[EM] Ordering defeats in Minimax
robert bristow-johnson
rbj at audioimagination.com
Thu Apr 27 22:23:20 PDT 2017
Markus, i am happy nearly every time you chip in with an idea. (Sorry, I just can't read all of the lit, including your writings.)
---------------------------- Original Message ----------------------------
Subject: Re: [EM] Ordering defeats in Minimax
From: "Markus Schulze" <markus.schulze at alumni.tu-berlin.de>
Date: Fri, April 28, 2017 12:46 am
To: election-methods at lists.electorama.com
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> Hallo,
>
> in section 2.1 and in section 6 of my paper, I discuss
> the different definitions for the strength of a defeat:
>
> http://m-schulze.9mail.de/verylong.pdf
>
> I my opinion, another natural definition is ratios:
>
> (W1, L1) > (W2, L2) if W1*L2 > W2*L1.
>
this is cool. it's like margins of the logarithms of the votes. or, if i may coin a term, the "Geometric Margins"
W1*L2 > W2*L1 implies
W1/L1 > W2/L2
log(W1) - log(L1) > log(W2) - log(L2)
maybe that is better than the simple "Arithmetic Margins":
W1 - L1 > W2 - L2
i've always been dissatisfied by using only Winning Votes (instead of Margins). i think the
number of Losing Votes should also say something in an election.
one thing that the Arithmetic Margins has over the Geometric Margins (if i may call it that) is that the Geometric Margins care only about the percentage defeat strength, and the number of people voting does not count.
Arithmetic Margins is the product of the percentage defeat strength times the number of voters weighing in on a race, which *should* (in my opinion) make that race more salient and another race with an equal percentage defeat strength but fewer voters who care about it.
> In any case, every definition for the strength of
> a defeat should satisfy at least the following
> properties:
>
> (2.1.1)
> ((W1 > W2 and L1 <= L2) or (W1 >= W2 and L1 < L2)) => (W1, L1) > (W2, L2).
> [In my paper, I call this property "positive responsiveness"
> because it says (a) that the strength of a defeat should
> respond to a change of its support or its opposition
> and (b) that it should respond in the correct direction.
> The term "monotonicity" would only imply that it
> responds in the correct direction.]
>
> (2.1.2)
> (W1, L1) > (W2, L2) => (L2, W2) > (L1, W1).
> [In my paper, I call this property "reversal symmetry".]
>
> (2.1.3)
> (c1*W1, c1*L1) > (c1*W2, c1*L2) => (c2*W1, c2*L1) > (c2*W2, c2*L2).
> [In my paper, I call this property "homogeneity"
> because this property is necessary to guarantee
> that the resulting election method satisfies
> homogeneity.]
those are all very reasonable requirements. seems to me that the simple Arithmetic Margins also have these properties. maybe not.
thanks for weighing in on this, Markus. still not *totally* convinced it's better than the conventional definition of
defeat strength with (arithmetic) margins. but it's something to think about.
--
r b-j rbj at audioimagination.com
"Imagination is more important than knowledge."
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