Talk:Shapley–Folkman lemma/Archive 2

Improvements to A-Class?
The article has improved greatly in the last weeks. Should we next try to improve it to satisfy the A-Class criteria (which are less demanding than the Featured Article criteria)? Kiefer.Wolfowitz (talk) 02:11, 20 January 2011 (UTC)

First set of comments

 * The sections "Mathematical Economics" and "Convex sets and economic equilibria" should be merged or the "Mathematical Economics" section should be expanded beyond a statement that economists were once bemused by supply and demand. :)
 * In the applications section, the subsection "Math Econ" (with a short lead) contains the sub-subsection "Convex sets ...". Kiefer.Wolfowitz (talk) 04:55, 22 January 2011 (UTC)
 * I did an extensive rewrite. Kiefer.Wolfowitz (talk) 17:41, 27 January 2011 (UTC)


 * The "Preceding papers" section might be better moved into a short narrative history of the lemma. As it stands the article is currently broken down in a very standard (not necessarily bad) way.  The mathematical basis for the lemma is asserted, the lemma itself is introduced, then applications are discussed.  However the applications section is a bit curious.  It reads both as applications and as historiography of progress on non-convex sets in economic theory.  I think the reader might benefit if we contextualized the lemma more.
 * Please review the updated version. Kiefer.Wolfowitz (talk) 17:42, 27 January 2011 (UTC)

Removed.Kiefer.Wolfowitz (talk) 17:41, 27 January 2011 (UTC) Restored, following the extensive discussion of demand sets (given prices and initial allocation). Kiefer.Wolfowitz (talk) 17:56, 30 January 2011 (UTC)
 * File:Price of market balance.gif seems largely unnecessary.
 * The article overlinks quite a bit. E.g. Convex Hull is wikilinked about 7 times.  Convex set is wikilinked about 6 times.  And so on.  An article like this demands some great linking, be it to basic concepts or more complex concepts.  Without those links the article would be 3-4 times longer.  But for an article of this length (about 2100 words) we shouldn't link to a concept like convex hull more than 4 times.  A great test is to look at the article on a monitor of about 1200X1068 resolution and see whether or not the same blue linked word or phrase shows up more than once in the screen window as you scroll down the article.
 * Thanks! I've started removing excessive links. Kiefer.Wolfowitz (talk) 04:55, 22 January 2011 (UTC)


 * A note, I started this review during what appears to be a big revamp of the article. I'm going to come back tonight and take another look when it is settled down. Protonk (talk) 18:03, 20 January 2011 (UTC)


 * Thanks for the great input! Kiefer.Wolfowitz (talk) 18:22, 20 January 2011 (UTC)
 * I've finished my major rewrite. Some things can be expanded, but the article is rather large now. I removed the section on non-convex sets in economics that was unrelated to the Shapley Folkman lemma.Kiefer.Wolfowitz (talk) 17:41, 27 January 2011 (UTC)
 * I'll take another look after class. One thing I was interested in.  Do you think an animation could be helpful here?  We have File:Shapley–Folkman lemma.svg, can you think of a good way to animate that to give some further illustration for the reader?  I know I could do this one in R with only a little bit of work, but I'm less clear on how I could improve on the main image.  Thoughts? Protonk (talk)
 * I have never uploaded images. Imho, the best image would illustrate the set
 * S = 1/2 ( [0,1]×[0,2] ∪ [0,2]×[0,1] )
 * and then
 * $$S_N = \frac{1}{2^N} \sum_{1\leq n \leq 2^N}{S} \qquad (N \geq 2)$$
 * for N = 0, 1, 2,3, ∞. A translate of this set appears in Mas-Colell's article on non-convex sets (etc.). The sequence is the "binary expansion" of the convex hull of S. (The New Palgrave no longer contains Mas-Colell's article. I suppose one could write to him and then the publisher and ask them to donate it to the public domain.)
 * Another interesting example appears early in Howe's paper, but it is not for animation.
 * David's example is too complicated for animation, imho (but it is the best picture explaining the Shapley-Folkman lemma in world history, as I've asserted many times). I should learn about uploading pictures sometime. (If I could only download Polymake, then life would be easier!)
 * Best regards, Kiefer.Wolfowitz (talk) 10:22, 28 January 2011 (UTC)

