The basis of the qualifying matches and of the first phase of world competitions is a draw in order to sort the participants into groups. Within a group everyone plays with everyone, and one or more teams qualify from every group. In some matches one participant or both participants have nothing at stake any more as they have already dropped out or qualified. For example: The chances in groups of four, supposing the first two teams qualify and victory means 3 points, draw means 1 point and no point is given for a defeat.
Participants: A, B, C, D.
1st round: A-B and C-D. The possible scoring is the following (there are 9 variations):
2nd round: A-C and B-D.
Possible scores after the 2nd round: there can be 9*9, that is 81 variations. In the 3rd round A-D and B-C play with each other, and the possible scores before the 3rd round are the following:
Green background: Already qualified before the match,
Yellow background: Qualifying even with a draw,
Grey background: Already fallen out before the match
Letters in bold: The participants of the match can agree on a result that would qualify both.
The most striking information from this chart is that out of the 81 variations there are only 3 in which
) all four teams still have a chance to qualify,
) none of them has qualifyed before the third round,
) for all four of them, only victory grants qualification.
) These 3 variations mean only 3.7% of the 81 cases!!!
In other words: Out of the 81 variations before the third round
) Not all teams have anything at stake in 38 matches, that is 46.9% - almost in half of the matches!!!
) In 16 variations it is possible to have an agreement by which both teams would qualify (that means 19.8%!!!)
) Every team has 9 variations to get 6 points, and from these
) 7 grant qualification (that means 77% of 9, 8.6% of 81),
) in one of the seven both teams have already qualified,
) in 3 of the 7 there is a potential agreement.
) The number of yellow-background cells is remarkably high. In 18 of the 81 variations, a draw is enough for the teams to qualify, this means 22.2% of the 81 variations.
) If a team wins the first two matches and gets 6 points, (9 variations), then they can either already be qualified (7) or a draw is enough for them to qualify (2).
) If a team has a victory and a draw in the first two matches, and therefore has 4 points (18 variations), then in 16 variations a draw is enough to qualify, and in 4 cases of the 16 there can be an agreement, a score which means qualification for both teams.
) Two defeats or a defeat and a draw in the first two matches would mean the opposite of the above with regard to chances.
) Out of the 81 cases a participant can have 0 points in 9 situations, in 7 of these 9 there is no chance for qualification. In 2 cases there is a mathematical chance for qualification, which means that the participant might win the third match in vain, they can still fall out because of the score of the other match of the group.
) If a participant has 1 point after two matches (1 draw and 1 defeat), that would mean 18 variations out of the 81, and in 6 cases out of the 18 it is possible that their two opponents make an agreement thus dropping out the team in question.
The chances for one team, with regard to the 81 variations:
) the competitor can have 6 points in 9 cases, out of which
) in 7 they have already qualified,
) in 3 of these they can have an agreement with the opponent so that both could qualify,
) in 2 cases a draw is enough,
) the competitor can have 4 points in 18 cases,
) in 16 of which a draw is enough for qualification,
) in 4 cases they can have an agreement with the opponent so that both could qualify,
) in 18 cases the competitor can get 3 points,
) in 9 cases they can get 2 points,
) in 18 cases they can have 1 point,
) in 9 cases they can have 0 points.
So if somebody wins the first two matches, or has one victory and one draw, they are sure to qualify in 25.9% of the potential cases, in 66.7% a draw is enough; this makes up for 92.6% of the 27 variations.
Reality, compared to mathematics, is even worse, as in these calculations we consider everybody equally powerful. In fact there are always stronger ones and weaker ones, so situations where all four teams still have a chance to qualify are even less frequent; it is much more likely that after the first two rounds the chances for qualification have already been settled for one or more teams.
My mathematician self is speaking to you now, and I am looking for contacts to top sports managers (on national, continental or world level) to whom I could present a brand new competition system.
