# Talk:Stable winner set

## Limitations

The example I added illustrates how there can be examples where the committee is in the core but it does not "seem fair". User:Kristomun added the further specification that it is not proportional. Being in the core is the strictest form of proportionality and implies more standard ones like Justified representation. I think it would be a mistake to say that the committee is not proportional. None of the underserved groups are a quota in size. Can we remove this claim? --Dr. Edmonds (talk) 16:07, 15 June 2022 (UTC)

- Sure; I was thinking of disproportionality in a setwise sense. In the example, as k->infty, the faction who gets every candidate contains almost none of the voters. If we consider there to be two factions: the c1...ck faction and the "anyone but these" faction, then the distribution of power is anything but proportional in these factions: the second faction's share can grow as large as you wish without getting any representation. No proportionality violation exists in the sense of a quota violation, but divisor methods would select some non-c_1..c_k candidates in this scenario, e.g.

- k = 3, L = 10,
- 10: A1 = A2 = A3
- 9: B1
- 9: B2
- 1: C1
- 1: C2

- Then Sainte-Laguë would elect one from A1, one from B1, and one from B2.

- The problem is analogous that the Droop proportionality criterion just says (if there were 10+epsilon voters): "the A group must get at least one candidate elected" without specifying anything about the representation of the other groups in aggregate.

- I'll rephrase by removing the term as the argument may require a different definition of proportionality. Kristomun (talk) 17:40, 15 June 2022 (UTC)

- Thanks Kristomun, this is better. Proportionality is not really a well defined term so it was ambiguous. While I have you. I did not get the size of the image right. It should be smaller to match the rest of the text better. Do you know how to fix this? --Dr. Edmonds (talk) 05:00, 16 June 2022 (UTC)

- I don't think there's a satisfactory solution because text in an image can't flow the way other text does. E.g. it's hard to find a relative size so that it looks right both in portrait and landscape. I'll try rephrasing it in text: the reference link is still available for anyone who would like to read the source. Feel free to revert if it doesn't look right. Kristomun (talk) 16:13, 16 June 2022 (UTC)

- I think the best solution would be to make it smaller so that it looks like a caption in all aspects. I thought about writing it out and only taking the actual image but the though of all that formatting scared me off. Can you just make it half size or something? --Dr. Edmonds (talk) 16:24, 16 June 2022 (UTC)

- Thinking about it a little more, the best solution is probably to use vector graphics and rescale to an appropriate size, while minimizing the amount of text in it. That is, restating/rephrasing as much as possible as article text and keeping the rest in the vector image itself. I tried to convert the relevant PDF page to SVG for this purpose, but inkscape consistently crashed on me. (Inkscape's option to render text glyphs directly worked, but wasted a lot of space and is quite inelegant anyway.) Kristomun (talk) 17:12, 27 June 2022 (UTC)

- I appreciate the effort but this is well outside my skill set so I cant help. --Dr. Edmonds (talk) 07:20, 29 June 2022 (UTC)

## Comments

First: is there any problem with just using "strictly greater than" instead of "greater than or equal to" in the definition of the stable winner set? Seems like "strictly" lets us avoid having to explain why it differs between the Hare and Droop quota.

Second:

- In the simplest model, voters have a certain quantity of "utility" for each candidate, and they strictly prefer set X over Y iff the sum of their utility for X is greater than the sum of their utility for Y. However, this definition, while simple, is problematic, because it can hinge on comparisons between "utilities" for winner sets of different sizes.

Anything wrong with taking the average instead? Here's a model to justify this. Say each voter's score represents their estimated probability that a legislator will cast a tiebreaking vote in their favor (weighted by importance of the vote). Then, the expected number of ties broken in a voter's favor will be proportional to the average utility of the legislature. —Closed Limelike Curves (talk) 17:46, 7 May 2024 (UTC)