[EM] Manipulation Resistant Voting
Susan Simmons
suzerainsimmons at outlook.com
Sun Jul 18 16:01:47 PDT 2021
>Now suppose there's a typical Condorcet cycle A>B>C>A and the agenda
ordering is A>B>C; so the method proceeds by matching B and C, and then
matching the winner with A, and then the outcome of this is the final
winner.
Good point ... just concatenating binary choices together is not enough. It is important that the outcome of each binary decision be final, not just the last decision in the sequence.
Suppose, for example, that the method, at every stage, either (1) accepts the next remaining agenda item X as the final winner or (2) eliminates X and applies the method recursively to the remainder of the list?
Then every C supporter would vote sincerely to stick with C, and the A supporters would vote sincerely to eliminate C. The B>C supporters would sincerely vote to keep C, while the B>A supporters would sincerely vote to eliminate C.
So the winner would be A or C depending on whether or not the B>A faction was larger than the B>C faction.
So given the agenda, and the special finality rule (once an option is chosen the winner must be a member of that option) all of the rational choices will be sincere in the sense that they are the choices whose rational outcomes are preferred.
.....
>In reference to an earlier post of mine, suppose we define honesty as
what a Random Ballot type method would return. In your method, this
would be a trace down the tree that ends at the candidate who the voter
prefers most of the candidates at the bottom level of the tree.
But a voter is not at liberty to choose an entire trace. You probably mean that as long as Favorite is a descendant of the current node, if you do not choose the branch that leads to Favorite, then your vote is insincere.
If that's your definition of "sincere" in this context, then you are right ... sincere strategy is not rational.
But it seems to me that when presented with only two choices, branchA or branchB, leading almost surely to final wins for X and Y, respectively, then voting for branchA should not be impunged as "insincere strategy," if you truly prefer X over Y.
It's a matter of definition. So perhaps we could distinguish between "naive sincerity" and "rational sincerity."
>Now suppose say, that the tree is
A
/
B C
/ /
D E F G
Technical point: in this tree the only leaves are D, E, F, & G, so A, B, & C can only be decision nodes, not candidates.
>There are some voters whose honest vote is
[more precisely, honest preferences are given by...]
G>C>A>B>D>E>F,
so their honest
[most hoped for trace? .. a trace is not one of their single choices, so can be neither honest nor dishonest]
trace would go down the C branch and end at G. However, they know
that if the C branch is chosen, then a vast majority will go for F. Thus
they falsify their ballots to indicate a preference for the B branch
But if they truly prefer the sure rational outcome of the B branch over the sure rational outcome of the C branch, why do you say they "falsify their ballots" by choosing the B branch, the one that leads to their preferred of the only two rationally certain final outcomes (conditioned on the outcome of the root node decision A).
So one can say naive sincerity is violated by this choose-a-branch method, but (it seems to me) realistic, rational, strategic sincerity that accepts the premise of the method is not violated.
The "premise" is that when you choose a branch, you are showing a preference for the likely winner of that branch over the likely winner of the other branch ... NOT implying support for any other preference. So it cannot be dishonest or "falsification" unless you actually prefer the likely winner of the other branch over the one you chose.
I think this notion of sincerity gives us some traction ... the other one is too stringent to lead to a manipulation proof Condorcet method.
Thanks!
Sent from my MetroPCS 4G LTE Android Device
-------- Original message --------
From: Kristofer Munsterhjelm <km_elmet at t-online.de>
Date: 7/18/21 1:43 AM (GMT-08:00)
To: Susan Simmons <suzerainsimmons at outlook.com>, election-methods at lists.electorama.com
Subject: Re: [EM] Manipulation Resistant Voting
On 7/17/21 11:12 PM, Susan Simmons wrote:
> I'm sure that Gibbard considers clone dependence a vulnerability to
> manipulation. In this case, that vulnerability is limited to the
> possibility of lying on the questionaires eliciting the information for
> the construction of the decision tree. If the decision tree is generated
> randomly, then we are leaving determinism behind.
IIRC, it doesn't consider clone dependence to be a vulnerability to
strategy; the alternatives are given and then the voters either submit a
ranking (G-S) or interact with the method in one way or another
(Gibbard's later theorem).
