At the start of every round of the FIFA world cup, there is always a "group of death" designated. Often, the group of death eventually does not turn out to be a great group to watch. Will there always be a "group of death?" A recent move in Rugby claims to kill off the group of death.
What is a group of death? Let us consider one possible definition. Assume that groups consist, as they do in the FIFA world cup, of 4 teams each. The top two after the first round qualify to the pre-quarters. In this case, an acceptable definition could be that all four teams are possibilities for the next stage.
Can we formalize this to show that any allocation of 24 teams into 6 groups will have a group of death? (The question can be generalized.)
When is a team clearly unlikely to proceed to the next stage? Without loss of generality, we can assume that the teams are seeded (ranked) from 1 to 24. Assuming the seeding is current and reliable, if A and B are teams with seed of A at least 6 more than B, then it is unlikely that B will defeat A. 6 is semi-arbitrary.
A simple criterion guarantees a safe group: if the third seed is ranked 6 below the second, then the group has clear winners.
We will produce an allotment where all groups are safe, hence a group of death is avoidable.
Greedily allocate
I II III IV V VI
-- -- --- -- --- --
1 3 5 7 9 11
2 4 6 8 10 12
13 15 17 19 21 23
14 16 18 20 22 24
Within each group, we have a rank difference of 11 between the second and the third seed. This arrangement does not have a group of death, far from it.
Some journalists propose a weaker criterion for the group of death. The definition is that at least 3 teams must be contenders for the next stage. This allows more configurations to be considered groups of death. Unfortunately, in the above arrangement, no group has 3 contenders for the next stage. So no group of death exists, even according to the more liberal definition.
Then there are the sentimental definitions. Suppose the previous victor, even though currently badly ranked, is either the 3rd or the 4th seed in a group. Fans would like this team to get through to the next stage, regardless of the current seeding. Will this always lead to a group of death? Slightly more tricky: if A is assured to be third or below in every arrangement, then by the above example, A's seed must be 12 or below. Let us assumed A is seeded 12. (Would you really want a 12th team to give trouble to top teams?) The above arrangement gives the possibility where A is least likely to contribute to a group of death scenario.
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