilities. The m od el also assum es that players w ho are not used (becau se the roster of team s consists of m ore than 11 players), receive zero w ages. Notice that, d ue to the integer constraint of players, the com position of the team can rem ain unchanged , even if the w ages change. t i is the efficiency param eter and ai is the d istribution param eter. Constraints (3a – 3d ) restrict the num ber of team players to 11. In reality, even for top and w ealthy team s, the num ber of S- is often low er than the num ber of N -players, w hich is given by constraints (4a – 4d ). Despite the fact, that such cond ition is not necessary, it is stated explicitly in ord er to speed up the solution of this com plex m od el. As is w ell know n, victories are w orth 3 points; d raw s are w orth 1 point and losses, zero points. Thus, the num ber of points collected to every team is given by constraints (5a – 5d ). 4
In reality, an S-p layer can su bstitu te an N -p layer m ore easily than the reverse. I assu m e that the elasticity of su bstitu tion is u nchanged , irresp ectively if the S- su bstitu tes the N -p layer, or if the N su bstitu tes the S-p layer.
4
4
P
max
(1)
i
i 1
s.t.
P t S P t S P t S
P1 t1 1 S1100 (1 1 ) N1100 (1 2 ) N 210
1 100
1 10
2
2
2
10 2
3
3
0.5 3
(1 3 ) N 30.5
3
4
4
0.1 4
(1 4 ) N 40.1
4
1 0.5
( 2a 2d )
1 0.1
S i N i 11
(3a 3d )
Si N i
( 4a 4d )
Pi 3wi d i
(5a 5d )
w w d
h i, j
1, pairwise,
(6a 6l )
w w d
a j ,i
1, pairwise,
( 7 a 7l )
h i, j
a j ,i
h i, j
a j ,i
di ,
(8a 8d )
w w wi ,
(9a 9d )
wi d i li 6,
(10a 10d )
Pi 18,
(11a 11d )
d
h i, j
d
a i, j
h i, j
a i, j
4
P 36, i 1
(12)
i
i 12 Pi 0.5Pi 2 wS S i wN N i 0, i
i
(13a 13d )
( Pi , S i , N i , wi d i , li ) Integer ( wih, j , w aj ,i , d ih, j , d aj ,i ) Binary i, j 1,..,4, i j ti 0 0 i 1 wSi , wNi wage parameters
Each one of the 12 m atches in the group of four team s, end s either w ith a hom e team victory, wih, j , w ith an aw ay team victory, w aj ,i , or w ith a hom e team d raw , d ih, j , w hich is obviously equal to the aw ay team d raw , d aj ,i . But, in ord er to id entify the correct pair of team s w hich play d raw at hom e and / or draw aw ay , w e m ust separate the hom e team d raw from the aw ay team d raw , i.e. w e need 24 ad d itional constraints (6a – 7l). The first 12 constraints (6a – 6l) relate each p air of team s to hom e team d r aw and the rem aining 12, (7a – 7l), relate to the aw ay team d raw . If for instance d1h, 2 1, (and consequently w1h, 2 w2a,1 0 ), from constraints (6a) and (7a) w e are ensured that
d 2a,1 1, as w ell. If that m atch end s w ith a hom e or aw ay victory, it im plies that d1h, 2 d 2a,1 0. Moreover, these constraints d o not exclud e im possible (non -binary) 5
m atch results, such as d1h, 2 d 2a,1 0. 