I aim to dispel the notion that the current seeding-pairing system is acceptable.
The highest priority for any pairing system based off of seeding is that the format should not, in any scenario, ever incentivize losing over winning. To satisfy this condition, (a) getting a higher seed must not be worse than a lower one, regardless of the true strengths of each seed. Furthermore, (b) getting a higher seed should ideally be better than a lower one, regardless of the true strengths of each seed. Meeting condition (a) means that the format is at least indifferent to winning and meeting condition (b) means that it incentivizes winning.
Therefore, the current format is in fact unacceptably broken because it incentivizes throwing in certain scenarios.
For example, suppose that the seeding is set aside from two players who are facing off in RBY Cup final for the 7th and 8th seeds. Suppose also that the 10th seed, who will play the 7th seed, is significantly stronger than the 9th seed, who will play the 8th seed. In this case, both players are incentivized to throw to get paired with the weaker 9th seed as opposed to the stronger 10th seed.
To construct a format which satisfies condition (a), we can simply pair completely randomly with no regard for seeding. The only incentive with respect to playoffs is to qualify, and therefore it never incentivizes losing. However, this format may not be ideal considering how it fares with respect to condition (b): under this format there is never any benefit to getting a higher seed beyond qualifying.
Therefore, we look to construct a format which still satisfies condition (a) while also meeting condition (b) as often as possible. To this end, McMeghan's original proposal is excellent (the highest remaining seed picks their opponent from all other remaining seeds iteratively until all players are paired); a player either prefers or is indifferent to being a higher seed in every scenario with this rule: Essentially, if a player gets picked by a higher seed, then their seed doesn't matter; and if a player doesn't get picked by a higher seed, then they get to pick before the seeds below them.
To clarify, the concern raised by Excal and Peng in posts #2 and #4 in this thread are not true: seed 9 is never preferable to seed 8. Suppose, for example, the seeds have strengths as follows:
Then, as Excal suggested in this scenario, the seeds will pick players as follows:
1 picks 15
2 picks 14
3 picks 13
4 picks 12
5 picks 11
6 picks 10
7 picks 8, 9, or 16 based off perceived skill, not seeding (so really between 8 and 9 since 16 is so strong)
8 or 9 is left to fact 16 based off which is perceived stronger by the 7 seed, not which is the higher seed
To extend this, note that dropping seeds would never result in the originally 8th seeded player not being paired with the strong 16 seed when they otherwise would have been. For example, if they sandbag to the 15th seed:
Then, the players will pick the same way:
1 picks 14
2 picks 13
3 picks 12
4 picks 11
5 picks 10
6 picks 9
7 picks 8, 15, or 16 based off perceived skill, not seeding (so really between 8 and 15 since 16 is so strong)
8 or 15 is left to face 16 based off which is perceived stronger by 7 seed, not which is the higher seed
You will find that any perceived counterexample fails similarly. This is because of the underlying logic I mentioned earlier: if a player gets picked by a higher seed, then their seed doesn't matter; and if a player doesn't get picked by a higher seed, then they get to pick before the seeds below them. The only way to avoid being picked by a player is to achieve a better seed than them. Higher seeding is often rewarded and never punished with this system.
As for alternative proposals:
Any system that does not have all pairings picked (like only the top 4 seeds pick for example) must pair the remaining players randomly as opposed to by seeding; else this regresses to the original problematic bracket in certain scenarios and can therefore also result in players being incentivized to lose.
As an example, take the following player strengths and seeding with a top 4 pick system:
This obviously results in 1-4 pairing with 13-16 and leaves the 5-12 seeds with the same messed up incentives as the status quo has. Suppose two players are facing off in RBY Cup final for the 7th and 8th seeds; both players are incentivized to throw to get paired with the weaker 9 seed as opposed to the stronger 10 seed.
Note that the seeding does not have to work out so nicely to result in the above scenario. I just chose it for the purposes of clear illustration.
However, if the limited number of pairings picked does pair the remaining players randomly, then it still meets condition (a), which makes sense because it is a combination of two systems that each individually satisfy condition (a). However, I currently believe that McMeghan's original proposal is preferable because it meets condition (b) more often than partially random formats.
As for immunity from being picked for top seeds, this can also incentivize throwing. For an example, suppose the top 4 players are immune from getting picked and each pick themselves with player strengths and seeding as follows:
Suppose two players are facing off in RBY Cup final for the 4th and 5th seeds. The player who loses will be the 5th seed and get picked by the relatively weak 1st seed; whereas the player who wins will be the 4th seed and, on account of being immune from getting picked by the other weak high seeds, will have to play a strong low seed. Thus, both players are incentivized to lose.
The first-round bye system also has these loss-incentivizing scenarios because of locked seeding, not to mention concerns regarding the size of advantage that a bye confers and the fact that it changes the number of players who reach playoffs.
What is left to discuss regarding McMeghan's original proposal is to establish what the format would be for pairings after round 1 (I haven't thought about this yet, so there may be an objectively best solution that I propose later) and to address concerns regarding 1) collusion, 2) bullying of highly seeded newcomers, 3) players coasting off prestige rather than earning the ability to avoid tough pairings based off of continued achievement, and 4) others concerns that have yet to be raised (these are all of the unique concerns that I found in the thread in the order that I found them; this is not intended to suggest what the absolute or relative merit of these concerns is or dismiss any concerns that I may have overlooked).
I may address these myself in a follow up post once I have had more time to think about them, but for now I will reiterate that whatever system we proceed with must not ever incentivize losing. I would advocate to default to purely random seeding until and unless a system that also satisfies condition (a) while being as good as possible with respect to condition (b) and not having significant, insurmountable flaws with respect to other concerns is agreed.
Thanks for reading.