Tie-breaking in round-robin soccer tournaments and its influence on the autonomy of relative rankings: UEFA vs. FIFA regulations

Abstract

Research question

Many sports tournaments, or tournament stages, are held in round-robin format. In establishing standings, tie-breaker criteria are often required to assign unique ranks to competitors that are equal on points. During the 2012 Union of European Football Associations (UEFA) European Football Championship, the relative ranking of two teams could possibly be influenced by a match in which neither team was involved, which we refer to as heteronomous relative ranking (HRR). We seek to shed light on the origin and the practical relevance of HRR in soccer competitions.

Research methods

We trace the appearance of HRR back to particularities in the UEFA EURO 2012 tie-breaker criteria, which favor head-to-head records over goal difference, and relate the concept of autonomous relative ranking to Arrow's Independence of Irrelevant Alternatives. Using historical and Monte–Carlo simulated data, we compare HRR occurrence rates under EURO 2012 regulations to those under Fédération Internationale de Football Association (FIFA) 2010 regulations, the latter of which give less importance to head-to-head records.

Results and findings

HRR is well explained by tie-breaker criteria, namely the priorities of head-to-head records and goal difference. HRR occurs in more than 10% of four-team soccer groups under EURO 2012 regulations; this rate is further increased in round-robin groups of six teams. FIFA 2010 regulations lead to less than 0.1% HRR.

Implications

Head-to-head records may result in counterintuitive side effects that should be avoided when designing ranking systems.

Keywords:

• ranking
• football
• soccer
• Independence of Irrelevant Alternatives
• UEFA
• FIFA

Notes

1. In alternative systems, such as double elimination tournaments participants join a losers’ bracket after being eliminated from the default winners’ bracket, thus, allowing more detailed ranks to be assigned, but requiring a higher number of matches.

2. We note that a fourth axiom could also prove useful in a round-robin tournament, which we tentatively term as scheduling invariance: the ranking should be invariant with respect to the order in which individual match results are determined.

3. In some competitions, participants may have incentives to lose a match if there is an elimination stage that follows, for example, in order to prevent having to face a specific opponent too early in the following stage. A recent example is the badminton women's doubles tournament at the 2012 Olympic Games, during which several participants have been disqualified as a consequence of attempting to lose their last group stage match. As this specific example shows, such behavior is considered detrimental to the sport and regulations are usually designed to prevent it, underlining the axiom's validity.

4. While the names of the first two axioms have been coined by Rubinstein (1980), we chose the third axiom's name.

5. This is in spite of some results showing that the goal difference may be a better measure of team fitness, for example, in soccer (Heuer & Rubner, 2009).

6. Section ‘d’ only appears in the amendments; see also UEFA (2012b). It is the only significant change regarding tie-breaking, compared to the preceding tournament's regulations (UEFA, 2005, p. 9).

7. For completeness, we note further peculiarities of the EURO 2012 regulations, which are less relevant than HRR and therefore outside the scope of this paper.

• Anonymity violations: Paragraph 8.07 g violates anonymity. It can easily be seen that all teams are ranked according to their UEFA national team coefficient in case of six goalless draws. Exchanging match results between any pair of participants does not influence the ranking, contrary to what is postulated by the anonymity axiom. (By contrast, the fair play conduct of the teams fulfills anonymity, as it is explicitly constrained to the final tournament and can be regarded part of the match results. Similarly, the drawing of lots is compatible with anonymity, as the average result does not depend on the teams’ labels and the random variable behaves identical for both parties.)

• Scheduling variance: The rules laid down in paragraph 8.08 of EURO 2012 regulation read (UEFA, 2012a):

  If two teams which have the same number of points, the same number of goals scored and conceded play their last group match against each other and are still equal at the end of that match, the ranking of the two teams in question is determined by kicks from the penalty mark […], provided no other teams within the group have the same number of points on completion of all group matches. Should more than two teams have the same number of points, the criteria listed under paragraph 8.07 apply.These rules may be understood as an attempt to obtain a decision on the playing field wherever possible, and this should be positively recognized. However, a consequence is that the order of matches (and not only their results) has an impact on the ranking: this can be seen by considering that two teams equal on points should decide their relative ranking by a penalty shoot-out in case they draw on their last group match, but by other rules if the order of match days is changed and the draw is a result of a different match day.

