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Utility of Gains and Losses: Measurement–Theoretical and Experimental Approaches
Journal of Economic Behavior & Organization Vol. 54 (2004) 133–136
Book review Utility of Gains and Losses: Measurement–Theoretical and Experimental Approaches R. Duncan Luce, Lawrence Erlbaum Associates, Mahwah, NJ, USA, 1999, 331 pp., Author Index, Subject Index, US$ 59.95, ISBN 0-8058-3460-5 This monograph sums up the understanding of R.D. Luce, a mathematician, on how people make selections among alternative possibilities under risk and uncertainty. The author, a well-known researcher in this field, has made many noteworthy contributions. Taking an approach much like that of most economists, the author first proposes and discusses the relevance of axioms that represent the behavioral properties people exhibit when they face choices. He then uses these properties to obtain a numerical representation of individual preferences. Although a good mathematical background is required to understand each development of its approach, this book can nonetheless be read by a wider audience since it is mostly devoted to the intuition behind each of the properties proposed and the links between them. We shall summarize and comment on all chapters. Chapter 1 introduces the notation and some of the basic concepts used in the book. The author explains and justifies his approach. He also points out a topic often neglected by economists: the difference between the various definitions of the certainty equivalent (CE) concept. Particularly, the CE obtained from the choice between a lottery and a series of sure monetary amounts is far from equal to the CE obtained from simply making a judgement about a lottery. For example, Tversky et al. (1990) obtains that, for lotteries (0, 1 − p; y, p) where p is small, the choice CEs will be smaller than the judged CEs. Chapter 2 defines and discusses such properties as idempotence, monotonicity, certainty and others that are used later in the book. Most of them are fairly weak and widely accepted in the literature. Transitivity is discussed at more length, because it could be the cause of the preference reversal paradox first tested by Lichtenstein and Slovic (1971). In many tests (Bostic et al., 1990; Tversky et al., 1990), a lottery 1 may be preferred to a lottery 2 in a direct choice by many people who also set a higher price (judged CE) for lottery 2 than for lottery 1. Luce concludes that this reversal of preferences occurs because, in the second case, the CE is obtained without making a choice. When people used a choice certainty equivalent, the number of reversals dropped sharply, because the choice CE of the lottery with a low chance of winning is a lot smaller than its judged CE. However, the author does not discuss very much the fact that even with the choice CE, many reversals clearly remain for two-point lotteries with a probability of winning close to 0 (Kahneman and Tversky, 1979; Hershey et al., 1982; Alarie and Dionne, 2001). 0167-2681/$ – see front matter © 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0167-2681(02)00198-1
134
Book review
In chapter 3, the author considers the simplest lotteries where both outputs are either positive or negative. He presents six axioms that lead to a joint structure and he shows two methods of obtaining the rank-dependent utility (RDU). The evaluation of lotteries (x, p; y, 1 − p), x > y, is w(p)u(x) + (1 − w(p))u(y) according to the RDU. The first method used to obtain the RDU is a straightforward application of the axioms, whereas the second method starts with a general representation that is reduced to the RDU under some restrictions. A large part of the chapter is devoted to the probability weighting function. The author has a preference for the inverse S-shape function. Even if, as Luce points out, a lot of tests do favor the inverse S-shape function, in our view this function with its fixed point seems to run counter to the data on some lottery choices. Tversky et al. (1990) propose a test where people face choices such as (0.71, 10) versus (0.08, 100). The lottery with a high probability of winning is chosen by the majority of subjects. So a weighting function that overvalues low probabilities and undervalues high ones seems to run counter to this result. Chapter 4 looks at the simultaneous evaluation of two elements such as a sure monetary amount and a lottery. The author adds the binary joint-receipt operation ⊕ to the structure and this leads to a V additive representation such that V(x ⊕ y) = V(x) + V(y). He then uses the binary segregation (x, p; y, 1 − p) ∼ (x − y, p; 0, 1 − p) ⊕ y to study the link between these concepts and the U representation. For example, he deduces some simple relations such as ␦U(f) = 1 − e−kV(f) . The rest of the section is concerned with various well-known concepts and their link with the RDU: these include the ones most frequently used in economics such as the Arrow–Pratt measure of risk aversion and the Rothschild and Stiglitz definition of mean preserving spreads (increases in risk). In chapter 5, Luce proposes an extension to gambles with several outputs for wins (and symmetrically for