Second set of comments

 * Love the heuristic warning but I can't imagine it will survive FAC.
 * Thanks! :) I simplified the section, which now plays it straight! ;)


 * I am probably alone on this, but the LEDE image seems appropriate. I might want the caption to be a little bit shorter, but not by much.
 * I shortened it. Kiefer.Wolfowitz (talk) 19:53, 22 January 2011 (UTC)

I am afraid that your suggestion would immediately threaten the GA status, and waste your time or my time, if I may worry out loud .... I trust that I am misunderstanding either the GA/FA criteria or misunderstanding your proposal .... Kiefer.Wolfowitz (talk) 19:53, 22 January 2011 (UTC) (emboldening added by K.W., to help today's discussion and spare weak-eyed K.W. Kiefer.Wolfowitz  (talk) 19:57, 1 February 2011 (UTC))
 * I will try to do some of this myself, but the references section ought to be broken into a notes section and a bibliography. We cite the same paper many times at different page numbers and it would make the presentation much more clear if we did so with harvnb notation.  That way individual cites to papers which only appear once can be found quickly in the references and identified by the reader.  Also some of the quotes left in the references may be better moved to an explanatory footnote section or exuded entirely. Protonk (talk) 20:51, 21 January 2011 (UTC) (emboldening added by K.W., to help today's discussion and spare weak-eyed K.W.  Kiefer.Wolfowitz  (talk) 19:57, 1 February 2011 (UTC))
 * Take a look at something like The Autobiography of Malcolm X or Tulip mania as a GA and FA example of what I am suggesting. The guidance to choose one system over another exists to ensure that people don't get into fights over which is "better" and that people don't go around inserting one reference of a certain style into an article using entirely the opposite style.  It is not meant as an impediment to making the references more easily accessible.  In fact, the article already uses harvnb templates to point between page numbered references and footnotes which contain full bibliographic information.  My suggestion is that those full bibliographic cites be placed in a separate section when/if we end up using a pointer to them in another reference. Protonk (talk) 20:33, 22 January 2011 (UTC) (emboldening added by K.W., to help today's discussion and spare weak-eyed K.W.  Kiefer.Wolfowitz  (talk) 19:57, 1 February 2011 (UTC))
 * That's very re-assuring, and makes sense. I'll try to work on content first, and gradually start the transition, as you suggested. I think that the economics section needs more explanation, as the biggest priority. Kiefer.Wolfowitz (talk) 21:11, 22 January 2011 (UTC)
 * I reformatted the references following your advice, and the featured-articel on Tulip Mania, which you suggested. Thanks again!
 * It would be good to go through Starr's article and ours again. I provided page numbers to every citation.) Kiefer.Wolfowitz (talk) 05:02, 25 January 2011 (UTC)


 * I am not an expert at reference formatting expectations for FA, but I think direct quotes from sources should be provided where the sourced statement demands some additional verification and the source material is not freely available. Protonk (talk) 20:51, 21 January 2011 (UTC)
 * I don't think that there is anything controversial in the article, so I doubt that this is a concern. (I changed the Borwein/O'Brein reference so that it became an secondary citation (via Schneider); the paper is inaccessible unless you have a good librarian or write Borwein's secretary.) All the remaining sources should be readily available to members of universities or even good colleges, who usually have to pay a club-fee tuition or subsidies indirectly. Is that what you mean by free? Kiefer.Wolfowitz (talk) 19:53, 22 January 2011 (UTC)
 * I'll try to articulate this a bit. A common practice (not widespread, but common enough) for FA class articles where there is some difficult to access source is to include a paragraph excerpt in the footnote of a particular claim.  So if I am writing an article on the Idaho Falls nuclear accident and I cite a claim to the commission's report on the incident I might want to include a quote in the footnote itself so that someone perusing the article can see how close or far my paraphrasing was from the original content.  I do this partially because the commission's report on Idaho Falls isn't exactly easy to get at and because I may be inserting a strong claim into the article.  By contrast an article like this may not require such excerpting.  On a closer review of the references I only see a few long parenthetical notes or excerpts so this isn't a bit deal. Protonk (talk) 20:33, 22 January 2011 (UTC)
 * Whew! That's a relief! I suppose that some French titles and Lemaréchal's preprint are hard to get, I had to ask him for it. But his preprint was cited by Ekeland directly and vaguely, so I don't think that it is a problem either. But your suggestion makes sense, and goes with the prime directive of writing to help the readers. Kiefer.Wolfowitz (talk) 21:11, 22 January 2011 (UTC)
 * Thanks for the examples: Emulating the FA tulip mania is easier than memorizing the MOS. I'm beginning to implement the suggested changes. I'll update the hyperlinks of the form after I've moved all the important references below. (Pardon the day or so of inconsistencies or redundancies in referencing, please.) Thanks again! Kiefer.Wolfowitz (talk) 23:28, 24 January 2011 (UTC)

