Your idea combines what feels like a significant percentage of randomness while still having enough participation to rationalize buy-in.
It's tough to want to yield control of an important choice to a random number generator... though that's not an inaccurate description for this.
I like this as a perspective on "the illusion of choice". Part of me hates the phrase because I want to believe that I can take meaningful action.
I think it was avoiding trusting a third party.
It was goofy, but satisfied them all without dissent. And it worked!
The least significant bits of a bitcoin PoW hash behave pretty randomly. You could either bet on a fixed future block height, or on the first block past a given future timestamp...
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[1] You can compensate for unequal economic situations by limiting the total amount that any one person can bid to a value that they can all afford.
(The film "Force of Evil" is about a plot to nevertheless rig the outcome.)
I came up with this: Each of them select a different stock market index. On the same calendar trading day, they choose two digits past the decimal. When the market's close, compare the two digits of the closing price of their chosen index to the two digits they selected and then choose heirlooms in the order of closeness.
It worked quite well. They were on board because they could make two selections in determining their order: Market and digits. Two of them could choose the same two digits if they wanted and still have different scores. No complaints.
Not random, but seemed fair to all.