In the 2020 Summer Olympics Sport Climbing was added as a new Olympic Sport. Competitors had to compete in three categories: lead climbing, bouldering and speed climbing.
Within each category the competitors were given a rank from $1$ to $n$. $1$ being the best in the category, $2$ being second-best and so on. The total points by which the contestants are then ranked is given by $$ \prod_{i=1}^{n} r_{i} $$ where $r_{i}$ denotes the competitor's order in category $i$. These are then sorted, and the lowest score wins gold, the second lowest silver, and so on.
While this is nice and simple, the system seemed unsatisfactory at the end of the last category based on the results.
What concerns me is that the results of one competitor can reorder other competitor.
To take a simplified example, consider the following situation:
| Contestant | Category A | Category B | Category C | Total Rank |
|---|---|---|---|---|
| Alice | 3 | 2 | 1 | 1 (6) |
| Bob | 2 | 1 | 4 | 2 (8) |
| Charlie | 1 | 3 | 3 | 3 (9) |
| Dan | 4 | 4 | 2 | 4 (32) |
If Dan had won the 3rd category, Alice and Bob's positions would be swapped like so:
| Contestant | Category A | Category B | Category C | Total Rank |
|---|---|---|---|---|
| Bob | 2 | 1 | 4 | 1 (8) |
| Charlie | 1 | 3 | 3 | 2 (9) |
| Alice | 3 | 2 | 2 | 3 (12) |
| Dan | 4 | 4 | 1 | 4 (16) |
Here we see that Dan's result has penalized Alice considerably. This seems to be an unfair way to combine ranks, as the partial order between Alice and Bob depends on the partial order between Dan and Alice.
The property the seems to be violated here appears to be very similar to the Independence of Irrelevant Alternatives.
What I'm looking for is whether there is a theory for functions that combine independent orderings into a global one, and what properties these rank-combining functions may have. For instance, whether excluding or adding a late contestant can change the relative order of others, whether the winner of all categories necessarily wins the whole event, etc.
It violated Independence of Irrelevant Alternatives
As you rightly point out, the violated property is very similar to Independence of Irrelevant Alternatives (IIA), because it is IIA. Indeed, because it's an ordinal voting system, Arrow's impossibility theorem applies, and because the rule is clearly not dictatorial, and it satisfies the efficiency requirement, it must necessarily violate IIA.
Specifically, the rule used is a positional voting system where the different formats are the voters and the contestants are the candidates. If the placings were summed instead of multiplied, we'd have the Borda count method. Instead we multiply the points, but this is equivalent in an important way, and we can fit them into the same framework by using logarithms as follows:
$$ \prod_{i=1}^{n} r_{i} = \exp(\sum_{i=1}^{n} \log(r_{i})) $$
Exponentiation is monotonic, so we can ignore it as we only care about the order of the candidates, which gives the final scoring method as:
$$\sum_{i=1}^{n} \log(r_{i})$$
Therefore the ranking system is a slightly disguised positional voting system where the coefficients are the logarithms of the ranks attained. It therefore also violates other properties you might want a system to have, as all voting rules do. Wikipedia has a list of some of some of the properties satisfied and violated by positional voting rules, in case you're further interested.