5 and w1h, 2 w2a,1 0 .25. Therefore w e require that all possible m atch resu lts are binary as w ell. Obviously, the d raw s (and the victories) for each team is the su m of all possible d raw s (and victories) against all other team s. Constraints (8a – 8d ) ensure that the num ber of d raw s in constraints is alw ays an integer (and actually even) num ber, or zero. For instance, w hen team 1 plays only one m atch d raw , it is d1 1. If the d raw m atch w as a hom e m atch against team 2, it m ust be d1h, 2 1, (from 8a), and d 2a,1 1, (from 8b), as w ell. In that case, if team 2 d oes not play another m atch d raw , it m u st be d 2 1 as w ell. Of course, if team 2 p lays three d raw m atches, i.e. d 2 3, it im plies that team 2 m ust have played d raw against the other team s as w ell. A sim ilar interpretation applies for the victory constraints (9a – 9d ). Since each team play six m atches, there are 6 possible results from its gam es, i.e. constraints (10a – 10d ) are required . Constraints (11a – 11d ) show that no team can collect m ore than 18 points. In ad d ition to that, constraint (12) show s that the m axim um num ber of points from all m atches is 36, com posed w ith 12 victories (for tw o team s) and 12 d efeats for the other tw o team s. Finally, constraints (13a – 13d ) ensure non-negative profits. The revenue function is quad ratic in the points collected . All team s pay the sam e, higher w ages to their Splayers, and the sam e, low er w ages to their N -players. The initial param eters are: wS 8 and wN 4.8, i.e. the N -players receive only 60% of the S-players’ w ages. i
i
Keeping wS 8 , the m od el is repeated and solved for higher and low er values of the i
param eter wN . Obviously the w age d ispersion or com pression d oes not influence the i
perform ance of the ow n team , but the perform ances of the other team s too. The nonnegative profits constraints are satisfied if each team collects at least 6 points. With less than 6 points there is no integer value of N - and S-players to ensure non-negative profits. The m od el is now com plete and w as solved in Lingo, using Global Solver. 2.2 The Solution I solved the mod el 124 tim es, i.e. 31 tim es per team , using 31 d ifferent wN , starting i
from as low as 3.4 up to 6.4, increased at a range of 0.1. When one team changed its wN , all other team s keep their ow n wN unchanged . When wN 6.4, there is no feasible i
i
i
(integer) solution, because the non -negative profits constraint(s) are violated .
6
5
Table 1 d epicts the ow n effects on performance and team com position, w here the policy variable is wN . Some points from the Table 1 are w orth m entioned . i
The initial solution, w ith wS 8 and wN 4.8, show s that the tournam ent is i
i
com pletely balanced , because each team w ins its three hom e m atches and collects 9 points. Despite the d ifferences in their “production” functions, all four team s have a sim ilar team com position, consisting of 1 S-player and 10 N -players and each team m akes a profit of 11.5. Table 1: Points m axim ization, su bject to non -negative p rofits
wNi
S 1 N 1 P1
3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
6 6 6 9 9 9 9 6 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 12 10 10 12 12
0.55 0.5 0.34 0.5 0.66 0.66 0.5 0.37 0.37 0.5 0.5 0.02 0.5 0.66 0.34 0.66 0.5 0.34 0.66 0.5 0.5 0.5 0.5 0.66 0.5 0.5 0.5 0.21 0.55 0.44 0.11
S 2 N 2 P2 2 1 1 1 1 1 2 2 4 1 1 1 1 1 1 1 4 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1
10 10 10 10 10 9 9 7 10 10 10 10 10 10 10 7 10 10 10 10 10 10 9 10 10 10 9 9 10 10 10
6 9 6 6 6 6 6 9 6 6 9 6 6 9 9 9 9 9 9 9 9 9 9 9 9 9 10 12 10 12 12
