# [EM] Ordering defeats in Minimax

Juho Laatu juho.laatu at gmail.com
Fri May 12 00:05:28 PDT 2017

```Some quick observations on the paper.

I agree that those criteria that do not allow electing the Minmax winner D of the example should not be considered as criteria that should always be true.

The paper aimed at using continuous and appropriately monotonic functions as preference functions. They fall within the families of natural preference functions that I described in a recent post. Or actually in the smaller family (W-L)*(W+L)^k or (W-L)/(W+L)*(W+L)^k. I.e. methods in a continuous range from proportions/ratio to the proposed square root based approach, and finally to Minmax, but not beyond that point, although methods beyond that point might be interesting too.

The approach of using the second preferences as tie breakers may be problematic because of problems with having or not having clones. Also the approach of using e.g. a square root based method to solve the ties of basic Minmax is a bit ad hoc. One simple approach to solving most of the ties in Minmax is simply to compare the W+L values of the tied candidates (to break ties in pairwise comparisons). That's what k values very close to 0 (e.g. -0,0001) would do anyway in (W-L)*(W+L)^k.

Don't take these comments too seriously since I skimmed the paper already few days ago, and this is about as much as I can remember of the comments that I had then :-).

Juho

> On 06 May 2017, at 12:33, Markus Schulze <markus.schulze at alumni.tu-berlin.de> wrote:
>
> Hallo,
>
> Darlington's paper ("Minimax is the Best Electoral System
> After All") is very interesting:
>
> https://arxiv.org/ftp/arxiv/papers/1606/1606.04371.pdf
>
> There are many papers where some author makes presumptions
> about the distribution of the voters and the candidates,
> about the used strategies, about how the performance of an
> election method is measured in concrete test cases, etc..
> The author then proves that his favorite election method
> performs better than every other known election method.
>
> However, this is not surprising. The purpose of an
> election method is to find the best candidate according
> to some heuristic. So you can always use test cases where
> this election method's heuristic happens to be met exactly
> and then claim that you have proven with random simulations
> that your favorite method is the best.
>
> Therefore, a more interesting question is which election
> method performs the second best. Here, Darlington writes
> that the Schulze method performs the second best, only
> slightly worse than his favorite method.
>
> Markus Schulze
>
> ----
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