Epistemic status: I don’t actually understand what strategic voting means, caveat lector
Suppose we have three voters, one who prefers A>B>C, another B>C>A, the third C>A>B. And suppose our preferences are that the middle one is 0.8 (on a scale where the top one is 1 and bottom 0).
Fred and George will be playing a mixed Nash equilibrium where they randomize between catering to A, B, or C preferrers—treating them as a black box the result will be 1⁄3 chance of each, all voters get utility 0.59.
But suppose I’m the person with A>B>C, and I can predict how the other people will vote. Should I change my vote to get a better result? What happens if I vote B>C>A, putting my own favorite candidate at the bottom of the list? Well, now the Nash equilibrium for Fred and George is 100% B, because 2 the C preferrer is outvoted, and I’ll get utility 0.8, so I should vote strategically.
Your right. This is a situation where strategic voting is effective.
I think your example breaks any sane voting system.
I wonder if this can be semi-rescued in the limit of a large number of voters each having an infinitesimal influence?
Edit: No it can’t. Imagine a multitude of voters. As the situation slides from 1⁄3 on each to 2⁄3 on BCA, there must be some point at which the utility for an ABC voter increases along this transition.
Yeah, “3 parties with cyclic preferences” is like the aqua regia of voting systems. Unfortunately I think it means you have to replace the easy question of “is it strategy-proof” with a hard question like “on some reasonable distribution of preferences, how much strategy does it encourage?”
Yep. And I’m seeing how many of the traditional election assumptions I need to break in order to make it work.
I got independence of irrelevant alternatives by ditching determinism and using utility scales not orderings. (If a candidate has no chance of winning, their presence doesn’t effect the election)
What if those preferences were expressed on a monetary scale and the election could also move money between voters in complicated ways?
The part that I don’t quite follow is about the structure of the Nash equilibrium in the base setup. Is it necessarily the case that at-equilibrium strategies give every voter equal utility?
The mixed strategy at equilibrium seems pretty complicated to me, because e.g. randomly choosing one of 100%A / 100%B / 100%C is defeated by something like 1/6A 5/6B. And I don’t have a good way of naming the actual equilibrium. But maybe we can find a lottery that defeats any strategy that priveliges some of the voters.
Yeah, I’m not actually sure about the equilibrium either. I just noticed that not privileging any voters (i.e. the pure strategy of 1⁄3,1/3,1/3) got beaten by pandering, and by symmetry there’s going to be at least a three-part mixed Nash equilibrium—if you play 1/6A 5/6B, I can beat that with 1/6B 5/6C, which you can then respond to with 1/6C 5/6A, etc.
Epistemic status: I don’t actually understand what strategic voting means, caveat lector
Suppose we have three voters, one who prefers A>B>C, another B>C>A, the third C>A>B. And suppose our preferences are that the middle one is 0.8 (on a scale where the top one is 1 and bottom 0).
Fred and George will be playing a mixed Nash equilibrium where they randomize between catering to A, B, or C preferrers—treating them as a black box the result will be 1⁄3 chance of each, all voters get utility 0.59.
But suppose I’m the person with A>B>C, and I can predict how the other people will vote. Should I change my vote to get a better result? What happens if I vote B>C>A, putting my own favorite candidate at the bottom of the list? Well, now the Nash equilibrium for Fred and George is 100% B, because 2 the C preferrer is outvoted, and I’ll get utility 0.8, so I should vote strategically.
Your right. This is a situation where strategic voting is effective.
I think your example breaks any sane voting system.
I wonder if this can be semi-rescued in the limit of a large number of voters each having an infinitesimal influence?
Edit: No it can’t. Imagine a multitude of voters. As the situation slides from 1⁄3 on each to 2⁄3 on BCA, there must be some point at which the utility for an ABC voter increases along this transition.
Yeah, “3 parties with cyclic preferences” is like the aqua regia of voting systems. Unfortunately I think it means you have to replace the easy question of “is it strategy-proof” with a hard question like “on some reasonable distribution of preferences, how much strategy does it encourage?”
Yep. And I’m seeing how many of the traditional election assumptions I need to break in order to make it work.
I got independence of irrelevant alternatives by ditching determinism and using utility scales not orderings. (If a candidate has no chance of winning, their presence doesn’t effect the election)
What if those preferences were expressed on a monetary scale and the election could also move money between voters in complicated ways?
The part that I don’t quite follow is about the structure of the Nash equilibrium in the base setup. Is it necessarily the case that at-equilibrium strategies give every voter equal utility?
The mixed strategy at equilibrium seems pretty complicated to me, because e.g. randomly choosing one of 100%A / 100%B / 100%C is defeated by something like 1/6A 5/6B. And I don’t have a good way of naming the actual equilibrium. But maybe we can find a lottery that defeats any strategy that priveliges some of the voters.
Yeah, I’m not actually sure about the equilibrium either. I just noticed that not privileging any voters (i.e. the pure strategy of 1⁄3,1/3,1/3) got beaten by pandering, and by symmetry there’s going to be at least a three-part mixed Nash equilibrium—if you play 1/6A 5/6B, I can beat that with 1/6B 5/6C, which you can then respond to with 1/6C 5/6A, etc.