1. On Carry Over Frits Spieksma joint work with Dries Goossens Leuven, May 14, 2013

2. Alanzinho

3. Receiving a carry-over effect

4. Giving a carry-over effect Walter Meeuws manager/coach SK Beveren, 2006-2007 “We always play against teams that play against Anderlecht (a top team in Belgium, fs) one week later. These teams think: we must make sure to give full measure against Beveren, because next week, chances are slim to collect points. In that regard, the schedule is against us.” Giving a carry-over effect to Anderlecht Round rRound r+1 Beveren vs. XX vs. Anderlecht Occured 29 times (out of 34)

5. Consider a round robin tournament.  Definition: Team B receives a carry-over effect from team A if A plays against X in round r X plays against B in round r+1 Also: team A gives a carry-over effect to team B.  Relevance: If match results are not independent, carry-over effects may have an influence on your result  The schedule itself gives you a(n) (dis)advantage Notice: it’s not about your previous match! The carry-over effect

6. Measuring carry-over effects  Single round robin tournament, 8 teams Let c ij be the number of carry-over effects that i receives from j Opponent schedule ABCDEFGH A-5001001 B0-121030 C30-00013 D003-2002 E1102-201 F20302-00 G100303-0 H0100123- Carry-over matrix (c ij ) The carry-over effects value = ∑ i,j (c ij ²) Russel (1980) 1234567 AHCDEFGB BCDEFGHA CBAFHEDG DEBAGHCF EDGBACFH FGHCBAED GFEHDBAC HAFGCDBE

7. Measuring carry-over effects  Single round robin tournament, 8 teams Let c ij be the number of carry-over effects that i receives from j Opponent schedule ABCDEFGH A-5001001 B0-121030 C30-00013 D003-2002 E1102-201 F20302-00 G100303-0 H0100123- Carry-over matrix (c ij ) 1234567 AHCDEFGB BCDEFGHA CBAFHEDG DEBAGHCF EDGBACFH FGHCBAED GFEHDBAC HAFGCDBE Fair schedule = balanced carry-over effects = minimize the carry-over effects value Balanced schedule = all non-diagonal c ij ’s = 1 = lower bound

8. Balancing carry-over effects: chronological n Lower bound: n x (n-1) Russell (1980) Anderson (1999) Miyashiro& Matsui (2006) Lower bound by Eggermont (2010) Eggermont (2010) 412 -- 63060-- 856 - 1090138108 12132196176 170 14182260234254 16240 - 18306428340400 20380520380488 22462- - 24552684644-

9. Balancing carry-over effects: current state-of-the-art # TeamsCarry-over Value 412* (Russell, 1980) 660* (Russell, 1980) 856* (Russell, 1980) 10108* (Andersson, 1999; Eggermont, 2011) 12170 (Eggermont, 2011) 14234 (Andersson, 1999) 16240* (Russell, 1980) 18340 (Andersson, 1999) 20380* (Andersson, 1999) 22462* (Andersson, 1999) 24644 (Andersson, 1999)

10. Do today’s schedules have a large coe-value? To answer this question, let us have a look at our experience with scheduling the Belgian soccer league.

11. Planning and scheduling’s achievement  The number of spectators in the Belgian League 2003-20042.992.700 2004-20052.911.800 2005-20063.132.000 2006-20073.233.950 2007-20083.458.000 2008-20093.360.000 2009-20102.694.850 (excl play-offs) 2010-2011 2.819.350 (excl play-offs) Source: Belgian Soccer Database

12. Planning and scheduling’s achievement  Costs of police involvement (in euro’s) 2004-20055.0 million 2005-20064.6 million 2006-20074.4 million 2007-20084.6 million 2008-20094.5 million 2009-20104.3 million Source: Vanhecke

13. Planning and scheduling’s achievement  Amount of money paid for the broadcasting rights in Belgium:  2005-2008 (three seasons): 36 million euro’s per year (Belgacom)  2008-2011 (three seasons): 45 million euro’s per year (Belgacom)  2011-2014 (three seasons): 55,2 million euro’s per year (Telenet) As an aside: in the UK broadcasting rights for 2010-2013 were sold for £ 1.780.000.000 to the British Sky Broadcasting Group.

