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The Continuity Postulate in Economic Theory: A Deconstruction and an Integration

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  • Metin Uyanik
  • M. Ali Khan

Abstract

This paper presents six theorems and ten propositions that can be read as deconstructing and integrating the continuity postulate under the rubric of pioneering work of Eilenberg, Wold, von Neumann-Morgenstern, Herstein-Milnor and Debreu. Its point of departure is the fact that the adjective continuous applied to a function or a binary relation does not acknowledge the many meanings that can be given to the concept it names, and that under a variety of technical mathematical structures, its many meanings can be whittled down to novel and unexpected equivalences that have been missed in the theory of choice. Specifically, it provides a systematic investigation of the two-way relation between restricted and full continuity of a function and a binary relation that, under convex, monotonic and differentiable structures, draws out the behavioral implications of the postulate.

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  • Metin Uyanik & M. Ali Khan, 2021. "The Continuity Postulate in Economic Theory: A Deconstruction and an Integration," Papers 2108.11736, arXiv.org, revised Jan 2022.
  • Handle: RePEc:arx:papers:2108.11736
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    1. Kim, Taesung, 1987. "Intransitive Indifference and Revealed Preference," Econometrica, Econometric Society, vol. 55(1), pages 163-167, January.
    2. Schmeidler, David, 1969. "Competitive Equilibria in Markets with a Continuum of Traders and Incomplete Preferences," Econometrica, Econometric Society, vol. 37(4), pages 578-585, October.
    3. M. Ali Khan & Metin Uyanık, 2021. "Topological connectedness and behavioral assumptions on preferences: a two-way relationship," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 411-460, March.
    4. Itzhak Gilboa & Fabio Maccheroni & Massimo Marinacci & David Schmeidler, 2010. "Objective and Subjective Rationality in a Multiple Prior Model," Econometrica, Econometric Society, vol. 78(2), pages 755-770, March.
    5. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, vol. 74(6), pages 1447-1498, November.
    6. Karni, Edi & Safra, Zvi, 2015. "Continuity, completeness, betweenness and cone-monotonicity," Mathematical Social Sciences, Elsevier, vol. 74(C), pages 68-72.
    7. Juan Dubra & Fabio Maccheroni & Efe A. Ok, 2004. "Expected Utility Without the Completeness Axiom," Yale School of Management Working Papers ysm404, Yale School of Management.
    8. Inoue, Tomoki, 2010. "A utility representation theorem with weaker continuity condition," Journal of Mathematical Economics, Elsevier, vol. 46(1), pages 122-127, January.
    9. Candeal-Haro, Juan Carlos & Indurain-Eraso, Esteban, 1995. "A Note on Linear Utility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 6(3), pages 519-522, November.
    10. Kaushik Basu & Tapan Mitra, 2003. "Aggregating Infinite Utility Streams with InterGenerational Equity: The Impossibility of Being Paretian," Econometrica, Econometric Society, vol. 71(5), pages 1557-1563, September.
    11. Anderson, Robert M. & Duanmu, Haosui & Khan, M. Ali & Uyanik, Metin, 2022. "On abstract economies with an arbitrary set of players and action sets in locally-convex topological vector spaces," Journal of Mathematical Economics, Elsevier, vol. 98(C).
    12. Stanca, Lorenzo, 2020. "A simplified approach to subjective expected utility," Journal of Mathematical Economics, Elsevier, vol. 87(C), pages 151-160.
    13. Kim, Taesung & Richter, Marcel K., 1986. "Nontransitive-nontotal consumer theory," Journal of Economic Theory, Elsevier, vol. 38(2), pages 324-363, April.
    14. Carlos Hervés‐Beloso & Hugo del Valle‐Inclán Cruces, 2019. "Continuous Preference Orderings Representable By Utility Functions," Journal of Economic Surveys, Wiley Blackwell, vol. 33(1), pages 179-194, February.
