O'Callaghan, Patrick
(2016):
*Parametric continuity from preferences when the topology is weak and actions are discrete.*

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## Abstract

Empirical settings often involve discrete actions and rich parameter spaces where the notion of open set is constrained. This restricts the class of continuous functions from parameters to actions. Yet suitably continuous policies and value functions are necessary for many standard results in economic theory. We derive these tools from preferences when the parameter space is normal (disjoint closed sets can be separated). Whereas we use preferences to generate an endogenous pseudometric, existing results require metrizable parameter spaces. Still, weakly ordered parameters do not form a normal space. We provide a solution and close with an algorithm for eliciting preferences.

Item Type: | MPRA Paper |
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Original Title: | Parametric continuity from preferences when the topology is weak and actions are discrete |

Language: | English |

Keywords: | Continuity, preferences, metrizability |

Subjects: | C - Mathematical and Quantitative Methods > C0 - General C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling |

Item ID: | 72356 |

Depositing User: | Mr Patrick O'Callaghan |

Date Deposited: | 04 Jul 2016 12:08 |

Last Modified: | 02 Oct 2019 20:27 |

References: | [1] Charalambos D. Aliprantis and Kim Border. Infinite dimensional analysis: a hitchhiker’s guide. 3rd. Springer, 2006. [2] Alessandro Caterino, Rita Ceppitelli, and Francesca Maccarino. “Continuous utility functions on submetrizable hemicompact k-spaces”. Applied General Topology 10 (2 2009), pp. 187–195. [3] Kate Connolly. Angela Merkel comforts sobbing refugee but says Ger- many can’t help everyone. The Guardian. 16 July 2015. url: https: //www.theguardian.com/world/2015/jul/16/angela-merkel- comforts-teenage-palestinian-asylum-seeker-germany. [4] Phoebus J. Dhrymes. Mathematics for econometrics. 4th. Springer, 1978. [5] Prajit K. Dutta, Mukul K. Ma jumdar, and Rangara jan K. Sundaram. “Parametric continuity in dynamic programming problems”. Journal of Economic Dynamics and Control 18.6 (1994), pp. 1069–1092. [6] Peter C. Fishburn. The foundations of expected utility. Theory & Decision Library V.31. Springer Science and Business Media, BV, 1982. [7] Peter C. Fishburn. Utility theory for decision making. Publications in operations research. R. E. Krieger Pub. Co., 1979. [8] David Gale. “Compact sets of functions and function rings”. Proceedings of the American Mathematical Society 1.3 (1950), pp. 303–308. [9] Itzhak Gilboa and David Schmeidler. “A derivation of expected utility maximization in the context of a game”. Games and Economic Behavior 44.1 (2003), pp. 172–182. [10] Itzhak Gilboa and David Schmeidler. “Act similarity in case-based decision theory”. Economic Theory 9.1 (1997), pp. 47–61. [11] Chris Good and Ian Stares. “New proofs of classical insertion theorems”. Comment. Math. Univ. Carolinae 41.1 (2000), pp. 139–142. [12] Juha Heinonen. “Nonsmooth calculus”. Bulletin of the American Mathematical Society 44.2 (2007), pp. 163–232. [13] Werner J. Hildenbrand. “On Economies with Many Agents”. Journal of Economic Theory 2 (1970), pp. 161–188. [14] Roy A. Johnson. “A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta”. The American Mathematical Monthly 77.2 (1970), pp. 172–176. [15] John L. Kelley. General topology. Van Nostrand, 1955. [16] David M. Kreps. Notes on the theory of choice. Underground classics in economics. Westview Press, 1988. [17] Vladimir L. Levin. “A Continuous Utility Theorem for Closed Preorders on a σ-Compact Metrizable Space”. Soviet Math. Doklady 28 (1983), pp. 715–718. [18] Andreu Mas-Colell. “On the Continuous Representation of Preorders”. International Economic Review 18 (1977), pp. 509–513. [19] Andrew McLennan. “Advanced fixed point theory for economics”. Unpublished manuscript, University of Queensland (2012). [20] Jean-Francois Mertens and Shmuel Zamir. “Formulation of Bayesian analysis for games with incomplete information”. International jounal of game theory 14 (1 1985). [21] Ernest Michael. “Continuous Selections I”. The Annals of Mathematics 63(2) (1956), pp. 361–382. [22] Ernest Michael. “Topologies on spaces of subsets”. Transactions of the American Mathematical Society 71.1 (1951), pp. 152–182. [23] Paul Milgrom and Ilya Segal. “Envelope theorems for arbitrary choice sets”. Econometrica 70.2 (2002), pp. 583–601. [24] James R. Munkres. Topology. 2nd. Prentice Hall, 2000. [25] Edward Nelson. “Feynman integrals and the Schrodinger equation”. Journal of Mathematical Physics 5.3 (1964), pp. 332–343. [26] David Nualart. The Malliavin calculus and related topics. Vol. 1995. Springer, 2006. [27] Kalyanapuram R. Parthasarathy. Probability measures on metric spaces. Vol. 352. American Mathematical Soc., 1967. [28] Raaj Kumar Sah and Jingang Zhao. “Some envelope theorems for inte- ger and discrete choice variables”. International Economic Review 39.3 (1998), pp. 623–634. [29] Arthur H. Stone. “Paracompactness and product spaces”. Bulletin of the American Mathematical Society 54.10 (1948), pp. 977–982. [30] Michael E. Taylor. Measure theory and integration. Vol. 76. Graduate Studies in Mathematics. American Mathematical Society, 2006. [31] Vladimir V. Tkachuk. A Cp-Theory Problem Book. Springer, 2011. [32] John von Neumann and Oscar Morgenstern. Theory of games and economic behavior. Sixtieth anniversary. Princeton and Oxford: Princeton University Press, 1944. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/72356 |