14. Planning and scheduling’s achievement  The deciding round (champion) 2003-200431 2004-200532 2005-200634 2006-200733 2007-200832 2008-200934+ Source: www.proleague.be

15. Problem characteristics  How many teams?  16  What format?  Double round robin (2RR): each team plays against each other team twice, once away, once home. It follows that there are 30 rounds, and 240 matches in total.  There is a playoff after the 2RR.

16. Problem characteristics  What about the venues?  Intermural: each team has a home- stadium (notice that Club Brugge and Cercle Brugge share a single stadium)

17. Problem characteristics  The problem is to find a schedule (a calendar) specifying: Who plays who? Where? When?

18. Schedule characteristics  The schedule needs to satisfy certain characteristics

19. Schedule characteristics  Example of a schedule corresponding to a league of six teams A, B, C, D, E, F R1R2R3R4R5R6R7R8R9R10 A-BB-EA-FC-BB-FB-AE-BF-AB-CF-B C-DD-AB-DE-AD-ED-CA-DD-BA-EE-D E-FF-CE-CF-DA-CF-EC-FC-ED-FC-A

20. Schedule characteristics  Example of a 6-team 2RR schedule R1R2R3R4R5R6R7R8R9R10 A-BB-EA-FC-BB-FB-AE-BF-AB-CF-B C-DD-AB-DE-AD-ED-CA-DD-BA-EE-D E-FF-CE-CF-DA-CF-EC-FC-ED-FC-A A property: two halves

21. Schedule characteristics  Example of a 6-team 2RR schedule R1R2R3R4R5R6R7R8R9R10 A-BB-EA-FC-BB-FB-AE-BF-AB-CF-B C-DD-AB-DE-AD-ED-CA-DD-BA-EE-D E-FF-CE-CF-DA-CF-EC-FC-ED-FC-A Another property: mirroring

22. Schedule characteristics  Example of a 6-team 2RR schedule R1R2R3R4R5R6R7R8R9R10 A-BB-EA-FC-BB-FB-AE-BF-AB-CF-B C-DD-AB-DE-AD-ED-CA-DD-BA-EE-D E-FF-CE-CF-DA-CF-EC-FC-ED-FC-A Yet another property: the so-called home-away patterns (HAPs)

23. Schedule characteristics  Example of a 6-team 2RR schedule R1R2R3R4R5R6R7R8R9R10 A-BB-EA-FC-BB-FB-AE-BF-AB-CF-B C-DD-AB-DE-AD-ED-CA-DD-BA-EE-D E-FF-CE-CF-DA-CF-EC-FC-ED-FC-A For instance, team A’s HAP is: H A H A H A H A H A, while team C’s HAP is: H A A H A A H H A H

24. Schedule characteristics Example of a 6-team 2RR schedule R1R2R3R4R5R6R7R8R9R10 A-BB-EA-FC-BB-FB-AE-BF-AB-CF-B C-DD-AB-DE-AD-ED-CA-DD-BA-EE-D E-FF-CE-CF-DA-CF-EC-FC-ED-FC-A For instance, team A’s HAP is: H A H A H A H A H A, while team C’s HAP is: H A A H A A H H A H Given a HAP, a break is defined as the occurrence of two consecutive away matches or two consecutive home matches

25. Schedule characteristics Example of a 6-team 2RR schedule R1R2R3R4R5R6R7R8R9R10 A-BB-EA-FC-BC-BB-FB-AE-BF-AB-CB-CF-B C-DD-AB-DE-AD-ED-CA-DD-BA-EE-D E-FF-CE-CF-DA-CF-EC-FC-ED-FC-A Consider team C’s HAP again: H A A H A A H H A H And team B’s HAP: A H H A H H A A H A. These two HAP’s are called complementary.

