[EM] round robin tournaments

[Jameson Quinn]

I can sympathize that win strength should be (strength of
victory)*(importance of race). But if you're going with that philosophy,
then "strength of victory" should not be percentage margin (W-L)/(W+L), but
rather statistical confidence (W-L)/sqrt(W+L). The sqrt(W+L) divisor is
proportional to the standard deviation of the margin. Multiply that
"statistical confidence" by the "importance" factor W+L, and you get

(W-L)*sqrt(W+L) = sqrt((W-L) * (W²-L²))

Since ordering is all that matters here, and the sqrt function is monotonic
over this domain, you can remove the sqrt:

(W-L) * (W²-L²)

Or play with the formula to keep the ordering but make the units come out as
votes:

sqrt( (W-L) * sqrt(W²-L²) )

I understand the attractiveness of the much simpler margins formula W-L. But
to argue that that formula is actually a fairly-weighted product of two
factors seems like a post-facto rationalization to me.

As for the [35 A>B, 25 B, 40 C] example and the (Margins, smith/max win)
suggestion: that doesn't really solve the problem, because it gives the
adjusted scenario [30 A>B, 25 B, 45 C] to C.

2011/6/27 robert bristow-johnson <rbj at audioimagination.com>

>
> On Jun 24, 2011, at 9:44 PM, Kevin Venzke wrote:
>
>  Hi Robert,
>>
>
> hi Kevin,
>
>
>
>> --- En date de : Ven 24.6.11, robert bristow-johnson <
>> rbj at audioimagination.com> a écrit :
>>
>>> my spin on why Margins makes the most sense is:
>>>
>>>   2.  the measure of importance of an
>>> election is proportional the number of voters participating
>>> in it.  if very few people weigh in on an election, it
>>> must not be very important.
>>>
>>
>> Knowing the larger of X and Y tells you more about X+Y than knowing X-Y.
>>
>
> but, statistically, both max(X,Y) and |X-Y| have mean pretty much
> proportional to X+Y, if X and Y have identical p.d.f. form, but different
> means.
>
>
>  Let's try an example. Suppose the counts are 4 and 3. If you know the
>> 4, you know that the number of participants was from 4 to 8. If you know
>> the difference, 1, all you know is that there was at least one
>> participant.
>>
>
> true, but razor-thin, close elections with a lot of voters doesn't indicate
> a strong popular case pro or con.  since (at least i believe) random lots
> should be used only for absolute ties where the popular support of both
> candidates is indistinguishable, then based on counting all the votes very
> carefully and verifying, if the number of votes for Candidate A exceeds
> those for Candidate B but only 1 vote, then a decision must be made and it
> must be for Candidate A.  but i still don't get how you are concluding that
> the *net* electoral sentiment for Candidate A is decisive (considering the
> alternative), particularly if a million votes are cast, if Candidate A gets
> only one vote more than B.
>
> Winning Votes has *no* salience attached to how close or decisive an
> election is.  i just cannot get past that.  a measure of how decisive an
> election is, is a factor and how many voters weigh in on the election is a
> factor also.  Margins represents the product of those two factors in a
> simple and elegant manner.
>
> just like in a simple binary election, decided by a "simple majority", the
> reason we grant preference to the candidate with more votes is that
> satisfying the greater number of voters who supported that candidate is more
> important than satisfying the lessor number of voters.  it's utility.  or
> utilitarianism.  the candidate with the simple majority has the mandate.
>
> likewise, as in Ranked Pairs or beatpath or MinMax-margins, the pairwise
> election with the greater margin has the mandate.  why give preference to
> the smaller set of *net* voters who prefer C over D than to the larger set
> of *net* voters who prefer A over B?  (the model i am using is that every
> vote for B over A is equal weight and cancels a vote for A over B.)  in both
> cases we are trying to minimize the number of disappointed voters.
>  (minimizing the number of disappointed voters is also what the Plurality
> guys are claiming, but they don't see the picture right).
>
>
>  This is one of my main criticisms of margins, that it doesn't try to
>> gauge the importance.
>>
>
> it *does*.  but it makes it *one* factor of two, the other factor being the
> *decisiveness*.
>
>   net salience  =   (popular import) x (decisiveness)
>
>                 = (number of voters) x (percent defeat margin)
>
> the main criticism of Winning Votes is that it doesn't try to gauge the
> decisiveness.
>
>
>  There's no fixing this either. If you want losing votes to count they
>> need to lessen the importance, or else you have a huge monotonicity
>> issue.
>>
>
> well, losing votes *should* count.  but with a negative weighting, and they
> do with Margins.
>
> it's like all of the winning voters are saying "This is what we want! And
> it's important!".  The losing voters may have lost the "This is what we
> want!" battle, but we don't totally remove their voice.  Their votes have
> the effect of reducing the importance of the "mandate" that the winners can
> claim.  and they *should* be able to do that, as the razor-thin example of
> 50000 to 49999 states.  that election doesn't really say diddley about
> whether we should pick Candidate A or Candidate B.  perhaps, there was a guy
> and his wife, both for Candidate B, who missed the election.  even with a
> big turnout, that election was too damn close to indicate the sentiment of
> the electorate.
>
>
>
>    4.  simplicity has its attraction.
>>>
>>
