Wikipedia:Peer review/Shapley–Folkman lemma/archive2

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Shapley–Folkman lemma[edit]

Previous peer review
(more info)

This peer review discussion has been closed.
I've listed this article for peer review because…

  • Several reviewers have commented (during GA) review that the article was nearly FA status. Since then, I have revised the lead and the applications to make the article more accessible and self-contained (and accurate). The economic applications section was extensively expanded, in particular.
  • Also, the referencing has been standardized and improved, to nearly FA status imho.
  • To reach FA status, concerns about reasonable accessiblity must be addressed in the review process. The previous PR and GA processes were conducted by the participants in the Mathematics Project; perhaps the social-science (economics) project(s) could best address the accessibility issues.

This is the only article that I have developed to GA status, and I'd like it to reach FA status: Thus I am unfamiliar with the next steps, and would like guidance. (Would it be helpful to apply for A-status?) Thanks, Kiefer.Wolfowitz 15:27, 31 January 2011 (UTC)

Comments by Geometry guy[edit]

  • As an initial remark (hoping that I will have time to comment further), it seems to me that this article is crying out to have separate (Foot)notes and citations sections, where the footnotes provide further explanations, and the citations link to the sources. Already in the lead there are two parenthetical references, which could be footnotes, and then expanded. The use of set builder notation in the second sentence limits the accessibility of the article dramatically: non-math-majors may not understand it. This is a pity, as the article could and should reach a broader audience than that.
The current formatting of citations could run into trouble at FAC, as some of the citations are parenthetical references, and some are footnotes. Geometry guy 23:18, 31 January 2011 (UTC)
I simplified the lead paragraph, providing the example of {0,1}+{0,1} and its average.  Kiefer.Wolfowitz  (talk) 22:19, 1 February 2011 (UTC)
That's an improvement. Geometry guy 22:56, 1 February 2011 (UTC)
Re Citations: The article emulates the referencing of the FA-status Tulip Mania article, following the advice of the GA review, during which I raised the concern about mixing footnotes and harvard-citations. I shall focus on content and accessibility before implementing consensus reformatting.)  Kiefer.Wolfowitz  (talk) 13:10, 1 February 2011 (UTC)
I like the referencing style of Tulip Mania (and contributed to popularizing it 2-3 years ago!) but the article does not entirely follow it. The text of the article should not contain parenthetical (Harvard style) references, but it does: "Guesnerie (1989, p. 138)" in the lead and "Contemporary economics", "Arrow & Hahn (1980), Cassels (1975), Ekeland (1999), Artstein (1980), and Schneider (1993)" in the section on "Proofs and computations", "Wold (1943b, p. 243)" in "Non-convex preferences", and "Starr (1969, pp. 35–38)" in "Starr's 1969 paper".
I agree with you though that content and accessibility is a more important issue. Geometry guy 19:20, 1 February 2011 (UTC)
In the next month, I shall replace the in-text harvard-style hypertext references with footnotes, following the Tulip Mania example, as suggested by both reviewers. I am glad that we can keep the references for the most important sources, and leave other sources in the footnotes, where each is used (at most) a few times among nearby footnotes.  Kiefer.Wolfowitz  (talk) 19:52, 1 February 2011 (UTC)
For "month", read "few hours"! Geometry guy 22:56, 1 February 2011 (UTC)
Thanks for noticing!  Kiefer.Wolfowitz  (talk) 00:10, 2 February 2011 (UTC)
  • Three other interrelated issues. These are in addition to the issue mentioned already: making the article more widely accessible.
    1. The lead should serve both as a summary of the article and a concise introduction to the topic. There is a (hopefully creative) tension between the need to be accessible and explain, and the need to summarize and be encyclopedic.
      Unless examples are given exemplifying convex sets and convex functions, the lead will be understandable only by those with mathematical background like advanced U.S. undergraduates in economics (perhaps those considering graduate M.A. study).  Kiefer.Wolfowitz  (talk) 15:29, 3 February 2011 (UTC)
    2. Several sections lack citations: most of the "Preliminaries" section, and the opening of the "Statements", "Economics" and "Mathematical optimization" sections. A lot of this material is "well-known": the Scientific citation guidelines provide advice on how to source such material using a few general references.
      Okay, I'll look for guidance.  Kiefer.Wolfowitz  (talk) 00:10, 2 February 2011 (UTC)
      I added references to the background sections. Rockafellar is the bible of convex analysis, and his early exposition is accessible. I added references to mathematical economics books, upon which a mathematician may glance "without a shudder" (in Cassels's words).  Kiefer.Wolfowitz  (talk) 01:55, 3 February 2011 (UTC)
    3. Wikipedia is not a textbook, and original exposition should be minimal, as no original research is allowed, including synthesis. I notice that the "griffin" has been discussed below. Is the eagle/lion example your own, or can you provide a source? If not, are there similar examples in the literature that you can use?
      The griffin example is mine. All the writers on concavities in consumer preferences discuss less memorable examples, e.g. pickles and cotton candy: Their problem is that they allow some convexity, where the lion & eagle don't. Others discuss indivisibilities (integral programming constraints), but I don't remember any living commodities, off hand.  Kiefer.Wolfowitz  (talk) 00:10, 2 February 2011 (UTC)
Any (or all!) of these issues could cause you problems at FAC. The lead is getting long, and is turning into an expository introduction to the article, rather than a summary. One approach would be to create a new first section called something like "Idea of the Shapley-Folkman lemma", which gives the gist of the result. However, this plays into the other two issues: are there some expository sources you can use as references for such introductory material? Geometry guy 23:22, 1 February 2011 (UTC)
In his "math for economists" book, CARTER (page 94) has an example of adding Conv({0,1}^{1000}) [where exponentiation implies homogeneous sumset (semigroup) addition 1000 times] which is more complicated than mine. Mine is the most obvious example (minimal Diophantine complexity!).  Kiefer.Wolfowitz  (talk) 00:10, 2 February 2011 (UTC)
I think an introductory section would help you handle the dual roles of the lead, as some material could be moved from the lead to the introductory section. Exposition doesn't need to be entirely based on existing text books: there is some editorial leeway in presenting facts and using simple examples, as long as you are not deviating significantly from published work. Geometry guy 21:31, 2 February 2011 (UTC)
Can you think of an exemplary article that has an introductory section, following the lead?  Kiefer.Wolfowitz  (talk) 12:23, 6 February 2011 (UTC)
I'm not sure it is exemplary, but take a look at Vector space. Geometry guy 00:38, 7 February 2011 (UTC)
I looked at Vector space and the discussion at its FA-nomination, where you made the same suggestion, which seems not to have been acted upon (at Vector space) however.
Nobody should confuse the WP article Shapley-Folkman lemma for a textbook, which should have both more examples and also some exercises (and proofs). I suppose that Starr's New Palgrave article "Shapley Folkman theorem" and Green & Heller's Handbooks of Economics article on "convexity. analysis, and economics" could serve as benchmarks---both of those sources are written for an economic student preparing to write an M.A. or Ph.D. thesis, so that they assume more knowledge. The WP article Shapley-Folkman lemma should be accessible to students in a first course in linear algebra or enrolled in a course in microeconomics, and the lead should be accessible to motivated high-school students.
I shall try to consult an old edition of Encyclopedia Brittanica; I vaguely remember that Harold Kuhn may have contributed to its article(s) on game theory or linear programming or mathematical economics, any of which should provide inspiration.  Kiefer.Wolfowitz  (talk) 09:46, 7 February 2011 (UTC)

