My voting system works like this. Each voter expresses their preferences for all candidates on a real numbered utility scale.
Then a Maximal lottery takes place over all lotteries over candidates. https://en.wikipedia.org/wiki/Maximal_lotteries
Lets describe this in more detail. Suppose there are 3 candidates. A,B,C.
The set of candidates is
A probability distribution over candidates looks like
This probability distribution is in the set of all probability distributions over .
A probability distribution over probability distributions looks like
Though note that there are infinitely many distributions, so most distributions-of-distributions will be assigning probability densities.
Also note that we can sample a candidate from this distribution over distributions by first sampling a distribution, and then sampling a candidate from that distribution. This is equivalent to integrating a distribution-of-distributions into a distribution over candidates and then sampling that.
A distribution is equivalent to a point in a triangle. A distribution over distributions is a probability density over a triangle, ie a non-negative function over the triangle (may include dirac deltas)
So the voters all mark their preferences on a numerical scale.
Then these votes get sent to Fred and George, 2 perfectly rational players in a 0 sum game.
Fred and George both propose probability distributions over the candidates.
Fred's utility is the number of candidates that strictly prefer Fred's proposed probability distribution over Georges, minus the number of voters that strictly prefer Georges distribution over Freds.
This game has a unique Nash equilibrium. This equilibrium is a distribution over distributions. Sample a candidate from this equilibrium to get the election winner.
I know that this has a few nice properties. If candidate is the first choice of the majority, then definitely wins. If everyone prefers A to B, then B has no chance of winning. If C has no chance of winning, the candidates existence doesn't influence the election.
Is this system strategy proof, or can it be gamed? Will voters ever be incentivized to lie about their preferences?
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.