# [EM] Minmax ranked method

Kristofer Munsterhjelm km_elmet at t-online.de
Mon Nov 6 04:36:13 PST 2017

```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. 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. 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.

In Minmax Approval, what you're minmaxing over is the voters'
preferences. It chooses the council that is minimally unrepresentative
to the voter to which it is maximally unrepresentative. E.g. consider an
Approval system like this:

100: A B
10: C D

Any proportional representation system would choose A and B for a
two-seat election. Minmax Approval would choose one of A and B for the
first seat and one of C and D for the second. This is because it's much
worse to leave the worst voter (one of the ten) not represented at all,
than it is to leave both only slightly represented.

In veto terms, if the 10 voters can say "nope", it doesn't matter how
well represented the 100 voters are by {A, B}. Either C or D must be on
the council because otherwise, it will be rejected. (By the same logic,
the method can't elect both C and D because then the majority of 100
would go nope).

Sometimes it isn't possible to please everybody, e.g.

100: A B
10: C D
5: E F

(two to elect)

but then it would try to come closest to actually pleasing everybody.
Most likely it would have the same outcome as the last example because
there's a greater chance that a minority of 10 can block the process
than that a minority of 5 can do so.
```