The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Shapley–Folkman Theorem

  • Ross M. Starr
Reference work entry


The Shapley–Folkman theorem places an upper bound on the size of the non-convexities (loosely speaking, openings or holes) in a sum of non-convex sets in Euclidean N-dimensional space, RN. The bound is based on the size of nonconvexities in the sets summed and the dimension of the space. When the number of sets in the sum is large, the bound is independent of the number of sets summed, depending rather on N, the dimension of the space. Hence the size of the non-convexity in the sum becomes small as a proportion of the number of sets summed; the non-convexity per summand goes to zero as the number of summands becomes large. The Shapley–Folkman theorem can be viewed as a discrete counterpart to the Lyapunov theorem on non-atomic measures (Grodal 2002).


Large economies Lyapunov th Non-convexity Shapley–Folkman th 

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  1. Anderson, R.M. 1978. An elementary core equivalence theorem. Econometrica 46: 1483–1487.CrossRefGoogle Scholar
  2. Anderson, R.M. 1988. The second welfare theorem with nonconvex preferences. Econometrica 56: 361–382.CrossRefGoogle Scholar
  3. Arrow, K.J., and F.H. Hahn. 1972. General competitive analysis. San Francisco: Holden-Day.Google Scholar
  4. Artstein, Z., and R.A. Vitale. 1975. A strong law of large numbers for random compact sets. Annals of Probability 3: 879–882.CrossRefGoogle Scholar
  5. Artstein, Z. 1980. Discrete and continuous bang-bang and facial spaces or: Look for the extreme points. SIAM Review 22: 172–185.CrossRefGoogle Scholar
  6. Aubin, J.-P., and I. Ekeland. 1976. Estimation of the duality gap in nonconvex optimization. Mathematics of Operations Research 1 (3): 225–245.CrossRefGoogle Scholar
  7. Cassels, J.W.S. 1975. Measure of the non-convexity of sets and the Shapley–Folkman–Starr theorem. Mathematical Proceedings of the Cambridge Philosophical Society 78: 433–436.CrossRefGoogle Scholar
  8. Chambers, C.P. 2005. Multi-utilitarianism in two-agent quasilinear social choice. International Journal of Game Theory 33: 315–334.CrossRefGoogle Scholar
  9. Ekeland, I., and R. Temam. 1976. Convex analysis and variational problems. Amsterdam: North-Holland.Google Scholar
  10. Green, J., and W.P. Heller. 1981. Mathematical analysis and convexity with applications to economics. In Handbook of mathematical economics, ed. K.J. Arrow and M. Intriligator, vol. 1. Amsterdam: North-Holland.Google Scholar
  11. Grodal, B. 2002. The equivalence principle. In Optimization and Operation Research, Encyclopedia of Life Support Systems (EOLSS), ed. U. Derigs. Cologne. Online. Available at Accessed 5 Apr 2007.
  12. Hildenbrand, W., D. Schmeidler, and S. Zamir. 1973. Existence of approximate equilibria and cores. Econometrica 41: 1159–1166.CrossRefGoogle Scholar
  13. Howe, R. 1979. On the tendency toward convexity of the vector sum of sets. Discussion Paper No. 538, Cowles Foundation, Yale University.Google Scholar
  14. Manelli, A.M. 1991. Monotonie preferences and core equivalence. Econometrica 59: 123–138.CrossRefGoogle Scholar
  15. Mas-Colell, A. 1978. A note on the core equivalence theorem: How many blocking coalitions are there? Journal of Mathematical Economics 5: 207–216.CrossRefGoogle Scholar
  16. Proske, F.N., and M.L. Puri. 2002. Central limit theorem for Banach space valued fuzzy random variables. Proceedings of the American Mathematical Society 130: 1493–1501.CrossRefGoogle Scholar
  17. Starr, R.M. 1969. Quasi-equilibria in markets with non-convex preferences. Econometrica 37: 25–38.CrossRefGoogle Scholar
  18. Starr, R.M. 1981. Approximation of points of the convex hull of a sum of sets by points of the sum: An elementary approach. Journal of Economic Theory 25: 314–317.CrossRefGoogle Scholar
  19. Tardella, F. 1990. A new proof of the Lyapunov Convexity Theorem. Applied Mathematics 28: 478–481.Google Scholar
  20. Weil, W. 1982. An application of the central limit theorem for Banach space valued random variables to the theory of random sets. Probability Theory and Related Fields 60: 203–208.Google Scholar
  21. Zhou, L. 1993. A simple proof of the Shapley–Folkman theorem. Economic Theory 3: 371–372.CrossRefGoogle Scholar

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Ross M. Starr
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