Is armageddon more fair if bids aren't 1-time only?

Question

As pointed out by GeneM here

Armageddon Chess is a fair tie-break system only if the two players bid for how much time Black should have.

Question: But is it actually more fair if the bids are the usual English auction instead of 1-time only?

My guess is yes but they do 1-time only bids for practical reasons like with plurality voting. Of course it could be that either 1-time only is actually more fair than bid or somehow, the results will end up the same regardless of the kind of auction.

Consider recently in 2023 champions chess tour Airthings Masters magnus vs hikaru, where the bids were Hikaru 8:59 and Magnus 8:58.

2 ways to look at it:

1. I think with 1-time only, they're bidding far lower than what they normally would bid in an English auction - something like they're bidding not how much they want black (or white, if you can choose colour) but how much they don't want their opponent to get black. I think if they did an English auction, then black would end up with maybe 10-12 minutes.

2. But here, there's only a 1 second difference. Maybe they'd bid even lower. If the bids would continue - let's say the next bid must be at least 10 seconds lower than your opponent's bid - then the might go as low as 6 minutes.

Consider also:

3. If I were to bid exactly 15 minutes because my opponent thinks I'd go as low as 9 minutes. Then I'd trick my opponent to do like 8:30 whereas my opponent wouldn't even have gone lower than 10 minutes otherwise. I think this can be fixed with Vickrey auction as pointed out actually by Jim Ratliff in the same thread with GeneM.

Answer 1

Auction theory says that, under pretty general conditions:

[A]ll four of the standard single unit auctions have the same expected sales price (or seller's revenue).The four standard single unit auctions are the English auction, the Dutch auction, first-price sealed-bid auction, and the second-price sealed-bid auction.

(See Revenue Equivalence for One-sided Markets, EconPort. See also the discussion in Math Stack Exchange.)

Here, the “sales price” corresponds to the amount of time on the clock a player is willing to sacrifice to obtain draw odds with the Black pieces rather than full time with the White pieces.

The reason this is achievable in a single round of sealed bids (as is the typical mechanism in Armageddon bidding) is that “truthful revelation” is the equilibrium, and the winner gets the loser’s bid.

Answer 2

Ok actually reading about the Vickrey auction made me think of the Jim Ratliff's answer made me think it's LESS fair:

If Hikaru wants to go for as low as 9 minutes, but Magnus wants to go for as low as 6 minutes, then maybe we'd just end up with that the final bid is - let's say there's a minimum of a 10 second difference between bids - Magnus with 8:50. Hikaru won't go lower, and then Magnus has a 3 minute advantage.