The S-Shaped Value Function as a Constrained Optimum
提出一种近似效用函数，在总敏感度受限的条件下，决策者通过最优分配敏感度来最小化误差，从而推导出S形价值函数，为前景理论提供了经济学最优性基础。
Almost since the Expected Utility Hypothesis (EUH) was first introduced, evidence has accumulated that decision makers systematically violate it in various ways. One early response to the evidence was to develop ad hoc alternatives, the best known of which is Prospect Theory (Daniel Kahneman and Amos Tversky, 1979).1 More recent responses have relaxed the independence axiom or other axioms underlying the EUH. See Mark Machina (1987) for a very readable survey of the evidence and responses. Despite the high intellectual caliber of much of this work, there is an important sense in which it has been retrograde: theory is adjusted to the evidence by weakening, not strengthening, its predictive power. In other areas (such as the theory of the firm or the theory of money) economists have followed a different research strategy: more subtle constraints (such as transactions costs or informational imperfections) are sought to explain data that seem anomalous according to received theory. When successful, such a strategy more clearly delineates the realm in which the older theory is applicable and makes new, testable predictions outside that realm. Thus theory is progressively strengthened. To my knowledge, only Jonathan Leland (1986, 1988) has adopted such a progressive strategy in a theoretical investigation of EUH anomalies. The key assumptions of his approximate expected utility theory (AEU) are (1) inexperience and/or cognitive limitations make the true utility function inaccessible, so some approximation must be used; but (2) decision makers efficiently allocate finite resources so as to minimize the resulting errors. He formalizes these assumptions (which he applies to probability assessments as well as to utilities) in terms of a constraint on the number N of steps allowed in a step-function approximation, with the optimal approximation defined as that minimizing expected squared error. His AEU theory is able to explain many of the major EUH anomalies, and reduces to ordinary EUH in areas in which relevant experience accumulates (for example, in competitive markets; as noted by Peter Knez, Vernon L. Smith, and Arlington W. Williams (1985); and Don L. Coursey, John L. Hovis, and William D. Schultze (1987), the prevalence and magnitude of anomalies seems to decline in such settings). The purpose of this note is to present a variant on Leland's AEU. I propose an approximate utility function (or function) in which utility increments are weighted by a sensitivity function. The resource constraint is that overall sensitivity is limited, but (prior to observable decisions) it can be allocated freely along the continuum of potential wealth increments. My approach has three advantages: it predicts an S-shaped value function when agents maximize expected sensitivity at actual choice opportunities, given very plausible assumptions regarding the distribution of such opportunities. Recall that the assumption that decision makers maximize an S-shaped value function, such as that depicted in Figure 1, is the centerpiece of Prospect Theory, *Economics Department, University of California, Santa Cruz, CA 95064. I appreciate the comments of Jonathan Leland. Amos Tversky, and two anonymous referees on earlier drafts of this paper. The usual caveat definitely applies. IThe authors of Prospect Theory call their model as opposed to normative, but do not consider it ad hoc because of its connections with the psychophysical literature. However, no substantial connections of Prospect Theory with generally accepted economic principles (for example, optimality) have previously been demonstrated. It is in this narrow economist's sense that I use the term ad hoc. The more recent work of Ariel Rubinstein (1988) and Tversky and Kahneman (1986) underscores the difficulty of reconciling such a descriptive approach to the expected utility hypothesis.