>> I find WV simpler because you don't need to do subtraction to determine
>> the defeat strengths. You find which side is higher and then you're done
>> with the losing side's figure. If you draw a triangle showing the wins,
>> you only have to write one number per win to say what happened.
>>
>
> but it doesn't say enough of what happened.
>
> the traditional vote-for-one ballot is simpler than a ranked ballot
> necessary for Condorcet (or IRV or Borda or Bucklin), but it doesn't say
> enough of what the reality is.  that's why i am for ranked-choice voting
> over the traditional ballot or for Approval voting (not enough information
> is collected from the voters).  WV doesn't sufficiently say what has
> happened when the election was very close.
>
>
>  Margins is probably easier to define, but is that much more desirable
>> than being easy to solve?
>>
>
> yes.  we want simple rules.  simple rules that make sense.  simple rules
> like simple majorities.  simple rules like more votes count for more than
> fewer votes.  the margins are just that (they are *net* votes).
>
> more *net* voters carry more weight than fewer *net* votes.
>
>
>    5.  Winning Votes communicates
>>> something regarding the number of voters participating, but
>>> says nothing about how close the election was.  an
>>> election with a razor-thin result, even with a lot of people
>>> voting, does not measure well the will of the people.
>>> if it's 99,999 voters and 50,000 said Candidate A and 49,999
>>> said Candidate B, that does *not* say that Candidate A has
>>> such a great mandate to lead.  Margins says his/her
>>> mandate to lead is 1 and Winning Votes says it's
>>> 50,000.  how can that make any sense?
>>>
>>
>> Because WV is measuring "importance," per your item #2, not closeness.
>> What "mandate" does a candidate have, who lost the most important race?
>>
>
> if he (let's call him "A") lost it by only one vote, perhaps more mandate
> than the candidate (let's call him "B") who beat him by just one vote, if
> that candidate (B) lost decisively to a candidate that A has beaten (or lost
> barely to a candidate that A has decisively beaten).
>
>
>  (And beat some other guy who had no chance of winning.)
>>
>
> now, how is that the case, if that other guy (call him "C") beat your guy
> ("B") decisively?  if C didn't beat B decisively, then maybe you gotta case
> for electing B.  but if B beats A by one vote (in the race with more
> participants), yet B is beaten by C decisively (with slightly fewer
> participants) who is then decisively beaten by A (again, with slightly fewer
> participants), i don't see how anyone would coronate B instead of A.
>
> if a vote for A>B cancels a vote for B>A (as it should for
> one-person-one-vote), it's the number of *net* votes for A (that have no
> canceling vote for B) that stands as the number of human beings with
> franchise that cry out to not be disappointed.
>
> think Utilitarian.
>
>
>  I would guess pretty much none, as far as voters would say.
>>
>
> an inescapable problem for elections that are very close, that could have
> gone either way, is that the winner does not emerge with a great mandate.
>  he'll do what agenda he campaigned on, if he can, but he cannot claim that
> the nation has clearly spoken.
>
> when there is a multicandidate election decided by Condorcet compliant
> ranked-choice voting, if there is a cycle, it is a lot like a tie, with
> nebulous results.  no candidate will come out of that with a big mandate
> either.  the motivation now is that of decisiveness (readily deciding a
> winner on election night, even under nebulous results), determinism (not
> deciding by coin flip, unless it just cannot be avoided), and some coupling
> to voter expression (elections with wider margins speak more loudly than
> those with narrow margins).
>
>
>  Methods getting confused by the presence of weak candidates (which is
>> basically what we are talking about here) is not good.
>>
>
> no, i'm talking about weak *results* a.k.a. close elections.
>
>
>  If a candidate can't win, he shouldn't be affecting the result.
>>
>
> but this is all what happens in a cycle.  the candidate in the Smith set
> *can* win, but so can the other candidates in the Smith set claim that.
>
>
>  Anyway, a margins proposal is DOA, from the moment anybody would point
>> out the 35 A>B 25 B 40 C scenario. Does anybody actually disagree with
>> that? One's EM postings will have to be very, very clever to persuade
>> the media, public, etc., that A should win that race.
>>
>
>
> and even more clever to persuade me.  lessee...
>
>   35 A>B
>   25 B>A     A>B by 10
>
>   40 C>A
>   35 A>C     C>A by 5
>
>   60 B>C
>   40 C>B     B>C by 20
>
> looks like B wins to me.  why shouldn't he?  (maybe because C's great
> defeat by B is not because B is so great, but that C is Sarah Palin: gets a
> lot of hard-core supports but a lot more of us know that she's a total
> doofus.)
>
> so maybe instead of Ranked Pairs or MinMax with margins (i presume Schulze
> would also elect A), it should go to the candidate in the Smith Set with the
> greatest winning margin over any other in the Smith Set.  but it still
> should be based on margins.
>
> but it *is* an interesting problem.  Marcus, can you comment?  Schulze
> beatpath method would also elect A, no?  can you persuade us that A should
> win in this scenario?
>
> bestest,
>
>
> --
>
> r b-j                  rbj at audioimagination.com
>
> "Imagination is more important than knowledge."
>
>
>
>
> ----
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>
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