[EM] Some thoughts on Condorcet and Burial
C.Benham
cbenham at adam.com.au
Tue Nov 7 20:07:57 PST 2023
In my last EM post I included Smith//DAC in a list of Condorcet methods
"that meets mono-raise".
A knowledgeable correspondent has cast doubt on this claim, and I admit
that I can't prove that it does.
But I am fairly sure that any failure example needs there to me more
than three candidates in the top cycle,
and if I'm right about that then I'm not concerned enough to scratch it
(at least) as a quite burial-resistant curiosity
that is far less absurd than "elect the member of the Smith set that is
voted the least desirable".
However it may be too difficult to sell and explain for it to be a
practical proposal for elections to public office.
So leaving that method idea aside for the time being, I have become more
convinced that Condorcet methods should
(if they are to compete convincingly with IRV/RCV) use ballots that
allow voters to enter an explicit approval cutoff, with the
default placement being just below the candidate/s ranked below no others.
My favourite in this category is Margins-Sorted Approval. It is the
most elegant, as monotonic as any Condorcet method can
be, and its explanation and counting has no need of any concept of
cycles or "Smith sets".
And like all acceptable methods that use that type of ballot (like
Smith//Approval) it meets "Double Defeat" ( which says that
a candidate that is pairwise beaten by a more approved candidate can't win).
This means that if the voters who prefer a purported sincere CW to a
likely Burier's favourite wish to protect the purported
sincere CW from that burial, they can do so (if the purported sincere CW
really does pairwise beat the likely BF) by simply approving
the purported sincere CW and not approving the likely BF.
No need for anyone to truncate or insincerely equal-rank.
If we want still more Burial resistance, and are prepared to put up with
something much less monotonic and more clunky,
I suggest "Double Defeat, Benham".
*Voters use ranked ballots with an explicit approval cutoff, default
placement just below first place. Above-bottom equal ranking
not allowed (otherwise Pushover strategising becomes easier and less risky).
Elect the CW if there is one. If not disqualify the Doubly Defeated
candidates. If more than one remains qualified then elect the
qualified candidate that pairwise beats all the others if there is one.
Otherwise proceed with IRV-style eliminations until one
qualified candidate pairwise beats any others remaining, and elect that
candidate.*
(That could probably put more clearly and succinctly).
So this method meets both Double Defeat and Unburiable Mutual Dominant
Third. I doubt that an acceptable Condorcet method
can get more Burial resistant than that.
Chris B.
On 4/11/2023 7:02 am, C.Benham wrote:
>
> Forest,
>
> Douglas Woodall in a 1996 article demonstrated that Condorcet is
> incompatible with Later-no-Help.
>
> By simple logic we know that any method that fails Later-no-Help must
> be vulnerable to Burial strategy.
>
> Therefore your quest to invent a completely burial-proof Condorcet
> method is futile.
>
> https://www.sciencedirect.com/science/article/pii/S0166218X9600100X
>
>> Theorem 2. (a) Even if truncated preference listings are not allowed,
>> CONDORCET is incompatible with
>> PARTICIPATION, MONO-RAISE-RANDOM and MONO-SUB-TOP.
>>
>> (b) In general, CONDORCET is incompatible with LATER-NO-HELP,
>> LATER-NO-HARM, MONO-RAISE-DELETE, MONO-SUB-PLUMP and,
>> in the presence of PLURALITY, MONO-ADD-TOP.
>>
>> (c) There is no election rule that sutisfies LATER-NO-HELP and
>> LATER-NO-HARM,
>> and that also satisfies CONDORCET whenever there are no truncated
>> preference listings.
>>
>> Election 3. (1 seat)
>> 3 A>B>C
>> 3 B>C>A
>> 3 C>A>B
>> 2 A>C>B
>> 2 B>A>C
>> 2 C>B>A
>>
>> Consider Election 3. By symmetry, the result must be a 3-way tie;
>> but, by the axiom of discrimination, there must be a profile P
>> arbitrarily close to this (in the proportions of ballots of each
>> type) that does not yield a tie. Without loss of generality, suppose
>> a is elected in P.
