User:Kiefer.Wolfowitz/FAforSFL

This is a draft of my responses to concerns about our article on the Shapley–Folkman lemma that were raised during the FA nomination, which is ongoing.

Geometry Guy and Ozob (and anybody else who commented previously) are welcome to edit this page, just as though it were in main space. Others may consider leaving comments on the discussion on this user-space page's discussion; however, such users should probably leave their comments on the nomination page.

Review by GG

Weak oppose. Let me first add to the praise above for the meticulous work that has gone into this well-referenced and carefully explained article on an important technical topic. The article has much improved since February (where I contributed a partial peer review). Nevertheless, on reading the article closely (having not done so since February), I find that it falls short of the demanding FA criteria in several respects, and so cannot support its promotion at present. Areas where I believe improvements could be made include: clarity of exposition, engaging/brilliant prose, comprehensiveness and organization. I will add detailed remarks shortly, most of which I hope can be easily addressed. Geometry guy 21:41, 7 October 2011 (UTC)

I thank you for your past comments. I shall address whatever suggestions you make to the best of my ability. (I am away from my office until next week, however.)  Kiefer.Wolfowitz 22:08, 7 October 2011 (UTC)

• Leading issues. The first few sentences of the lead already illustrate some of the issues (numbered so that it should not be necessary to interleave replies).
  1. The repetition "In geometry and economics... In mathematics..." is clunky; despite comments made above about readers not knowing what a lemma is, an entire sentence is overkill – why not simply wikilink "Shapley-Folkman Lemma"?
  2. The article does not explain what the Shapley-Folkman-Starr theorem is: it discusses their results, including a Shapley-Folkman theorem and Starr's corollary, but no theorem with that name.
  3. The lead sentence does not define the topic: the Shapley-Folkman(-Starr) results do not describe the Minkowski addition of sets in a vector space (the definition of Minkowski addition does that); they describe the extent to which the Minkowski sum of many sets is approximately convex.
  4. The distinction between "addition" and "sum" is important. "Addition" is a synonym for "summation", the process of adding, not a synonym for "sum", the result of the addition. (We do not say "5 is the addition of 2 and 3".) The lead needs to make this distinction clear for Minkowski addition/sums, so that the terms can be selected and used for maximum clarity in the article.
  5. The lead uses terms such as "summand(-)set" and "sumset" without defining or wikilinking them. A particularly problematic example is "average sumset". I was completely unclear about what this meant until I read section 3.2.
  6. Theorems do have ("hypotheses" and) "conclusions", and I am relaxed about the idea that a theorem may "address" or "concern" a particular question. "The Shapley–Folkman–Starr results suggest..." is a bit too loose for me, however. A naive reviewer might ask "suggest to whom?", but the point of the sentence is to provide an intuitive summary of the results, not make suggestions.
  7. "The Shapley–Folkman–Starr theorem states an upper bound on the distance between the Minkowski sum and its convex hull—the convex hull of the Minkowski sum is the smallest convex set that contains the Minkowski sum." Unnecessary repetition: "its convex hull (the smallest convex set containing it)".
  8. "Their bound on the distance..." Antecedent missing/unclear.
  9. Final paragraph: here and elsewhere, "The Shapley-Folkman do-dah..." is used far too much as the subject of the sentence. Try turning sentences around by looking for other subjects, and cut down on the tiresome "also"s.

"The topic of non-convex sets in economics has been studied by many Nobel laureates..."

I left this to last, as it may be a more substantial issue. This segment of the lead is repeated in the article, but does not really summarize anything. The comprehensiveness criterion really bites here: "it neglects no major facts or details and places the subject in context". The legacy of the Shapley-Folkman Lemma is that results previously confined to convex economics and optimization (relatively easy) could be extended to the non-convex domain (much harder) by averaging (e.g., assuming many agents); this needs to be discussed to place the article in context. The applications section contains some such discussion, but is primarily pedagogical/technical and mixes mathematical, historical and evaluative material. The segment "Starr's 1969 paper and contemporary economics" then ends with a list which cries out for elaboration. Overall the treatment of the economics background, history and legacy for the results falls short of what I would hope for in a featured article. Geometry guy 23:18, 7 October 2011 (UTC)

Reply to GG

• Leading issues. The first few sentences of the lead already illustrate some of the issues (numbered so that it should not be necessary to interleave replies).
• 1. The repetition "In geometry and economics... In mathematics..." is clunky; despite comments made above about readers not knowing what a lemma is, an entire sentence is overkill – why not simply wikilink "Shapley-Folkman Lemma"?

