[EM] Honest equal-rank/truncation?

[Forest Simmons]

El sáb., 18 de jun. de 2022 6:29 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> A thought about how honest equal-rank might be defined. Earlier I've
> said that a good way to define a honest ballot is to find a randomized
> strategyproof system that induces it (e.g. Random Ballot for
> single-mark, Random Pair for strict ranked, possibly some transformation
> of Hay for VNM utility ballots).
>
> How about this as a starting point?
>
> "Random Approval": Voters provide Approval-style ballots. Choose a
> ballot at random. If this ballot approves a single candidate, then elect
> that candidate. Otherwise eliminate every non-approved candidate and
> draw another ballot (without replacement). Ignore ballots only approving
> eliminated candidates. If every ballot is visited, choose at random a
> candidate from the winning set.
>
> The optimal strategy seems to be to just designate your favorite,


Not neccessarily.

Suppose honest preferences are

x: A>C>>>B
y: B>C>>>A,

where x-y is the voter's subjective random variable with estmated mean near
zero and estimated standard deviation at about 2 percent of x+y.

If that by itself is not enough to make the voter approve C, what if less
than 51 percent approval for the winner required fallback from random
approval to random favorite?

In other words, suppose the method is to first figure out who the random
approval winner RAW is.

Then check to see if the RAW has more than 51 percent approval according to
the marked approval cutoffs on the ballots.

If so, then RAW has been ratified. Otherwise, elect random favorite (e.g.
from the set of ballots already drawn to determine the Random Approval
Winner).


for
> the same reason that it's optimal in Random Ballot. However, suppose
> you've got a limited amount of time available and two candidates are
> nearly equal. Then it might be worth it to equal-rank them (approve
> both) instead of taking the effort to determine which candidate is ever
> so slightly better than the other.
>
> So according to this interpretation, honest equal-rank is an indication
> that you don't know which of the candidates is better and/or it's not
> worth the chance of getting it wrong.
>
> This idea could presumably be extended to Random Pair with equal-rank.
> Suppose that d[A,B] is true if more people rank A over B than vice
> versa, i.e. it doesn't count equal-rankers at all (and if everybody
> equal-ranks A and B, set it to true at random) Then similarly, if you
> equal-rank A and B, you choose to let the other voters decide.
>
> Perhaps there is a model similar to a Condorcet jury where jurors who
> know that they don't know are better off equal-ranking two candidates
> than trying to force an outcome. E.g. the certainty of getting the
> comparison right is a function of time, you're time limited, and then
> equal-ranking reduces the variance compared to just guessing. But then
> again, if everybody did that, then the variance of a simple coin flip is
> worse than the combined noisy guesses, which suggests your best ballot
> depends on others', which isn't strategy-proof.
>
> The simpler version for Approval is just "elect an approved candidate at
> random". But that's harder to generalize to Random Pair.
>
> -km
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