0.43 0.5 0.5 0.01 0.29 0.48 0.01 0.44 0.30 0.66 0.37 0.44 0.44 0.5 0.36 0.5 0.44 0.01 0.34 0.37 0.5 0.65 0.76 0.5 0.65 0.65 0.65 0.44 0.5 0.5 0.2
S 3 N 3 P3 3 3 1 1 3 2 1 3 5 1 1 1 5 1 4 1 1 1 1 2 1 2 1 1 2 1 1 1 1 2 1 1
5
8 10 10 8 9 10 8 6 10 10 10 6 10 7 10 10 10 10 9 10 9 10 10 9 10 10 10 10 9 10 10
6 9 9 6 6 9 9 9 6 6 6 9 6 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 12 12 12
0.5 0.89 0.65 0.65 0.5 0.48 0.65 0.5 0.03 0.72 0.01 0.65 0.5 0.5 0.57 0.89 0.5 0.99 0.02 0.01 0.5 0.43 0.65 0.5 0.5 0.63 0.62 0.99 0.71 0.65 0.5
S 4 N 4 P4 4 1 1 3 1 2 2 1 1 1 1 1 5 1 4 1 1 4 3 1 1 1 1 1 1 1 1 1 1 1 1 1
10 10 8 10 9 9 10 10 10 10 10 6 10 7 10 10 7 8 10 10 10 10 10 10 10 10 10 10 10 10 10
9 9 9 9 6 6 6 9 6 6 6 9 6 9 9 9 9 9 9 9 9 9 9 9 9 9 12 12 10 12 12
0.78 0.74 0.01 0.28 0.5 0.65 0.01 0.45 0.4 0.5 0.61 0.5 0.5 0.65 0.67 0.74 0.65 0.8 0.03 0.62 0.01 0.5 0.78 0.03 0.68 0.02 0.5 0.99 0.72 0.21 0.5
To save sp ace, I have exclu d ed the cross effects, i.e. effects on o ther team s. The fou r ad d itional Tables can be p rovid ed by the au thor at a requ est.
7
As expected , team 1 (closed to Leontief), never changes its team com position and uses 1 S- and 10 N -players. On the other hand , it is team 3 that changes its team com position m ore frequent (in 11 out of 31 tim es) and not team 4, (closed to CobbDou glas). The m ost striking result is that m ore S-players d o not alw ays increase the perform ance of team s! This is d ue to tw o reasons. First, other teams m ight also play w ith m ore S-players too. Second the efficiency 6 of the team s changes too. For instance, w hen wN 4 4.7, both team 4 and team 2 use 7 N -players and all team s collect 9 points and their respective efficiencies are alm ost sim ilar, t4 43.6, t2 38.6. When wN 4 6.0, both team s use 10 N -players, team 4 collects 12 points, w hile team 2 collects 7 points, because their new efficiencies are t4 35.5, t2 7.5. Obviously, w hen the perform ance d oes not increase, the profits of the team s are red uced w ith m ore S-players. And the points collected per team are not related to their d istribution param eter of players’ ai. Finally, the tournament appears to be frequently balanced . For instance, it is com pleted balanced in 23/ 31 cases w hen team 1 selects w ages; it is also com pletely balanced in 16/ 31 cases w hen team 2 selects w ages, and in 19/ 31 cases w hen team 3 or team 4 select w ages. And w hile in m ost cases the total num ber of points selected equals to 36, there are som e d raw m atches w ith a total num ber of points equal to 35 (alw ays w hen wN 6.0 ). i
2.2.1 (i)
The regressions from the optimal solutions
The ow n effects
In ord er to find out how the w age d ispersion influences the perform ance of the team s, I run the follow ing tw o regression equations, based on all 124 optim al solutions.
Pi 1 (8 wN i ) Pi 1 (8 wN i ) 2 (8 wN i ) 2 . The right hand sid e variable is the w age d ifference betw een the respective fixed wS 8 and wN , w hile the d epend ant variable is points collected . N egative (p ositive) i
i
-estim ates im ply that the perform ance of team s im proves if the w age equality (inequality) betw een the S- and N -players increases. Sim ilarly, negative -estim ates 6
Du e to sp ace constraints, the efficiency p aram eters are not p resented .