> The Gibbard–Satterthwaite theorem states roughly that every
> deterministic voting rule is manipulable, except possibly in two cases:
> if there is a distinguished voter who has a dictatorial power, or if the
> rule limits the possible outcomes to two options only.
>
> Once the decision tree has been constructed all decisions are of this
> binary type.
>
> It may be true that Gibbard-Satterthwaite applies more or less to any
> deterministic democratic process that could be used to generate a
> decision tree.
Gibbard-Satterthwaite probably wouldn't, because the tree construction
would fail universal domain. However, there is a more general theorem
that doesn't require universal domain:
https://en.wikipedia.org/wiki/Gibbard%27s_theorem
And I think not just the tree composition, but the method as a whole
would fail this.
Consider a mix of exhaustive runoff and agenda, where the agenda is set
beforehand, and then there are (n-1) one-on-one elections after which
the candidate standing at the end is the winner.
Now suppose there's a typical Condorcet cycle A>B>C>A and the agenda
ordering is A>B>C; so the method proceeds by matching B and C, and then
matching the winner with A, and then the outcome of this is the final
winner.
Under honesty, B beats C pairwise and then A beats B pairwise. But the
voters whose honest ranking is B>C>A have an incentive to misreport
their B vs C preference in the first round so that C beats B pairwise
and then goes on to beat A pairwise. Even though every round is a duple
(only one of two outcomes are possible), this is not true for the
composite rule, and so it fails Gibbard.
I would suspect your method also is vulnerable to strategy for the same
reason.
In reference to an earlier post of mine, suppose we define honesty as
what a Random Ballot type method would return. In your method, this
would be a trace down the tree that ends at the candidate who the voter
prefers most of the candidates at the bottom level of the tree. Now
suppose say, that the tree is
A
/ \
B C
/ \ / \
D E F G
There are some voters whose honest vote is G>C>A>B>D>E>F, so their
honest trace would go down the C branch and end at G. However, they know
that if the C branch is chosen, then a vast majority will go for F. Thus
they falsify their ballots to indicate a preference for the B branch
instead (a compromising strategy) because they prefer both D and E to F.
> Suppose as a first approximation we ask the candidates themselves to
> estimate the distances between the various candidates on the various
> issues. They might be tempted to distort the truth to influence the
> structure of the decision tree to their advantage.
>
> Not to worry ...instead of taking their estimates at face value by
> averaging them together ... suppose we pay more attention to how similar
> they are in their responses.
>
> Then it doesn't matter so much if they try to distort the truth to their
> advantage, the closer they are to each other in candidate space, the
> more similar their responses, whether sincere or feigned.
I imagine you could end up with "false issues" that don't really matter,
but that the candidates agree to play up. Imagine a country that admits
about a hundred immigrants a year. Both major parties are seeking to
weaken the rule of law, but to distinguish themselves, they make a big
fuss about whether even fewer immigrants should be admitted to the
country or not - even though the number, as is, is insignificant in the
grand scheme of things.
It would be much harder to do so in a ranked voting setup because of the
(relatively) easier entry by newcomers. However, it's possible that
everybody will have an incentive to accept an extra dimension of
differentiation, for the purpose of strategically engineering the tree
structure.
> So the distance function is not based per se on their distance
> estimates, but rather on the distance between their patterns of answers.
>
> Their answer patterns are recorded as arrays of numbers called "data
> vectors." There are many possible "norms" for measuring the "magnitude"
> of a data vector.
>
> The distance between two candidates is taken to be the magnitude of the
> difference between their pattern-of-response vectors.
>
> This is the kind of psychometrics used by the NSA and other data miners
> to map out your precise location in political/consumer space without
> even listening in on your phone conversations.
>
> The TV series Bull is based on this kind of psychometric analysis of
> jurors for finding "mirror jurors" that are very close in psychological
> distance from the actual jurors. Bull is a trial scientist who can
> predict the reaction of the actual jurors on the basis of the respective
> mirror juror reactions.
>
> Do you find this to be interesting?
Possibly, but if we're going in a jury direction, I think a method like
sortition or even a mixed process like the Venetian Doges' would be more
appropriate. Maybe all this finding flaws in methods have turned me into
a "nah, that won't work" kind of guy :-)
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