losses). This leads him to introduce some new properties which are, however, strongly linked to those in earlier chapters. For example, the property of coalescing where we add two probabilities assigned to the same monetary amount is very similar to the idempotence where, for binary gambles, we have (x, p; x, 1 − p); x. Another property is segregation that contains binary segregation as a particular case. So these extra hypotheses are not really strong. However, the RDU is not very good at explaining some tests (Birnbaum and Navarrete, 1998; Wu, 1994). The chapter ends with reference to a series of models which are more or less closely related to the RDU. Included are the most widely known in economics such as von Neumann Morgenstern’s expected utility and Savage’s subjective expected utility. Chapters 6 and 7 consider cases such as lotteries where one output is a loss and the other one is a gain. In chapter 6, the author assumes that the U function is additive with ⊕. General segregation and duplex decomposition are discussed. Chapter 7 considers that the V function is additive with ⊕ and under very simply hypotheses the author obtains a representation that seems to fit the tests (Chechile and Butler, 1999) fairly well, as compared with the first hypothesis of chapter 6. Finally, chapter 8 discusses the main ideas of the book and some open problems. According to the author, the consequence monotonicity property appears to be sustained in choice experiments. However, for lotteries (x, p; y, 1 − p), x, y > 0, Birnbaum (1997) pointed out that the choice CE for this lottery when p is close to 0 is smaller than the one for (0, p; y, 1 − p) when the choice CE is the amount selected from a list and where the subject is indifferent as to the choice between it and the lottery. In another study, Von Winterfeldt
Book review
135
et al. (1997) use the same lotteries (but with larger monetary amounts) to obtain results opposed to those of Birnbaum for the choice CE. Luce concludes that the PEST procedure (Parameter Estimation using Sequential Testing) is best for obtaining choice CEs. Another conclusion could be that both procedures will lead to almost the same results and the use of larger monetary amounts (100y instead of y) is the really important difference. If one agrees that the certainty of obtaining at least some amount of money decreases risk aversion when dealing with a lottery (a very plausible hypothesis), then the choice CE of (x, p; y, 1 − p) will increase when the value of x increases: both tests are thus explained but consequence monotonicity does not hold for the first one. The book concludes with five useful appendices that the reader can consult regularly in order to keep the book’s major concepts clearly in mind. Though the book is not intended to be a survey it does cover all the major fields (empirical, theoretical) in the literature. Furthermore, even for a reader who follows a totally different approach such as bounded rationality, for example, the book remains useful. This is because the goal of any theory is to explain how people make choices and this book is a significant study of the properties that shape choices and of the many links between these properties and the choices made.
ERRATA ¯ g). Page 60: Complementary, Eq. (2.3): (g, D; h) ∼ (h, D; Page 153: Line 9, 4.48 instead of 4.47. So the text becomes: when (4.48) holds, U is said to be exponential in V. Page 178: Line 3, Appendix B and not A. Page 185: Two square parentheses are missing in the expression W [E(k − 1)]. Page 213: Two lines from below, Section 4.4.3 instead 4.4.1. Page 321: Birnbaum 262 instead of 727.
References Alarie, Y., Dionne, G., 2001. Lottery decisions and probability weighting function. Journal of Risk and Uncertainty 22 (1), 21–33. Birnbaum, M.H., 1997. Violations of monotonicity in judgment and decision making. In: Marley A.A.J. (Ed.), Choice, Decision, and Measurement: Essays in Honor of R. Duncan Luce. Erlbaum, Mahwah, NJ, pp. 73–100. Birnbaum, M.H., Navarrete, J., 1998. Testing descriptive utility theories: violation of stochastic dominance and cumulative independence. Journal of Risk and Uncertainty 17, 49–78. Bostic, R., Herrnstein, R.J., Luce, R.D., 1990. The effect on the preference reversal phenomenon of using choice indifferences. Journal of Economic Behavior and Organisation 13, 193–212. Chechile, R.A., Butler, S.F., 1999. Is Generic Utility Theory a Suitable Theory of Choice Behaviour for Gambles with Mixed Gains and Losses. Mimeo. Hershey, J.C., Kunruther, H.C., Schoemaker, P.J.H., 1982. Sources of bias in assessment procedures for utility function. Management Science 28, 936–954. Kahneman, D., Tversky, A., 1979. Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291. Lichtenstein, S., Slovic, P., 1971. Reversal of preferences between bids and choices in gambling decisions. Journal of Experimental Psychology 89, 46–55.
136
Book review
Von Winterfeldt, D., Chung, N.-K., Luce, R.D., Cho, Y., 1997. Test of consequence monotonicity in decision making under uncertainty. Journal of Experimental Psychology: Learning Memory and Cognition 23, 406– 426. Wu, G., 1994. An empirical test of ordinal independence. Journal of Risk and Uncertainty 9, 39–60.
