JOHN C. BUTLER. UU. Department of Accounting ... Benjamin Franklin. Based on our risk-value .... mean X Jia, Dyer, and Butler, 1999 . In this case, preference ...
Journal of Risk and Uncertainty, 22:1; 59᎐78, 2001 䊚 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.
Generalized Disappointment ModelsU JIANMIN JIA Department of Marketing, Faculty of Business Administration, The Chinese Uni¨ ersity of Hong Kong, Shatin, N.T., Hong Kong JAMES S. DYER Department of Management Science and Information Systems, The Graduate School of Business, Uni¨ ersity of Texas at Austin, Austin, TX 78712 JOHN C. BUTLERUU Department of Accounting and MIS, Fisher College of Business, The Ohio State Uni¨ ersity, Columbus, OH 43210
Abstract ‘‘Blessed is he who expects nothing, for he shall never be disappointed’’ ᎏBenjamin Franklin Based on our risk-value framework, this paper presents extensions for the disappointment models that were originally proposed by Bell Ž1985. and Loomes and Sugden Ž1986.. We provide explicit functional forms for modeling the effect of disappointment on risky choice behavior that generalizes Bell’s work and lends tractability to the efforts of Loomes and Sugden. Our generalized disappointment models can explain a number of decision paradoxes, and offer additional insights into nonexpected utility preferences based on the intuitive notions of disappointment and risk-value tradeoffs. Key words: nonexpected utility, risk-value models, disappointment JEL Classification: D82
In our previous studies ŽJia and Dyer, 1995, 1996; Dyer and Jia, 1997., we proposed risk-value models such that decisions can be made based on the intuitively appealing idea of risk-value tradeoffs. We demonstrated that these risk-value models are very flexible in modeling preferences, and provide new resolutions for observed risky choice behavior and some decision paradoxes. In this paper, we investigate a specific class of risk-value model typically referred to as disappointment models. This class of models was first proposed by Bell Ž1985. and Loomes * This work has benefited from the comments of one anonymous referee and the editor. This research was supported by Research Grants Council of Hong Kong ŽGrant No. CUHK 4098r98H. and the National Science Foundation of the U.S. ŽGrant No. SES-98-19354.. ** Corresponding author: Department of Accounting and MIS, Fisher College of Business, The Ohio State University, Columbus, OH 43210.
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and Sugden Ž1986.. These preference models were able to explain some common decision paradoxes, but the use of our risk-value structure allows us to extend these developments and gain new insights into the implications and descriptive abilities of disappointment models. We also demonstrate that many of the commonly used models of finance are special cases of disappointment models. Thus, applications of these models should be promising. The paper is organized as follows. We begin with a review of Bell’s Ž1985. original disappointment formulation. In Section 2, we briefly review the basics of our risk-value theory ŽJia and Dyer, 1995, 1996. and provide a simple extension of Bell’s model. We use this linear disappointment model to predict the behavior commonly known as the Allais paradox, and discuss the general properties of our disappointment model in Section 3. In Section 4, we develop non-linear disappointment models that offer more flexibility in describing risky choice, particularly regarding the concept of decision weights. Finally, we make some concluding remarks.
1. Bell’s disappointment model The intuition that motivates the disappointment models is that a decision maker will form an expectation when considering a risky prospect, and use that expectation as a reference point when evaluating what is eventually received. Bell Ž1985. defined disappointment as a psychological reaction to an outcome that does not meet a decision maker’s a priori expectation. Of course, when the decision maker does better than expected, he will experience ‘‘elation,’’ the opposite of disappointment. More formally, consider a simple lottery Ž x, p, y ., i.e., the decision maker receives $ x with probability p and $ y with probability Ž1 y p ., where x ) y. Bell used the mean of these two outcome lotteries, X s px q Ž1 y p . y, as the decision maker’s psychological expectation. The measure of disappointment is assumed to be proportional to the difference between the lottery’s expectation and the less preferable outcome y, so disappointment s d Ž X y y . s d w px q Ž1 y p . y y y x s dpŽ x y y ., where d G 0 is a constant. Similarly, receipt of x leads to elation s eŽ x y X . s ew x y px y Ž1 y p . y x s eŽ1 y p .Ž x y y ., where e G 0 is a constant. Bell took the expectation of disappointment and elation, and combined them into the decision maker’s expected overall psychological satisfaction: Ž e y d . pŽ1 y p .Ž x y y .. Bell also interpreted the opposite of psychological satisfaction, Ž d y e . pŽ1 y p .Ž x y y ., as a measure of the psychological risk involved in the simple lottery for the decision maker. Finally, he assumed that the decision maker’s preference is based on the expected economic payoff and this psychological satisfaction, and that the preference relation is additive, leading to three equivalent forms of disappointment models: px q Ž 1 y p . y q Ž e y d . p Ž 1 y p . Ž x y y .
