Net
ranking national football teams @ sportrankings.net
1. The difficulty of ranking
football teams
There are more than 200 national
teams in the World, very diverse geographically and in their ability
and ambition. They usually play about a dozen games every year, and
many teams have never played each other. Every four years the FIFA
World Cup brings together 32 teams and the best one becomes World
Champion. But even here, half the teams meet only three opponents, and
no teams play more than seven games. Apart from the World Cup, there
are Continental Cups whose levels considerably vary. The Oceanian
champion is not necessarily on a par with its African counterpart. It
is usually accepted that European and South American levels of play are
comparable. But here also comparisons are tricky, because all South
American games have outstanding level, while Europeans enjoy several
one-sided meetings between large and small nations.
A honest ranking scheme should handle these differences fairly. A team
should not benefit from having weak opposition. But neither should
there be a penalty. Arguably Mexico and the USA have an easier path to
qualifying for the World Cup. And the organizer of the World Cup plays
only friendlies for two years. The ranking scheme needs to assign their
proper ranks to them nonetheless.
Any ranking formula assigns a number to each team, that is based on
past results. Games can be officials of friendlies, they can be recent
or rather old, and they need to be weighed accordingly. A fair ranking
scheme should have the following properties:
- A victory earns points; a tie may earn or lose points; a
defeat loses points.
- A win may increase the number of points for many teams, not
only the winner; but the winner should benefit the most.
- There should be no advantage to choose a strong or a weak
opponent, the average gain (or loss) of points should be zero for each
team.
In all reasonable ranking schemes
are victories better than defeats. However, with several ranking
formulas a victory against a weak opponent results in a lower score! The actual FIFA
ranking procedure suffers this defficiency, albeit to a low level.
2. About the net formula, used here
The ranking for the current month
takes into account all international games of the past 48 months. Games
are weighed according to their type, namely
- 1 for a game in the World Cup final;
- 3/4 for a game in the Continental Cup final;
- 1/2 for a qualifying game, either for the World Cup or the
Continental cup;
- 1/4 for a friendly game.
In addition, there is a time coefficient: a game played n months ago receives a factor (49
- n) / 48. Scores are not
taken into account, and victories obtained after extra time or after
penalties count as a regular victory (same for defeats). A tie counts
like half a victory and half a defeat. Apart from these weights, the
net formula does not have
free parameters.
The results are encoded in a matrix A
indexed by the teams. Aab
(resp. Aba) is
equal to the sum of all victories (resp. defeats) of team a versus team b, the games being weighed by the
coefficients described above.
The key idea behing the net formula is to compare two teams using
indirect games. Namely, if team a
has defeated team b who has
defeated team c, then a is given a virtual win over team c. The coefficient of this win is
the product of the coefficients of each actual game. We can compare
teams a and b by looking at all indirect games
mediated by teams c1,
c2, ... cn. It is natural to
divide the contribution of a path of n teams by n!. Allowing for teams to be
compared with themselves, this leads to the following elegant formula
for the average score of team a
versus team b:
sab
= [(e A) ab - (e A) ba] / [(e A) ab + (e A) ba]
The score for team a is given by the average of sab over all teams such
that the denominator above differs from zero. On the page team comparison you can find the average
scores between any two teams, for the last month. Precisely, the
comparison page gives sab
+ 1, which is a number between 0 and 2 and is therefore more natural to
football fans.
Finally, we shift and scale the score so that it is a number between 0
and 1000. A team with 1000 points must has won all its games in the
past 48 months --- this is unlikely to happen any time soon. A team
with 0 points has lost all its games in the past 48 months --- several
teams are in this situation, but they certainly enjoyed playing this
great game nonetheless!
One can show that the property 1. above holds true, namely that wins
are always favorable and defeats are always unhelpful. The property 2.
may well be true, although we do not have a full mathematical proof of
it. Property 3. could be formulated mathematically with the help of
some probability model, and it should be true. In words, let us just
mention that a victory is more likely against a weak opponent rather
than against a strong one. But the reward for a victory is small in the
former case and big in the latter case. And also, the cost of a defeat
is big in the former case and small in the latter case.
Teams who met less than 10 opponents in the last four years are
penalized, in a linear way.
3. And what about ranking ranking schemes?
Well, we do not have a formula
that assigns a score to ranking schemes, instead of teams... yet! Soon
to be discussed: FIFA formula, World Elo Ratings, and Elephant Rankings.
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