Math A-lympiad: Final
1994-1995 Math A-lympiad: Final
1994-1995Tournaments are held in many sports where matches organised in different groups bring about a ranking between the teams. There have been many cases in history where the favourite in the group matches meets with disaster and does not even get a place in the (semi-)final. The smaller the differences between the teams, and the greater the role luck plays in the results of the group matches, the more likely it is that such a situation will arise. It is clear that luck plays a greater role in one sport than in another.
In this assignment we want to examine the role of luck in sport. By examining
how the luck factor plays a role in the final results of a group, (by using
the results of past tournaments) we can perhaps make an assessment of the degree
to which a sport can be classed as a lucky sport.
Further note: on the one hand the results are influenced by relative strengths,
the differences in the strengths of the participating teams, but they also depend
on luck to a greater or lesser extent.
You can imagine two extreme situations. In one case, only the relative strength
matters. There is a ranking of the strengths between the participating teams
and a stronger team will always beat a weaker one. Luck plays no role.
The converse to this is where everything is governed by luck. In every match
the two participants have an equal chance of winning.
Reality lies somewhere between these two extremes. One sport is closer to the
first extreme, another closer to the second. The stronger team does not necessarily
win but the probability will of course be greater (depending on the relative
strength). A greater influence of luck often gives more spectacular matches
and tournament outcomes, and thereby a higher entertainment value. A greater
influence of the relative strength is perhaps fairer, however.
In many sports, tournaments are held in which a large part of the tournament
is played in groups, and then perhaps continued in quarter finals, semi-finals
and a final.
The examination will for the time being be on the possible events in groups,
where each contender plays every other contender in the group once.
The appendices show the results from the soccer world cup and the ladies volleyball
world cup.
In group D of the 1994 soccer world cup we see that Argentina beat Nigeria,
Nigeria beat Bulgaria and Bulgaria beat Argentina. Such an event in a group
is called a 3-cycle. We can illustrate this here:
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the arrow points at the loser
In theory it is possible for there to be two 3-cycles in a group result, for example:
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Exercise 1
Examine all the group matches of the 1994 soccer world cup and the
1994 (ladies) volleyball world cup. In the two tournaments see how many group
results there are without 3-cycles, how many with one 3-cycle and how many
with two 3-cycles. Are there any differences between soccer and volleyball?
Cycles are contrary to the notion of “the strongest wins”. They
do perhaps fit in with the luck model. To examine whether the luck model can
explain the occurrence of cycles, you have two possibilities:
a Simulation.
b Theoretical.
The number of 3-cycles in the group results can be an indication of the level to which a sport is governed by luck, or by mutual differences in strength. The question is then how many 3-cycles can be expected in a group result where luck alone plays a role. One way of answering this question is by simulating a (large) number of group results. Simulation means that the result (win, lose or draw) of each match in the group is determined by throwing dice or tossing a coin, or by using random numbers, etc.
Exercise 2
a) Assume in the first instance that draws are not allowed. Determine a number
of group outcomes with a simulation, where each team has an equal chance of
winning each match.
Go through the simulated group results to see how many results there are without
3-cycles, how many with one 3-cycle and how many with two 3-cycles.
Do this for at least 25 groups.
b) 4-cycles can also occur in a group result: A beats B, B beats
C, C beats D and D beats A. It turns out that the occurrence of a 4-cycle
in a group result is the same situation as the occurrence of two 3-cycles
in a group result.
Show that that is indeed the case.
c) In the second instance you can do a simulation where draws
are allowed. Take the probability of a draw as 2/9. (This corresponds to the
percentage of draws in the 1994 soccer world cup, 1st round). Again assume
that in each match the teams have an equal chance of winning.
Simulate a number of group results here.
Search the simulated group results for how many there are without 3-cycles,
how many with one 3-cycle, and how many with two 3-cycles.
By systematically going through all possibilities, the number of 3-cycles to be expected in group competitions where the results are determined purely by chance can be calculated more accurately.
Exercise 3
a) In how many ways can you set the six results in a group of four teams,
if a draw is not allowed?
b) Again assume that no draws are allowed. If one point is awarded for each
match that is won, how will the points distribution look (the rankings) in
a group, ranked according to decreasing number of points? With what point
distributions do you have cycles, with which do you not? How great is the
chance of a result without cycles? How great is the chance that you have no
cycles in four groups?
c) Now assume that a draw is allowed. Suppose that both teams have a probability
of p (p < 1/2) of winning, then the probability of a draw is
q = 1 – 2p. What is the probability of a cycle occurring in
a group with only three teams?
