Competitive balance in football: a new approach

Conventional measures of competitive balance – the Noll-Scully index based on actual versus idealised standard deviation of team winning percentages – are flawed in many ways with football because of the low number of games per season. A low number of games per season means that the idealised standard deviation of team winning percentages is relatively large since there is more room for random variation, while the Noll-Scully index ignores how the maximum standard deviation is consistent regardless of how few games each team plays. In low-scoring sports like soccer, gridiron, ice hockey and rugby where a lucky score can easily change the result of a match, this is not important, because even large differences in team qualities will not result in absolutely extreme win percentages.

In football, however, the rules of play allow for teams to score easily and, unlike basketball or netball, teams do not take “turns” with the ball to have opportunities to score. This means that propensities of teams to score or concede points can deviate much more than when the rules give each alternate “turns” – in football, each team requires skill to gain a “turn” at scoring. In fact, as Loek Groot shows in ‘Some Determinants of the Natural Level of Competitive Balance in European Football (Soccer) and US Team Sports: The Role of the Referee, the Scoring Context and Overtime’, a team with half the propensity to score of an competition’s average team would expect a winning equivalent no higher than 2 percent (a practically certain winless season in real-world football schedules) as against 20 percent in rugby and 35 percent in soccer. To put it another way, for the same disparity in win percent team qualities would need to deviate twice as much in rugby as in football.

The problem of not considering an upper limit for standard deviation was noted by P. Dorian Owen in ‘Limitations of the relative standard deviation of win percentages for measuring competitive balance in sports leagues’. Football, crucially, differs from baseball, ice hockey or soccer in that plausible differences in team qualities could easily produce probabilities of winning equal to or negligibly different from 0 or 1, as observed when Hawthorn during its early years as a League club faced the “big three” of Carlton, Collingwood and Richmond. However, at the same time one should not ignore the fact that if teams were equal in winning probability they would not all win the same number of games – especially in short-season sports like football.

Thus, as a new measure of competitive balance I propose the following steps:

1. Calculate what Dorian Owen symbolises ‘ASD^ub’ or the standard deviation of a perfectly unbalanced league
• Owen demonstrates for us that ASD^ub = ((n+1)/(12*(n-1)))^½ where n is number of teams
2. Subtract the idealised standard deviation from ASD^ub (ASD^ub-ISD)
• The ISD as based on the simple binomial distribution is give by (4l)^½/4l where l is number of games per season or average number for unequal schedules
3. Divide this value obtained in (2) into the actual standard deviation

Thus we have a formula for a ‘relative index of competitive balance’ of:

(ASD^actual-ISD)/(ASD^ub-ISD)

which can take values from below zero when the actual standard deviation is less than the ideal to positive unity for the perfectly unbalanced league.

(ASD^actual-ISD)/(ASD^ub-ISD) ratios for the three largest (Australian rules) football leagues between 1898 and 2013.
Note: the zeroes for the SANFL are seasons without regular play during the World Wars (1916, 1917, 1918, 1942, 1943, 1944).

What is notable is that the diagrams show football as competitively unbalanced as theories of competitive balance predict a sport with very high scoring and a very restricted talent pool to be.

The fact that the pre-World War I period without equalisation by zoning or revenue sharing was – despite much lower scoring than later eras – quite close to the hypothetical perfectly unbalanced league implies:

1. that without these regulations football would with higher scoring have acquired (ASD^actual-ISD)/(ASD^ub-ISD) ratios consistently negligibly different from positive unity, and/or
2. that reduced variation in team qualities after World War I led stronger teams to play more attacking football and thus increased scoring
• a proposition supported by more defensive tactics since Docklands supplanted Waverley. This change eliminated opportunities for shorter players of value in wet or windy conditions, made very tall people of limited supply more valuable, and almost certainly increased discrepancies in team qualities.

It’s also notable that the NBA, discussed much for its competitive imbalance, has an ASD^ub of:

• ((30+1)/(12*(30-1)))^½ 
• = (31/(12*29))^½ 
• = (31/348)^½

Thus, the Noll-Scully for a perfectly unbalanced NBA equals:

• (((31/348)^½)/((4*82)^½))/(4*82)
• = ((31/348)^½)/((328)^½)/328)
• = ((31/348)^½)*(328/(328)^½))
• = ((31/348)^½*(107584/328)^½)
• = ((31/348)^½*(328)^½)
• = (10168/348)^½
• = (2542/87)^½
• ≈ 5.40540385

This suggests that, in fact, football is at least as competitively unbalanced within-season as basketball, since a Noll-Scully of 1.90 (typical for football over the past 115 years) corresponds to a (ASD^actual-ISD)/(ASD^ub-ISD) ratio of about 0.65 – larger than the 0.351 of the NBA (for details see pages 65 to 67 from David Berri’s of The Wages of Wins). There are several periods when none of the three leagues graphed ever achieved a value as low as 0.35 (1908-1915, 1946-1953, since Docklands), and in the first period there were two values under 0.5 out of 24.

These are the raw Noll-Scully ASD/ISD ratios for the “major” football leagues since 1898 (again, the zeroes in the SANFL data are wartime seasons without regular football)

The next step needed will be a more detailed analysis of what these figures reveal about competitive imbalance in football and what has driven changes over time.