Valuing Health Risks, Costs, and Benefits for Environmental Decision Making: Report of a Conference (1990)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

7 Choice Under Uncertainty: Problems Solved and Unsolved MARK ]. MACHINA Fifteen years ago, the theory of choice under uncertainty could be considered one of the 'Success stories" of economic analysis: it rested on solid axiomatic foundations;) it had seen important breakthroughs in the analytics of risk and risk aversion and their applications to economic issues;2 and it stood ready to provide the theoretical underpinnings for the newly emerging "information revolution" in economics.3 1bday, choice under uncertainty is a field in flux: the standard theory and, implicitly, its public policy implications are being challenged on several grounds from both within and outside the field of economics. The nature of these challenges, and of economists' responses to them, is the topic of this paper. The following section provides a brief but self-contained description of the economist's canonical model of individual choice under uncertainty, the expected Slid model of preferences over lotteries. I shall describe this model from two different perspectives. The first perspective is the most familiar and has traditionally been the most useful for addressing standard economic questions. However, the second, more modern perspective will be the most useful for illustrating some of the problems that have beset this model, as well as some of the proposed responses. Each of the following sections is devoted to one of these problems. All are important; some are more completely "solved" than others. In each Mark J. Machina is professor in the Department of Economics at the University of California, San Diego. 1 See, for example, van Neumann and Morgenstern (1947), Marschak (1950), and Savage (1954~. 2See, for example, Arrow (1963, 1974), Pratt (1964) and Rothschild and Stiglitz (1970, 1971~. For surveys of applications, see Lippman and McCall (1981) and Hey (1979~. 3See, for example, Akerlof (1970) and Spence and Zeckhauser (1971~. For overviews of the subsequent development of this area, see Stiglitz (1975, 1985~. 134

Al4RK ~ ~4 CHINA 135 case, I begin with a specific example or description of the phenomenon in question. I then review the empirical evidence regarding the uniformity and extent of the phenomenon. Finally, I shall report on how these findings have changed, or are likely to change, or should change, the way economists view and model private and public decisions under uncertainty. On this last topic, the disclaimer that "my opinions are my own" has more than the usual significance. THE EXPECTED UTILITY MODEL The Classical Perspective: Cardinal Utility and Attitudes Toward Risk In light of current trends toward generalizing this model, it is useful to note that the expected utility hypothesis was itself first proposed as an alternative to an earlier, more restrictive theory of risk-bearing. During the development of modern probability theory in the 17th century, such mathematicians as Blaise Pascal and Pierre de Fermat assumed that the attractiveness of a gamble offering the payoffs (at, ..., an) with probabilities (Pi, ~Pn) was given by its expected value x (i.e., the weighted average of the payoffs where each payoff is multiplied by its associated probability, so that x = Up + . .. + Alps). The fact that individuals consider more than just expected value, however, was dramatically illustrated by an example posed by Nicholas Bernoulli in 1728 and now known as the St. Petersburg Paradox: Suppose someone offers to toss a fair coin repeatedly until it comes up heads, and to pay you $1 if this happens on the first toss, $2 if it takes two tosses to land a head, $4 if it takes three tosses, $8 if it takes four tosses, and so on. What is the largest sure payment you would be willing to forgo in order to undertake a single play of this game? Because this gamble offers a 1/2 chance of winning $1, a 1/4 chance of winning $2, and so forth, its expected value is (1/2~$1 + (1/4~$2 + (1/8~$4 + . . . = $1/2 + $1/2 + $1/2 + .. . = boo; thus, it should be preferred to any finite sure gain. However, it is clear that few individuals would forgo more than a moderate amount for a one-shot play. Although the unlimited financial backing needed to actually make this offer is somewhat unrealistic, it is not essential for making the point: agreeing to limit the game to at most a million tosses will still lead to a striking discrepancy between a typical individual's valuation of the modified gamble and its expected value of $500,000.