In a group with four teams it is more difficult to calculate the probability
of no, one, or two cycles. These probabilities can be calculated, however.
The probability of no cycles is 1 – 8p3 + 12p5
The probability of one cycle is 8p3 – 24p5
The probability of two cycles is 12p5
Can you compare this to the results of the real matches and with the simulation
that you did above for matches in groups of four teams where draws are allowed?
Exercise 4
a) Volleyball is a faster sport than soccer. More happens in a match and more
sets are played. If it were a slower sport you can imagine that only one set
would be played in a match.
Examine how the results of the four groups of ladies volleyball (and how many
cycles occur) would be if only the first set was played.
We will now look at the reality: ‘best of five’. (Who was the
first to win three sets?)
b) If team A is stronger than team B, such that in each set A has a probability
of p (for example 2/3) of winning (and B thus 1 – p), how great
is the probability that A will win the match?
Exercise 5
We will now look at matches in groups with only three clubs A, B and C, without
draws. From the results of a former, bigger competition in which other clubs
played in addition to A, B and C, it was concluded that the probability of
A beating B was 3/4 and the probability of B beating C was 3/4. Can you say
anything about the probability of A beating C?
Suppose that the probability of A beating C is equal to p, or select
a suitable value for p. What is the probability of a cycle occurring
now? What is the probability of A winning?
Exercise 6
Sometimes the strength of a club is given a hypothetical figure: the higher
the figure, the stronger the club. If this is done for a number of clubs,
you get a kind of ranking for the clubs in which cycles supposedly cannot
occur. But it may be that these figures only say something about the probability
of a stronger club beating a weaker club.
By allocating the hypothetical figure a to club A and b to club B, can you
devise a way to say something about the probability that A or B will win a
certain match?
Specify how you yourself would organise a tournament with 24 teams. Give arguments based on the results of the examinations you have done. Also consider factors such as entertainment value, limited duration of the tournament, fairness of the results, any distinction between different sports, etcetera.
| USA - Switserland | 1 - 1 | Roemenia - Colombia | 3 - 1 |
|---|---|---|---|
| Switserland - Roemenia | 4 - 1 | USA - Colombia | 2 - 1 |
| Colombia - Switserland | 2 - 0 | Roemenia - USA | 1 - 0 |
| Cameroun - Sweden | 2 - 2 | Brazil - Russia | 2 - 0 |
|---|---|---|---|
| Brazil - Cameroun | 3 - 0 | Russia - Sweden | 1 - 3 |
| Brazil - Sweden | 1 - 1 | Cameroun - Russia | 1 - 6 |
| Germany - Bolivia | 1 - 0 | Spain - Korea | 2 - 2 |
|---|---|---|---|
| Germany - Spain | 1 - 1 | Korea - Bolivia | 0 - 0 |
| Germany - Korea | 3 - 2 | Spain - Bolivia | 3 - 1 |
| Argentina - Greece | 4 - 0 | Nigeria - Bulgary | 3 - 0 |
|---|---|---|---|
| Argentina - Nigeria | 2 - 1 | Bulgary - Greece | 4 - 0 |
| Nigeria - Greece | 2 - 0 | Argentina - Bulgary | 0 - 2 |