>>
>> But c becomes the Condorcet winner, and so must be elected by
>> CONDORCET, ...if all the abc ballots are replaced by a (contrary to
>> LATER-NO-HELP),
>> or if all the bac ballots are replaced by a (contrary to
>> MONO-RAISE-DELETE and MONO-SUB-PLUMP), or if all the abc ballots are
>> replaced by acb
>> (contrary to LATER-NO-HELP and LATER-NO-HARM together).
>>
>> This proves (a), (c) and three parts of (b).
>
> The three relatively burial-resistant Condorcet methods that meet
> mono-raise and don't ask or allow voters to enter an explicit approval
> cutoff that I like are
> Smith//Approval (ranking), Margins-Sorted Approval (ranking), and
>
> Smith//Descending Acquiescing Coalitions.
>
> Not allowing voters to rank among unapproved candidates increase the
> risk for Burial strategists that they will simply elect their "bus".
>
> By itself DAC (one of Woodall's inventions) meets Later-no-Help and
> Participation (both incompatible with Condorcet) but can behave very
> oddly and badly in the presence
> of one or two weak should-be-irrelevant candidates (which is why I'm
> wary of Smith,DAC).
>
> So defining it thus: Voters rank from the top, equal ranking and
> truncation is fine. Eliminate and drop from the ballots all the
> candidates not in
> the Smith set.
>
> Ballots "acquiesce" to a candidate or set or subset of candidates (a
> "coalition") if they vote no other (outside the set or subset)
> candidate strictly above any of them.
>
> Number all the possible coalitions according to how may ballots
> acquiesce to them. Start with the highest-numbered and disqualify all
> the candidate not in it.
> Proceed to the next-highest numbered that contains any not-yet
> disqualified candidates and disqualify those not in it, and so on
> until one candidate is left undisqualified.
>
> 46 A
> 44 B>C (sincere might be B or B>A)
> 10 C
>
> In this example all candidates are in the Smith set. The acquiescing
> coalitions are AC 56, BC 54, AB 46, A 46, B 44, C 10.
>
> AC disqualifies B, BC disqualifies A so C wins.
>
> That would be great for fans of the Minimal Defense criterion, but
> would be sad if the B>C voters are sincere and the C voters are
> truncating against B.
>
> Chris B.
>
>
> On 4/11/2023 12:05 am, Forest Simmons wrote:
>>
>>
>> On Mon, Oct 30, 2023, 6:21 PM C.Benham <cbenham at adam.com.au> wrote:
>>
>>
>> Why do we support the Condorcet criterion? For me there are
>> three reasons:
>>
>> (1) Failure to elect a voted CW can give the voters who voted the
>> CW over the actual winner
>> a potentially very strong, difficult (if not impossible ) to
>> answer complaint.
>>
>> And those voters could be more than half the total.
>>
>> (2) Always electing a voted CW is (among methods that fail
>> Favorite Betrayal) is the best way to minimise
>> Compromise incentive.
>>
>> (3) Limited to the information we can glean for pure ranked
>> ballots (especially if we decide to only refer
>> to the pairwise matrix), the voted CW is the most likely utility
>> maximiser.
>>
>> If there is no voted CW , then the winner should come from the
>> Smith set. Condorcet is just the logical
>> consequence of Smith and Clone Independence (specifically
>> Clone-Winner).
>>
>> Some methods are able to meet Condorcet but not Smith, but
>> hopefully they get something in return.
>> (For example I think Min Max Margins gets Mono-add-Top and maybe
>> something else).
>>
>> So coming to the question of which individual member of the Smith
>> set should we elect, I don't see that a
>> supposed, guessed-at "sincere CW" has an especially strong claim,
>> certainly nothing compared to an actual
>> voted CW.
