Earliest Inspiration: The Kendall-Wei ranking scheme and the FIFA World Cup 2010

This is a reposting of some of the initial work done in 2010. It still makes and interesting read!

Dr. Gangan Prathap

The 2010 FIFA World Cup tournament needed to choose 16 teams from 32 finalists for the knock-out stage. The 32 were first grouped into 8 lots of 4 teams each (Groups A to H). The 4 teams within each group had to play round-robin matches (i.e. each team had to play 3 matches for a total of 6 matches) and the two best teams were chosen using a simple scoring system, 3 points for a win and 1 point for a draw. Where there was a tie, this was resolved by considering the goal difference (the difference between goals scored and goals against). In most cases, this was a simple exercise (for further reference, we shall call this the FIFA scheme). 

The simplest case was that of Group E. The Netherlands won all three matches and down the line, Japan and Denmark won 2 matches and 1 match each, while Cameroon lost all three.

The case of Group B is a little less simple. Argentina won all three matches and earned 9 points, by giving the sameweightageto each opponent it won against, irrespective of their “strength”. Korea won 1 match and drew the match against Nigeria for 4 points, while Greece won only against Nigeria for 3 points. Nigeria, by virtue of their draw against Korea earned a point. 

At this stage we shall introduce the idea of recursive iteration. Assume that points are to be given according to the strengths of the team beaten. Thus Argentina should earn more points for beating Korea than for beating Nigeria. Ab initio, this is impossible, as the strengths of the individual teams are not known. However, mathematically, such recursive computations can be set up as an eigenvalue problem. The idea of using such an approach to do ranking is due to Kendall and Wei [1, 2], more than 50 years ago, applied first to the sporting field  (http://www.math.utsc.utoronto.ca/b24/KendallWei.pdf).

This idea has since then been used by bibliometricians to rank papers and journals [3] according to the strength of the citations they receive, and to rank web-pages according to the importance of the pages that are linked to them [4]. The use of such a recursive approach will lead to weighted points (Points**) as shown in Table 2 above. Note now that both Argentina and Nigeria have improved their points score, but the ranking has not changed, Korea continuing to be ranked above Greece and thus both Argentina and Korea proceed to the next round of the World Cup.

It is interesting to display the relationship between the raw points scored and the weighted points after application of the Kendall-Wei approach. One can assume that the best-fit line indicates how far out-of-joint the mixing up process (upsets in the ranking) has been. We see a slope of 1.08 and a correlation of 0.98 in Figure 1, showing that in Group B, a mild upset was registered when Nigeria held Korea to a draw. 

This process can be continued for all the other groups. We can use the linear regression (best-fit) metaphor to determine in which group the greatest “upset of ranking” has taken place. It turns out to be Group D (slope 0.58 and correlation 0.88). The conventional FIFA scheme has Germany through to the next round by virtue of the two matches it won against Ghana and Australia (6 points). When rescaled, it leads the table both on raw points (0.353) and weighted points (0.322). Australia and Ghana are tied on raw points (both have defeated Ghana and drew against each other). The tie was resolved noting that Ghana performed better on goals difference and so went through to the next round. On purely an unweighted scheme, Serbia has been ranked last. But the surprise now is that when the Kendall-Wei algorithm is applied, its losses to Australia and Ghana are counter-weighed by its win against the strongest team of this group and so on weighted points alone, it ranks ahead of both Australia and Ghana and should have gone through to the next round.

For the remaining groups, the FIFA scheme and the Kendall-Wei scheme let the same teams go through to the next round. In Group C, the FIFA scheme ranks USA ahead of England on goal difference but the Kendall-Wei weighting ranks England ahead of USA.

Acknowledgements

The author is grateful to Prof N V Joshi of IISc, Bangalore for drawing his attention to the Kendall-Wei scheme for ranking teams in tournaments.

[1] T.H. Wei, The algebraic foundations of ranking theory, Cambridge University Press, London, 1952.

[2] M.G. Kendall, Further contributions to the theory of paired comparisons, Biometrics 11 (1955), p. 43.

[3] G. Pinski and F. Narin, Citation influence for journal aggregates of scientific publications, with applications to the literature of physics, Information Processing and Management, 1976, 12, 297-312.

[4] Sergey Brin and Lawrence Page, The anatomy of a large-scale hypertextual web search engine, Computer Networks and ISDN Systems, 30 (1998), 107-117. 

http://www.math.utsc.utoronto.ca/b24/KendallWei.pdf