8
and positive -estim ates im ply a U-type success, i.e. either highly unequal or highly equal w ages w ill im prove perform ance. Table 2a sum m arizes the regression estim ates for every team . Based on the average num ber of points, team 1 and 4 are qualified . It is clear that in the linear m od el, all team s im prove their perform ances w hen the w age d ispersion d ecreases (all estim ates are strongly negative). Consequently, the find ings supp ort the Milgrom (1988) and Lazear (1989) hypothesis. Table 2a: OLS ow n estim ates Team 1 Team 2 Team 3
R2
R2
Team 4
P 8.97 1.47 12.65** (19.24) -1.19** (-5.84) 0.52
P 8.39 1.89 13.6** (18.52) -1.68** (-7.39) 0.64
P 8.68 1.72 12.75** (15.64) -1.32** (-5.20) 0.46
P 8.74 1.80 12.77** (14.16) -1.30** (-4.65) 0.41
12.97** (5.44) -1.41 (-0.87) 0.036 (0.14) 0.51
17.25** (6.73) -4.25* (-2.43) 0.414 (1.48) 0.66
18.36** (6.70) -5.26** (-2.82) 0.637* (2.13) 0.52
21.7** (7.88) -7.58** (-4.04) 1.01** (3.38) 0.56
** significant at 0.01 level; * significant at 0.05 level; t -statistics is in p arentheses
Regard ing the quad ratic function, the U-type success for team s 3 and 4 is not rejected , over the relevant range wN (3.4,..., 6.4). For instance, team 3 w ill m inim ize i
its perform ance and collect about 7.5 points if it pays wN 3.9, (i.e. if its N -players 3
receive slightly less than 50% of the S-players w ages), it w ill collect about 8.5 points if it pays the low est w age wN 3.4 and it w ill collect about 11.5 points if it pays the 3
upper-lim it w age wN 6.4. Sim ilarly, team 4 w ill m inim ize its selected points (7.5), if 3
it pays wN 4.25 . 4
(ii)
The cross effects
In ord er to find out the perform ance effects to the other team s, I run the follow ing tw o regression equations:
Pi 1 (8 wN j ) Pi 1 (8 wN j ) 2 (8 wN j ) 2 , i j 9
The estim ates are given in Table 2b. In the linear m od el, the estim ates are com pletely consistent w ith those in Table 2a. A d ecrease of w age d ispersion in team i red uces the perform ance of team j (all -estim ates are strongly positive). Table 2b: OLS cross estim ates Linear
Team 1
Team 2
Team 3
Team 4
R2
Quadratic
R2
Effect on Team 2 7.19** 0.64** 0.27 9.08** -0.69 0.22 0.27 (12.2) (3.48) (4.3) (-0.5) (1.14) Effect on Team 3 8.20** 0.22* 0.13 5.87** 1.85** -0.26* 0.26 (27.1) (2.32) (5.89) (2.74) (-2.43) Effect on Team 4 6.99** 0.59** 0.34 4.57** 2.3* -0.27 0.37 (14.8) (4.1) (2.78) (2.05) (-1.53) Effect on Team 1 6.95** 0.66** 0.34 7.5** 0.28 0.06 0.32 (13.2) (4.05) (3.93) (0.2) (0.3) Effect on Team 3 6.87** 0.66** 0.30 2.61 3.66** -0.48* 0.39 (11.9) (3.73) (1.37) (2.81) (-2.32) Effect on Team 4 7.60** 0.62* 0.16 5.14 2.35 -0.28 0.16 (9.95) (2.60) (1.88) (1.27) (-0.94) Effect on Team 1 6.83** 0.85** 0.30 5.88* 1.51 -0.11 0.27 (9.21) (3.69) (2.20) (0.83) (-0.37) Effect on Team 2 8.15** 0.23* 0.15 5.24** 2.28** -0.33** 0.37 (27.3) (2.53) (5.72) (3.66) (-3.3) Effect on Team 4 7.3** 0.49** 0.23 3.0 3.52** -0.49* 0.37 (14.3) (3.13) (1.82) (3.13) (-2.7) Effect on Team 1 7.91** 0.34* 0.14 6.59** 1.27 -0.15 0.14 (17.5) (2.45) (4.08) (1.16) (-0.8) Effect on Team 2 7.57** 0.49* 0.13 1.37 4.86** -0.7** 0.31 (11.0) (2.30) (0.63) (3.28) (-2.97) Effect on Team 3 6.78** 0.73** 0.28 2.84 3.5* -0.45 0.33 (10.3) (3.56) (1.26) (2.28) (-1.82)
