Many major sporting competitions, including the Wimbledon tennis championship, Olympic boxing and even the Women’s World Chess Championship, have a similar elimination structure. After each match the winner advances further while the loser leaves the tournament. In such competitions, the initial opponent pairings and the order of all remaining games are determined before the tournament by a procedure known as ‘seeding’. There are, however, many different ways to seed the competitors in tournaments. Seeding is therefore a matter of policy. We discuss how the organisers may influence spectator demand by choosing optimal seeding.
The economic objective of tournament organisers
Empirical research establishes the special role of two characteristics of the teams participating in a match for the match attractiveness. The first one is the teams’ quality and the second one is the difference between teams’ quality (also known as ‘competitive intensity’). Forrest and Simmons (2002) collected the results and betting odds for 872 matches played in the 1997-1998 season on Saturdays in the top three divisions of the English football league. They chose the number of points gained before the match as the proxy for the team quality and thus excluded several first match days from the dataset. The authors found that match attendance correlates positively with home team quality, whereas there is no significant relationship with away team quality. Also, a lower difference between the quality of the two teams generates higher spectator demand. Taking into consideration this result, policymakers may be interested in developing an optimal tournament format that maximises the sum of qualities and competitive intensities over all matches of the tournament. In the class of elimination tournaments with a fixed number of participants, seeding is an important policy mechanism for the organisers.
Seedings in elimination tournaments
Initially, seedings were introduced in order to protect the best teams from meeting each other at the early stages of a tournament by putting the favourites in different strands of the tournament bracket. As Wright (2014) put it, the reason for seeding “is clearly to ensure high competitive intensity, especially at the latter stages of a tournament”. Such a design is justified because a loss of a strong, well-known team in the first rounds may reduce spectator interest in the whole tournament. However, by simply separating the favourites in the bracket, organisers make the first rounds very unbalanced. The essence of the optimisation problem is simply the trade-off between many unbalanced matches at the early stages and several unbalanced matches closer to the final. This dilemma has been studied in the literature mainly for a small number of participants (e.g. for four to eight participants). Vu (2010) constructs a revenue function that includes two terms reflecting quality and competitive intensity, though the construction is different from our own. He then simulates the parameters of the model, including the matrix of pairwise winning probabilities – for every realisation of parameters he conducts an exhaustive search and finds the optimal seeding in a tournament with eight players. The author finds that the traditional seeding (where the first seeded plays against the eighth, fourth against fifth, third against sixth, second against seventh, and the winners of the first and last two games play each other in the semi-finals) is optimal in 23% of cases and achieves, on average, more than 99% of the optimal value. No seeding turns out to be optimal more often than the traditional one.
In contrast, we provide results for an arbitrarily large number of participants.
We consider a standard knockout tournament with 2n teams, where n is the total number of rounds. The teams are strictly ranked by their strength (for example, in the European football club competitions the UEFA club ranking may be used). Without loss of generality, let Team 1 be the strongest, Team 2 be the second strongest, and so on. In our model, we start from the following two assumptions:
- A stronger team always beats a weaker one;
- Spectator interest in a single match depends positively and linearly on the strength of the teams involved and on the competitive intensity.
The organisers seek to maximise the overall spectator interest in watching the matches of the tournament.
Two groups of seedings play a special role in our analysis, as they turn out be the only possible solutions to the problem.
Close seedings are seedings such that under the first assumption, in every round any team faces an opponent closest to it in rank out of the participants remaining in the tournament. Hence in the first round the, Team 1 is paired with the Team 2, Team 3 with Team 4, and so on. Moreover, these pairs are placed within the bracket in such a way that in the second round Team 1 faces Team 3, Team 5 faces Team 7, and so on.
By contrast, distant seedings mean that, under our first assumption, a team from the strongest half of remaining teams meets a team from the weakest half of remaining teams in each match of the tournament. Figure 1 below shows the examples of a close and a distant seeding.
