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Xachariah comments on Game Theory As A Dark Art - Less Wrong
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All of them are defeated by the opposition coordinating and precommitment.
In democrats v republicans you can also do it without cross-aisle coordination. Whoever had a majority could just say unilaterally that 0 members will vote for the bill and ensure that it stays dead. The plutocrat would have no incentive to ever offer the deal in the first place, because it would always fail and he'd end up wasting a huge amount of money.
The family business could instead vote 4-to-1 to change the rules themselves. I mean, if the board is allowed to pass motions that divvy up 51% of other people's shares, they could just pass a motion to take that money back anyhow. Or, since they're a family they could just coordinate to vote 3-2 and split the money, gain back majority, and re-vote the two board members back.
As MBlume noted coordination obliterates the pay-all auction (or all auction systems, really). And as you note, a strong solo strategy wins even without coordination.
Finally, all of the pirates can get greater shares via precommitment. There's a reason humans evolved rejection to the ultimatum game, afterall.
Trustworthy coordination obliterates the pay-all auction. Anyone in the room can still defect and bid $2, and the question is whether you let them get away with it, or whether you commit to chasing them in a destructive bidding war.
The group either coordinated or it didn't. Trustworthy only comes into play when you're looking at a human participant's expectations, not the actions they're taking.
Also, 'letting them get away with it' isn't necessarily a binary question. Unless you will literally never see them again, you punish them outside the bidding system, and they know you will punish them outside the bidding system, so they do not defect unless they don't trust your willingness or ability to punish them sufficiently.
And this was a group of housemates, so...
I need to gather a group of complete strangers and auction a dollar to them. Maybe in a waiting room somewhere. Are there any laws against holding auctions in waiting rooms or long lines at the sandwich shop?