Aspirational preferences and their representation by risk measures (Online Appendix)
نویسندگان
چکیده
Though the general setup does not require a particular probability measure, there may be situations in which the decision maker makes choices according to some subjective, probabilistic beliefs. For such decision makers, it is useful to understand the implied stochastic ordering properties of their choices relative to the underlying probability measure P ∈ P that they are using. In this section, we characterize these properties for aspiration measures. We show that aspiration measures share the stochastic dominance properties of their underlying risk family. Moreover, we show that under the mild assumption that the aspiration measure is indifferent to all acts with the same distribution under P, then the aspiration measure preserves first-order stochastic dominance (FSD) for all acts, second-order stochastic dominance (SSD) for all acts in the diversification favoring set, and risk-seeking stochastic dominance (RSSD) for all acts in the concentration favoring set. We first recall the definition of the stochastic orders just mentioned. Note that in this section, if not specified explicitly, expectations are taken with respect to the probability measure P. We say that f dominates g by FSD if and only if E [u(g)] ≥ E [u(g)] for all nondecreasing functions u; in this case we write f ≥(1) g. Similarly, f dominates g by SSD (respectively RSSD) if and only if E [u(f)] ≥ E [u(g)] for all u nondecreasing and concave (respectively convex); in this case we write f ≥(2) g (respectively f ≥(−2) g). Equivalent definitions of first order, second order and risk-seeking stochastic dominance can be found in Levy (2006). We first note the following.
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Aspirational Preferences and Their Representation by Risk Measures
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