Figure 1. Close (left) and distant (right) seedings
We show that under our assumptions only two types of seedings – close seedings or distant seedings – can possibly maximise the objective function. The answer depends on the relative importance of the next round in comparison with current round (this parameter may be interpreted as profitability as the ticket price may vary across rounds). Close seedings are optimal when the importance of the next round is relatively low, while distant seedings are optimal when it is relatively high. These findings underscore why the existing popular and intuitive seeding procedure is optimal in the sense of spectator interest maximisation.
More seeding problems
Maximisation of revenues is important but not the only goal of tournament organisers. Sometimes the flaws and inaccuracies in the seeding procedures generate notable scandals that underline major seeding weaknesses.
One of the key problems is that the outcome of a tournament depends too much on the seeding. Back in 2005, UEFA made an irregular decision to admit Liverpool FC to the Champions League – a major international European football club tournament. The English side won the previous Champions League (2004/2005), but failed to qualify for the next Champions League (2005/2006) through the national championship. After long discussions, UEFA made the exceptional decision to allow Liverpool to defend the title. However, since the total number of participants of Champions League group stage is fixed, UEFA was forced to send Liverpool to a qualification tournament. This, in turn, shifted the `regular’ seeding of the qualifiers because Liverpool had the best team coefficient among all participants of the qualifiers. The team that suffered most was Slavia Prague. In the right circumstances, they could have been the last seeded team in the third qualifying round and could have expected opponents ranging from Real Betis (the highest ranking) to Vеlerengen (the lowest ranking). However, Liverpool’s admission moved Slavia to the unseeded pot, making it possible for them to be drawn against Liverpool themselves, Manchester United, Inter Milan or other stronger competitors. Slavia submitted an appeal on this decision, which was later rejected by UEFA. Though Slavia escaped the grand names during the draw and were paired with Anderlecht, even the Belgian side was thought to be stronger than any possible unseeded opponent. Slavia lost the tie and, thus, were eliminated from the Champions League.
Another problem often arising in elimination tournaments is the emergence of perverse incentives. In the 2012 Olympics badminton tournament, four women pairs were disqualified for "not using one's best efforts to win". Tournament rules suggested that before the elimination stage the pairs are reseeded according to the group stage results. However, an accidental loss of the tournament favourite in one of the group stage matches means that the favourite won’t be the first-seeded in the play-offs. This shifts other pairs’ incentives, making it profitable for them to lose the final group stage match in order to avoid the favourite in the play-off. The different nature of perverse incentives is outlined in the work of Baumann et al. (2010). Analysing data from the NCAA March Madness basketball tournaments, they found statistical evidence for violation of monotonicity of the team’s expected outcome with respect to the seeding number.
Since seeding procedures are often not determined precisely, organisers may manipulate the seeding in their interests. One of the boxing qualifying tournaments for the 2012 Olympics was held in Azerbaijan. It appeared that all nine participating Azerbaijani sportsmen (in different weights) were seeded among the top eight boxers that simplified their tournament path substantially. The interesting part of the story is that some of them were seeded despite a relatively low world ranking. The International Boxing Association (AIBA) was forced to give the clarifications on the seeding procedures, which appeared to be highly controversial.
These examples illustrate that tournament organisers should not confine themselves to demand maximisation. Choosing the proper scheme is a hard but feasible goal for tournament designers.
Baumann, R, V A Matheson and C Howe (2010), “Anomalies in tournament design: the madness of March madness”, Journal of Quantitative Analysis in Sports 6(2).
Dagaev, D, A Suzdaltsev (2015), “Seeding, Competitive Intensity and Quality in Knock-Out Tournaments”, Higher School of Economics Research Paper WP BRP 91/EC/2015, available at SSRN.
Forrest, D, R Simmons (2002), “Outcome uncertainty and attendance demand in sport: the case of English soccer”, Journal of the Royal Statistical Society Series D: The Statistician 51, Part 2. 229–241.
Vu, T D (2010), “Knockout tournament design: a computational approach”, PhD dissertation, Stanford University, Department of Computer Science.
Wright M (2014), “OR analysis of sporting rules – A survey”, European Journal of Operational Research 232: 1–8.