(1) DONE!

• 2. The article does not explain what the Shapley-Folkman-Starr theorem is: it discusses their results, including a Shapley-Folkman theorem and Starr's corollary, but no theorem with that name.

(2) DONE! The mistake "SFS theorem" conventionally refers to Starr's corollary to the SF theorem. This can be explained, and the SFS theorem clearly labeled an alternative to Starr's corollary.  Kiefer.Wolfowitz 09:25, 8 October 2011 (UTC)

• 3. The lead sentence does not define the topic: the Shapley-Folkman(-Starr) results do not describe the Minkowski addition of sets in a vector space (the definition of Minkowski addition does that); they describe the extent to which the Minkowski sum of many sets is approximately convex.

(3) On the contrary, the SF lemma does describe the Minkowski addition of sets. However, it does not describe all aspects of Minkowski addition. Concern for our reading audience makes me want to provide a simple introduction. KW

• 4. The distinction between "addition" and "sum" is important. "Addition" is a synonym for "summation", the process of adding, not a synonym for "sum", the result of the addition. (We do not say "5 is the addition of 2 and 3".) The lead needs to make this distinction clear for Minkowski addition/sums, so that the terms can be selected and used for maximum clarity in the article.

(4) Excellent diagnosis and proposed cure. I shall examine this. (I have removed all use of "sumset".)  Kiefer.Wolfowitz 14:28, 8 October 2011 (UTC)

• 5. The lead uses terms such as "summand(-)set" and "sumset" without defining or wikilinking them. A particularly problematic example is "average sumset". I was completely unclear about what this meant until I read section 3.2.

(5) Well diagnosed. I unhid a symbolic statement of the average, which should clarify and highlight the average.

• 6. Theorems do have ("hypotheses" and) "conclusions", and I am relaxed about the idea that a theorem may "address" or "concern" a particular question. "The Shapley–Folkman–Starr results suggest..." is a bit too loose for me, however. A naive reviewer might ask "suggest to whom?", but the point of the sentence is to provide an intuitive summary of the results, not make suggestions.

6 More to come ....

• 7. "The Shapley–Folkman–Starr theorem states an upper bound on the distance between the Minkowski sum and its convex hull—the convex hull of the Minkowski sum is the smallest convex set that contains the Minkowski sum." Unnecessary repetition: "its convex hull (the smallest convex set containing it)".

(7) I rephrased this sentence. Please review it. This is a matter of taste. I favored saving the reader from hunting for the referent of "it", particularly because most readers of the lede will be meeting Minkowski sum and convex hull for the first time (or renewing their acquaintance after many years).

• 8. "Their bound on the distance..." Antecedent missing/unclear.

(8) Good suggestion. Please review the new version.

• 9 Final paragraph: here and elsewhere, "The Shapley-Folkman do-dah..." is used far too much as the subject of the sentence. Try turning sentences around by looking for other subjects, and cut down on the tiresome "also"s.

(9) The "also"'s can be reduced, agreed. I shall look at uses of the SF lemma. (However, new journalists avoid repetition excessively---a new journalist once wrote an article about bananas; to avoid repeating "bananas", the journalist referred to "elongated yellow fruits"!)

"The topic of non-convex sets in economics has been studied by many Nobel laureates..."

I left this to last, as it may be a more substantial issue. This segment of the lead is repeated in the article, but does not really summarize anything.

The article should run on the day that the Nobel prize is awarded. The main-page lede and the article should welcome readers interested in the news of the day. KW

The comprehensiveness criterion really bites here: "it neglects no major facts or details and places the subject in context". The legacy of the Shapley-Folkman Lemma is that results previously confined to convex economics and optimization (relatively easy) could be extended to the non-convex domain (much harder) by averaging (e.g., assuming many agents); this needs to be discussed to place the article in context. The applications section contains some such discussion, but is primarily pedagogical/technical and mixes mathematical, historical and evaluative material. The segment "Starr's 1969 paper and contemporary economics" then ends with a list which cries out for elaboration. Overall the treatment of the economics background, history and legacy for the results falls short of what I would hope for in a featured article.

