Honest equal-rank/truncation?
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
As an aside, "Random Approval" is the single-winner version of COWPEA Lottery. https://electowiki.org/wiki/COWPEA_Lottery
On Saturday, 18 June 2022, 14:29:44 BST, Kristofer Munsterhjelm km_elmet@t-online.de wrote:
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.
-km
Election-Methods mailing list - see https://electorama.com/em for list info
On 18.06.2022 15:43, Toby Pereira wrote:
As an aside, "Random Approval" is the single-winner version of COWPEA
Lottery. https://electowiki.org/wiki/COWPEA_Lottery
https://electowiki.org/wiki/COWPEA_Lottery
A possible way to determinize a lottery is to take the mode (the council
elected most often). That generally preserves monotonicity. Would it
preserve IIB and ULC?
I once thought about a semiproportional method that goes: pick a random
ballot, elect the candidate on it, eliminate this candidate from every
other ballot, and repeat. The determinized version is not PR, but it
gets closer the more seats there are. Possibly the determinized COWPEA
method also loses its proportionality and becomes only semiproportional,
but I haven't checked.
(I wonder if the determinization of a DAC/DSC version would be
Droop-proportional... something like "take random ballot,
round-eliminate the uneliminated candidate ranked last, repeat taking
random ballots until only one candidate remains, then elect and
eliminate (for good) this candidate, then restore the round-uneliminated
candidates and repeat".)
-km
If you determinize COWPEA it would lose independence of clones (cloned candidates would struggle to get elected), so it could give quite disproportional results because of that. Off the top of my head, not sure about IIB and ULC.
On Saturday, 18 June 2022, 17:33:40 BST, Kristofer Munsterhjelm km_elmet@t-online.de wrote:
On 18.06.2022 15:43, Toby Pereira wrote:
As an aside, "Random Approval" is the single-winner version of COWPEA
Lottery. https://electowiki.org/wiki/COWPEA_Lottery
https://electowiki.org/wiki/COWPEA_Lottery
A possible way to determinize a lottery is to take the mode (the council
elected most often). That generally preserves monotonicity. Would it
preserve IIB and ULC?
I once thought about a semiproportional method that goes: pick a random
ballot, elect the candidate on it, eliminate this candidate from every
other ballot, and repeat. The determinized version is not PR, but it
gets closer the more seats there are. Possibly the determinized COWPEA
method also loses its proportionality and becomes only semiproportional,
but I haven't checked.
(I wonder if the determinization of a DAC/DSC version would be
Droop-proportional... something like "take random ballot,
round-eliminate the uneliminated candidate ranked last, repeat taking
random ballots until only one candidate remains, then elect and
eliminate (for good) this candidate, then restore the round-uneliminated
candidates and repeat".)
-km
El sáb., 18 de jun. de 2022 6:29 a. m., Kristofer Munsterhjelm <
km_elmet@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
Election-Methods mailing list - see https://electorama.com/em for list
info
On 6/19/22 1:09 AM, Forest Simmons wrote:
El sáb., 18 de jun. de 2022 6:29 a. m., Kristofer Munsterhjelm
<km_elmet@t-online.de mailto:km_elmet@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?
I think I see your point. If your preference is A>C>>B and the other
guy's is B>C>>A, then if the other guy gets picked first, then a
favorite-only ballot of yours won't be counted because A intersect {B,C}
is empty.
So perhaps I was being too clever. What I was thinking of was that
equal-rank honestly (regardless of other votes) makes sense if:
- You have dichotomous preferences (Mike Ossipoff's u/a model),
- you have tiered preferences (e.g. these candidates are all excellent,
these candidates are all good, these candidates are all poor), or - you don't have time to find the exact ordering, e.g. your preferences
are something like A>B>>>>C>>>>>>>>D and you vote A=B>C>D.
I think just random favorite with equal rank would preserve this on the
top end (e.g. preserve equal-rank among the top candidates). That's the
method that picks a random candidate among a random voter's top-voted ones.
But the method is secondary. Are there other reasons to honestly
second-rank? If so, then the mechanism should be adapted to reveal those
as well.
-km
On 6/18/22 11:29 PM, Toby Pereira wrote:
If you determinize COWPEA it would lose independence of clones (cloned
candidates would struggle to get elected), so it could give quite
disproportional results because of that. Off the top of my head, not
sure about IIB and ULC.
Yeah, that's a good point. Random Ballot is cloneproof but its
determinization (Plurality) definitely isn't.
-km
Another interpretation of determinization of random ballot would be to
elect a weighted vote assembly ... possibly as a the beginning of a proxy
scheme ... or simple asset voting, which could preserve clone independence
if done properly.
Other than that, Martin Harperization of Equal Top Approval would be a
clone proof method preserving the spirit of random favorite.
Your plurality vote goes to the candidate that you ranked equal top having
equal top approval on the most ballots.
It can be thought of as a reasonable DSV Pluralty that satisfies clone
independence and monotonicity, ... because it is an equivalent formulation
of single winner approval ... as Martin Harper pointed out 20 years ago on
this EM list.
El dom., 19 de jun. de 2022 3:48 a. m., Kristofer Munsterhjelm <
km_elmet@t-online.de> escribió:
On 6/18/22 11:29 PM, Toby Pereira wrote:
If you determinize COWPEA it would lose independence of clones (cloned
candidates would struggle to get elected), so it could give quite
disproportional results because of that. Off the top of my head, not
sure about IIB and ULC.
Yeah, that's a good point. Random Ballot is cloneproof but its
determinization (Plurality) definitely isn't.
-km
Election-Methods mailing list - see https://electorama.com/em for list
info