Yves Alarie∗ , Georges Dionne Risk Management Chair, HEC Montreal 3000 Chemin de la Cote-Sainte-Catherine Montreal, Que., Canada, H3T2A7 ∗ Corresponding author E-mail addresses: [email protected] (Y. Alarie) [email protected] (G. Dionne) Accepted 13 September 2002
Book review Utility of Gains and Losses: Measurement–Theoretical and Experimental Approaches R. Duncan Luce, Lawrence Erlbaum Associates, Mahwah, NJ, USA, 1999, 331 pp., Author Index, Subject Index, US$ 59.95, ISBN 0-8058-3460-5 This monograph sums up the understanding of R.D. Luce, a mathematician, on how people make selections among alternative possibilities under risk and uncertainty. The author, a well-known researcher in this field, has made many noteworthy contributions. Taking an approach much like that of most economists, the author first proposes and discusses the relevance of axioms that represent the behavioral properties people exhibit when they face choices. He then uses these properties to obtain a numerical representation of individual preferences. Although a good mathematical background is required to understand each development of its approach, this book can nonetheless be read by a wider audience since it is mostly devoted to the intuition behind each of the properties proposed and the links between them. We shall summarize and comment on all chapters. Chapter 1 introduces the notation and some of the basic concepts used in the book. The author explains and justifies his approach. He also points out a topic often neglected by economists: the difference between the various definitions of the certainty equivalent (CE) concept. Particularly, the CE obtained from the choice between a lottery and a series of sure monetary amounts is far from equal to the CE obtained from simply making a judgement about a lottery. For example, Tversky et al. (1990) obtains that, for lotteries (0, 1 − p; y, p) where p is small, the choice CEs will be smaller than the judged CEs. Chapter 2 defines and discusses such properties as idempotence, monotonicity, certainty and others that are used later in the book. Most of them are fairly weak and widely accepted in the literature. Transitivity is discussed at more length, because it could be the cause of the preference reversal paradox first tested by Lichtenstein and Slovic (1971). In many tests (Bostic et al., 1990; Tversky et al., 1990), a lottery 1 may be preferred to a lottery 2 in a direct choice by many people who also set a higher price (judged CE) for lottery 2 than for lottery 1. Luce concludes that this reversal of preferences occurs because, in the second case, the CE is obtained without making a choice. When people used a choice certainty equivalent, the number of reversals dropped sharply, because the choice CE of the lottery with a low chance of winning is a lot smaller than its judged CE. However, the author does not discuss very much the fact that even with the choice CE, many reversals clearly remain for two-point lotteries with a probability of winning close to 0 (Kahneman and Tversky, 1979; Hershey et al., 1982; Alarie and Dionne, 2001). 0167-2681/$ – see front matter © 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0167-2681(02)00198-1
134
Book review
In chapter 3, the author considers the simplest lotteries where both outputs are either positive or negative. He presents six axioms that lead to a joint structure and he shows two methods of obtaining the rank-dependent utility (RDU). The evaluation of lotteries (x, p; y, 1 − p), x > y, is w(p)u(x) + (1 − w(p))u(y) according to the RDU. The first method used to obtain the RDU is a straightforward application of the axioms, whereas the second method starts with a general representation that is reduced to the RDU under some restrictions. A large part of the chapter is devoted to the probability weighting function. The author has a preference for the inverse S-shape function. Even if, as Luce points out, a lot of tests do favor the inverse S-shape function, in our view this function with its fixed point seems to run counter to the data on some lottery choices. Tversky et al. (1990) propose a test where people face choices such as (0.71, 10) versus (0.08, 100). The lottery with a high probability of winning is chosen by the majority of subjects. So a weighting function that overvalues low probabilities and undervalues high ones seems to run counter to this result. Chapter 4 looks at the simultaneous evaluation of two elements such as a sure monetary amount and a lottery. The author adds the binary joint-receipt operation ⊕ to the structure and this leads to a V additive representation such that V(x ⊕ y) = V(x) + V(y). He then uses the binary segregation (x, p; y, 1 − p) ∼ (x − y, p; 0, 1 − p) ⊕ y to study the link between these concepts and the U representation. For example, he deduces some simple relations such as ␦U(f) = 1 − e−kV(f) . The rest of the section is concerned with various well-known concepts and their link with the RDU: these include the ones most frequently used in economics such as the Arrow–Pratt measure of risk aversion and the Rothschild and Stiglitz definition of mean preserving spreads (increases in risk). In chapter 5, Luce proposes an extension to gambles with several outputs for wins (and symmetrically for losses). This leads him to introduce some new properties which are, however, strongly linked to those in