# [EM] Ordering defeats in Minimax

Markus Schulze markus.schulze at alumni.tu-berlin.de
Thu Apr 27 21:46:09 PDT 2017

```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.

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.]

Markus Schulze

```