(←) In the vector space FAC, I suggested spinning out an entire "Introduction to..." article, whereas here I am just proposing a section. The purpose is to take stress out of the lead. I disagree with "the lead should be accessible to motivated high-school students", as the lead has an additional role. Instead I would say that "the introductory section and most of the lead should be accessible to motivated high-school students". Otherwise, I am quite in agreement with you that WP is not a textbook. Geometry guy 21:48, 7 February 2011 (UTC)

Okay, that makes sense. (I didn't understand what you were suggesting before.)
I did a bit of work on the lead, addressing accessibility concerns, following TCO's suggestions. It shall be 2 weeks before I can resume intensive editing.  Kiefer.Wolfowitz  (talk) 16:07, 8 February 2011 (UTC)
Okay. I'm unlikely to have time to make any further peer review comments in the interim. Geometry guy 22:06, 8 February 2011 (UTC)
Following your suggestion, I moved part of the introduction/lead to to an "introductory example" example. Now the lead has only the examples exemplifying Minkowski addition and convex sets, which are needed for public understanding of the problem. The examples illustrating ideas of the lemma and theorem don't appear in the lead.  Kiefer.Wolfowitz  (Discussion) 19:03, 21 February 2011 (UTC)

Comments by SunCreator[edit]

  • Inconsistency in references. Date: (3 November 1979) and 2011–01–15
I'll try to re-do the access dates for consistency.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
Done, I think.  Kiefer.Wolfowitz  (talk) 17:42, 2 February 2011 (UTC)
  • Same with page numbers. experiments on page 373, (1993, p. 131), pp. 1–5, pages 306–310. Some of the references start with page numbers i.e Pages 93–94. I wonder if these are suppose to be notes, but anyhow the page should not be at the beginning of any reference and the reference want to be consistent about whether they are in brackets are not and starting "p." or "page"
Fixed, I think. I tried to use templates everywhere, and I removed commentary, which might have been OR, also.  Kiefer.Wolfowitz  (talk) 17:42, 2 February 2011 (UTC)
  • Some bare links in the references. Ref 60(Vind, Karl) has one.
DONE!  Kiefer.Wolfowitz  (talk) 17:42, 2 February 2011 (UTC)
  • Some journals don't seem to be indicated as such. Use {{cite journal}}?
The article consistently misuses the cite_article template. I should switch to the cite_journal template. THANKS!  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • Some websites links don't have access dates.
Are you referring to on-line editions of journals?  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
Striked, my mistake Regards, SunCreator (talk) 14:54, 1 February 2011 (UTC)
  • Above points suggests a review of the method of referencing to get consistency.
As noted above (replying to GG), the current referencing implements the suggestions from the GA review, and emulates the article "Tulip Mania", which has FA status.  Kiefer.Wolfowitz  (talk) 13:56, 1 February 2011 (UTC)
As noted above, the clean-up of prosaic page references has begun.  Kiefer.Wolfowitz  (talk) 22:23, 1 February 2011 (UTC)
DONE! There may be minutia remaining to be corrected, e.g., using the template field for translations (and using either () or [] elsewhere consistently), giving page numbers for the whole book (or not), formatting of original-edition information in reprint editions, etc.  Kiefer.Wolfowitz  (talk) 17:42, 2 February 2011 (UTC)
  • Is 'nonnegative' a word? Maybe non-negative
I replaced "nonnegative" with "non-negative" in the alternative text for a picture. Thanks!  Kiefer.Wolfowitz  (talk) 17:26, 1 February 2011 (UTC)
  • Maths is heavy in this article can some thing be made easier for the general reader. Some of the words I'm not sure of the meaning, consider linking them if there is some suitable links. Possible words summand-set, circumradius, convexified sum. Is sumset the same as summand-set? Both are used.
The operated set is the output from the operation on the operands. The sumset is the output of Minkowski addition on the summand-sets (or summands). I'll say more later.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • You don't need to say the year and page number in the prose like (1989, p. 138) and (1989, p. 138). Such info goes in the references.
The clean-up of prosaic page references has begun.  Kiefer.Wolfowitz  (talk) 22:21, 1 February 2011 (UTC)
Done, I think. ~~
  • The use of "(or a griffin)!" is interesting. I'm not sure it's encyclopedic, but get others opinion because I maybe wrong on that.
