[EM] Ordering defeats in Minimax

[Andrew Myers]

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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