The stability of Lonely Vickrey, or One's Choice of Neighbourhood

See Ausubel….

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.

We want to ask the question, how much do you have to cheat at a Vickrey auction to profit more than you otherwise would? Let's build it as follows. Our pie is the set of "Vickrey-seeming" auctions - nominally second-price sealed-bid auctions whether single-unit or multiunit; this means we include auctions that have some players sharing information amongst themselves and coordinating strategies, unbeknowns tto the oher players (and presumably the game-proprietor). Express the nature of these auctions, incuding the cheaters, and arrange them into a "pie" . How many cuts do you have to make before every game in the slice is incentive compatible? That's one way to characterise how unique the property is. I think this the original thrust of the Lonely Vickrey paper. Even a little rules violation pays off in multiunit Vickrey. But we can also ask, how big is the payoff for a certain level of cheating? In other words, we have a "distance" and a "force" in this space of similar games; can we write ourselves a "stability"? For each cut (i.e. each further narrowing of the game description) we can identify a subset of "players affected". Which way do their incentives point across that cut and how much is the difference in payoff between games in each slice? This is the "potential profit of conspiracy" for that cut. Now we can price cuts! Which means we have a metagame. It makes the difference between whether a given game is merely unique or actively being deserted some number of its participants. The primary question, now, is the specs of the pie. We are trying to make a choice of "game expression" and "nearby games" that accurately describes how players think about the game - what they have at least coalitional consensus about what the alternatives among games even are. This might be productively narrowed down by enforcing the protocol by which the game ensues, or we can as is traditional make educated guesses about the spec within well-chosen exemplars. Why is this interesting? Well, because cheating is interesting! In some cases, it's punished instantly; it's often tolerated to various degrees in large systems; it's often incorporated into the gameplay if it's considered benign to most other players; and sometimes it results in a perversion of the entire game to unaligned ends.