In the quad ratic m od el, som e estim ates are consistent w ith those in Table 2a, because a reversed U-type success appears. For instance, w hen team 3 pays wN 6.4, w hile 3
team 2 keeps its initial w ages at wN 4.8, team 2 collects about 8 points, w hile team 3 2
10
w ould collect 11.5 points. The pair of team s (first entry d enotes w age d ispersion of team i and second entry d enotes the perform ance of team j) w ith a reversed U-type perform ance are: (1, 3), (2, 3), (3, 2), (3, 4) and (4, 3), w hile in the rem aining seven pairs, there is no effect. N otice that team 3, which im proves its ow n perform ance by low or high w age d ispersion (Table 2a), is affected negatively by all other team s too. To sum m arize the estim ates, w e conclud e that team 1 (w ith the alm ost Leontief type), w hich collects the highest average num ber of points (8.97), im proves its ow n perform ance linearly w ith low er w age d ispersion. And d espite the fact that the Uform ed w age d ispersion d oes not im prov e its ow n perform ance, it is the only team that is not influenced from low or high w age d ispersions of other team s. 3. Empirical study Obviously, to test the sam e hypotheses w ith real d ata is alm ost im po ssible, becau se there are m any problem s w ith the observed players’ w ages and of course the team form ation. It w ill require a huge am ount of tim e to refine all available d ata set by observing each ind ivid ual player, to exam ine if he w as injured or punished and unavailable for som e m atches, or if he played only a sm all part in a m atch. And even if such inform ation w ere relatively easy to collect, the w age d isp ersion of the team w ould vary d epend ing upon the various w eights one uses for each player participation in the m atches. To ad hoc exclud e som e players (often the youngsters w ith low w ages) m ight be erroneous too, because som etim es m anagers d o not play the “expensive” players and d eliberately use “cheap” players in som e m atches. Tw o very sm all d ata sets have been collected , assu m ing sim p ly that all players, for w hom w age observations exist, are available to play, and all players’ w ages have th e sam e w eights, irrespectively if they played or not. The team spirit or the envy of players exists for the entire team roster and not necessarily for those players w ho are field ed . The first d ata set consists of 32 European team s (496 players) from 48 UCL group m atches, played in 2011. The aggregate statistics are show n in Append ix (Table A). As is show n, for som e team s there are accurate observations for at m ost 10-13 players. The second d ata set consists of 40 Italian team s from the Serie A (about 1015 players), over two seasons 2010-11/ 2011-12, obtained from the follow ing sites: http:/ / w w w .football-m arketing.com / 2010/ 09/ 07/ italian -serie-a-w age-list/ , http:/ / w w w .xtratim e.org/ forum / show thread .php?t=261972. N otice that 34 of these Italian team s appear in both seasons, w hile the rem aining 6 team s appear only in one season (3 of them relegated in the first season and 3 of them ad vanced in the second season). Since the team roster of these 34 team s change and / or som e w age contracts are re-negotiated , the w age d ispersion w ithin the sam e team varies from year to year.
