The stability of Lonely Vickrey, or One's Choice of Neighbourhood
Why is the Vickrey auction "lonely" Ausubel…? In what space is it shown to be isolated?
The paper demonstrates that even one instance of collusion between players is game breaking. This suggests a space in which you can show the movement from "legal" Vickrey to "two cheaters" Vickrey, as a small perturbation; where the outcome of the game is a function on that space, and the image of the legal->2 cheaters movement is large. In this view, Vickrey isn't merely "lonely", but unstable.
Another way to think about it (older draft):
In the traditional cut-and-choose game one player cuts a cake, and the other player chooses the piece they want. To a mathematician, the iteration of this game is the ideal way to find "empty cake pieces". The faster the cutter is forced to cut an empty piece, the simpler a function is.
Our pie is the set of all possible outcomes of Vickrey auctions, including the ones that have players sharing information amongst themselves and coordinating strategies. It is difficult to isolate an MSW outcome in this pie, making Vickrey unstable. I would argue that there is a bigger pie, of which MSW is a yet smaller slice. It consists of all the possible ways players could choose to play sub-games in order to generate a coordinated strategy. There are likely more dominant sub-game strategies than dominant collusion strategies, and never less (because collusion is a special case of subgame).