earlier chapters. For example, the property of coalescing where we add two probabilities assigned to the same monetary amount is very similar to the idempotence where, for binary gambles, we have (x, p; x, 1 − p); x. Another property is segregation that contains binary segregation as a particular case. So these extra hypotheses are not really strong. However, the RDU is not very good at explaining some tests (Birnbaum and Navarrete, 1998; Wu, 1994). The chapter ends with reference to a series of models which are more or less closely related to the RDU. Included are the most widely known in economics such as von Neumann Morgenstern’s expected utility and Savage’s subjective expected utility. Chapters 6 and 7 consider cases such as lotteries where one output is a loss and the other one is a gain. In chapter 6, the author assumes that the U function is additive with ⊕. General segregation and duplex decomposition are discussed. Chapter 7 considers that the V function is additive with ⊕ and under very simply hypotheses the author obtains a representation that seems to fit the tests (Chechile and Butler, 1999) fairly well, as compared with the first hypothesis of chapter 6. Finally, chapter 8 discusses the main ideas of the book and some open problems. According to the author, the consequence monotonicity property appears to be sustained in choice experiments. However, for lotteries (x, p; y, 1 − p), x, y > 0, Birnbaum (1997) pointed out that the choice CE for this lottery when p is close to 0 is smaller than the one for (0, p; y, 1 − p) when the choice CE is the amount selected from a list and where the subject is indifferent as to the choice between it and the lottery. In another study, Von Winterfeldt
Book review
135
et al. (1997) use the same lotteries (but with larger monetary amounts) to obtain results opposed to those of Birnbaum for the choice CE. Luce concludes that the PEST procedure (Parameter Estimation using Sequential Testing) is best for obtaining choice CEs. Another conclusion could be that both procedures will lead to almost the same results and the use of larger monetary amounts (100y instead of y) is the really important difference. If one agrees that the certainty of obtaining at least some amount of money decreases risk aversion when dealing with a lottery (a very plausible hypothesis), then the choice CE of (x, p; y, 1 − p) will increase when the value of x increases: both tests are thus explained but consequence monotonicity does not hold for the first one. The book concludes with five useful appendices that the reader can consult regularly in order to keep the book’s major concepts clearly in mind. Though the book is not intended to be a survey it does cover all the major fields (empirical, theoretical) in the literature. Furthermore, even for a reader who follows a totally different approach such as bounded rationality, for example, the book remains useful. This is because the goal of any theory is to explain how people make choices and this book is a significant study of the properties that shape choices and of the many links between these properties and the choices made.
ERRATA ¯ g). Page 60: Complementary, Eq. (2.3): (g, D; h) ∼ (h, D; Page 153: Line 9, 4.48 instead of 4.47. So the text becomes: when (4.48) holds, U is said to be exponential in V. Page 178: Line 3, Appendix B and not A. Page 185: Two square parentheses are missing in the expression W [E(k − 1)]. Page 213: Two lines from below, Section 4.4.3 instead 4.4.1. Page 321: Birnbaum 262 instead of 727.
References Alarie, Y., Dionne, G., 2001. Lottery decisions and probability weighting function. Journal of Risk and Uncertainty 22 (1), 21–33. Birnbaum, M.H., 1997. Violations of monotonicity in judgment and decision making. In: Marley A.A.J. (Ed.), Choice, Decision, and Measurement: Essays in Honor of R. Duncan Luce. Erlbaum, Mahwah, NJ, pp. 73–100. Birnbaum, M.H., Navarrete, J., 1998. Testing descriptive utility theories: violation of stochastic dominance and cumulative independence. Journal of Risk and Uncertainty 17, 49–78. Bostic, R., Herrnstein, R.J., Luce, R.D., 1990. The effect on the preference reversal phenomenon of using choice indifferences. Journal of Economic Behavior and Organisation 13, 193–212. Chechile, R.A., Butler, S.F., 1999. Is Generic Utility Theory a Suitable Theory of Choice Behaviour for Gambles with Mixed Gains and Losses. Mimeo. Hershey, J.C., Kunruther, H.C., Schoemaker, P.J.H., 1982. Sources of bias in assessment procedures for utility function. Management Science 28, 936–954. Kahneman, D., Tversky, A., 1979. Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291. Lichtenstein, S., Slovic, P., 1971. Reversal of preferences between bids and choices in gambling decisions. Journal of Experimental Psychology 89, 46–55.
136
Book review
Von Winterfeldt, D., Chung, N.-K., Luce, R.D., Cho, Y., 1997. Test of consequence monotonicity in decision making under uncertainty. Journal of Experimental Psychology: Learning Memory and Cognition 23, 406– 426. Wu, G., 1994. An empirical test of ordinal independence. Journal of Risk and Uncertainty 9, 39–60.
Yves Alarie∗ , Georges Dionne Risk Management Chair, HEC Montreal 3000 Chemin de la Cote-Sainte-Catherine Montreal, Que., Canada, H3T2A7 ∗ Corresponding author E-mail addresses: [email protected] (Y. Alarie) [email protected] (G. Dionne) Accepted 13 September 2002