A simple preference foundation of cumulative prospect theory with power utility
Introduction
Under expected utility, attitudes to risk are modeled solely through the curvature of utility, i.e. the nonlinear sensitivity towards outcomes. A number of phenomena are, however, hard to explain in this manner. Examples are the classical Allais (1953) and Ellsberg (1961) paradoxes, the simultaneous existence of gambling and insurance (Friedman and Savage, 1948), and the equity premium puzzle (Mehra and Prescott, 1985). Alternative, nonexpected utility, models have been developed to accommodate these phenomena; for a survey, see Starmer (2000).
A popular nonexpected utility model is rank-dependent utility (RDU) (Quiggin, 1981; Schmeidler, 1989). The model received sound preference foundations, is mathematically tractable, and its empirical performance is promising. RDU adopts a useful concept in addition to the nonlinear evaluation of outcomes: a nonlinear weighting of probabilities, modeled through a probability transformation function. The classical paradoxes can be explained by probability transformation. Gambling and insurance are not only reconciled, but even have the same cause: the overweighting of small probabilities, for gains and losses, respectively.
Traditional rank-dependent models do not incorporate the important empirical phenomenon that, in most situations, agents do not perceive monetary outcomes as absolute wealth but as changes with respect to their status quo. In the last five decades, many authors have emphasized the importance of the status quo outcome in empirical decision making (Edwards, 1954; Harless and Camerer, 1994; Kahneman and Tversky, 1979; Markowitz, 1952; Tversky and Kahneman, 1991; Yaari, 1965). Agents are especially sensitive to losses, i.e. outcomes below the current status quo. This characteristic, known as loss aversion, is one of the major factors in human risk attitude and underlies much of the empirically observed risk aversion. Cumulative prospect theory (CPT) incorporates the different treatment of gains and losses (Luce and Fishburn, 1991; Tversky and Kahneman, 1992). The theory combines the theoretical soundness of RDU with the empirical realism of original prospect theory (Kahneman and Tversky, 1979).
Empirical studies of CPT have commonly assumed power utility. The implied infinite marginal utility at the origin reflects an extreme sensitivity of subjects towards changes near the status quo. Promising empirical results have been obtained. Fetherstonhaugh et al. (1997) and Stevens (1959) gave psychological explanations for the prevalence of power perception functions. For these reasons, the CPT model with power utility is generally assumed in parametric tests and is currently the most used nonexpected utility form. For a theoretical definition of loss aversion, the extreme derivatives of power utility at zero may cause some problems (Köbberling and Wakker, 2000).
As a price to pay for the empirical success of CPT, its model and preference foundation are more complex than they are for RDU. Both CPT and RDU invoke a comonotonic generalization of expected utility's independence condition, namely comonotonic independence (or, equivalently, tail independence). This preference condition does not entail the separation of utility and probability weighting that is characteristic of CPT and RDU. To obtain this separation, more complex preference conditions are commonly added.2 Such preference conditions are more complex for CPT than for RDU because of the separate treatment of gains and losses (Luce and Fishburn, 1991; Tversky and Kahneman, 1992). Only when utility has been derived is constant proportional risk aversion3 invoked so as to imply power utility (Tversky and Kahneman, 1992). The line of reasoning just sketched can be found in the traditional derivations of CPT.
CPT, and its predecessor prospect theory, are the most used models in empirical studies and applications of decision under risk, but have received little attention in theoretical economic studies (Wakker, 1998). The aim of this paper is to simplify the theoretical tractability of CPT by means of an appealing technique used before by Ebert (1988). Our analysis will show that CPT necessarily results from some natural preference conditions.
To obtain the desired simplification, we interchange two steps in the traditional line of reasoning, as follows. As in the traditional approach, we first invoke tail independence to obtain the rank-dependent additivily decomposable functional of Green and Jullien (1988), in which utility and probability weighting are not yet separated. With this functional obtained, we immediately invoke constant proportional risk aversion. This condition turns out to have a surprising extra merit at this early stage: It implies the separation of utility and probability weighting. That is, it implies the additional more complex preference conditions described before. These conditions can therefore simply be dropped. An additional surprise is that this approach, with constant proportional risk aversion imposed both on gains and on losses, leads to CPT without further modification. That is, the separate treatment of gains and losses (sign dependence), and their CPT aggregation, are likewise implied by constant proportional risk aversion when applied to gains and losses. This natural fit of CPT and constant proportional risk aversion can be attributed to the special role of the status quo in both. Finally, with utility established, constant proportional risk aversion characterizes power utility as it did in the traditional approach.
In summary, both the separation of utility and probability weighting and the different treatment of gains and losses are obtained free of charge under constant proportional risk aversion. We hope that this natural and elementary foundation of CPT will increase its interest for economic theory. An extension of our results to decision under uncertainty and to multiattribute outcomes will be provided by Zank (2000).
Section snippets
Definitions
Outcomes are monetary and is the set of outcomes. A lottery P=(p1,x1;…;pn,xn) is a finite probability distribution over the set of outcomes, assigning probability pj to outcome xj, j=1,…,n. The probabilities pj are nonnegative and sum to one. Lotteries are written in a rank-ordered form, i.e. it is implicitly assumed that the outcomes are rank-ordered (x1⩾⋯⩾xn) in the above notation.
Positive outcomes are gains and negative outcomes are losses. The status quo is the zero outcome. A lottery P
A preference characterization
The central condition in this paper is constant proportional risk aversion. It is usually studied when only gain outcomes are present. It then requires invariance of preference with respect to outcome multiplication by a common positive factor. On our domain with both gains and losses present, two extensions are conceivable. First, strong constant proportional risk aversion holds if preferences are invariant with respect to outcome multiplication by a common positive factor. That is, for all
Conclusion
Our result constitutes the simplest preference foundation of CPT that is presently available for the special case of power utility, the most popular utility specification in empirical studies. Only elementary conditions are used. CPT necessarily follows as soon as tail independence, a natural weakening of von Neumann–Morgenstern independence that lies at the heart of rank dependence, and constant proportional risk aversion are assumed. We hope that this result will increase the tractability and
Acknowledgements
Chris Birchenhall, Paul Madden, and three anonymous referees made useful comments.
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