[EM] Some thoughts on Condorcet and Burial

[Michael Ossipoff]

Chris--

No, the goal wasn't to achieve complete burial-proofness.

The goal is merely to achieve *probabilistic* deterrence.

On Fri, Nov 3, 2023 at 4:32 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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