Economics applications

 * Pay special attention to the section titled "Non-convex sets in contemporary economics". The first paragraph is nearly all a quotation, the second paragraph is a mass of blue links (there to support the claims of the first paragraph) and the third paragraph is an attempt at contextualization which doesn't completely succeed.  The thesis of the section is that Shapley & Folkmann (and later Starr) influenced mid 20th century mathematical economics by expanding the bounds within which optimizing agents could be used to describe an economy.  they did this by showing that convex hulls could be operated on with non-convex preferences if the number of agents exceeded the dimensionality of the problem (basically).  The section itself should clearly lay this out by first presenting the problem of non-convexity of preferences and picking salient examples where economic analysis was unable to provide clear answers prior to Starr's work.  Then you can give a very brief precis of how the work allowed economists to reach a solution.  This may seem clear to you but picture a hypothetical educated reader interested in economics but unable to see immediately where the limitations exist and why they are important.  the benefit of restructuring this section is that you can probably fold in some of the "Convex sets and economic equilibria" parts.  As it stands the explanation is that convexity allows us to exploit the fixed point theorem.  That's great, but I would submit a large number of readers don't know the fixed point theorem or can't explain all of its implications (AMM had a great interview with a mathematician who surveyed faculty at various math departments and found that only a small minority said they could prove the fixed point theorem--I have lost the PDF but I will try to find it). Protonk (talk) 22:12, 21 January 2011 (UTC)


 * I worked on the economics: Please consider helping! I removed the paragraph on non-convexity unrelated to the Shapley-Folkman lemma, because the article has 78 K, and 80+K triggered warning messages. Kiefer.Wolfowitz (talk) 12:51, 26 January 2011 (UTC)
 * Reformatting and removing redundancies in the notes reduced the article from 78 to 72 K. Kiefer.Wolfowitz (talk) 18:56, 26 January 2011 (UTC)
 * I did a major rewrite of the economics, providing the context. The convexification of preferences is much older than indicated previously. Following Diewalt, I cite Wold. I also provided references to Lyapunov's theorem, Vind (1964) and then Aumann.
 * I shall look another day at providing more narrative: However, I am afraid of OR violations if I give too much of a narrative. The previous versions made Starr's paper a work of genius. The present writing makes his approach a natural development, following Wold, Farrell & Rothenberg, Vind, Aumann, Shapley & Shubik, etc. Starr and Arrow/Hahn certainly credit all of these (but Wold). Kiefer.Wolfowitz (talk) 18:02, 30 January 2011 (UTC)

Images

 * There are also a few too many images in this article. I would drop (in ascending order of importance to the article): File:Kenneth Arrow, Stanford University.jpg, File:Unit circle.svg, and the pair File:Convex polygon illustration1.png File:Convex polygon illustration2.png.  The last set might stay in because they give a concise graphical explanation of both what a convex set is and how to test for convexity, but they aren't absolutely critical.  I would also differentiate the various reuses of File:Shapley–Folkman lemma.svg visually.  At time of writing it is used 4 times in the article to illustrate the basic idea of the lemma and three steps along the way.  Each of the three steps might be better served by some coloring of points differently or otherwise a stronger visual hint to the reader.  Otherwise we don't gain much by seeing them again.
 * David Eppstein created another great graphic, better distinguishing the circum-radius and inner-radius. (However, I still think that his original graphic should be savored.) Kiefer.Wolfowitz (talk) 05:07, 22 January 2011 (UTC)
 * Regarding pictures, I'd ask for a third opinion: I heard Victor Lomonosov lecture on his (juvenalia) invariant-subspace theorem (which used Schauder's generalization of Brouwer's theorem), and he drew a picture (5 seconds). (I recognize that my feelings are strong here: My mind is very geometrical, and so I like pictures. Also, Ken Arrow is a hero & mensch, and seeing his picture gives me a warm feeling, honestly.) Kiefer.Wolfowitz (talk) 05:07, 22 January 2011 (UTC)
 * I love pictures myself, and econ concepts generally don't produce too many. But keep in mind that the page is going to get visually crowded quickly and soon you will have to be making decisions at the margin about where to position pictures and which may be removed. Protonk (talk) 05:59, 22 January 2011 (UTC)
 * I also love pictures and think that too many is better than too few. But I suspect that using the same one three times is probably suboptimal. If you want to refer to it from later text, why not just refer to it? —David Eppstein (talk) 06:53, 22 January 2011 (UTC)
 * (Having learned to program with Pascal and having read Dijkstra's "Notes on programming", I like to reduce cognitive demands at all times! ;) The short captions focused on different aspects of the picture, which illustrated the nearby text. Kiefer.Wolfowitz (talk) 08:55, 22 January 2011 (UTC))
 * Your comment about the budget of article size (especially pictures) was correct. The article is at 78K, and 80+K generated warning messages about length. I removed some stuff, which is stored on this page. Kiefer.Wolfowitz (talk) 12:53, 26 January 2011 (UTC)