>>
>> Suppose sincere looks like:
>>
>> 49 A>>>C>B
>> 48 B>>>C>A
>> 03 C>A>>>B
>>
>> My favorite burial proof method is to elect the nemesis of the
>> nemesis of the (repeated) Submidway Approval Elimination winner.
>>
>> Max Approval is 52
>> Min Approval is 3
>> Midway is 27.5
>>
>> So we eliminate C.
>>
>> Updating approvals we have 52 for A and 48 for B. Midway is 50. B is
>> the only Sub Midway candidate so A is the SubMidway Approval winner.
>>
>> The nemesis of A is C, and C has no nemesis because it not
>> pairbeaten. We respect the Condorcet Criterion and elect C, a very
>> weak CW.
>>
>> There is no way out of it if we want a Condorcet Criterion Compliant
>> method.
>>
>>
>>
>> Suppose that all voters get about the same utility from electing
>> their favourites. In that case A is the big utility
>> maximiser.
>>
>> Now suppose that this is say the first post-FPP election, and the
>> voters are all exhorted to express their full
>> rankings, no matter how weak or uncertain some of their
>> preferences may be, because we don't want anything
>> that looks like the (shudder) "minority rule" we had under FPP.
>>
>>
>> This reminds me of the practice omanipulative judges in the US
>> warning jurors to ignore their sacred right of "Jury Nullification"
>> (inherited from English Common Law).
>>
>> Those lying (by intimidation) judges are a disgrace to their office
>> ... and should be stripped of their holy robes ... imho.
>>
>>
>> So they vote:
>>
>> 49 A>C
>> 48 B>C
>> 03 C>A
>>
>> C is the voted CW. For some pro-Condorcet zealots, this is ideal.
>> No sincere preferences were reversed or
>> "concealed", resulting in the election of the "sincere CW".
>>
>> (In passing I note that in most places if the non-Condorcet
>> method IRV/RCV were used, A would be uncontroversially
>> elected probably without anyone even noticing that C is the CW.)
>>
>> Backing up a bit, suppose that instead of the voters being
>> exhorted to fully rank no-matter-what, they are given the
>> message "this election is for a serious powerful office, so we
>> don't want anything like GIGO ("garbage in, garbage out")
>> so if some of your preferences are weak or uncertain it is quite
>> ok to keep them to yourself via truncation or equal-ranking."
>>
>>
>> That warning should be mandatory ... as should an honest effort to
>> "fully inform" a jury of their most sacred rights predatimg even the
>> Magna Carta.
>>
>>
>> So they vote:
>>
>> 49 A
>> 48 B
>> 03 C>A
>>
>> Now the voted CW is A. Should anyone be seriously concerned
>> that, due to so many voters truncating, that some other
>> candidate might actually be the "sincere CW"?
>>
>>
>> Your example shows the wisdom of electing the ballot CW when there is
>> one.
>>
>> In fact, all of our burial resistant methods elect the voted CW when
>> one exists. But when one does not exist we suspiciously suspect that
>> its absence was most likely caused by subversion of a CW, since that
>> is by far the easiest way to create a beat cycle ... whether
>> innocently or with "malice aforethought."
>>
>>
>> For me, if voters have the freedom to fully rank but for whatever
>> reason choose to truncate (and/or equal-rank, assuming that
>> is allowed) a lot of that is fine and the voting method should
>> prefer not to know about weak and uncertain preferences.
>>
>> The type of insincere voting that most concerns me is that which
>> produces outrageous failure of Later-no-Help, achieving by
>> order-reversal
>> Burial what could not have been done by simple truncation.
>>
>> 46 A
>> 44 B>C (sincere is B or B>A)
>> 10 C
>>
>>
>> Let's see what our Sub Midway Approval Elimination burial resistant
>> method does here:
>>
>> The respective max and min implicit approval scores are 54 for C and
>> 44 for B. So Midway is 49. Both A and B have Sub Midway Approval
>> leaving C as the candidate whose nemesis's nemesis we should elect.
>>
>> C's nemesis is B, and B's nemesis is A.
