Every “Game Form” (which is like a “Game” in the Game Theory sense, except instead of having the results be utilities for each outcome, it just has labels for the outcomes) which is “straightforward” (meaning, for each participant, what utility that participant alone assigns to each outcome label (not depending on the utilities anyone else assigns) is sufficient to determine what choices they should make to maximize their expected utility) must be either a simple vote between exactly two options (as in, everyone votes for one of the two, and an outcome label happens if it gets enough votes), or dictatorial (a single person picks an outcome), or a probability mixture of games of that form (for example, pick a random voter’s ballot, and pick whoever they picked. Or, another example, pick a random pair of candidates, and hold a vote between the two).
That is to say : any deterministic voting system with more than 2 candidates, and where no voter is a dictator, is subject to tactical voting.
This is Gibbard's 1978 theorem.
Though, I wonder,
are there game forms which are not “straightforward” but which are a Pareto improvement (in the sense of “regardless of their assignment of utilities to the possible outcomes, every voter’s expected utility is at least as high as it would in the alternate option) over game forms that are straightforward?
I think probably yes?
If each voter puts their utilities, and then the system computes each voter’s utility under random ballot voting, and then checks whether any probability distribution over candidates would result in a strict Pareto improvement on the expected utilities of the voters, and if there is, pucks one, and if not, defaults to the distribution from “pick a randomly selected voter’s first choice”,
Well, we know that this wouldn’t be a straightforward game form, because it isn’t (equivalent to) the form required by Gibbard’s theorem,
So, strategic misrepresentation of one’s (normalized vNM) utilities must be sometimes useful in it (because if it were not, then one would only need to put the same utilities each time, making it a straightforward game),
but perhaps the degree of misrepresentation that could be useful would always be small enough that it would never select an option as being a Pareto improvement of the reported utilities from the random ballot method, unless it was also an actual Pareto improvement over it, using the true utility assignments?
Another thing that I think is relevant is the "revelation principle", which says that any "social choice function" that can be implemented by some mechanism, can be implemented by a truthful social choice mechanism.
But it seems clear that the thing that I proposed as being an improvement on random ballot can't count as a mechanism which implements a "social choice function", because there can't be an equivalent mechanism which is truthful (... well, I assume that no probability mix of pairwise votes and random ballot stuff is equivalent to the thing I described. Can't see how it could be.).
And I think the reason why is that the thing that I described (or, any fully specified version of it) would have to have multiple Nash equilibria.
So, for it to make sense to call it a Pareto improvement, I think would need to have that _all_ of the Nash equilibria would be Pareto improvements on the thing.
I am now less confident than I was that such an improvement exists.
That is to say : any deterministic voting system with more than 2 candidates, and where no voter is a dictator, is subject to tactical voting.
This is Gibbard's 1978 theorem.
Though, I wonder, are there game forms which are not “straightforward” but which are a Pareto improvement (in the sense of “regardless of their assignment of utilities to the possible outcomes, every voter’s expected utility is at least as high as it would in the alternate option) over game forms that are straightforward?
I think probably yes? If each voter puts their utilities, and then the system computes each voter’s utility under random ballot voting, and then checks whether any probability distribution over candidates would result in a strict Pareto improvement on the expected utilities of the voters, and if there is, pucks one, and if not, defaults to the distribution from “pick a randomly selected voter’s first choice”,
Well, we know that this wouldn’t be a straightforward game form, because it isn’t (equivalent to) the form required by Gibbard’s theorem, So, strategic misrepresentation of one’s (normalized vNM) utilities must be sometimes useful in it (because if it were not, then one would only need to put the same utilities each time, making it a straightforward game), but perhaps the degree of misrepresentation that could be useful would always be small enough that it would never select an option as being a Pareto improvement of the reported utilities from the random ballot method, unless it was also an actual Pareto improvement over it, using the true utility assignments?
But, I’m really not sure.