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Locally Complete Path Independent Choice Functions and Their Lattices

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  • Richard A. Dean

    (California Institute of Technology)

  • Mark R. Johnson

    (Tulane University)

Abstract

The concept of path independence (PI) was first introduced by Arrow (1963) as a defense of his requirement that collective choices be rationalized by a weak ordering. Plott (1973) highlighted the dynamic aspects of PI implicit in Arrow's initial discussion. Throughout these investigations two questions, both initially raised by Plott, remained unanswered. What are the precise mathematical foundations for path independence? How can PI choice functions be constructed? We give complete answers to both these questions for finite domains and provide necessary conditions for infinite domains. We introduce a lattice associated with each PI function. For finite domains these lattices coincide with locally lower distributive or meet-distributive lattices and uniquely characterize PI functions. We also present an algorithm, effective and exhaustive for finite domains, for the construction of PI choice functions and hence for all finite locally lower distributive lattices. For finite domains, a PI function is rationalizable if and only if the lattice is distributive. The lattices associated with PI functions that satisfy the stronger condition of the weak axiom of revealed preference are chains of Boolean algebras and conversely. Those that satisfy the strong axiom of revealed preference are chains and conversely.

Suggested Citation

  • Richard A. Dean & Mark R. Johnson, 2000. "Locally Complete Path Independent Choice Functions and Their Lattices," Econometric Society World Congress 2000 Contributed Papers 0622, Econometric Society.
    • Handle: RePEc:ecm:wc2000:0622
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    References listed on IDEAS

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    1. Johnson Mark R., 1995. "Ideal Structures of Path Independent Choice Functions," Journal of Economic Theory, Elsevier, vol. 65(2), pages 468-504, April.
    2. Johnson, Mark R., 1990. "Information, associativity, and choice requirements," Journal of Economic Theory, Elsevier, vol. 52(2), pages 440-452, December.
    3. Plott, Charles R, 1973. "Path Independence, Rationality, and Social Choice," Econometrica, Econometric Society, vol. 41(6), pages 1075-1091, November.
    4. Malishevski, Andrey V., 1994. "Path independence in serial--parallel data processing," Mathematical Social Sciences, Elsevier, vol. 27(3), pages 335-367, June.
    5. Sertel, Murat R & Van der Bellen, Alexander, 1979. "Synopses in the Theory of Choice," Econometrica, Econometric Society, vol. 47(6), pages 1367-1389, November.
    6. Kelly, Jerry S., 1984. "The Sertel and Van der Bellen problems," Mathematical Social Sciences, Elsevier, vol. 8(3), pages 287-290, December.
    7. Koshevoy, Gleb A., 1999. "Choice functions and abstract convex geometries," Mathematical Social Sciences, Elsevier, vol. 38(1), pages 35-44, July.
    8. Sertel, Murat R., 1988. "Characterizing fidelity for reflexive choices," Mathematical Social Sciences, Elsevier, vol. 15(1), pages 93-95, February.
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    Cited by:

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    2. Alva, Samson, 2018. "WARP and combinatorial choice," Journal of Economic Theory, Elsevier, vol. 173(C), pages 320-333.

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