• Incompleteness: Table 10 shows possible match results with the EURO 2012 ranking displayed in Table 11. By parsing paragraph 8.07, we observe that a complete ranking cannot be established: rules ‘a’ to ‘c’ apply to teams A–D but do not break the tie between teams A–C. Consequently, rule ‘d’ does not apply, since it references ‘two teams,’ which is not defined in this case. Consequently, since rules ‘e’ and ‘f’ do not lead to a decision, teams A–C are ranked according to rule ‘g’ and their UEFA national team coefficient. This is despite availability of a unique ranking that can be established using three-team head-to-head records after four-team head-to-head records, as can be seen from the right part of Table 11. Since the intention of rule ‘d’ is clear by the rankings in the examples to follow, this problem is easily fixable by changing the restrictive formulation. This has been considered in more recent UEFA regulations, such as those for the 2013 UEFA European Women's Championship (‘two or more teams’; UEFA, 2012c, paragraph 8.05, second sentence) or for the 2013/2014 UEFA Champions League (‘teams’; UEFA, 2013d, paragraph 7.06 e). Our implementation uses this corrected formulation: head-to-head records are reapplied if two teams are equally ranked within a group of three or four teams equal on points.

Table 10. Match results yielding incomplete ranking under EURO 2012 regulations.

Table 11. Standings with incomplete ranking (head-to-head records in the right panel do not apply when interpreting EURO 2012 regulations strictly).

8. In this situation, the same ranking would result if head-to-head records had not been applied, and the teams had simply been ranked by points and goal difference.

9. For EURO and FIFA final tournaments, held in one tournament region, all participants except the hosts can be considered ‘away team,’ so no discrimination between home and away teams was made.

10. For historical data, current regulations are essentially different from the ones used when the respective tournaments were held; for example, wins were awards two points only in tournaments before 1996. Therefore, standings for historical match results are not necessarily identical to original standings.

11. We note that these rules give room for interpretation since, unlike UEFA rules, they do not explicitly state whether rules ‘d’ to ‘f’ shall be applied repeatedly. As this is necessary to completely rank participants in situations such as the one exemplified by Tables 12 and 13, we implemented the FIFA ranking with up to twofold evaluation of head-to-head rankings.

Table 12. Match results yielding incomplete ranking under strictly interpreted FIFA 2010 regulations.

Table 13. Standings with incomplete ranking under strictly interpreted FIFA 2010 regulations if head-to-head records in the right-most panel do not apply.

12. Repeated evaluation of head-to-head records is only required if the number of tied teams in a head-to-head group is lower than the size of that group and greater than 1. Hence, 4-to-3 following by 3-to-2 transitions are the only way that might require a third evaluation of head-to-head records between the remaining two teams. We show that under the points system awarding three points for a win and one point for a tie in a group of four teams, these two teams necessarily need to draw their match.Each match awards 3 + 0, 1 + 1, or 0 + 3 points to the participants, that is, either two or three. The final total number of points in the group (after six matches) is thus bounded by 12 and 18. For four teams to be equal on (integer) points, 16 and 12 points are the only remaining possibilities. With 16 points, which equals 4 per team, each team must have won, drawn, and lost one game, respectively. Hence, when removing any one participant from a group of four teams equal on points to establish a group of three teams, these three teams cannot be equal on points; 4 points per team are thus impossible. Hence, the group needs to count 12 points in total, which is the minimum possible; as a consequence, each match only generates the minimum number of points (two) by ending in a draw.

13. This comprises, for example, the case of the 1974 FIFA World Cup Group 2, where Scotland failed to proceed to the elimination stage under (both historical and current) FIFA regulations, but would have done so under EURO 2012 regulations. However, despite the differences, the UEFA ranking is consistent in that it does not manifest HRR in this example.