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

� 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@alumni.tu-berlin.de> Date: Fri, April 28, 2017 12:46 am To: election-methods@lists.electorama.com -------------------------------------------------------------------------- > 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@audioimagination.com "Imagination is more important than knowledge."