26. Required schedule characteristics The schedule should have a minimum number of breaks (this number equals 42 (3n-6) in case of a mirrored schedule) A schedule should start and end with H A or A H for each team A A A or H H H should not occur. Complementary HAP-set (*Mirroring: the second half (rounds 16 up to 30) is identical to the first half except that home and away is inverted.*)

27. Internationally? Country#teamsFormatHalves?H A as end/start Min # breaks England202RRNoYesNo Spain202RRYes (M)NoYes Italy202RRYes (M)No Germany182RRYes (M)NoYes France202RRYes (M’)Yes Russia162RRYes (M’)Yes USA15Subleagues The Netherlands 182RRNo Scotland123RRYes (M)No Switzerland104RRYesNoYes Austria104RRYes (M’)No

28. The old approach By hand:  The secretary of the calendar committee departed from the so-called number model (a basic match schedule)

29. The old approach 1234567891011121314151617 1-32-41-72-81-112-121-152-161-22-31-62-71-102-111-142-151-18 4-173-183-54-63-94-103-134-143-174-13-44-53-84-93-124-133-16 6-155-16-25-185-76-85-116-125-156-165-26-35-66-75-106-115-14 8-137-168-177-38-47-187-98-107-138-147-178-17-48-57-88-97-12 10-119-1410-159-110-29-510-69-189-1110-129-1510-169-210-39-610-79-10 12-911-1212-1311-1612-1711-312-411-712-811-1811-1312-1411-1712-111-412-511-8 14-713-1014-1113-1414-1513-114-213-514-613-914-1013-1813-1514-1613-214-313-6 16-515-816-915-1216-1315-1616-1715-316-415-716-815-1116-1215-1815-1716-115-4 18-217-618-417-1018-617-1418-817-118-1017-518-1217-918-1417-1318-1618-1717-2 A basic match schedule:

30. The old approach 1234567891011121314151617 1-32-41-72-81-112-121-152-161-22-31-62-71-102-111-142-151-18 4-173-183-54-63-94-103-134-143-174-13-44-53-84-93-124-133-16 6-155-16-25-185-76-85-116-125-156-165-26-35-66-75-106-115-14 8-137-168-177-38-47-187-98-107-138-147-178-17-48-57-88-97-12 10-119-1410-159-110-29-510-69-189-1110-129-1510-169-210-39-610-79-10 12-911-1212-1311-1612-1711-312-411-712-811-1811-1312-1411-1712-111-412-511-8 14-713-1014-1113-1414-1513-114-213-514-613-914-1013-1813-1514-1613-214-313-6 16-515-816-915-1216-1315-1616-1715-316-415-716-815-1116-1215-1815-1716-115-4 18-217-618-417-1018-617-1418-817-118-1017-518-1217-918-1417-1318-1618-1717-2 A basic match schedule: How is a basic match schedule constructed??

31. About the basic match schedule  When you view each team as a node, and given some specific round, you connect two teams when they meet, you get a matching (one factor).  So, the basic match schedule is a set of one factors, aka a one-factorization.

32. 2 3 1 4 17 16 15 14 13 5 6 7 8 9 1011 12 18

33. Round 1 2 3 1 4 17 16 15 14 13 5 6 7 8 9 1011 12 18

34. Round 2 2 3 1 4 17 16 15 14 13 5 6 7 8 9 1011 12 18

35. The old approach 1234567891011121314151617 1-32-41-72-81-112-121-152-161-22-31-62-71-102-111-142-151-18 4-173-183-54-63-94-103-134-143-174-13-44-53-84-93-124-133-16 6-155-16-25-185-76-85-116-125-156-165-26-35-66-75-106-115-14 8-137-168-177-38-47-187-98-107-138-147-178-17-48-57-88-97-12 10-119-1410-159-110-29-510-69-189-1110-129-1510-169-210-39-610-79-10 12-911-1212-1311-1612-1711-312-411-712-811-1811-1312-1411-1712-111-412-511-8 14-713-1014-1113-1414-1513-114-213-514-613-914-1013-1813-1514-1613-214-313-6 16-515-816-915-1216-1315-1616-1715-316-415-716-815-1116-1215-1815-1716-115-4 18-217-618-417-1018-617-1418-817-118-1017-518-1217-918-1417-1318-1618-1717-2 Example: Anderlecht = 2, Club Brugge = 12, Standard = 3, Genk = 6 1. Assign a number to each topteam Make sure that the corresponding matches satisfy police requirements and are balanced over the season