    15. Diewert, W. E. & Avriel, M. & Zang, I., 1981. "Nine kinds of quasiconcavity and concavity," Journal of Economic Theory, Elsevier, vol. 25(3), pages 397-420, December.
    16. Dubra, Juan & Maccheroni, Fabio & Ok, Efe A., 2004. "Expected utility theory without the completeness axiom," Journal of Economic Theory, Elsevier, vol. 115(1), pages 118-133, March.
    17. Gerasimou, Georgios, 2015. "(Hemi)continuity of additive preference preorders," Journal of Mathematical Economics, Elsevier, vol. 58(C), pages 79-81.
    18. Schmeidler, David, 1971. "A Condition for the Completeness of Partial Preference Relations," Econometrica, Econometric Society, vol. 39(2), pages 403-404, March.
    19. M. Ali Khan & Metin Uyanik, 2021. "The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 799-840, April.
    20. Basu, Kaushik & Mitra, Tapan, 2007. "Utilitarianism for infinite utility streams: A new welfare criterion and its axiomatic characterization," Journal of Economic Theory, Elsevier, vol. 133(1), pages 350-373, March.
    21. Peleg, Bezalel, 1970. "Utility Functions for Partially Ordered Topological Spaces," Econometrica, Econometric Society, vol. 38(1), pages 93-96, January.
    22. Neuefeind, Wilhelm & Trockel, Walter, 1995. "Continuous Linear Representability of Binary Relations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 6(2), pages 351-356, July.
    23. Wakker, Peter, 1988. "Continuity of Preference Relations for Separable Topologies," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 29(1), pages 105-110, February.
    24. Baucells, Manel & Shapley, Lloyd S., 2008. "Multiperson utility," Games and Economic Behavior, Elsevier, vol. 62(2), pages 329-347, March.
    25. Tapan Mitra & M. Ozbek, 2013. "On representation of monotone preference orders in a sequence space," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 473-487, September.
    26. Lee, Lung-Fei, 1972. "The Theorems of Debreu and Peleg for Ordered Topological Spaces," Econometrica, Econometric Society, vol. 40(6), pages 1151-1153, November.
    27. Philip J. Reny, 2020. "Nash Equilibrium in Discontinuous Games," Annual Review of Economics, Annual Reviews, vol. 12(1), pages 439-470, August.
    28. Dubra, Juan, 2011. "Continuity and completeness under risk," Mathematical Social Sciences, Elsevier, vol. 61(1), pages 80-81, January.
    29. Brown, Donald J. & Robinson, Abraham, 1974. "The cores of large standard exchange economies," Journal of Economic Theory, Elsevier, vol. 9(3), pages 245-254, November.
    30. McCarthy, David & Mikkola, Kalle, 2018. "Continuity and completeness of strongly independent preorders," Mathematical Social Sciences, Elsevier, vol. 93(C), pages 141-145.
    31. Aniruddha Ghosh & M. Ali Khan & Metin Uyanık, 2023. "Continuity postulates and solvability axioms in economic theory and in mathematical psychology: a consolidation of the theory of individual choice," Theory and Decision, Springer, vol. 94(2), pages 189-210, February.
    32. Yi, Gyoseob, 1993. "Continuous extension of preferences," Journal of Mathematical Economics, Elsevier, vol. 22(6), pages 547-555.
    33. Brock, William A, 1970. "An Axiomatic Basis for the Ramsey- Weizsacker Overtaking Criterion," Econometrica, Econometric Society, vol. 38(6), pages 927-929, November.
    34. Epstein, Larry G, 1987. "The Unimportance of the Intransitivity of Separable Preferences," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 28(2), pages 315-322, June.
    35. Grant, Simon & Kajii, Atsushi & Polak, Ben, 1992. "Many good risks: An interpretation of multivariate risk and risk aversion without the Independence axiom," Journal of Economic Theory, Elsevier, vol. 56(2), pages 338-351, April.
    36. Galaabaatar, Tsogbadral & Khan, M. Ali & Uyanık, Metin, 2019. "Completeness and transitivity of preferences on mixture sets," Mathematical Social Sciences, Elsevier, vol. 99(C), pages 49-62.
    37. Shafer, Wayne J, 1974. "The Nontransitive Consumer," Econometrica, Econometric Society, vol. 42(5), pages 913-919, September.