Ž 1a .
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or px q Ž 1 y p . y y Ž d y e . p Ž 1 y p . Ž x y y .
Ž 1b .
or y q Ž p. Ž x y y .
Ž 1c .
where Ž p . s p y Ž d y e . pŽ1 y p . is a decision weight and Ž d y e . measures the relative influence of disappointment and elation on preference. In general, disappointment has a greater impact on preference than elation Ži.e., d ) e ., which implies risk aversion for the decision maker. The form of Ž1b. reflects the tradeoff between mean value and the psychological risk for constructing preference. Bell showed that this simple model of disappointment can explain the common ratio effect, Ellsberg’s paradox, and other observed risky choice behaviors. Bell’s development of the disappointment model Ž1. has intuitive appeal. However, since his model only applies to lotteries with two outcomes, its application is limited. Following Bell’s basic ideas, we offer more general disappointment models based on our risk-value framework.
2. Risk-value models and disappointment risk In this section we provide a logical extension of Bell’s original work by allowing disappointment and elation to impact preference when there are more than two outcomes. We begin with a brief review of our risk-value of preference theory ŽJia and Dyer, 1995, 1996; Dyer and Jia, 1997. and then present a more general disappointment formulation. 2.1. A brief re¨ iew of risk-¨ alue theory Let X be a random variable representing a lottery, and X the mean of that lottery; then X X s X y X is a standard risk variable with zero expected value. We generalize the traditional utility framework by using a two attribute structure of the mean and standard risk for the evaluation of a lottery, Ž X, X X .. Based on two attribute utility axioms and a risk independence condition, we can have the following basic form of risk-value model: f Ž X , X X . s V Ž X . y Ž X . R Ž X X . y R Ž 0.
Ž 2.
where V Ž X . is an increasing function of the subjective value of the mean X, Ž X . ) 0 is a tradeoff factor that may depend on the mean X, RŽ X X . s yEw u 0 Ž X X .x s yEw u 0 Ž X y X .x is a standard measure of risk ŽJia and Dyer, 1996.,
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the symbol E represents the expectation operator, and RŽ0. s yu 0 Ž0. is a constant, where u 0 Ž⭈. is a utility function Žvon Neumann and Morgenstern, 1947. defined on the set of lotteries with zero expectation. Model Ž2. can accommodate both risk averse and risk prone behavior based on whether RŽ X X . ) RŽ0. or RŽ X X . - RŽ0.. To see this, we consider the certainty equivalent, CE, of a lottery X. As the CE is a degenerate lottery, its standard measure of risk is 0. Applying Ž2. yields f Ž CE, 0 . s V Ž CE . s V Ž X . y Ž X . R Ž X X . y R Ž 0 . or, V Ž X . y V Ž CE . s Ž X . R Ž X X . y R Ž 0 . . Since V Ž X . is increasing, RŽ X X . ) RŽ0. implies X ) CE, which we call lottery-risk aversion for the lottery considered. Conversely, RŽ X X . - RŽ0. implies X - CE, which we call lottery-risk seeking. When u 0 Ž⭈. is concave, RŽ X X . ) RŽ0. will be true for all lotteries, and we call the risk-value model Ž2. globally risk averse. Note that the concept of lottery-risk aversion is different from that of global risk aversion unless the utility function for the standard measure of risk is concave. Lottery-risk aversion and lottery risk seeing are determined based on both the nature of the lottery considered and the nature of the standard measure of risk. An individual’s standard measure of risk may be based on a utility function that is both locally concave and locally convex with respect to different ranges of outcomes, but the individual will be lottery-risk averse to the lottery considered if RŽ X X . ) RŽ0.. The term Ž X .w RŽ X X . y RŽ0.x in the risk-value model Ž2. plays a role in measuring the perceived risk of a lottery if Ž X . is a decreasing function of the mean X ŽJia, Dyer, and Butler, 1999.. In this case, preference can be represented by the difference between value and perceived risk. In our risk-value framework, the utility function for the standard measure of risk, the value function, and the tradeoff factor can be treated independently. Thus, we can choose appropriate functional forms for each of them based on different theoretical and empirical considerations. This makes the risk-value model very flexible in modeling risky choice behavior.
2.2. A simple extension of Bell’s disappointment model The concept of disappointment risk is associated with a particular type of the standard measure of risk. Consider the following piece-wise linear utility model: u0 Ž x . s
½
ex dx
when x G 0 when x - 0
Ž 3.
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where d, e ) 0 are constant. When d ) e we can model decision makers who are averse to downside risk or losses using a concave piece-wise linear utility function, as illustrated in Fig. 1. In this case model Ž3. implies risk aversion for lotteries with both gain and loss outcomes.1 The associated standard measure of risk for this piece-wise linear utility model can be obtained as follows: R Ž X X . s dEy < X y X < y eEq < X y X
