There is a unique consistent voting system in cases where the system may return a stochastic distribution of candidates!
(where consistent means: grouping together populations that agree doesn’t change the result, and neither does duplicating candidates)
What is the rule? Take a symmetric zero-sum game where each player picks a candidate, and someone wins if their candidate is preferred by the majority to the other, winning more points if they are preferred by a larger majority. This game’s Nash equilibrium is the distribution.
Scott pointed out to me that the condorcet criterion makes more sense if we include stochastic outcomes. In the cases where the Condorcet winner is the utilitarian-worst candidate, a mixture of other candidates will win over the Condorcet winner. (So that candidate won’t really be the Condorcet winner, if we include stochastic outcomes as “candidates”.)
But that’s not what’s going on here, because this technique always selects a Condorcet winner, if there is one.
So (apparently) it’s not including stochastic outcomes in the right way.
We can do better by modifying the game:
We specify a symmetric two-player zero-sum game where each player selects a distribution over candidates. You score points based on how many more votes your proposed distribution would get against the other player’s. The game’s Nash equilibrium (a distribution over distribution over candidates) is the output distribution.
However, I’m a bit suspicious of this, since I didn’t especially like the basic proposal and this is the same thing one level up.
Since this is the unique voting system under some consistency conditions, I must not agree with some of the consistency conditions, although I’m not sure which ones I disagree with.
Curious what you think of Consistent Probabilistic Social Choice.
My summary:
There is a unique consistent voting system in cases where the system may return a stochastic distribution of candidates!
(where consistent means: grouping together populations that agree doesn’t change the result, and neither does duplicating candidates)
What is the rule? Take a symmetric zero-sum game where each player picks a candidate, and someone wins if their candidate is preferred by the majority to the other, winning more points if they are preferred by a larger majority. This game’s Nash equilibrium is the distribution.
OK, I basically don’t like the voting system.
Scott pointed out to me that the condorcet criterion makes more sense if we include stochastic outcomes. In the cases where the Condorcet winner is the utilitarian-worst candidate, a mixture of other candidates will win over the Condorcet winner. (So that candidate won’t really be the Condorcet winner, if we include stochastic outcomes as “candidates”.)
But that’s not what’s going on here, because this technique always selects a Condorcet winner, if there is one.
So (apparently) it’s not including stochastic outcomes in the right way.
We can do better by modifying the game:
We specify a symmetric two-player zero-sum game where each player selects a distribution over candidates. You score points based on how many more votes your proposed distribution would get against the other player’s. The game’s Nash equilibrium (a distribution over distribution over candidates) is the output distribution.
However, I’m a bit suspicious of this, since I didn’t especially like the basic proposal and this is the same thing one level up.
Since this is the unique voting system under some consistency conditions, I must not agree with some of the consistency conditions, although I’m not sure which ones I disagree with.
I agree that moving to distributions and scalar utility is a good way of avoiding Pareto suboptimal outcomes.
Sounds awesome!! I’ll need to evaluate it to get a better idea of what’s going on.
I’m not necessarily expecting it to be utilitarian-good or good in a bargaining sense, but still, it sounds really interesting.