36. The old approach 1234567891011121314151617 1-32-41-72-81-112-121-152-161-22-31-62-71-102-111-142-151-18 4-173-183-54-63-94-103-134-143-174-13-44-53-84-93-124-133-16 6-155-16-25-185-76-85-116-125-156-165-26-35-66-75-106-115-14 8-137-168-177-38-47-187-98-107-138-147-178-17-48-57-88-97-12 10-119-1410-159-110-29-510-69-189-1110-129-1510-169-210-39-610-79-10 12-911-1212-1311-1612-1711-312-411-712-811-1811-1312-1411-1712-111-412-511-8 14-713-1014-1113-1414-1513-114-213-514-613-914-1013-1813-1514-1613-214-313-6 16-515-816-915-1216-1315-1616-1715-316-415-716-815-1116-1215-1815-1716-115-4 18-217-618-417-1018-617-1418-817-118-1017-518-1217-918-1417-1318-1618-1717-2 What kind of carry-over effect do we get here? Take, for instance, teams 1 and 3 from round 2 onwards

37. The old approach 1234567891011121314151617 1-32-41-72-81-112-121-152-161-22-31-62-71-102-111-142-151-18 4-173-183-54-63-94-103-134-143-174-13-44-53-84-93-124-133-16 6-155-16-25-185-76-85-116-125-156-165-26-35-66-75-106-115-14 8-137-168-177-38-47-187-98-107-138-147-178-17-48-57-88-97-12 10-119-1410-159-110-29-510-69-189-1110-129-1510-169-210-39-610-79-10 12-911-1212-1311-1612-1711-312-411-712-811-1811-1312-1411-1712-111-412-511-8 14-713-1014-1113-1414-1513-114-213-514-613-914-1013-1813-1514-1613-214-313-6 16-515-816-915-1216-1315-1616-1715-316-415-716-815-1116-1215-1815-1716-115-4 18-217-618-417-1018-617-1418-817-118-1017-518-1217-918-1417-1318-1618-1717-2 What kind of carry-over effect do we get here? Take, for instance, teams 1 and 3 from round 2 onwards … Or take teams 2 and 4 from round 3 onwards

38. The old approach 1234567891011121314151617 1-32-41-72-81-112-121-152-161-22-31-62-71-102-111-142-151-18 4-173-183-54-63-94-103-134-143-174-13-44-53-84-93-124-133-16 6-155-16-25-185-76-85-116-125-156-165-26-35-66-75-106-115-14 8-137-168-177-38-47-187-98-107-138-147-178-17-48-57-88-97-12 10-119-1410-159-110-29-510-69-189-1110-129-1510-169-210-39-610-79-10 12-911-1212-1311-1612-1711-312-411-712-811-1811-1312-1411-1712-111-412-511-8 14-713-1014-1113-1414-1513-114-213-514-613-914-1013-1813-1514-1613-214-313-6 16-515-816-915-1216-1315-1616-1715-316-415-716-815-1116-1215-1815-1716-115-4 18-217-618-417-1018-617-1418-817-118-1017-518-1217-918-1417-1318-1618-1717-2 What kind of carry-over effect do we get here? Take, for instance, teams 1 and 3 from round 2 onwards … Or take teams 2 and 4 from round 3 onwards Basic match schedules have a huge carry-over effect!

39. Is there room for improvement?  The basic match schedule: disadvantages Huge carry-over effect There are many, many other potential schedules!

40. Balancing carry-over effects n Lower bound: n*(n-1) In Practice: Circle method Best known value 412 63060 85619656 1090468108 12132924170 141821612234 16240 2580240 183063876340 203805538380 224627644462 2455210212644

42. Is there room for improvement?  Yes.  Let us take a three-phase approach

43. A 3-phase approach (seasons 2007-2008, 2008-2009, …)  Phase 0: find feasible HAP-sets  Phase 1: given a HAP-set, assign each team to a HAP  Phase 2: compute a schedule, given the assignment from Phase 1

44. Does the carry-over effect exist statistically in practice? 1. Is it an (dis)advantage to receive a carry- over effect from a big team? Stabaek (Alanzinho)  advantage? 2.Is it a (dis)advantage to give a carry-over effect to a big team? Beveren  disadvantage?