    38. Lorenz Hartmann, 2020. "Savage's P3 Is Redundant," Econometrica, Econometric Society, vol. 88(1), pages 203-205, January.
    39. Beardon, Alan F & Mehta, Ghanshyam B, 1994. "The Utility Theorems of Wold, Debreu, and Arrow-Hahn," Econometrica, Econometric Society, vol. 62(1), pages 181-186, January.
    40. Kopylov, Igor, 2016. "Canonical utility functions and continuous preference extensions," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 32-37.
    41. Duggan, John, 1999. "A General Extension Theorem for Binary Relations," Journal of Economic Theory, Elsevier, vol. 86(1), pages 1-16, May.
    42. Giarlotta, Alfio & Greco, Salvatore, 2013. "Necessary and possible preference structures," Journal of Mathematical Economics, Elsevier, vol. 49(2), pages 163-172.
    43. Bosi, Gianni & Herden, Gerhard, 2012. "Continuous multi-utility representations of preorders," Journal of Mathematical Economics, Elsevier, vol. 48(4), pages 212-218.
    44. David S. Ahn & Haluk Ergin, 2010. "Framing Contingencies," Econometrica, Econometric Society, vol. 78(2), pages 655-695, March.
    45. Nakamura, Yutaka, 1990. "Subjective expected utility with non-additive probabilities on finite state spaces," Journal of Economic Theory, Elsevier, vol. 51(2), pages 346-366, August.
    46. Dekel, Eddie, 1986. "An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom," Journal of Economic Theory, Elsevier, vol. 40(2), pages 304-318, December.
    47. Herden, G., 1995. "On some equivalent approaches to Mathematical Utility Theory," Mathematical Social Sciences, Elsevier, vol. 29(1), pages 19-31, February.
    48. Georgios Gerasimou, 2013. "On continuity of incomplete preferences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 157-167, June.
    49. Uyanık, Metin & Khan, M. Ali, 2019. "On the consistency and the decisiveness of the double-minded decision-maker," Economics Letters, Elsevier, vol. 185(C).
    50. Wakker, Peter, 1993. "Additive representations on rank-ordered sets : II. The topological approach," Journal of Mathematical Economics, Elsevier, vol. 22(1), pages 1-26.
    51. Yamazaki, Akira, 1983. "On Open Preferences," Hitotsubashi Journal of Economics, Hitotsubashi University, vol. 24(2), pages 149-152, December.
    52. Abdellaoui, Mohammed & Wakker, Peter P., 2020. "Savage for dummies and experts," Journal of Economic Theory, Elsevier, vol. 186(C).
    53. Bergstrom, Theodore C. & Parks, Robert P. & Rader, Trout, 1976. "Preferences which have open graphs," Journal of Mathematical Economics, Elsevier, vol. 3(3), pages 265-268, December.
    54. Simone Cerreia-Vioglio & Efe A. Ok, 2018. "The Rational Core of Preference Relations," Working Papers 632, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    55. Hong Chew Soo & Karni Edi, 1994. "Choquet Expected Utility with a Finite State Space: Commutativity and Act-Independence," Journal of Economic Theory, Elsevier, vol. 62(2), pages 469-479, April.
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    Cited by:

    1. Aniruddha Ghosh & M. Ali Khan & Metin Uyanık, 2023. "Continuity postulates and solvability axioms in economic theory and in mathematical psychology: a consolidation of the theory of individual choice," Theory and Decision, Springer, vol. 94(2), pages 189-210, February.
    2. Metin Uyanik & Aniruddha Ghosh & M. Ali Khan, 2023. "Separately Convex and Separately Continuous Preferences: On Results of Schmeidler, Shafer, and Bergstrom-Parks-Rader," Papers 2310.00531, arXiv.org.
    3. Marie-Louise Vierø, 2022. "Lost in objective translation: Awareness of unawareness when unknowns are not simply unknowns," Economics Working Papers 2022-06, Department of Economics and Business Economics, Aarhus University.
    4. Aniruddha Ghosh & Mohammed Ali Khan & Metin Uyanik, 2022. "The Intermediate Value Theorem and Decision-Making in Psychology and Economics: An Expositional Consolidation," Games, MDPI, vol. 13(4), pages 1-24, July.

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