Subjective risk and disappointment
If an investor does care for utilities -and not for monetary outcomes- stochastic dominances should be expressed in terms of utility units ("utils"). If so, any "rational" investor may be characterized by an elementary utility function -called canonical utility function- which is such that the partial weak order induced by stochastic dominance over utils is as "close" to the weak order of preferences as possible. As a consequence, the random utilities of the available prospects do not violate the second-order stochastic dominance property. Substituting utils for monetary units leads to substitute "subjective" risk for "objective" risk à la Rothschild and Stiglitz (1970). A weakened independence axiom may them be set over comparable prospects, i.e. those which exhibit the same canonical expected utility. This leads to a fully choice-based theory of disappointment. The functional is lottery-dependent (Becker and Sarin 1987). When constant marginal utility is assumed, it is but the opposite to a convex measure of risk (Föllmer and Schied 2002). It may be viewed as a theoretical justification for choosing this measure of risk.
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