In most domestic football leagues, a round robin tournament (i.e. each teams plays against each other team a fixed number of times) forms the core of the competition format. Any schedule for a round robin tournament involves an order in which each team meets its opponents.
If some team A plays against team B in one round, and against team C in the next round, we say that team C receives a carry-over effect from team B. If team B is a very strong, tough-playing side, one can imagine that its opponent, team A, is weakened by injuries or fatigue, which could be an advantage for its next opponent, team C. Alternatively, if team A, confronted with a very strong team B, decides to throw the game, team A will be fit for its next opponent. Moreover, the carry-over effect could also be relevant in a strictly psychological interpretation, when team A loses confidence and morale after a severe loss against the strong team B, again to the benefit of their next opponent, team C. The opposite may be true if team B is a weak team. There may also be an influence due to the next opponent of A, say D. For instance, if A meets a strong team D in the next round, it may be more motivated to collect points against their current opponent B. In this case, we can say that team B receives a carry-over effect from team D.
Cases from Norway and Belgium
Although the perceived influence of the carry-over effect on football results was never studied, its impact on the practice of football should not be underestimated. We present two examples that received some attention in popular media.
- In the Norwegian Tippeligean, the 2007 schedule resulted in Brann Bergen playing against Stabaeck’s opponent of the previous round in 22 of the 26 rounds. Not only did Stabaeck regularly defeat their opponent, they also had a talented Brazilian player who forced his adversary into making fouls and incurring yellow and/or red cards and resulting suspensions. At the end of the season, the Norwegian media explained the frequent receiving of a carry-over effect from Stabaeck as one of the reasons why Brann Bergen eventually won the league title. This was taken quite seriously by the league organizers, who hired a private company to produce a schedule with more balanced carry-over effects for the next season.
- In Belgium, the 2006-07 schedule had Beveren playing against the team whose next opponent was the top team Anderlecht, on 29 of the 34 rounds. The coach of Beveren claimed that the schedule was unfair, since opponents were giving full measure against Beveren, because they realized that in the next game, their chances were slim to collect points. At the end of the season, Beveren were relegated to the second division and their coach was sacked. He mentioned this carry-over effect as an important reason for the team’s relegation.
The question arises whether these carry-over effects really have an impact on the outcome of football matches. Is the carry-over effect the culprit for Brann Bergen winning the Norwegian championship, and the Beveren coach being sacked? In other words: does the previous (or next) opponent of your opponent affect your result?
Measuring the carry-over effect
To measure the influence of the carry-over effect, we used data from the highest division in Belgian football, the Jupiler ProLeague, from season 1976-77 till season 2008-09, which includes 10,098 games. In this period, the Jupiler ProLeague consisted of 18 teams, playing a double round robin tournament with a mirrored schedule, which is the most common setup in football leagues. Furthermore, Haan, Koning and van Witteloostuijn (2008) measured the quality difference between the best four teams and the other teams in various competitions, and found that for the Jupiler ProLeague, this measure is close to average and stable over the last three decades. Therefore, it is safe to say that the Jupiler ProLeague is a fair representative of football in Europe (see also Goossens and Spieksma, 2012b).
The idea of our approach was to compare the result of each game that is influenced by a carry-over effect with the result that could be expected when no carry-over effects were involved. From the difference between these results, we obtained insight in the significance of the carry-over effect.
We estimated a probability distribution for the chances for a win, a draw, or a loss in an ideal world without carry-over from the games in our dataset. We reckon that the result of a particular match is determined by the respective strength of the two opponents, and by the home advantage. For each season, we assigned each team to one of 10 strength groups, depending on the total number of points the team collected in that season. The result is given in Figure 1, which gives the expected number of points for the home team in a match between two opponents from any combination of strength groups. We used the “old” rule that a win brings in 2 points, and that 1 point is accorded to a draw, since the majority of our data was played under this rule.
Figure 1 Expected number of points for the home team; the strength of the teams increases in the direction of the arrow
To test whether the actual results differ significantly from the probability distribution we expect, we performed a chi-square test. Notice that our approach assumes that the total number of points obtained by a team at the end of the season is not substantially influenced by the carry-over effect. If this is not the case, then carry-over effects may be underestimated. Until the season 2007-08, the type of schedule used in Belgium (as in most other countries) was such that for some teams the schedule was completely dominated by the carry-over effect received from one particular team. For instance, the Beveren coach could claim that the reason that Beveren ended up in the lowest strength group is precisely the fact that they received carry-over effects from Anderlecht 29 times in 34 rounds. Therefore, their results should not be compared with results from the lowest strength group, but with a higher one, which would show an increased influence of carry-over effects. It is difficult to predict in which strength group a team would have ended up without such pronounced carry-over effect. Therefore, our analysis only took into account teams that receive carry-over effects a limited number of times from the same team, so that we could safely assume that positive and negative carry-over effects cancel out each other over the course of a whole season.
Results
We distinguished between carry-over effects from the previous and the next round, and we estimated the influence of carry-over effects caused by teams from each of the ten strength groups. For each of these carry-over effects, we compared the actual result with the expected result. Table 1 summarizes our results.
Table 1 Influence of carry-over from the previous and the next round caused by teams of various strength groups
| Previous round | Next round | |||||
|---|---|---|---|---|---|---|
| s | Extra points | p-value | #effects | Extra points | p-value | #effects |
| 1 | 0.020 | 0.875 | 56 | 0.084 | 0.737 | 64 |
| 2 | -0.056 | 0.528 | 64 | 0.008 | 0.939 | 69 |
| 3 | -0.058 | 0.390 | 161 | 0.077 | 0.278 | 166 |
| 4 | 0.007 | 0.837 | 167 | -0.042 | 0.799 | 169 |
| 5 | 0.036 | 0.735 | 134 | -0.065 | 0.491 | 145 |
| 6 | 0.096 | 0.303 | 124 | -0.029 | 0.057 | 105 |
| 7 | -0.071 | 0.108 | 96 | -0.040 | 0.586 | 113 |
| 8 | -0.144 | 0.029 | 44 | 0.008 | 0.877 | 61 |
| 9 | 0.080 | 0.510 | 50 | -0.016 | 0.937 | 55 |
| 10 | 0.030 | 0.746 | 51 | -0.083 | 0.696 | 42 |
The values in the second column are the average number of points gained (or lost, in the case of a negative number) per game because of receiving carry-over effects through an opponent that played against a team from the strength group s (where group 1 is the weakest strength group, group 10 is the strongest one) in the previous round.
The table shows, for example, that a team may expect to collect 0.058 points less in a match against an opponent that played against a team from strength group 3 in its previous match, than in normal circumstances. The third column gives the p-values for the chi-square test. The number of carry-over effects these values are based on are mentioned in the fourth column. Similar values are presented for carry-over effects from the next round.
The table shows that for a significance level of 1%, the p-values are not small enough to conclude that the distribution of wins, draws and losses is any different from the distribution we would expect if no carry-over effects were present. Moreover, the influence of the carry-over effects does not seem to depend on the strength of the team playing against your opponent in the previous or next round.
Conclusions
We found that the influence of carry-over on the outcome of a football match is negligible. Therefore, a football team cannot rightfully claim to be at a disadvantage because of a schedule that does not balance carry-over effects. This does however not exclude that there may exist specific circumstances where the carry-over effect may have an influence. For instance, a team with a small number of players may be more susceptible to the carry-over effect than a team with a lot of players, which can more easily deal with injuries or suspensions, or even simply put a completely fresh team on the pitch after a difficult match. However, our research shows that carry-over effects are not an explanation for winning or losing a match, let alone for winning the title or for relegation.