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Table 3 sum m arizes the OLS estim ates for these d ata sets. In UCL, the w age d ispersion is m easured by Coefficient of Variation (CV) and by Mean Absolute Deviation (MAD) as w ell; as control variables, tw o d ifferent proxies are used , the team value (V) and the respective UEFA ranking (R); in Serie A, the w age d ispersion is also m easured by CV and by Coefficient of Dispersion (CD) w hile as a control variable the average wage (W) per team is used . In case (i), the num ber of points d oes not autom atically reflect qualification or elim ination from the tournam ent, because it d epend s on the group. For instance in group D, the elim inated third team ’s points (Ajax) are id entical to the second team ’s (Lyon), but Lyon had a better goal d ifference against Ajax. Sim ilarly, w hile Manchester City elim inated w ith 10 points, four other team s in other groups qualified w ith less p oints. Consequently, in case (ii), a d um m y variable w as introd uced w ith the follow ing values: 1 for the tw o qualified team s per group; 0.5 for the third elim inated team and 0 for the fourth team . Table 3: OLS estim ates for UCL and Serie A UCL: (i) Point s
R2
R2
MAD 1.775 (0.78) MAD 3.056 (1.93)
V CV V 4.205** 0.014 0.45 4.66** -0.45 0.02** 0.44 (4.50) (1.89) (2.61) (-0.09) (4.91) R CV R 3.066** 0.044* 0.47 2.14 2.865 0.07** 0.41 (2.66) (2.16) (1.12) (0.6) (4.60) UCL: (ii) (Dummy )*(Point s) MAD V CV V 2.24 4.37 0.008 0.35 1.72 2.76 0.02** 0.30 (1.79) (1.45) (0.75) (0.74) (0.42) (3.61) MAD R CV R 0.64 3.52 0.051 0.41 -1.13 5.55 0.08** 0.37 (0.43) (1.73) (1.95) (-0.47) (0.92) (4.17) It alian Serie A: Point s CD W CV W 42.73** -0.08 14.34** 0.50 43.49** -0.08 14.44** 0.50 (6.81) (-0.68) (5.99) (6.49) (-0.62) (6.01) ** significant at 0.01 level; * significant at 0.05 level; t -statistics is in p arentheses.
Both d ata sets show sim ilar estim ates, w ith rather low explana tory pow er. N one of the d ispersion variables is statistically significant from zer o. On the other hand , as expected , stronger or richer team s perform better. Moreover, since both sam ples are very sm all and the players’ statistics are not refined , it is d ifficult to draw certain conclusions.
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4. Conclusions The purpose of this paper w as to d evelop a kind of “general equilibrium ” m od el to investigate if team s p erform better or w orse w hen they pay rather com press ed or m ore d ispersed w ages to their S- and N - quality players. In the m od el, four d ifferent football team s com pete, in a tournam ent like the U CL group m atches, to m axim ize their points and qualify in the next ground . Som e key features of the m od el are the non-linearity of “prod uction” functions (w hich are of CES type w ith d ifferent elasticity of substitution am ong players) and a num ber of various integer variables (like players, points, victories and d raw s). Assum ing a high w age level for the im plicitly solved S-qu ality players in every team and a rather large range of low er w ages for the im plicitly solved N -quality players, global optim al solutions w ere obtained . In m ost cases, the team form ation is com posed by 1 S- and 10 N -players. Moreover, as it often happens in football, team s w hich field m ore S- players d o not alw ays perform better than team s w ith just 1 S-player! This is m ainly d ue to d ifferences in absolute and / or relative efficiency the team s. Despite the fact that all four team s perform alm ost equally w ell and the w age param eters lead to a rather balanced tournam ent, team 1, w ith the low est elasticity of substitution betw een S- and N - players, collects on average, slightly m ore points than all others (follow ed by team 4, w ith the highest elasticity of substitution). Using the collected points from the respective optim al solutions, linear and quad ratic regressions w ere run to exam ine the w age dispersion on (i) the ow n effects and (ii) the cross effects. All four team s im prove their ow n perform ance if the w age d ispersion d ecreases. In m ost cases, the d ecrease in w age d ispersion d eteriorates the perform ance of other team s as w ell. In ad d ition, som e evid ence on the U-type ow n success and a reverse U-type cross su ccess appears. Consequently, w hile highly d epressed w ages im p rove perform ances linearly, interm ed iary w age d ispersion is w orse than a highly d ispersed one. On the other hand , using tw o sm all d ata sets, from the UCL and the Italian Serie A, the observed w age d ispersion from all players of the team s w as not statistically significant from zero, w hen other controlled variables (like the ranking or the value of team s) w ere includ ed in the regressions. The m od el can be easily extend ed to catch other im portant aspects. For instance, instead of m axim izing points, subject to non -negative profits, team s can m axim ize profits. One can also increase the “price” param eter of points from 12 to 15, so that team s can collect m ore than 12 points that som e global solutions provid ed . Another 13
interesting extension can be to allow d ifferent team s com positions in d ifferent m atches, or com bine the “prod uction” functions per m atch, to find out if the “Leontief team ” beats or is d efeated by the “Cobb-Douglas team ”, at hom e or aw ay, and try to collect appropriate m atch d ata to test the d erived hypothesis. Finally, since the selected CES functions d o not let team s u se various tactical d ispositions of players, it w ould be d esirable to stress the field positions of the S- and N -players. That can be sim ilar in spirit to H irotsu & Wright (2006), w ho applied a N ash -Cournot gam e to figure out the w in probabilities of the 4-4-2 strategy over the 4-5-1 one.
