[EM] Honest equal-rank/truncation?

[Kristofer Munsterhjelm]

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, 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