[EM] Ordering defeats in Minimax

Andrew Myers andru at cs.cornell.edu
Fri May 5 12:10:05 PDT 2017

```Thanks for all the useful feedback from everyone on this topic.

I decided to implement Minimax with the following ordering: defeats are
ordered by margins (W-L), and by ratios (W/L) when margins are tied.
Further, two candidates are compared first by their weakest defeats;
then, if tied on those, by their 2nd weakest defeats, and so on. This
comparison method was suggested by Richard Darlington. It seems to work
well in practice -- the results seem intuitive and reasonable, and the
algorithm is also efficient.

-- Andrew

Juho Laatu wrote:
> That was a good summary of the weaknesses of the Plurality criterion.
> I wonder if the Plurality criterion could be reformulated so that it
> would not refer to the first preferences at all, and it would make its
> message on the implicit approvals clearer.
>
> Juho
>
>
>> On 28 Apr 2017, at 15:45, Toby Pereira <tdp201b at yahoo.co.uk
>> <mailto:tdp201b at yahoo.co.uk>> wrote:
>>
>> I think I basically agree with Juho on this. The plurality criterion
>> sounds like a reasonable criterion on the surface, but think about it
>> more and it's arguably less so. To summarise, in a pairwise method,
>> first place on a ballot doesn't hold any special status, nor does
>> indeed last place (or joint last place or "unranked"). And a
>> criterion shouldn't be used to impose an approval cut-off on a method
>> that doesn't have one in its definition.
>>
>> So while it sounds like a good criterion, removing the special status
>> of these positions means that we are left with just saying that a
>> candidate who pairwise beats another candidate should finish ahead in
>> the overall ranking. Which is what all Condorcet methods do - except
>> when there's a cycle.
>>
>> Toby
>>
>>
>> ------------------------------------------------------------------------
>> *From:* Juho Laatu <juho.laatu at gmail.com <mailto:juho.laatu at gmail.com>>
>> *To:* Election Methods <election-methods at lists.electorama.com
>> <mailto:election-methods at lists.electorama.com>>
>> *Sent:* Thursday, 27 April 2017, 22:57
>> *Subject:* Re: [EM] Fwd: Ordering defeats in Minimax
>>
>> > On 27 Apr 2017, at 10:25, Kristofer Munsterhjelm
>> <km_elmet at t-online.de <mailto:km_elmet at t-online.de>> wrote:
>>
>> > Do you consider Plurality a strategic criterion? It seems to me to be
>> > more of a "natural behavior" criterion: if A gets more first
>> preferences
>> > than B gets any preferences, then B shouldn't win. This seems
>> reasonable
>> > from a natural behavior perspective because A dominates B in some
>> > Approval-ish sense.
>> >
>> > If that's a "natural behavior" criterion, then you could say that
>> > margins is more natural from a descriptive point of view (no
>> > discontinuities) while wv is more natural from a criterion point of
>> > view. Though, if we're to go by the apparent popularity of IRV, it
>> seems
>> > that descriptive clarity weighs heavier than criterion clarity.
>>
>> First of all, my thinking when it comes to practical election methods
>> is not very criterion oriented. I tend to see criteria and criterion
>> compatibility as important theoretical results that are mostly too
>> far from practical election method considerations to be applied
>> directly on them as viability criteria. The relevance of different
>> criteria to practical election methods is almost as low as the
>> relevance of latest mathematical inventions to practical everyday
>> economic calculations (well, not quite, but something in that
>> direction). I'm about at the level of accepting Condorcet criterion
>> if we seriously want to have a neutral majority based method for
>> consensus oriented single winner elections. One has to take also into
>> account the fact that all election methods are bound to break some
>> potentially useful criteria. All this means that I classify Plurality
>> and most other criteria as an interesting discussion points but not
>> something to be followed categorically. There are many
>>   criteria that are useful in the sense that most elections should
>> have strong orientation in the described direction, but no need, or
>> possibly with strong reasons to deviate from some criteria in some
>> special (usually marginal) situations.
>>
>> I think Plurality is a bit strange. Actually it is not even a
>> criterion of of ranked methods. It is a criterion for ranked methods
>> with implicit approval cutoff. It makes the assumption that a voter
>> that casts a short vote has somehow approved those candidates that he
>> marked, and not approved the others. In different elections the
>> behaviour of voters with respect to which candidates will be marked
>> on the ballot may vary a lot, and that may have nothing to do with
>> how much the voters support or approve those candidates. In order to
>> make any sense of the Plurality criterion we are thus tied to having
>> an assumption of implicit approval in the ballots, where marking a
>> candidate means approving that candidate at some level.
>>
>> One reason why I don't like implicit approval in general (as a fact
>> that is known by the voters) is that it encourages voters not to rank
>> candidates that they don't like. Ranked methods work well only if
>> most voters do rank explicitly at least all the potential winners (or
>> all of them except one). If there is an approval cutoff, it would be
>> better if it was an explicit one (this comment is not Plurality
>> criterion specific but a general one).
>>
>> Plurality criterion is a "heuristic" criterion in the sense that its
>> message somehow sounds good (e.g. to people that do not regularly
>> deal with election methods and their peculiarities). People would
>> like also criterion "if voters would prefer A to B, then B should not
>> win". But EM experts know that this criterion would not be a very
>> good one, although it states something that we all would like to be
>> true in all elections. What I'm trying to say here is only that we
>> should be careful with cyclic group opinions. They will contain some
>> nasty features. Instead of trying to pick a set of criteria that
>> should be met 100%, my preferred approach is to see what kind of
>> problems each method would be likely to face in real elections
>> (typically but not necessarily large public elections with many
>> different kind of voters that the strategists can not control), and
>> evaluate them based on their performance in such real life situations.
>>
>> I'm not well prepared to comment how margins can handle Plurality
>> criterion but I'll address one basic (but theoretical and extreme,
>> i.e. unlikely to happen in typical elections) example. 35:A, 34:B>C,
>> 31:C. A has more first preferences than B has ballots where B is
>> marked. B's worst defeat margin is however smallest (1), so it will
>> win in typical margins based methods. Plurality criterion says that B
>> should not win. B is however two votes short of being a Condorcet
>> winner, so it can't be the worst of the worst. What if A would win?
>> Plurality criterion pays special attention to A's high number of
>> first preferences. But on the other hand voters would like to elect C
>> instead of A with large majority (C>A voters would not be happy with
>> the result). How about C then? Plurality criterion accepts C too, but
>> using the high number of first preference votes of A as an argument
>> that supports C does not make much sense. My conclusion is that this
>> is a typical mess that we can get with c
>> ircular preferences. Our voters were quite stupid when they didn't
>> sufficiently rank the potential winners. There are many different
>> possible scenarios on what the truncated opinions might have been,
>> and different results emerging from that. In this example my
>> recommendation would be to tell to the voters that they should rank
>> all potential winners (except maybe the worst one). I don't see any
>> need to start blaming (or praising) margins on what happened. Maybe
>> you have some realistic examples in your mind, that would give better
>> justification to the Plurality criterion.
>>
>>
>> Juho
>>
>>
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>>
>>
>

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