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

Alanzinho

Receiving a carry-over effect

Giving a carry-over effect Walter Meeuws manager/coach SK Beveren, “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)

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

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 B C D E F G H Carry-over matrix (c ij ) The carry-over effects value = ∑ i,j (c ij ²) Russel (1980) AHCDEFGB BCDEFGHA CBAFHEDG DEBAGHCF EDGBACFH FGHCBAED GFEHDBAC HAFGCDBE

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 B C D E F G H Carry-over matrix (c ij ) 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

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)

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) (Eggermont, 2011) (Andersson, 1999) 16240* (Russell, 1980) (Andersson, 1999) 20380* (Andersson, 1999) 22462* (Andersson, 1999) (Andersson, 1999)

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.

Planning and scheduling’s achievement  The number of spectators in the Belgian League (excl play-offs) (excl play-offs) Source: Belgian Soccer Database

Planning and scheduling’s achievement  Costs of police involvement (in euro’s) million million million million million million Source: Vanhecke

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

Planning and scheduling’s achievement  The deciding round (champion) Source:

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.

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

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

Schedule characteristics  The schedule needs to satisfy certain characteristics

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

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

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

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)

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

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

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.

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.*)

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

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

The old approach A basic match schedule:

The old approach A basic match schedule: How is a basic match schedule constructed??

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.

Round

Round

The old approach 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

The old approach What kind of carry-over effect do we get here? Take, for instance, teams 1 and 3 from round 2 onwards

The old approach 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

The old approach 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!

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

Balancing carry-over effects n Lower bound: n*(n-1) In Practice: Circle method Best known value

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

A 3-phase approach (seasons , , …)  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

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?

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

Step 1: Collect data  Highest Belgian football division Double round robin, 18 teams Match results from 10,098 league games, from season till  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

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: points … Group 10: ≥ 53 points  For each season, teams are assigned to the corresponding strength group. Example: season : AA Gent – Anderlecht2-3 AA Gent (07-08) scored 38 points  group 6 Anderlecht (07-08) scored 49 points  group 9

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

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

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

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 )

Step 3. Compare the carry-over results with normal results  Example ( ) 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 = points more than expected Is this effect significant?

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

Step 3. Compare the carry-over results with normal results Can we use every game influenced by carry-over?  Beveren scored 20 points in (=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

Results 1. Carry-over effects received from teams in strength group s weakstrong s extra pointsp-value#effects 10,0200, ,0560, ,0580, ,0070, ,0360, ,0960, ,0710, ,1440, ,0800, ,0300,14151

Results 2. Carry-over effects given to teams in strength group s weakstrong s extra pointsp-value#effects 10,0840, ,0080, ,0770, ,0420, ,0650, ,0290, ,0400, ,0080, ,0160, ,0830,70842

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, ,1310, ,1570, ,0050, ,0280, ,1800, ,0560, ,1730, ,1540, ,1750,14151

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, ,0350, ,1450, ,2970, ,1170, ,0600, ,1120, ,1000, ,1440, ,0510,70842

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.

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%

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 no influence on strength group maximal influence on strength group

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 no influence on strength group maximal influence on strength group

Literature Fundamental paper:  De Werra, D. (1980), Geography, Games and Graphs, DAM 2, Surveys:  Rasmussen, R., M. Trick (2008), Round robin scheduling - a survey, EJOR 188,  Drexl, A., S. Knust (2007), Sports League Scheduling: graph- and resource-based models, Omega 35,  Goossens, D., F. Spieksma, (2012) Soccer schedules in Europe: an overview, Journal of Scheduling 15, Stories:  Bartsch, T., A. Drexl, S. Kröger (2006), Scheduling the professional soccer leagues of Austria and Germany, COR –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,  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,  Kendall, G. (2008), Scheduling English football fixtures over holiday periods, JORS 59, Bibliography Sigrid Knust’s website: 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,  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,

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