 * Apologies that these comments are nearly exclusively non-mathematical. I'm the sort of economist who shies away from talking about the finer points of set theory and measure theory due to a severe skill deficit. :) Protonk (talk) 01:09, 22 January 2011 (UTC)
 * You've done a great job! (K.W.)

Revamping: Inner radius and optimization
User:Protonk noted that I was making a major revamp of the article.

David's sumset graphic
I recognized that David's graphic sufficed for an explanation of the inner radius and the circumradius, so I re-used it with another caption. Kiefer.Wolfowitz (talk) 18:22, 20 January 2011 (UTC)
 * It will do for now but I can produce a more targeted graphic later. (Probably not today, I have a conference submission deadline looming.) —David Eppstein (talk) 18:28, 20 January 2011 (UTC)


 * Tell me about it! ;) I would have thought you were used to it by now. Good luck on your conference. Let other people patrol the NP and P page while you work on the submission.


 * It would be useful to have an example of non-convex preferences, and then give an example of summing them, and showing the approximate decrease in non-convexity. Now, the economics doesn't provide motivation and insight into the mathematics. I would suggest using the example from the Shapley-Shubik article, which is the max of two Leontieff functions.


 * (Maybe it would be original research to make up an example of a zoo-keeper deciding between one lion and one eagle, and being bummed because no griffin be available!) Both Shapley-Shubik and Starr (1969) have clear and interesting examples to drive home the idea that non-convexities are important.
 * Best regards, Kiefer.Wolfowitz (talk) 19:39, 20 January 2011 (UTC)

WOW! David produced another great graphic. (BTW, I simplified the captions and used his earlier picture a few times.) Kiefer.Wolfowitz (talk) 05:09, 22 January 2011 (UTC)

Optimization
I tried to make the Optimization section intelligible to a reader who remembered the definition of the Euclidean metric from a sophomore course in linear algebra (USA). I added pictures of convex functions, using the epigraphal definition for simplicity (and maximum connection with the earlier sections), rather than tangential or secant criteria.

I took the risky decision to give a heuristic description of Ekeland's analysis first, and then give the truth (with another slew of topological notions). I would appreciate feedback, and probably damage control from other editors. (Rockafellar and Ekeland/Teman or Ioffe/Tickhomirov have the main results in accessible form: I consulted Ekeland for the lsc equivalence with a closed epigraph). Sincerely, Kiefer.Wolfowitz (talk) 18:22, 20 January 2011 (UTC)


 * Editor Protonk commented above.
 * I shall continue to edit the optimization section, aiming for additional simplifications. Kiefer.Wolfowitz (talk) 21:08, 21 January 2011 (UTC)


 * I simplified it as much as possible, focusing on the SF lemma, and down-playing the (human interest story about the) young mathematicians Lemaréchal and Ekeland. I should probably cite Bertseka's Lagrange-multipliers book more, which is rather similar to Ekeland's appendix. I should add page numbers to Rockafellar's "bible", which gives an unbounded closed set and bounded closed set having a sum that is not closed. Kiefer.Wolfowitz (talk) 05:14, 22 January 2011 (UTC)