[EM] Some thoughts on Condorcet and Burial

[Forest Simmons]

Woodall has been a great EM pioneer .... and has done us a great favor by
making clear some of the limitations of the UD box, as well has showing us
by his Solid Coalitions methods some of the previously unknown
possibilities of working within that box.

I have suggested some of the "pockets of predictability" within the UD
domain that have not been fully exploited for burial resistant methods ...
at the same time coloring outside the UD lines as you have (eg with
approval cutoffs) to surpass some of the UD limitations.

I'm glad that you did not give up In helpless futility .... nut have
pragmatically pressed on.

In general, impossibility proofs are valuable by making clear the
assumptions that entail impossibility ... so we know which lines we have to
color outside of if we want to avoid dead endends.

The three candidate two stage fresh ballot runoff is one non UD tool tailor
made for finishing elections that surpass the boundaries of UD
possibilities.

Sometimes with manipulated ballot sets we can narrow down to two candidates
that form a satisfactory basis for a two candidate fresh ballot runoff.

But when you push the boundaries further, a two candidate runoff isn't good
enough.

But in the case of a three candidate Landau set ... the right kind of fresh
ballots runoff is adequate for recovering the buried candidate.

I doubt there has ever been (or ever will be) a real public election ballot
set with more than three uncovered candidates.

It is easy to show that when a CW is buried, it will still be a member of
(not only the Smith set), but also a member of Landau.

Smith may have a large clone set of "buses", but (with statistical
certainty) one of these buses  will cover all of the others.

These tools are outside the scope of Woodall ... who would surely
appreciate their usefulness were he still in the loop.

fws




On Fri, Nov 3, 2023, 1:32 PM C.Benham <cbenham at adam.com.au> 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
>> <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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