45. How to measure the influence of the carry-over effect? Our approach Step 1. Collect data: which games are influenced by the carry-over effect? Or: does it matter whether the opponent of your current opponent in the previous round was a strong/weak team? Step 2. Determine what a normal match result would be Step 3. Compare 1 & 2

46. Step 1: Collect data  Highest Belgian football division Double round robin, 18 teams Match results from 10,098 league games, from season 1976-1977 till 2008-2009  Assumptions: A carry-over effect within the league can only be present if at most 10 days are between the matches We hold on to 2 points for a win, 1 for a draw, 0 for a loss

47. Step 2. Determine what a normal match result would be  We need a probability distribution that gives the chance of a win, a draw, or a loss in a world without carry-over.  estimate this from our dataset  We distinguish 10 strength groups, depending on the number of points scored in the season. Group 1: ≤ 20 points Group 2: 21-24 points … Group 10: ≥ 53 points  For each season, teams are assigned to the corresponding strength group. Example: season 2007-2008: AA Gent – Anderlecht2-3 AA Gent (07-08) scored 38 points  group 6 Anderlecht (07-08) scored 49 points  group 9

48. Step 2. Determine what a normal match result would be  Construct a matrix with the average proportions of home team wins w ij for teams from each pair of strength groups (i,j).  The same for draws (d ij ), losses (l ij ).  Regularity properties: Higher chance of winning against a weaker team Higher chance of losing against a stronger team However: the estimates for w ij, and l ij do not satisfy these regularity properties

49. Step 2. Determine what a normal match result would be Irregular shape: peaks and valleys It happens that Less home team points for stronger home teams More home team points against stronger away teams weakstrong Illustration: compute matrix A, with average results in terms of home team points, i.e. a ij = 2w ij + d ij. Change the probability estimates w ij, d ij, l ij such that: Regularity properties are satisfied Changes are as limited as possible

50. Step 2. Determine what a normal match result would be LP model Parameters: b ij = new estimate of result of games between strength groups i and j n ij = number of games between strength groups i and j p i, q i = lower and upper bound of strength group i’s point range Variables: x ij = correction term to add to w ij y ij = correction term to add to d ij z ij = correction term to add to l ij

51. Step 2. Determine what a normal match result would be Home team points weakstrong Matrix A (with old estimates a ij ) Matrix B (with new estimates b ij )

52. Step 3. Compare the carry-over results with normal results  Example (1976-1977) List of matches with a team receiving carry-over from a team from strength group 1  receiving this carry-over effect leads to (7-5.33)/5 = 0.334 points more than expected Is this effect significant?

53. Step 3. Compare the carry-over results with normal results  Example (1976-1977) List matches that with a team receiving carry-over from a team from strenght group 1 Use a χ² test to determine significance

54. Step 3. Compare the carry-over results with normal results Can we use every game influenced by carry-over?  Beveren scored 20 points in 2006-2007 (=group 1) gave a carry-over effect to Anderlecht 29 times in 34 rounds Can we compare Beveren’s 29 carry-over affected games with results of an average group 1 team? Beveren may argue that without the carry-over effect, they would have performed much better, and thus we should compare them with a better group… Impossible to say how Beveren would have scored without giving carry-over to Anderlecht, as only 5 games are not influenced by this effect. Solution: only consider teams that have balanced carry-over effects  we can assume that carry-over did not influence their strength group

55. Results 1. Carry-over effects received from teams in strength group s weakstrong s extra pointsp-value#effects 10,0200,87556 2-0,0560,52164 3-0,0580,769161 40,0070,778167 50,0360,543134 60,0960,912124 7-0,0710,48896 8-0,1440,04344 90,0800,39350 100,0300,14151

56. Results 2. Carry-over effects given to teams in strength group s weakstrong s extra pointsp-value#effects 10,0840,44664 20,0080,92269 30,0770,833166 4-0,0420,561169 5-0,0650,835145 6-0,0290,078105 7-0,0400,832113 80,0080,68561 9-0,0160,80055 10-0,0830,70842

57. Maybe the carry-over effect is too small to be measured by match result Does the carry-over effect have an effect on goal difference? weakstrong Receving carry-over effect Results s extra goalsp-value#effects 1-0,2460,74856 2-0,1310,52164 3-0,1570,769161 4-0,0050,778167 50,0280,543134 60,1800,912124 7-0,0560,48896 8-0,1730,04344 9-0,1540,39350 100,1750,14151

