I. What players have said:
Fabi says the white advantage is lower. But Fabi's answer might assume the players don't bid and are under
Danya says the white advantage is lower. Robert Hess doesn't seem to disagree but prefers White in chess960 (as in chess?). Their answers are dependent on that the players bid in the 15min vs X min case (where X is the lower bid). Robert's answer also depends on that the armageddon has no increment (as usual?).
Black wins more based on the 2019 & 2022 WFRCC and the 2022 St Louis Chess 9LX. White win rate is only 1/3.
Total games: 9
If the analogous win rate for white is 50%, then white definitely has a lower advantage, namely 1/3 lower because -1/3=(1/3-1/2)/(1/2).
But these aren't necessarily meaningful statistics because (besides the low sample size) they aren't quite with the same amount of time because some had bidding.
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.
For predetermined 5min vs 4min, white win rate is only 20%.
Total games: 5
Putting aside the obvious elephant in the room of low sample size, if the analogous win rate for white in chess for 5min vs 4min is 50%, then white definitely has a lower advantage, namely 60% lower because -60% = (1/5-1/2)/(1/2).
Also we could look at the bid only games to see if there might be cause to have a Vickrey auction instead of a blind auction.
For bid, white win rate is exactly 50%.
Total games: 4
Of course the sample size is low, but there's no a posteriori evidence here that shows a need for a Vickrey auction or something.
III. Comparing winning bids:
I guess it doesn't matter if it's chess or chess960 if the players bid even if the auction is unfair, but I think one's bid might change depending on chess or chess960. Eg You'd want black (or white) all the more in chess compared to chess960 because in chess there's theory (theory helps both white and black I guess) and more draws if you get black.
Consider 2 cases where the losing bid gets 15min:
Recently in the 2023 champions chess tour Airthings Masters magnus vs hikaru that the bids were Hikaru 8:59 and Magnus 8:58.
However in the 2022 (rapid) WFRCC finals between hikaru and nepo, the winning bid by Nepo was 13min.
There's a 4 minute gap. The players seem to want black 31% more in chess than in chess960 because -31% = (9-13)/13.
III.1. Actually let's do more
Armageddon: How much more desirable is black in chess than chess960 based on airthings masters 2023 vs 2022 WFRCC?
It's still around 30%. Average is 12.1458333...min = 12min 8.75sec in chess960 (still 2022 WFRCC) vs 9.3875min = 9min 23.25sec in chess (still 2023 Airthings).