>>
>> So A is the winner of our burial resistant method.
>>
>> How does this work?
>>
>> Well, Nanson is very good at fingering the Burial Faction Favorite,
>> but Poor man's Nanson aka SubMidApproval Elimination is even better
>> at electing the BFF than ordinary Nanson or RP wv, although they are
>> both pretty good at it as this example shows.
>>
>> Once you have the likely BFF you can follow the cycle forward one or
>> backward two candidates to the "bus" under which the buried candidate
>> was buried.
>>
>> [Electing this "bus" is what makes the method backfire on the burying
>> faction ... so electing the strongest bus (their could be a clone set
>> of them) is the proper aim of a good burial proof method designer ...
>> to completely deter potential subverters from even flirting with the
>> idea.]
>>
>> Like I said you can go in either direction around the cycle to get to
>> a "bus" from the BFF candidate .... but it's better psychologically
>> to elect the syrongest wv defeater of the strongest wv defeater of
>> the Nanson winner .... as opposed to saying "Elect the candidate with
>> the most losing votes against the Nanson winner" .. they are almost
>> always the same candidate even when Smith has more than three members.
>>
>> Like I say, SubMidAolroval Elimination is more reliable than even
>> ordinary Nanson or RP wv at finding the BFF candidate. But for the
>> curjous, you can always do a sincere runoff between the likely
>> strongest Bus (the nemesis of the nemesis of the likely BFF), and the
>> Smith candidate with the fewest losing votes against it (i.e.against
>> the bus) the same as the weakest Smith weakest Smith candidate to
>> defeat the BFF.
>>
>> [It is easy to show that a subverted CW always defeats the BFF, which
>> beats the bus that beat the buried candidate that precipitated the
>> insincere cycle resulting from the burial.]
>>
>> In this example the sincere runoff (were that option to be taken)
>> would be between the Bus A and the Smith candidate B with the fewest
>> losing votes to it A.
>>
>> For the sincere winner of this runoff we have to go clear back to the
>> original scenario. And it turns out that A is the sincere winner of
>> the runoff, because C, the sincere CW was acting more like a BFF ...
>> so not a finalist in the runoff!
>>
>>
>> Electing B here is completely unacceptable.
>>
>>
>> Right, and now we have a method designed to automatically avoid that
>> kind of mistake.
>>
>> Regardless of whether or not the B>C voters are sincere, there
>> isn't any case that B has a stronger
>> claim than A.
>>
>> I don't like (but it can sometimes be justified) a larger faction
>> being stung by a successful truncation Defection strategy of a
>> smaller one, but apart
>> from that I consider a lot of truncation to be normal, natural
>> and mostly desirable.
>>
>> More later.
>>
>> Chris Benham
>>
>>
>>
>>
>>
>>> *Forest Simmons*forest.simmons21 at gmail.com
>>> <mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Benefit%20of%20a%20doubt%20runoff%20challenge&In-Reply-To=%3CCANUDvfru_xs%2BEE6kd7Xbb4p%2Bsh3Zijqy-yCmBwNPOdwLP1emgQ%40mail.gmail.com%3E>
>>> /Sun Oct 29 21:30:58 PDT 2023/
>>> ------------------------------------------------------------------------
>>> Are the beatcycles that sometimes arise from expressed ballot preferences
>>> ... are these cycles more likely to arise from occasional inevitable
>>> inconsistencies inherent in sincerely voted ballots? ... or from ballots
>>> that reflect exaggerated preferences from attempts to improve the election
>>> outcome over the one likely to result from honest, unexagerated ballots (?)
>>>
>>> Should Condorcet methods be designed on the assumption that most ballot
>>> cycles are sincere? .... or on the assumption that most are the result of
>>> insincere ballots (?)
>>>
>>> Some people think that the question is irrelevant ... that no matter the
>>> answer, the best result will be obtained by assuming the sincerity of the
>>> voted ballots. Others think healthy skepticism is necessary for optimal
>>> results. What do you think?
>>
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