This is my initial proposal, but I am going to think about it more:
There are n people. Everyone privately names a number, where the prompt is “What number do you think the answer has probability 1/n of being below.” Lowest bidder gets the territory (−∞,b1), where b1 is the second lowest bid.
The remaining n−1 people divide up the [b1,∞) using the same process, with the prompt “What number do you think the answer has probability 1/(n−1) of being below conditioned on being at least b1. Lowest bidder getting the territory [b1,b2), where b2 is the second lowest bid.
Continue this pattern until there is only one person left, who gets the remaining region.
This is not symmetric with respect to negating the answers, but it does have the property that everyone has a strategy (honest reporting) that guarantees that they win with subjective probability at least 1/n.
I am not sure that the second price part of this algorithm is actually doing anything useful, but it feels more fair, and closer to incentivizing honest reporting than the first price version. I am not actually sure whether it is better to leave at the beginning or end if everyone reports honestly for the second price version. The first price version is equivalent to:
Move the knife slowly across the cake from left to right. Anyone can say “cut” at any time and get the piece to the left of the knife and leave the game. Repeat.
This is my initial proposal, but I am going to think about it more:
There are n people. Everyone privately names a number, where the prompt is “What number do you think the answer has probability 1/n of being below.” Lowest bidder gets the territory (−∞,b1), where b1 is the second lowest bid.
The remaining n−1 people divide up the [b1,∞) using the same process, with the prompt “What number do you think the answer has probability 1/(n−1) of being below conditioned on being at least b1. Lowest bidder getting the territory [b1,b2), where b2 is the second lowest bid.
Continue this pattern until there is only one person left, who gets the remaining region.
This is not symmetric with respect to negating the answers, but it does have the property that everyone has a strategy (honest reporting) that guarantees that they win with subjective probability at least 1/n.
I am not sure that the second price part of this algorithm is actually doing anything useful, but it feels more fair, and closer to incentivizing honest reporting than the first price version. I am not actually sure whether it is better to leave at the beginning or end if everyone reports honestly for the second price version. The first price version is equivalent to:
Move the knife slowly across the cake from left to right. Anyone can say “cut” at any time and get the piece to the left of the knife and leave the game. Repeat.
It is not exactly equivalent, because people get to see each other’s bids from the previous round.