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Unformatted text preview: Incomplete Menu Preferences and Ambiguity * Madhav Chandrasekher Current Draft: February 28, 2011 Abstract: This paper models decision problems where the decision-maker (DM) may not be able to compare two risky acts. Incomparability of acts can, in prin- ciple, stem from two sources. First, perception of risk is subjective and the DM might have only a coarse assessment of the probability measure over the space of ex post outcomes that determine the values of the acts. Second, there may be co- dependence on the realized outcome and the set of the consumption choices that are available conditional on that outcome. Existing models of incomplete preferences, which build on Bewleys (1986) model of Knightian Uncertainty, accommodate the first consideration, but foreclose the second possibility as a source of incompleteness. This paper presents an axiomatic model of incomplete preferences that accounts for both of these possibilities as a source of incompleteness. Keywords : Menu Choice, Incomplete Preferences, Knightian Uncertainty. * This paper has benefitted from the comments of the audiences of the Spring 2010 Midwest Theory meeting at Northwestern and at SWET 2011 at CalPoly, San Luis Obispo. I am also grateful to Peter Klibanoff, Fabio Maccheroni, Ed Schlee, and Kyoungwon Seo for helpful com- ments. This paper was completed while I was a visitor at the Math Center of the Kellogg School at Northwestern University. I thank them for their hospitality. Comments are welcome. Mailing address: Department of Economics, W.P. Carey School of Business, P.O. Box 873806, Tempe, AZ 85287-3806; E-mail address: firstname.lastname@example.org 1 Introduction This paper generalizes Bewleys classic formulation of Knightian ambiguity to a set- ting where the decision-makers domain of choice is the set of subsets of ( X ), the space of lotteries on a finite set X . Preferences over this domain have been given the moniker preferences over subjective contingencies after the seminal paper by DLR (2001) that analyzed such preferences and provided an important represen- tation result. The DLR paper shows that preferences on 2 ( X ) that satisfy the Herstein-Milnor mixture space axioms admit a utility representation that takes the form of a (state-dependent) expected utility functional. Moreover, the utility kernel and the probability space over which the expectation operator is defined are shown to be essentially unique. 1 The DLR Theorem provides a beautiful counterpart (on both the axiomatic and functional form side) of the Anscombe-Aumann Theorem. The model interprets a menu as an implicit Anscombe-Aumann act, where the domain of the act (the subjective state space) is in the mind of the decision-maker. An important axiom implied by the DLR representation is the order axiom, which requires preferences over menus to be complete. The starting point of this paper is to analyze what hap- pens when we relax this assumption. Incomplete preferences were famously analyzedpens when we relax this assumption....
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This note was uploaded on 04/28/2011 for the course BUS 315 taught by Professor Powell during the Spring '11 term at University of Hawaii, Manoa.
- Spring '11