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

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  • M. Ali Khan
  • Kali P. Rath

Abstract

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.

Suggested Citation

  • M. Ali Khan & Kali P. Rath, 2011. "The Shapley-Folkman Theorem and the Range of a Bounded Measure: An Elementary and Unified Treatment," Economics Working Paper Archive 586, The Johns Hopkins University,Department of Economics.
    • Handle: RePEc:jhu:papers:586
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    References listed on IDEAS

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    1. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, September.
    2. 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.
    3. Carmona, Guilherme & Podczeck, Konrad, 2009. "On the existence of pure-strategy equilibria in large games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1300-1319, May.
    4. 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 371-372, April.
    5. Claude Henry, 1972. "Market Games with Indivisible Commodities and Non-convex Preferences," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 39(1), pages 73-76.
    6. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
    7. Robert M. Anderson & William R. Zame, 1997. "Edgeworth's Conjecture with Infinitely Many Commodities: L1," Econometrica, Econometric Society, vol. 65(2), pages 225-274, March.
    8. Robert M. Anderson & M. Ali Khan & Salim Rashid, 1982. "Approximate Equilibria with Bounds Independent of Preferences," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 49(3), pages 473-475.
    9. Carmona, Guilherme, 2004. "On the purification of Nash equilibria of large games," Economics Letters, Elsevier, vol. 85(2), pages 215-219, November.
    10. Cassels,J. W. S., 1981. "Economics for Mathematicians," Cambridge Books, Cambridge University Press, number 9780521286145, January.
    11. 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.
    12. Rashid, Salim, 1985. "The approximate purification of mixed strategies with finite observation sets," Economics Letters, Elsevier, vol. 19(2), pages 133-135.
    13. William Geller, 1986. "An Improved Bound for Approximate Equilibria," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(2), pages 307-308.
    14. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
    15. 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.
    16. 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.
    17. Anderson, Robert M., 1987. "Gap-minimizing prices and quadratic core convergence," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 1-15, February.
    18. Anderson, Robert M, 1988. "The Second Welfare Theorem with Nonconvex Preferences," Econometrica, Econometric Society, vol. 56(2), pages 361-382, March.
    19. Anderson, Robert M, 1978. "An Elementary Core Equivalence Theorem," Econometrica, Econometric Society, vol. 46(6), pages 1483-1487, November.
    20. Broome, John, 1972. "Approximate equilibrium in economies with indivisible commodities," Journal of Economic Theory, Elsevier, vol. 5(2), pages 224-249, October.
    21. Anderson, Robert M, 1982. "A Market Value Approach to Approximate Equilibria," Econometrica, Econometric Society, vol. 50(1), pages 127-136, January.
    22. Heller, Walter Perrin, 1972. "Transactions with set-up costs," Journal of Economic Theory, Elsevier, vol. 4(3), pages 465-478, June.
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