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Table A: MAD, CV, UEFA Ranking, Team valu e in m . of € and Points Teams per group FC Bayern N apoli Man. City Villarreal Inter CSKA Trabzonspor Lille Benfica Basel Man. United Otelui Galati Real Mad rid Lyon Ajax Dinam o Zagreb Chelsea Bayer Leverkussen Valencia Genk Arsenal Marseille Olym piacos Borussia Dortm und Apoel Zenit Porto Shakhtar Barcelona AC Milan Plzen Bate Borisov
Observed players 24 24 16 16 26 16 13 13 16 13 16 13 16 15 13 13 16 13 13 13 15 13 15 16 13 13 13 13 16 25 13 13
MAD (1) 1.8196 0.4203 1.3656 0.2875 1.3231 0.2625 0.2793 0.3562 0.2672 0.2414 0.9422 0.1402 1.5078 0.9173 0.4911 0.2734 1.3500 0.5609 0.3030 0.2367 0.8089 0.5882 0.3067 0.6031 0.2604 0.3172 0.3976 0.4391 1.6711 1.2758 0.2166 0.1112
CV (2) 0.5517 0.6453 0.3051 0.3647 0.6479 0.2153 0.3130 0.2580 0.2189 0.2704 0.3329 0.2970 0.4148 0.3394 0.3211 0.2972 0.3127 0.2907 0.1755 0.2882 0.3250 0.2520 0.3118 0.4752 0.3221 0.2421 0.2730 0.4539 0.3680 0.6483 0.3445 0.2403
Rank (3) 122.507 39.853 61.507 78.551 102.853 80.566 20.115 38.802 86.835 53.360 141.507 7.764 110.551 94.802 57.943 24.774 122.507 59.403 83.551 12.480 113.507 84.802 61.420 30.403 33.599 79.066 97.835 83.894 151.551 88.853 14.070 29.641
Value (4) 359.95 194.2 467.0 142.6 246.85 137.3 87.8 119.75 168.2 48.3 415.0 18.15 542.0 152.2 97.35 44.8 381.0 137.0 180.0 47.9 299.25 140.65 81.4 179.75 14.95 155.2 210.0 143.45 579.0 266.3 163.0 17.05
Points (5) 13 11 10 0 10 8 7 6 12 11 9 0 18 8 8 0 11 10 8 3 11 10 9 4 9 9 8 5 16 9 5 2
Notes: (1) & (2): For the three Italian teams, the annual wages are found in: http://www.xtratime.org/forum/showthread.php?t=261972 ; for all other teams, searching extensively various sport sites, forums and the teams’ official sites; most values for the following teams are uncertain and for the non- Euro teams, estimated with various exchanges rates into €: Trabzonspor, Otelui Galati, Dinamo Zagreb, Genk, Apoel, Plzen, Bate Borisov. (3): http://kassiesa.home.xs4all.nl/bert/uefa/data/method4/trank2012.html (4): http://www.transfermarkt.co.uk/ (5): http://www.uefa.com/uefachampionsleague/season=2012/matches/round=2000263/index.html
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