Talk:Shapley–Folkman lemma/Archive 4

EdJohnston's comments
Hello Kiefer. Congrats on this article, which seems very well done! I gather it has been nominated for GA, a process I know little about. If I'm allowed to give an unstructured comment, I'd say that it's very good right up to the point where it's trying to explain the economic significance of the result. Maybe a further sentence or two would supply the final motivation. (It points over to General equilibrium theory as the main article, but that article doesn't provide much illumination). "The derivation of these results in general form has been one of the major achievements of postwar economic theory". Hmm.. It suggests that the theorem is a success because it has been able to get into textbooks. Except for that minor disappointment, I am happy to see this work, which is well-motivated. The name of the article is not easy to type because it contains a funny dash. Perhaps a redirect could be created using a normal hyphen. EdJohnston (talk) 02:52, 3 December 2010 (UTC)


 * Dear Ed,


 * Thanks for your encouraging words. In fact, I nominated the article for good article (GA) status, to get some comments on how to improve it (following helpful feedback from the peer-review process): I hope that it was okay for me to nominate the article for GA status. (I know that I cannot review it for GA status.)


 * Your specific comments are also useful. I shall try to provide some more context. The quote from Guesnerie is there because it provides an overall evaluation, and because Guesnerie has been one of the world's leading mathematical economists (e.g. a President of the Econometric Society).


 * I don't like Wikipedia's policy of preferring large dashes (which don't appear on my keyboard) over small dashes. (Before, on Windows IE, I couldn't see the difference when I was editing.) There is a redirect, Shapley-Folkman lemma, as you suggested.


 * Thanks again for your great suggestions.


 * Best regards, Kiefer.Wolfowitz (talk) 09:03, 3 December 2010 (UTC)


 * Following your suggestions, I wrote this more friendly version. Thanks again. (I'm sorry for forgetting to credit you in the comments.) Best regards, Kiefer.Wolfowitz (talk) 20:49, 16 December 2010 (UTC)

Mathematical economics


The non-convexity of the Minkowski sum of possibly non-convex sets is important in the microeconomics of consumption and production. Non-convex sets are widely associated with market failures. Indeed, in the era before Starr's paper, non–convex sets seemed to stump economists from proving that that, with several consumers and several goods, supply and demand could be "balanced" — in economic terms, so that a market equilibrium exists. The study of economic equilibria of complicated markets occurs as the "theory of general equilibrium", perhaps the most mathematically advanced branch of mathematical economics.

Before Starr's paper, Arrow and Gérard Debreu proved the existence of general equilibria by invoking Kakutani's theorem on the fixed points of a continuous function from a compact, convex set into itself. In the Arrow-Debreu approach, convexity is essential, because such fixed–point theorems are inapplicable to non–convex sets: The rotation of the unit circle by 90 degrees lacks fixed points, although this rotation is a continuous transformation of a compact set into itself; although compact, the unit circle is non–convex.

In his paper, Starr studied the general equilibria of the artificial economy in which non–convex preferences were replaced by their convex hulls. Starr was investigating the existence of economic equilibria when some consumer preferences need not be convex. Applying the Shapley–Folkman lemma, proved that the "convexified" economy has general equilibria that are closely approximated by some "quasi–equilbrium" of the original economy. Using his corollary, Starr derived a bound on the distance from a "quasi–equilbrium" to an equilibrium of a "convexified" economy, when the number of agents exceeds the dimension of the goods. With his 1969 paper, Starr extended the scope of general equilibrium theory beyond convex sets: Thus, in the aggregate, the discrepancy between an allocation in the fictitious economy generated by [taking the convex hulls of all of the consumption and production sets] and some allocation in the real economy is bounded in a way that is independent of the number of economic agents. Therefore, the average agent experiences a deviation from intended actions that vanishes in significance as the number of agents goes to infinity. Starr began his research while he was an undergraduate at Stanford University, where he had enrolled in the (graduate) advanced mathematical economics course of Kenneth J. Arrow, who provided him with an extensive and annotated reading list. The Shapley–Folkman results are named after Lloyd Shapley and Jon Folkman, who proved both the Shapley–Folkman lemma and a weaker version of the Shapley–Folkman–Starr theorem in an unpublished report, "Starr's problem" (1966), which was cited by. . Before Starr's work, the approximate convexity of sums of non–convex sets had been discussed in the Journal of Political Economy from 1959 to 1961 by F. M. Bator, M. J. Farrell, T. C. Koopmans, and T. J. Rothenberg; these earlier economics papers lacked the mathematical propositions and proofs of Starr's paper.

Economic textbooks
I provided uses of SF-lemma to help readers find notable, reliable applications, which would be too detailed to be discussed individually in this article. Most of these textbooks are world leading, imho.