58. Maybe the carry-over effect is too small to be measured by match result Does the carry-over effect has an effect on goal difference? weakstrong Receving carry-over effect Results s extra goalsp-value#effects 10,2060,44664 20,0350,92269 30,1450,833166 4-0,2970,561169 5-0,1170,835145 60,0600,078105 7-0,1120,832113 8-0,1000,68561 9-0,1440,80055 100,0510,70842

59. Conclusions  In the Belgian competition, there is no evidence for an influence from receiving a carry-over effect giving a carry-over effect on the match result nor on the goal difference.  The carry-over effect has no significant influence on the outcome of a football match.  Clubs cannot claim to be at a disadvantage because of a schedule with unbalanced carry-over effects.

60. The European carry-over effect Standard - Liverpool0-0 Liverpool – Standard1-0 Everton – Standard2-2 Standard – Everton2-1 Standard – Sevilla1-0 Partizan – Standard0-1 Standard – Sampdoria3-0 Stuttgart – Standard3-0 Braga – Standard3-0 Standard – Braga 1-1 Dender - Standard1-3 Roeselare – Standard1-1 Tubize – Standard0-1 Cercle – Standard4-1 G.Beerschot - Standard1-3 KV Mechelen – Standard0-0 Standard – Z.Waregem1-2 Standard – AA Gent2-1 Anderlecht – Standard4-2 Standard – Cercle4-0 League games played after a European game Did the European campaign cost Standard points? Rest of the season: 60/72 points  83% League games played after a European game: 17/30 points  57%

61. The European carry-over effect 3. Carry-over effects from a midweek European game on the results of teams in strength group s in their next league match s#ca 12 217 326 430 551 645 793 897 9150 10132 no influence on strength group maximal influence on strength group

62. The European carry-over effect 3. Carry-over effects from a midweek European game on the goal difference of teams in strength group s in their next league match s#ca 12 217 326 430 551 645 793 897 9150 10132 no influence on strength group maximal influence on strength group

63. Literature Fundamental paper:  De Werra, D. (1980), Geography, Games and Graphs, DAM 2, 327-337. Surveys:  Rasmussen, R., M. Trick (2008), Round robin scheduling - a survey, EJOR 188, 617-636.  Drexl, A., S. Knust (2007), Sports League Scheduling: graph- and resource-based models, Omega 35, 465- 471.  Goossens, D., F. Spieksma, (2012) Soccer schedules in Europe: an overview, Journal of Scheduling 15, 641- 651. Stories:  Bartsch, T., A. Drexl, S. Kröger (2006), Scheduling the professional soccer leagues of Austria and Germany, COR 33 1907–1937.  Della Croce, F., D. Oliveri (2006), Scheduling the Italian Football League: an ILP-based approach, COR 33, 1963–1974.  Duran, G., M. Guajardo, J. Miranda, D. Sauré, S. Souyris, A. Weintraub (2007), Scheduling the Chilean soccer league by integer programming. Interfaces 37, 539–552.  Rasmussen, R. (2008), Scheduling a triple round robin tournament for the best Danish soccer league, EJOR 185, 795–810.  Ribeiro C., S. Urrutia (2006), Scheduling the Brazilian Soccer Championship, Proceedings of PATAT’06, 481- 483.  Schreuder, J. (1992), Combinatorial aspects of construction of competition Dutch Professional Football Leagues, DAM 35, 301–312.  Goossens, D., F. Spieksma, (2009) Scheduling the Belgian soccer league, Interfaces 39, 109-118.  Kendall, G. (2008), Scheduling English football fixtures over holiday periods, JORS 59, 743-755. Bibliography Sigrid Knust’s website: http://www.inf.uos.de/knust/sportssched/ Other work:  Griggs, T., A. Rosa (1996), A tour of European soccer schedules, or testing the popularity of GK 2n, Bulletin of the Institute of Combinatorics and its Applications 18, 65-68.  Briskorn, D. (2008), Sport leagues scheduling: models, combinatorial properties, and optimization algorithms, Lecture Notes in Economics and Mathematical Systems 603, Springer, Berlin.  Post, G., G. Woeginger (2006), Sports tournaments, home-away assignments, and the break minimization problem, Discrete Optimization 3, 165-173.

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65. Alanzinho