# Thread: A tough mathematical question in a sport competition

1. ## A tough mathematical question in a sport competition

It is a difficult question but only can be solved with mathematical tools.

In the 2 most important Australian competition (AFL and NRL) use a final stage that gives more advantage to those teams that ends better the regular season. The debate is that every spot you get better in regular season, more advantages you must have in the final stage. According to many people, it is better end 2nd than 1st, but to other not. The only way to solve that is using maths tool.

Then, my mathematical question is:

-Is This final stage draw mathematically flawed (comparing top two spots) or is mathematically fair (it is better finish 1st than 2nd in the regular season)?

This is the system (top 4 teams have a double chance, and 5th-8th go to knockout stage:

Day 1

A-1st vs 4th
B-2nd vs 3rd
C-5th vs 8th
D-6th vs 7th

Losers C-D are eliminated

Day 2

E-Loser A vs Winner C
F-Loser B vs Winner D

Losers are eliminated

Day 3

G.-Winner A vs Winner F
H.-Winner B vs Winner E

Losers are eliminated

Day 4 (Final)

Winner G vs Winner H

Look that the point is that 1st has an easy opponent than 2nd in Day 1 (play against 4 instead 3) that is desirable; but if the favourites win their matches in Day 3 (semifinals) has a tougher opponent (3rd instead 4th). But on same time this match can be not produced if one the favourites lose in the first two days, even can play 1st against 2nd if one of them lose Day 1

I would like if somebody can solve this. I´m not Australian but there is a question than no math could solve until now

Salutations and Thanks in advance

2. ## Re: A tough mathematical question in a sport competition

I went through some brute force and it appears to be quite fair. In the end of my three scenarios which overall presumes that the ranking from the regular season is ordered.

(i)
Consider the ideal situation, where ranking is preserved / reflects the regular season.

DAY 1
A - 1st wins. 4th loses
B - 2nd wins. 3rd loses
C - 5th wins. 8th loses
D - 6th wins. 7th loses

DAY 2
E - 4th wins. 5th loses
F - 3rd wins. 6th loses

DAY 3
G - 1st wins. 3rd loses.
H - 2nd wins. 4th loses.

DAY 4
1st wins over 2nd

Hence,
1st has to face 2nd, 3rd and 4th place teams (disregarding order).
2nd has to face 1st, 3rd and 4th place teams (disregarding order).

This is fair. 1st and 2nd face off each other in the top 4.

(ii)
What if 1st loses the first match, but ranking is preserved otherwise?

DAY 1
A - 1st loses. 4th wins. (upset)
B - 2nd wins. 3rd loses.
C - 5th wins. 8th loses
D - 6th wins. 7th loses

DAY 2
E - 1st wins. 5th loses
F - 3rd wins. 6th loses

DAY 3
G - 4th loses. 3rd wins.
H - 2nd loses. 1st wins.

FINAL
3rd loses. 1st wins.

1st has to face 2nd, 3rd, 4th and 5th.
2nd has to face 1st, 3rd.

The fact that 1st had to win over 5th to make up for the loss to 4th is fair to me.
The fact that 2nd would have to win over 1st and 3rd only seems fair too, since beating 3rd under an assumption of preserved ranking means they're likely to beat 4th.

(iii)
What if 2nd loses the first match, but ranking is preseved otherwise?

DAY 1
A - 1st wins. 4th loses
B - 2nd loses. 3rd wins (upset)
C - 5th wins. 8th loses
D - 6th wins. 7th loses

DAY 2
E - 4th wins. 5th loses
F - 2nd wins. 6th loses

DAY 3
G - 1st wins. 2nd loses.
H - 3rd wins. 4th loses.

DAY 4
1st wins over 3rd.

1st has to face 2nd, 3rd, 4th
2nd has to face 1st, 3rd, 6th

For 2nd to make up for an upset against 3rd against 6th is almost comparable to case (ii), but the important match is still against 1st. (one could even say 5th and 6th place are about equal in skill anyway)
1st does have to face the other three best teams in this case, but this would directly show they're the best team.
--

Notes: Even if a tournament is set to be in a seemingly unfair way, consider that organizers would want both: exciting matches with possibility of upsets, and the two best teams to be in the finals. The configuration above seems to do this well.

If I didn't assume ranking was preserved, then situations can be much more complex, especially if winning took more of a rock-paper-scissors scheme. For situations like those, I'd recommend researching electoral mathematics to see elimination style tournament constructed in such a way to control winners.