Some (or all) of the mathematical methods for economists books could be trimmed, certainly. Kiefer.Wolfowitz (talk) 20:54, 16 December 2010 (UTC)

Current state
Following Starr's 1969 paper, the Shapley–Folkman–Starr results were "much exploited in the theoretical literature", according to Guesnerie (p. 112), who wrote, "The derivation of these results in general form has been one of the major achievements of postwar economic theory". In particular, the Shapley–Folkman–Starr results were incorporated in the theory of general economic equilibria and in the theory of market failures and of public economics. The Shapley–Folkman–Starr results are introduced in graduate-level textbooks in microeconomics, general equilibrium theory, game theory, and mathematical economics. References

Nonessential books
The article need not list the following five books, which may however suggest further reading for somebody: * See Ellickson (page xviii), especially Chapter 7 "Walras meets Nash" (especially section 7.4 "Nonconvexity" pages 306–310 and 312, and also 328–329) and Chapter 8 "What is Competition?" (pages 347 and 352):
 * Theorem 1.6.5 on pages 24–25:
 * Theorem 1.6.5 on pages 24–25:

*

Thanks, Kiefer.Wolfowitz (talk) 20:59, 16 December 2010 (UTC)


 * Carter's book has the simplest example of the Shapley Folkman lemma, which is cited many times, and so it must stay. Kiefer.Wolfowitz  (Discussion) 20:07, 1 March 2011 (UTC)

Comments on splitting off Shapley-Folkman-Starr theorem?
The only way to improve this article to FA status seems imho to require removing the material on the Shapley-Folkman theorem and Starr's corollary to it, which would then be transferred to a new article on the SFS theorem. When I floated this idea on the Peer Review page, Geometry Guy suggested that such a split-off would improve the summary style of the article. David Eppstein also liked the proposal and suggested that the name "SFS theorem" would be better than "Starr's theorem" or "Starr's corollary to the SF theorem".

Please comment! Thanks! Kiefer.Wolfowitz (Discussion) 00:18, 2 March 2011 (UTC)

Closure of a set
The previous discussion followed Ekeland's notation, using the closure of a set. Sequential convergence allowed an elementary treatment: The need to consider the closure of a set is noted in the footnote. Kiefer.Wolfowitz (Discussion) 15:54, 5 March 2011 (UTC)
 * This revision is discussed at User talk:Kiefer.Wolfowitz with concern to distinguish original research and original exposition, when following wikipedia OR policy.
 * It is jarring for me to read "For example, ... For a separable problem, we consider an optimal solution ... For a separable problem, we consider an optimal solution ..." Maybe say "Given a separable problem". --P64 (talk) 18:22, 9 April 2011 (UTC)
 * That's a good suggestion. Thanks for your thoughtful comments and again thanks for alerting WP editors about the discussion on my talk page. Kiefer.Wolfowitz  (Discussion) 11:57, 10 April 2011 (UTC)

Vector measure
The article gives an example, stating the measures be defined on the same probability space. This is sloppy: they should be defined on the same finitely measurable space (that is a space with a sigma algebra on which a finite measure is definable). Kiefer .Wolfowitz 17:51, 29 September 2011 (UTC)


 * UPDATE Kiefer .Wolfowitz 03:32, 1 October 2011 (UTC)

In advanced measure-theory, the Shapley–Folkman lemma has been used to prove Lyapunov's theorem, which states that the range of a vector measure is convex. Here, the traditional term "range" (alternatively, "image") is the set of values produced by the function. A vector measure is a vector-valued generalization of a measure; for example, if p1 and p2 are probability measures defined on the same measurable space, then the product function (p1, p2) is a vector measure, where (p1, p2) is defined for every event ω by
 * ( p1, p2 ) (ω)= ( p1(ω), p2(ω) ).

Nobel Prize in Economics: Increased readership
I identified Nobel laureates associated with non-convex sets and also convex sets. Most of these have been mentioned in association with linear programming; many have already been discussed here (Arrow, Debreu, Samuelson, Koopmans, Aumann, etc.). Krugman received the prize for his work on increasing returns, which is a topic of non-convexity in economics (discussed e.g. by Partha Dasgupta).

One effect of the Nobel prize identification should be an increased readership, particularly if this page be featured on the 10th (of October) this year, when the Nobel Prize is announced. Kiefer .Wolfowitz 14:11, 3 October 2011 (UTC)

Nonconvexified versus unconvexified
This article uses "unconvexified", following our masters: Boris Mordukhovich's Variational analysis and generalized differentiation and Terry Rockafellar's and Roger Wets's Variational analysis.

The alternative "nonconvexified" is non grata.

Kiefer .Wolfowitz 18:04, 7 October 2011 (UTC)