True. But then, what does anyone have to gain from duplicity? And #5's offer can be made to #2 and #3 instead. "including me in the triplet that says yes and it's worth an extra 50 for each of you. Still a lot better than me ending up with zero."
So.. #1 makes a bid and it seems clear that he can get away with 720 exactly. Any more and he's voted down, gets 720 and a bullet, and the voting resumes.
Some vote yes and get a share of the rest and the rest.. nothing? But if the yes-votes ask for more than 720, one of the nays can say "I'll do it for less." So if yes-votes are auctioned, then it would seem that asking for more than 720 won't fly.
Guess it comes down to information and timing.
Eldest: "everyone, write on a piece of paper the lowest figure for your own share that you accept to say "yes". I'll pay the lowest three figures of the loot, the three greediest get nothing, and I'll keep the rest of the loot, if any. Commit that you follow through with a yes-vote, and I commit to honor that division if it is possible, as long as I get at least 720. If the sum of the three lowest and my own share is over 5040, just shoot me and good luck for the next guy to make the attempt. If there's a tie, I'll choose randomly among those with the same but feasible request."
Is that the same or different problem? There is no information disparity here, as all get the same data at the same time. Since situation is similar to all, it makes sense they all write same number on the paper. They can all write 1,440 for a 50% chance of getting that and a 50% chance of getting nothing for an expected value of 720. But knowing that, it's tempting to write 1,439 for a 100% chance to get that, or any value in the 721-1439 range, as long as it is just barely in the bottom three.
Asking for 720 might actually be the best strategy.
Or. "We all know that we can do this 0-1680-1679-1678-2-1-0. Let's agree that no one gets the same share of the loot, that's nickles and dimes anyhow. And then we go around, and the one with the least share can propose a new way to divide the share of the one with the most loot, increasing his share and mine, as long as all the numbers differ. When the one with the least share refuses to do that, that's how we split the loot."
Even that would diverge to roughly 720 each.