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A lower bound on computational complexity given by revelation mechanisms (*)

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  • Kenneth R. Mount
  • Stanley Reiter

Abstract

This paper establishes a lower bound on the computational complexity of smooth functions between smooth manifolds. It generalizes one for finite (Boolean) functions obtained (by Arbib and Spira [2]) by counting variables. Instead of a counting procedure, which cannot be used in the infinite case, the dimension of the message space of a certain type of revelation mechanism provides the bound. It also provides an intrinsic measure of the number of variables on which the function depends. This measure also gives a lower bound on computational costs associated with realizing or implementing the function by a decentralized mechanism, or by a game form.

Suggested Citation

  • Kenneth R. Mount & Stanley Reiter, 1996. "A lower bound on computational complexity given by revelation mechanisms (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(2), pages 237-266.
    • Handle: RePEc:spr:joecth:v:7:y:1996:i:2:p:237-266
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    References listed on IDEAS

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    1. Jordan, J. S., 1982. "The competitive allocation process is informationally efficient uniquely," Journal of Economic Theory, Elsevier, vol. 28(1), pages 1-18, October.
    2. Reichelstein, Stefan, 1984. "Incentive compatibility and informational requirements," Journal of Economic Theory, Elsevier, vol. 34(1), pages 32-51, October.
    3. Reichelstein, Stefan & Reiter, Stanley, 1988. "Game Forms with Minimal Message Spaces," Econometrica, Econometric Society, vol. 56(3), pages 661-692, May.
    4. Neyman, Abraham, 1985. "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma," Economics Letters, Elsevier, vol. 19(3), pages 227-229.
    5. Kalai, Ehud & Stanford, William, 1988. "Finite Rationality and Interpersonal Complexity in Repeated Games," Econometrica, Econometric Society, vol. 56(2), pages 397-410, March.
    6. Mount, Kenneth & Reiter, Stanley, 1974. "The informational size of message spaces," Journal of Economic Theory, Elsevier, vol. 8(2), pages 161-192, June.
    7. Kenneth R. Mount & Stanley Reiter, 1983. "On the Existence of a Locally Stable Dynamic Process With a Statically Minimal Message Space," Discussion Papers 550, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    8. Chen, Pengyuan, 1992. "A lower bound for the dimension of the message space of the decentralized mechanisms realizing a given goal," Journal of Mathematical Economics, Elsevier, vol. 21(3), pages 249-270.
    9. Sonnenschein, Hugo, 1974. "An Axiomatic Characterization of the Price Mechanism," Econometrica, Econometric Society, vol. 42(3), pages 425-433, May.
    10. Stefan Reichelstein, 1981. "On the Informational Requirements for the Implementation of Social Choice Rules," Discussion Papers 507, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    11. Saari, Donald G & Simon, Carl P, 1978. "Effective Price Mechanisms," Econometrica, Econometric Society, vol. 46(5), pages 1097-1125, September.
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    Cited by:

    1. Van Zandt, Timothy, 2003. "Real-Time Hierarchical Resource Allocation with Quadratic Costs," CEPR Discussion Papers 4022, C.E.P.R. Discussion Papers.
    2. Ehud Kalai, 1995. "Games," Discussion Papers 1141, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    3. Tierney, Ryan, 2021. "Simple Social Choice Rules for Exchange," Discussion Papers on Economics 4/2021, University of Southern Denmark, Department of Economics.
    4. Marschak, Thomas, 2006. "Organization Structure," MPRA Paper 81518, University Library of Munich, Germany.
    5. Mehmet Barlo & Nuh Aygün Dalkıran, 2022. "Computational implementation," Review of Economic Design, Springer;Society for Economic Design, vol. 26(4), pages 605-633, December.

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    JEL classification:

    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles
    • O11 - Economic Development, Innovation, Technological Change, and Growth - - Economic Development - - - Macroeconomic Analyses of Economic Development
    • O47 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - Empirical Studies of Economic Growth; Aggregate Productivity; Cross-Country Output Convergence

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