Prose doesn't need to be boring. In this example, the "contemporary" qualification is necessary to exclude medieval beast-masters and apothecaries who would have purchased a griffin. Explicitly mentioning a griffin makes this example memorable. (It is not necessary, though. I did restrain myself from adding an image of a griffin!)  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • brackets are used a lot in prose "then (for some prices) ", "baskets (maximally preferred and meeting their budget constraints).", "set (with respect to Starr's measurement)". Prose is normally written in such a way to remove them.
Usually. However, mathematical sentences require more parsing and re-reading than other forms of prose. The parentheses are intended to help on the first reading. However, you are correct: I agree that many and perhaps all can be removed.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • Redundancy words, "some" re-occurs again and again "some prices", "some regions", "some discontinuous", "some allocation", the prose should be tightened on that, either to specify what or removal of "some" if used in a redundant fashion.
The existential quantifier "some" explicitly signals existence but makes no claim of uniqueness; in this article, "some" is used where multiple discontinuities, etc., do exist (for some examples). Many readers misread the default elided "some" as "the" or "every", which confuses them, unfortunately. (For example, I misread a sentence without an explicit some as having an elided "every" on my mathematics GRE!) Thus, "some" is used frequently in mathematics.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • A lot of small sections, consider combining some.
A matter of taste, perhaps?  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
I combined two sections in economics, per your suggestion.  Kiefer.Wolfowitz  (talk) 12:28, 6 February 2011 (UTC)
The downside of little sections is a larger TOC and associated white space right of TOC which in this case is not filled by an infobox, another downside is the inflexibility of images as moving them to the left on small sections looks awkward. Consider {{TOC limit}} with limit=3 as a method to shorten the TOC. Regards, SunCreator (talk) 14:54, 1 February 2011 (UTC)
DONE! The TOC looks better following your suggestion. Thanks!  Kiefer.Wolfowitz  (talk) 17:21, 1 February 2011 (UTC)
  • The Shapley–Folkman_lemma image is used twice. Not a good idea in my view.
The caption differs in each case. The image is unusually rich, and was previously used to explain the inner-radius and circumradius, before David added his new picture.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • Many images don't have WP:ALT. The alt on the main image is long.
I'll add alt to the new pictures.
DONE!  Kiefer.Wolfowitz 22:10, 27 April 2011 (UTC)
Again, the main image is unusually rich, and the alt's details are needed to explain the image to a blind person. Some repeated wording exists to help a blind person keep track of the 5 images.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • Lots of images on right and together, consider moving some to the left.
I tried experiments, and they all looked funny or weird imho.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • The top image was a reference on the caption. That is unusual. Maybe acceptable but just a second opinion on it.
I was trying to document all claims. It can go, imho.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • Is there a suitable infobox. Some editors like them.
There are many images now. There is no suitable infobox, imho, and any additional one would distract or exclude informative images. (The economics sidebar was removed from its position with the footnotes.)  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
  • Some terms are linked more then once. This is unusual even when linked from different words. A reader would be surprised to click and find the same page? Perhaps prose can be changed to the same wording? convex function, Convex optimization, convex hull are examples. I didn't make a detailed check for more.
"Convex minimization" is correct, while the ambiguous "convex optimization" is misused at Wikipedia. (Convex maximization is NP hard, I believe.) However, sometimes the correct "minimization" sounds stiff and pedantic. Kiefer.Wolfowitz  (talk) 17:42, 2 February 2011 (UTC)
  • Why so much in "See also", if these are important to the topic then consider bringing them into the prose somehow. If they are related but different terms consider making a navigation template for them.
Good suggestion.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
I removed the see also section, most of whose items are linked from other articles linked here, e.g. convex hull, etc.
  • I know nothing about the topic so make so comment regarding it's contents or completeness.
This WP article is much more comprehensive than any source known to me. Including a proof or two would be useful.  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)

