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The Shapley-Folkman Theorem and the Range of a Bounded Measure: An Elementary and Unified Treatment

  • M. Ali Khan
  • Kali P. Rath

We present proofs, based on the Shapley-Folkman theorem, of the convexity of the range of a strongly continuous, finitely additive measure, as well as that of an atomless, countably additive measure. We also present proofs, based on diagonalization and separation arguments respectively, of the closure of the range of a purely atomic or purely nonatomic countably additive measure. A combination of these results yields Lyapunov's celebrated theorem on the range of a countably additive measure. We also sketch, through a comprehensive bibliography, the pervasive diversity of the applications of the Shapley-Folkman theorem in mathematical economics.

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Paper provided by The Johns Hopkins University,Department of Economics in its series Economics Working Paper Archive with number 586.

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Date of creation: Dec 2011
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Handle: RePEc:jhu:papers:586
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  1. Robert M. Anderson & M. Ali Khan & Salim Rashid, 1982. "Approximate Equilibria with Bounds Independent of Preferences," Review of Economic Studies, Oxford University Press, vol. 49(3), pages 473-475.
  2. Anderson, Robert M, 1982. "A Market Value Approach to Approximate Equilibria," Econometrica, Econometric Society, vol. 50(1), pages 127-36, January.
  3. Anderson, Robert M, 1988. "The Second Welfare Theorem with Nonconvex Preferences," Econometrica, Econometric Society, vol. 56(2), pages 361-82, March.
  4. Carmona, Guilherme, 2003. "On the Purification of Nash Equilibria of Large Games," FEUNL Working Paper Series wp436, Universidade Nova de Lisboa, Faculdade de Economia.
  5. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, November.
  6. Shaked, A., 1976. "Absolute approximations to equilibrium in markets with non-convex preferences," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 185-196, July.
  7. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
  8. Starr, Ross 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, Elsevier, vol. 25(2), pages 314-317, October.
  9. Carmona, Guilherme & Podczeckz, Konrad, 2008. "On the Existence of Pure-Strategy Equilibria in Large Games," FEUNL Working Paper Series wp531, Universidade Nova de Lisboa, Faculdade de Economia.
  10. Anderson, Robert M, 1978. "An Elementary Core Equivalence Theorem," Econometrica, Econometric Society, vol. 46(6), pages 1483-87, November.
  11. Broome, John, 1972. "Approximate equilibrium in economies with indivisible commodities," Journal of Economic Theory, Elsevier, vol. 5(2), pages 224-249, October.
  12. Carmona, Guilherme, 2008. "Purification of Bayesian-Nash equilibria in large games with compact type and action spaces," Journal of Mathematical Economics, Elsevier, vol. 44(12), pages 1302-1311, December.
  13. Anderson, Robert M. & Zame, William R., 1995. "Edgeworth's Conjecture with Infinitely Many Commodities," Department of Economics, Working Paper Series qt5kb2x3cd, Department of Economics, Institute for Business and Economic Research, UC Berkeley.
  14. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
  15. Zhou, Lin, 1993. "A Simple Proof of the Shapley-Folkman Theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(2), pages 371-72, April.
  16. Anderson, Robert M., 1987. "Gap-minimizing prices and quadratic core convergence," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 1-15, February.
  17. Heller, Walter Perrin, 1972. "Transactions with set-up costs," Journal of Economic Theory, Elsevier, vol. 4(3), pages 465-478, June.
  18. William Geller, 1986. "An Improved Bound for Approximate Equilibria," Review of Economic Studies, Oxford University Press, vol. 53(2), pages 307-308.
  19. Claude Henry, 1972. "Market Games with Indivisible Commodities and Non-convex Preferences," Review of Economic Studies, Oxford University Press, vol. 39(1), pages 73-76.
  20. Yannelis, Nicholas C., 1983. "Existence and fairness of value allocation without convex preferences," Journal of Economic Theory, Elsevier, vol. 31(2), pages 283-292, December.
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