| Ireland - Italy | 1 - 0 | Norway - Mexico | 1 - 0 |
|---|---|---|---|
| Italy - Norway | 1 - 0 | Ireland - Mexico | 1 - 2 |
| Ireland - Norway | 0 - 0 | Mexico - Italy | 1 - 1 |
| Belgium - Marocco | 1 - 0 | Netherlands - Saudi-Arabia | 2 - 1 |
|---|---|---|---|
| Belgium - Netherlands | 1 - 0 | Marocco - Saudi-Arabia | 1 - 2 |
| Netherlands - Marocco | 2 - 1 | Belgium - Saudi-Arabia | 0 - 1 |
| Belo Horizonte - Sph. Mineirinho | ||||
|---|---|---|---|---|
| Group A: Germany, Brazil, Roemenia, South-Korea Group C: Italy, Oekraïny, Russia, China |
||||
| 21.10 | A | GER - KOR | 3 - 0 | (16-14; 15-10; 16-14) |
| A | BRA - ROM | 3 - 0 | (15-2; 15-2; 15-3) | |
| B | AZE - NETH | 1 - 3 | (8-15; 5-15; 15-11; 10-15) | |
| B | PER - CUB | 0 - 3 | (9-15; 1-15; 5-15) | |
| 22.10 | A | KOR - ROM | 3 - 0 | (15-2; 15-3; 15-7) |
| A | GER - BRA | 0 - 3 | (5-15; 7-15; 5-15) | |
| C | CHN - RUS | 3 - 0 | (15-13; 15-12; 15-3) | |
| C | ITA - OEK | 2 - 3 | (4-15; 15-5; 15-4; 6-15; 12-15) | |
| 23.10 | A | GER - ROM | 3 - 0 | (15-2; 15-2; 15-4) |
| A | BRA - KOR | 3 - 1 | (10-15; 15-10; 15-7; 15-8) | |
| C | RUS - ITA | 3 - 0 | (15-7; 15-9; 15-5) | |
| C | OEK - CHN | 1 - 3 | (8-15; 10-15; 15-3; 11-15) | |
| Sao Paulo - Sph. Ibirapuera | ||||
| Group B: Azerbeidzjan, Peru, Cuba, Netherlands Group D: Kenia, Japan, Tsjechia, USA |
||||
| 21.10 | C | ITA - CHN | 1 - 3 | (15-8; 3-15; 14-16; 5-15) |
| C | OEK - RUS | 1 - 3 | (15-8; 14-16; 13-15; 11-15) | |
| D | KEN - USA | 0 - 3 | (2-15; 7-15; 4-15) | |
| D | JPN - TSJ | 3 - 0 | (15-7; 15-9; 15-2) | |
| 22.10 | D | USA - TSJ | 3 - 0 | (15-2; 15-1; 15-1) |
| D | KEN - JPN | 0 - 3 | (1-15; 3-15; 4-15) | |
| B | NET - CUB | 0 - 3 | (4-15; 9-15; 10-15) | |
| B | AZE - PER | 3 - 0 | (15-12; 15-13; 15-6) | |
| 23.10 | D | TSJ - KEN | 3 - 0 | (15-7; 15-3; 15-0) |
| D | USA - JPN | 3 - 1 | (15-8; 6-15; 15-7; 15-7) | |
| B | CUB - AZE | 3 - 0 | (15-9; 15-9; 15-12) | |
| B | PER - NETH | 0 - 3 | (8-15; 7-15; 5-15) | |
| Belo Horizonte - Sph. Mineirinho | ||||
|---|---|---|---|---|
| 25.10 | seeding group
|
|||
| CUB - USA | 3 - 0 | (16-14; 15-9; 15-12) | ||
| BRA - CHN | 3 - 0 | (15-12; 15-4; 15-9) | ||
| Sao Paulo - Sph. Ibirapuera | ||||
| 25.10 | elimination group
|
|||
| NET-KOR | 1 - 3 | (11-15; 4-15; 15-6; 6-15) | ||
| GER - AZE | 3 - 1 | (15-4; 15-3; 8-15; 15-9) | ||
| JAP - OEK | 3 - 0 | (15-10; 15-11; 15-8) | ||
| RUS - TSJ | 3 - 1 | (16-14; 9-15; 15-8; 15-11) | ||
| Sao Paulo - Sph. Ibirapuera | ||||
|---|---|---|---|---|
| 28.10 | CUB - GER | 3 - 0 | (15-9; 15-5; 15-5) | |
| BRA - JPN | 3 - 0 | (15-10; 17-15; 15-7) | ||
| KOR - CHN | 3 - 1 | (15-5; 4-15; 15-5; 15-11) | ||
| RUS - USA | 3 - 1 | (15-9; 9-15; 15-9; 16-14) | ||
| 29.10 | BRA - RUS | 3 - 2 | (15-7; 14-16; 12-15; 15-8; 15-10) | |
| CUB - KOR | 3 - 0 | (15-4; 15-9; 15-5) | ||
| GER - CHN | 3 - 1 | (13-15; 15-13; 15-10; 15-13) | ||
| USA - JPN | 3 - 1 | (8-15; 15-7; 15-7; 15-7) | ||
| 30.10 | pl. 7/8 | JPN - CHN | 3 - 2 | (9-15; 15-12; 10-15; 15-10; 15-7) |
| pl. 5/6 | GER - USA | 3 - 1 | (16-14; 15-9; 13-15; 15-10) | |
| pl. 3/4 | RUS - KOR | 3 - 1 | (14-16; 15-11; 15-6; 15-8) | |
| pl. 1/2 | CUB - BRA | 3 - 0 | (15-2; 15-10; 15-5) | |