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Modeling large societies: Why countable additivity is necessary

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  • Khan, M. Ali
  • Qiao, Lei
  • Rath, Kali P.
  • Sun, Yeneng

Abstract

The economic literature with an atomless countably additive measure space to model the interaction of agents in a large society is enormous. However, there have been a number of attempts to relax the countable additivity assumption by working with a finitely (but non-countably) additive measure space (such as the set of natural numbers with a density charge). The main purpose of this paper is to demonstrate the necessity of countable additivity in modeling a large society in terms of the existence of equilibrium and its idealized limit property in both general equilibrium and game theory as illustrations. In addition, we point out that in the setting of atomless finitely additive agent spaces, even approximate equilibria may not exist in general, but do so only with additional assumptions.

Suggested Citation

  • Khan, M. Ali & Qiao, Lei & Rath, Kali P. & Sun, Yeneng, 2020. "Modeling large societies: Why countable additivity is necessary," Journal of Economic Theory, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:jetheo:v:189:y:2020:i:c:s0022053120300958
    DOI: 10.1016/j.jet.2020.105102
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    as
    1. Armstrong, Thomas E. & Richter, Marcel K., 1986. "Existence of nonatomic core-walras allocations," Journal of Economic Theory, Elsevier, vol. 38(1), pages 137-159, February.
    2. Ebrahimy, Ehsan & Shimer, Robert, 2010. "Stock-flow matching," Journal of Economic Theory, Elsevier, vol. 145(4), pages 1325-1353, July.
    3. Richard McLean & Andrew Postlewaite, 2002. "Informational Size and Incentive Compatibility," Econometrica, Econometric Society, vol. 70(6), pages 2421-2453, November.
    4. Joao Gomes & Leonid Kogan & Lu Zhang, 2003. "Equilibrium Cross Section of Returns," Journal of Political Economy, University of Chicago Press, vol. 111(4), pages 693-732, August.
    5. Xavier Vives, 2017. "Endogenous Public Information and Welfare in Market Games," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 84(2), pages 935-963.
    6. Anderson, Robert M, 1985. "Strong Core Theorems with Nonconvex Preferences," Econometrica, Econometric Society, vol. 53(6), pages 1283-1294, November.
    7. HILDENBRAND, Werner, 1970. "On economies with many agents," LIDAM Reprints CORE 61, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. Williams, Jeffrey, 1987. "Futures Markets: A Consequences of Risk Aversion or Transactions Costs?," Journal of Political Economy, University of Chicago Press, vol. 95(5), pages 1000-1023, October.
    9. Felix J. Bierbrauer & Pierre C. Boyer, 2016. "Efficiency, Welfare, and Political Competition," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 131(1), pages 461-518.
    10. Martin Hellwig, 2010. "Utilitarian mechanism design for an excludable public good," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 44(3), pages 361-397, September.
    11. Sargent, Thomas J., 1991. "Equilibrium with signal extraction from endogenous variables," Journal of Economic Dynamics and Control, Elsevier, vol. 15(2), pages 245-273, April.
    12. Mordecai Kurz & Mukul Majumdar, 1972. "Efficiency Prices in Infinite Dimensional Spaces: a Synthesis," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 39(2), pages 147-158.
    13. Volij, Oscar & Serano, Roberto & Vohra, Rajiv, 2001. "On the Failure of Core Convergence," Staff General Research Papers Archive 5098, Iowa State University, Department of Economics.
    14. Pablo Casas-Arce & F. Asís Martínez-Jerez, 2009. "Relative Performance Compensation, Contests, and Dynamic Incentives," Management Science, INFORMS, vol. 55(8), pages 1306-1320, August.
    15. Darrell Duffie, 2012. "Over-The-Counter Markets," Introductory Chapters, in: Dark Markets: Asset Pricing and Information Transmission in Over-the-Counter Markets, Princeton University Press.
    16. Roberto Serrano & Rajiv Vohra & Oscar Volij, 2001. "On the Failure of Core Convergence in Economies with Asymmetric Information," Econometrica, Econometric Society, vol. 69(6), pages 1685-1696, November.
    17. Maxwell B. Stinchcombe, 1997. "Countably Additive Subjective Probabilities," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 64(1), pages 125-146.
    18. Gregory D. Hess & Athanasios Orphanides, 2001. "War and Democracy," Journal of Political Economy, University of Chicago Press, vol. 109(4), pages 776-810, August.
    19. McLean, Richard & Postlewaite, Andrew, 2005. "Core convergence with asymmetric information," Games and Economic Behavior, Elsevier, vol. 50(1), pages 58-78, January.
    20. Khan, M. Ali & Sun, Yeneng, 2002. "Non-cooperative games with many players," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 46, pages 1761-1808, Elsevier.
    21. Brown, Donald J & Robinson, Abraham, 1975. "Nonstandard Exchange Economies," Econometrica, Econometric Society, vol. 43(1), pages 41-56, January.
    22. Trockel, Walter, 1976. "A limit theorem on the core," Journal of Mathematical Economics, Elsevier, vol. 3(3), pages 247-264, December.