Regards, SunCreator (talk) 00:51, 1 February 2011 (UTC)

Thank you very much for your kind and helpful suggestions, which will occupy me for some weeks. I shall first work on content and readability questions. Later, I shall work on citation style questions. Thanks again for great help. Best regards,  Kiefer.Wolfowitz  (talk) 13:37, 1 February 2011 (UTC)
I emboldened your suggestions that still require work, to help me plan future clean-up. Thanks!  Kiefer.Wolfowitz  (talk) 17:48, 2 February 2011 (UTC)
Your welcome. One thing I just noticed. You have a full URL linking to, I imagine there is a neat way to make that link to the book, but I confess I don't know how. Regards, SunCreator (talk) 19:23, 2 February 2011 (UTC)
I suppose that you are referring to my location of the Milton quote, from Arrow & Hahn (whose mis-capitalization I corrected; they didn't cite the lines, btw, but this is trivial OR)?  Kiefer.Wolfowitz  (talk) 19:34, 2 February 2011 (UTC)
I'm not checking for original research. H:IW says how to link to wikisource. Regards, SunCreator (talk) 19:57, 2 February 2011 (UTC)
I sourced Milton, my hero! Thanks!  Kiefer.Wolfowitz  (talk) 01:56, 3 February 2011 (UTC)

Comments by Kiefer.Wolfowitz[edit]

  • I simplified the lead further, creating an introductory example section, following a suggestion of Geometry Guy. I simplified the caption, following TCO's suggestion.
  • I suggest moving the material on the metric theorems of Shapley-Folkman and Shapley-Folkman-Starr to its own article. This would make the lead less intimidating and simplify everything.  Kiefer.Wolfowitz  (Discussion) 19:51, 24 February 2011 (UTC)
    That might be good use of Summary style. Geometry guy 21:58, 24 February 2011 (UTC)