[EM] Minmax ranked method

[Kristofer Munsterhjelm]

On 11/07/2017 03:55 AM, robert bristow-johnson wrote:
>
>
> ---------------------------- Original Message ----------------------------
> Subject: Re: [EM] Minmax ranked method
> From: "Kristofer Munsterhjelm" <km_elmet at t-online.de>
> Date: Mon, November 6, 2017 7:36 am
> To: rbj at audioimagination.com
> "EM" <election-methods at lists.electorama.com>
> --------------------------------------------------------------------------
>
>> On 11/06/2017 08:02 AM, robert bristow-johnson wrote:
>>>
>>>
>>>
>>>
>>> ---------------------------- Original Message
> ----------------------------
>>> Subject: Re: [EM] Minmax ranked method
>>>
> From: "Kristofer Munsterhjelm" <km_elmet at t-online.de>
>>> Date: Sun, November 5, 2017 3:39 pm
>>> To: rbj at audioimagination.com
>>> "EM" <election-methods at lists.electorama.com>
>>>
> --------------------------------------------------------------------------
>>>>
>>>> For each potential council, there's a voter that's most displeased with
>>>> having that council elected. A minmax method minimizes how displeased
>>>> this most displeased voter is (which may be a different voter for
>>>> different proposals).
>>>>
>>>> In veto situations, if a minority can say "nope", it's more important
>>>> that no such minority can be annoyed enough that they do so than just
>>>> how annoyed the rest of the voters get.
>>>>
>>>
>>> and, for a Smith set of size 3, Minmax picks the same candidate as does
>>> Ranked-Pairs (margins) as does Schulze (margins), right? i just wanna
>>> make sure i got that right.
>>>
>>> so how is the rule worded differently for these three methods in this
>>> context of 3 candidates?
>>
>> That's because the word "minmax" is used in two different contexts.
>
> i understand that now.  but i mean in the ranked-ballot context.  (sorry
> to poke in this question out of context.)

To reduce the confusion, let's call the method Simpson (which is another 
name for Minmax Condorcet), and call the other types of methods 
"veto-friendly" (from the example I gave).

>> In the Minmax Condorcet method, what you're taking the minimum of the
>> maximum of is the Condorcet matrix. The Minmax method chooses the
>> candidate with the weakest (minimal) greatest (maximal) defeat, i.e. the
>> candidate who loses the least one-on-one to the candidate he loses the
>> most to.
>
> and that is the same candidate who is chosen by  RP (margins) and by
> Schulze (margins).

Only for three candidates. For instance, Schulze and RP are cloneproof, 
but Simpson is not; e.g. 
https://en.wikipedia.org/wiki/Independence_of_clones_criterion#Minimax

>> When there's a Condorcet winner, that CW doesn't lose to
>> anybody, and so he's the winner of the Minmax method since you can't do
>> better than not losing at all.
>
> i understand there is no issue (with any of those 3 methods: Minmax, RP,
> Schulze) when there is a CW.  my specific question was about the case
> that there is no CW and a Smith set of 3 candidates, which i think is
> the Rock-Paper-Scissors scenario.

When there are only three candidates, then RP and Schulze give the same 
result as Simpson. When there are more candidates but a Smith set of 
three, then RP and Schulze might give a different result from Simpson. 
In Wikipedia's clone example above, the Smith set is {B1, B2, B3} (three 
candidates), but Simpson elects A.

Also, Simpson probably isn't a good candidate for a veto-friendly method 
since it's more majoritarian than even a proportional method. E.g.

51: A>B>C>D>E
25: F
24: G

Simpson's social ordering would be something like A>B>C>D>E>F>G, but 
electing the four highest ordered for a multiwinner election would elect 
{A, B, C, D}. In this particular election, a better result for a method 
that's aiming to represent everybody would be {A, B, F, G}, which is 
what a proportional representation method would provide.