    23. He, Hua & Wang, Jiang, 1995. "Differential Information and Dynamic Behavior of Stock Trading Volume," The Review of Financial Studies, Society for Financial Studies, vol. 8(4), pages 919-972.
    24. Daron Acemoglu & Martin Kaae Jensen, 2015. "Robust Comparative Statics in Large Dynamic Economies," Journal of Political Economy, University of Chicago Press, vol. 123(3), pages 587-640.
    25. Weiss, Ernst-August Jr., 1981. "Finitely additive exchange economies," Journal of Mathematical Economics, Elsevier, vol. 8(3), pages 221-240, October.
    26. Peter J. Hammond, 1979. "Straightforward Individual Incentive Compatibility in Large Economies," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 46(2), pages 263-282.
    27. Voorneveld, Mark, 2010. "The possibility of impossible stairways: Tail events and countable player sets," Games and Economic Behavior, Elsevier, vol. 68(1), pages 403-410, January.
    28. Rath, Kali P, 1992. "A Direct Proof of the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(3), pages 427-433, July.
    29. Hildenbrand, Werner, 1970. "On economies with many agents," Journal of Economic Theory, Elsevier, vol. 2(2), pages 161-188, June.
    30. Nicholas Yannelis, 2009. "Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 38(2), pages 419-432, February.
    31. Dubey, Pradeep & Shapley, Lloyd S., 1994. "Noncooperative general exchange with a continuum of traders: Two models," Journal of Mathematical Economics, Elsevier, vol. 23(3), pages 253-293, May.
    32. Anderson, Robert M, 1978. "An Elementary Core Equivalence Theorem," Econometrica, Econometric Society, vol. 46(6), pages 1483-1487, November.
    33. Gerard Debreu, 1963. "On a Theorem of Scarf," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 30(3), pages 177-180.
    34. Arnaud Costinot & Dave Donaldson & Ivana Komunjer, 2012. "What Goods Do Countries Trade? A Quantitative Exploration of Ricardo's Ideas," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 79(2), pages 581-608.
    35. Vives, Xavier, 1988. "Aggregation of Information in Large Cournot Markets," Econometrica, Econometric Society, vol. 56(4), pages 851-876, July.
    36. Khan, M. Ali & Sun, Yeneng, 1999. "Non-cooperative games on hyperfinite Loeb spaces1," Journal of Mathematical Economics, Elsevier, vol. 31(4), pages 455-492, May.
    37. M Ali Khan & Yeneng Sun, 1996. "Non-Atomic Games on Loeb Spaces," Economics Working Paper Archive 374, The Johns Hopkins University,Department of Economics, revised Aug 1996.
    38. Armstrong, Thomas E. & Richter, Marcel K., 1984. "The core-walras equivalence," Journal of Economic Theory, Elsevier, vol. 33(1), pages 116-151, June.
    39. Stinchcombe, Maxwell B., 2016. "Objective and subjective foundations for multiple priors," Journal of Economic Theory, Elsevier, vol. 165(C), pages 263-291.
    40. Cole, Harold L & Mailath, George J & Postlewaite, Andrew, 1992. "Social Norms, Savings Behavior, and Growth," Journal of Political Economy, University of Chicago Press, vol. 100(6), pages 1092-1125, December.
    41. Gul, Faruk & Postlewaite, Andrew, 1992. "Asymptotic Efficiency in Large Exchange Economies with Asymmetric Information," Econometrica, Econometric Society, vol. 60(6), pages 1273-1292, November.
    42. Bierbrauer, Felix & Sahm, Marco, 2010. "Optimal democratic mechanisms for taxation and public good provision," Journal of Public Economics, Elsevier, vol. 94(7-8), pages 453-466, August.
    43. Pradeep Dubey & Lloyd S. Shapley, 1977. "Noncooperative Exchange with a Continuum of Traders," Cowles Foundation Discussion Papers 447, Cowles Foundation for Research in Economics, Yale University.
    44. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    45. Feldman, Mark & Gilles, Christian, 1985. "An expository note on individual risk without aggregate uncertainty," Journal of Economic Theory, Elsevier, vol. 35(1), pages 26-32, February.
    46. Majumdar, Mukul, 1972. "Some general theorems on efficiency prices with an infinite-dimensional commodity space," Journal of Economic Theory, Elsevier, vol. 5(1), pages 1-13, August.
    47. SCHMEIDLER, David, 1973. "Equilibrium points of nonatomic games," LIDAM Reprints CORE 146, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    48. Ennis, Huberto M. & Keister, Todd, 2010. "Banking panics and policy responses," Journal of Monetary Economics, Elsevier, vol. 57(4), pages 404-419, May.
    49. Sun, Yeneng, 2006. "The exact law of large numbers via Fubini extension and characterization of insurable risks," Journal of Economic Theory, Elsevier, vol. 126(1), pages 31-69, January.
    50. BEWLEY, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," LIDAM Reprints CORE 122, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    51. Khan, M Ali, 1974. "Some Remarks on the Core of a "Large" Economy," Econometrica, Econometric Society, vol. 42(4), pages 633-642, July.
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    More about this item

    Keywords

    Countable additivity; Finite additivity; Competitive equilibrium; Nash equilibrium; Existence of equilibrium; Idealized limit property;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D50 - Microeconomics - - General Equilibrium and Disequilibrium - - - General
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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