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.
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.