You can wish for more, of course, but this article does a far better job than anything in the literature. An article need not be perfect to attain FA status. The article is at 87 kilobytes, which is at the upper end of recommended article lengths. (The footnotes take up almost half of the memory, so some expansion would be possible, without violating the spirit of the length guidelines ....) KW

Review by O

• Oppose. Here are some issues with the article as it stands:
  1. The article never defines "sumset". It begins using the term in the lead, but the non-expert reader may not realize this is meant to be the Minkowski sum.
  2. In the paragraph where the lead discusses the Shapley–Folkman–Starr theorem, the claim is first made that "[t]heir bound on the distance" does not depend on "the number of summand-sets N, when N > D". It goes on to say that "as the number of summands increases to infinity, the bound decreases to zero". My first thought was that these could not both be true, and that the article was in error. It turns out that I was wrong, because I missed the word "average". I think I won't be the only reader to make this error, so I think the paragraph needs to make a bigger contrast between sumsets and average sumsets.
  3. Still in the lead: It's not clear to me what a convexified economy is, not even vaguely. Nor is it clear what kind of equilibrium is meant (a Nash equilibrium, I guess?) or how that's different from a quasi-equilibrium.
  4. In general, the lead spends a lot of time trying to explain the statement of the Shapley–Folkman lemma (and its relatives). That's not the right direction. What the reader needs to learn is, "Why do I care?" Imagine, for example, that I'm a layman who has read a pop economics book, and I know what supply and demand are and what economies are, but I don't know what a convex set or a Minkowski sum is. Why should I learn about the Shapley–Folkman lemma? The lead does not answer this question. The closest it comes is near the end, where it explains that a lot of famous and successful economists have said that the study of non-convex situations is important (the lead has already made it clear that the Shapley–Folkman lemma is important for these). But essentially it's a proof by authority: All these Nobel laureates think it's important, so you should, too! This will not entice a novice reader to continue.
  5. In general, the article is structured like (I almost hate to say this) a math article. First it describes some preliminary notions (real vector spaces, convex sets, Minkowski sums, etc.). Then it states a theorem. Then it states applications. Kind of like "Definition–Theorem–Proof–Corollary". I realize that we all write this way (including me), but we shouldn't, and we especially shouldn't in an encyclopedia article.
  6. Right now, the article lacks a history section. I am going to suggest (this is only a suggestion, and there may be better ways of doing this) that you move some of the material from the Economics subsection of the Applications section into a new "History and Motivation" section immediately following the lead. This could put foundational material (like convexity) into a historical context: Before the Shapley–Folkman lemma, economists studied convex economies. Convexity is ..., and these are economies in which ... and they were important because of ..., but non-convex economies, which are ..., were important because ..., and prior to the Shapley–Folkman lemma nothing was known about non-convex economies. If you combine the foundational material with historical context, you make it more interesting and easier to grasp: It comes with vivid examples of what used to be cutting-edge research. By the time you are done with the historical context, you should have managed to introduce the prerequisite material for the Shapley–Folkman lemma. Then you can state it (and its corollaries and variations). Once that's done, you can move on to other applications.
  7. The optimization section has the same "Definition–Theorem–Proof–Corollary" feel. Again, I think it would be more effective to weave together the history and the prerequisite material.
  8. I was surprised at how short the section on probabilistic applications is. I don't know how important the Shapley–Folkman lemma is in such work, but you mention that it can be used to prove some analogs of standard results for real-valued random variables (like a law of large numbers and a central limit theorem). It would be good to include some more detail about these so that the reader at least knows how the Shapley–Folkman lemma is relevant (you don't necessarily have to state the theorems to prove this).
  9. Also, it might be good to explain in more detail how Lyapunov's theorem is related to the Shapley–Folkman lemma.
• Ozob (talk) 02:33, 8 October 2011 (UTC)