### I am looking for...

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.
I would like to show them what I have, to talk about it, and if the are interested I am ready to traverse the world if needed. So if any of you knows sports managers who might be interested, do not hesitate to send me a message.

Thank you very much,
Andras

### The basic rules of the new competition management system

1st The participians qualify with a certain number of wins and drop out with a certain number of defeats.
2nd The system does not know the notion of draw, a win must be reached (through one or more matches).
3rd Every team plays with opponents from their own category.
4th Same category = Teams with the same amount of win and defeats.
5th Any deviation from point 3 is only possible in special cases.
6th If, according to pure mathematics, not enough teams qualify, one or more or the drop-out teams can be invited to remain in the competition (using a rule that selects the best of them).

### Prolog

Dear Sporstfellows,
I would like to call your attention to a so far unknown competition management system. This system is suitable for the management of world and continental level competitions too. As opposed to the currently applied systems, this one amends their mistakes and has some further advantages.

There are many problems with the currently used systems, for instance:
) There are many matches where nothing is at stake for a team or for either of the teams,
) It is easy to achieve results that are favourable for both teams,
) The strongest teams play with the weakest teams,
) The weakest teams can never meet teams with a similar power in important matches.
) Everybody has to play the same amount of matches, regardless of whether they still have the chance to qualify or whether they have already been qualified,
) The power-based classification is determined prior to the competition period, thus the actual condition cannot be taken into consideration,
) There is, in my opinion, a logical mistake in the management of the European qualifying matches of the World and European Cups.

How about a management model in which these problems do not exist? A model in which no team would play a match with the attitude of 'it does not matter, we have already qualified / fallen out', where everybody plays matches until they have the chance to qualify, and thus there would be no pre-arranged matches.
This new competition management system would fulfil the three most important requirements, namely:
) It would be mathematically valid,
) It would be easy to understand for the people,
) It would not mean more (or significantly more) matches than in the systems applied so far.

) Competitions would be realisable for example with 48 participants,
) This competition system could manage an optional number of starters and qualifiers (e.g. 53 starters, 13 qualifiers.)
Besides that
) As usual, every participant plays several matches,
) There could be no results which would be favourable for both teams.

If you prefer to hear more about the new system, I would gladly present it in more details with the help of examples, modelling and mathematical calculations.

Yours sincerely,
AndrĂ¡s Halasi

### Chance calculation with regard to the mistakes of the team system, to the chances of having matches with nothing at stake or agreement matches

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.

### The basic rules of the new competition management system (explanation)

1st The participians qualify with a certain number of wins and drop out with a certain number of defeats.
2nd The system does not know the notion of draw, a win must be reached (through one or more matches).
3rd Every team plays with opponents from their own category.
4th Same category = Teams with the same amount of win and defeats.
5th Any deviation from point 3 is only possible in special cases.
6th If, according to pure mathematics, not eenough teams qualify, one or omre or the drop-out teams can be invited to remain in the competition (using a rule that selects the best of them).
The new competition system is as simple as these 6 points. And of these the audience only has to understand the first three points.

An example: on a competition with 32 participants, after the first round there will be 16 with a win and 16 with a lose.
I will use these notes to mark this    >>>                1-0 / 16
0-1 / 16.

In the second round the 16 winners will play each-other (1-0)
and the result will be    >>>                                    2-0 / 8,
1-1 / 8.
Furthermore, the 16 teams with a single defeat will also play with each other (0-1)
and the result will be    >>>                                   1-1 / 8,
0-2 / 8,
2-0 /  8,
1-1 / 16,
0-2 /  8,
In the third round there will be four matches between the 2-0 teams, eight between the 1-1 teams and four between the 0-2 teams. After this round we will have:
3-0 /  4,
2-1 / 12,
1-2 / 12,
0-3 /  4,
etc.

When a team reaches enough wins they qualify, if they collect enough defeats they drop out, but in between the two thresholds they constantly have a chance to qualify. It can not happen that a team with a prospect to qualify plays with a team that has already dropped out. The matches can be drawn at each round, for several rounds beforehand, or for the whole competition in advance. This system has the further advantages that the teams, except in the very first round, always get opponents with similar actual performance (their performance on the actual competition) and also that at the end of the competition it is possible to set up a full hierarchy of every participant based on their achievements.

### Logical mistakes of the qualifying rounds of World and European Championships

In the qualifying rounds of these world competitions the participating national teams are sorted into 9 groups.   As the number of the participant teams (53) and that of the qualifying teams (14 and 13) did not give the organisers a mathematical chance for a clean arrangement, they chose the following method:
In the 8 groups there were 6 participant teams, in one there were 5,
The winners qualified (this meant 9 qualifying teams),

In case of 14 qualifying teams the winner teams qualified (this meant 9 qualifying teams),
and also the best second team (as the 10th qualifying team),
and four more from the other 8 second best teams (determined by pair matches),
thus a total of 14 teams qualified.

In case of 13 qualifying teams the winners,
from the 9 second best the worst team drops out, and from the remaining eight four more qualify based on the results of pair matches, this means 4 more besides the 9 winners.

The logical mistake is the following:
Out of the 9 groups there were 6 participant teams in 8 of the groups, every one of these teams played 10 matches, and a maximum of 30 points could be reached by the winners and a maximum of 27 points for the second bests. There were 5 teams in the ninth group, every team played 8 matches, so the winners could reach 24 points, the second ones could reach 21 points.  The second team in this group would be at a disadvantage - according to the organisers, and finally they are right in that  - therefore, in order to establish a ranking between the second teams, they decreased the score of the second teams in the other groups by the score of the match against the weakest team of the group, as if they had not played against that team.
The result of this measure was that the score of the team that lost points against the weakest team of the group was decreased by fewer points, which is - in my view - incorrect.  The team that could not win against the weakest team of the group in both matches is privileged.

This mistake is clearly visible in the qualifying rounds of current world championships:
Iceland’s matches against the weakest team of the group, Cyprus: 2-1 and 0-1, thus their score was decreased by only 3 points, and they got into a better position than Denmark, who won 2 matches (6 points) against the weakes team of the group.  Originally Denmark had 16 points, Iceland had 15 points, yet it is Iceland who can play on to qualify for the world championship.

Soon...