136 CHOICE UNDER UNCERTAINTY The resolution of this paradox was proposed independently by Gabriel Cramer and Nicholas's cousin Daniel Bernoulli.4 Arguing that a gain of $2,000 was not necessarily "worth" twice as much as a gain of $1,000, they hypothesized that individuals possess what is now termed a von Neumann- Morgenstem utility of wealth Scion U f ). Rather than evaluating gambles on the basis of their expected value ~ = Up + . . . + ~nPn, individuals will evaluate them on the basis of their expected utility u = U(x~)p~ + ... + U(Xn)Pn This value is calculated by weighting the utility of each possible outcome by its associated probability, and it can therefore incorporate the fact that successive increments to wealth may yield successively diminishing increments to utility. Thus, if utility took the logarithmic form U(x) = Infix) (which exhibits this property of diminishing increments) and the individual's wealth at the start of the game were, let us say, $50,000, the sure- gain that would yield just as much utility as taking this gamble (i.e., the individual's certainty equivalent of the gamble), would be about $9, even though the gamble has an infinite expected value.5 Although it shares the name "utility," this function Ural is quite distinct from the ordinal utility function of standard consumer theory. Although the latter can be subjected to any monotonic transformation, a von Neumann- Morgenstern utility function is cardinal in that it can only be subjected to transformations that change the origin point or the scale (or both) of the vertical axis, but do not affect the "shape" of the function. The ability to choose the origin and scale factor is often exploited to normalize the utility function-for example, to set U(O) = 0 and U(M) = 1 for some large value M. ~ see how this shape determines risk attitudes, let us consider Figures la and lb. The monoton~city of the curves in each figure reflects the property of stochastic dominance preference, by which one lottery Is said to stochastical) dominate another if it can be obtained from it by shifting probability from lower to higher outcome levels.6 Stochastic dominance preference is thus the probabilistic extension of the attitude that "more Is better." Consider a gamble offering a 2/3 chance of a wealth level of a' and a 1/3 chance of a wealth levels of x". The amount x = (2/3)x' + (1/3)x" in the figures gives the expected value of this gamble; Ua = (2/3)Ua~x') + 4Bernoulli (1738~. For a historical overview of the St. Petersburg paradox and its impact, see Samuelson (1977~. 5Algebraically, the certainty equivalent of the Petersburg gamble is given by the value ~ that solves U(W+~) = (1/2)U(W+1) + (1/4)U(W+2) + (1/~)U(W+4) + ..., where W denotes the individual's initial wealth (i.e., wealth going into the gamble). 61bus, for example, a 2/3:1/3 chance of $100 or $20 and a 1/2:1/2 chance of $100 or $30 both stochastically dominate a 1/2:1/2 chance of $100 or $20.

CLARK J. MA CHINA A Ua(x ) c a cn c a ~Uafx') z c o ua B Ub(X ) a) ~n c - o c ~Ub(X) Z Ub(X ) 137 7 . /~ /' {, f 1 1 1 Ua(~)  . , I I X' X ub x Wealth 1 Ub(~) ,"'/ ~ '~' / I ~1 ~1 ''1' 1 ~^-_' 1 '' ~1 ___ ~1 1 _, 1 1 1 1 1 1 1 ~1' ~ J I l 1 1 1 1 1 1 X X X Weatth FIGURE 1 Utility functions of risk. A: Concave utility function of a risk averter. B: Convex utility function of a risk lover.

138 CHOICE UNDER UNCERTAINTY ( 1/3) Ua (x" ) and Ub = ( 2/3) Ub (x' ) + ( 1/3) Ub (x") give its expected utility for the utility functions UP ) and Ub( ). For the concave (i.e., bowed upward) utility function Ua(~), we have Ua(x) > ha, which implies that this individual would prefer a sure gain of x twhich would yield utility Ua(x)] to the gamble. Because someone with a concave utility function will in fact always rather receive the expected value of a gamble than receive the gamble itself, concave utility functions are termed risk averse. For the convex (bowed downward) utility function Ubt ), we have ub > USA). Because this preference for bearing the risk rather than receiving the expected value will also extend to all gambles, Ub(~) is termed risk-loving. In their famous article, Friedman and Savage (1948) showed how a utility function that was concave at low-wealth levels and convex at high-wealth levels could explain the behavior of individuals who both incur risk by purchasing lottery tickets as well as avoid risk by purchasing insurance.7 Algebraically, Arrow (1963, 1974), Pratt (1964) and others have shown that the degree of concavity of a utility function, as measured by the curvature index-U"(x)/U'~x), can lead to predictions of how risk attitudes, and hence behavior, will vary with wealth or across individuals in a variety of- situations.8 Because a knowledge of UP ~ would allow the prediction of preferences (and hence behavior) in any risly situation, experimenters and applied decision analysts are frequently interested in eliciting or recovering their subjects' (or clients') von Neumann-Morgenstern utility functions. One means of doing this is the.fiactile method. This approach begins by adopting the normalization U(0) = 0 and U(M) = 1 for some positive amount M and fixing a "mixture probability" ~say, p = 1/2. The next step involves obtaining the individual's certainty equivalent (~ of a gamble yielding a 1/2 chance of M and a 1/2 chance of 0, which will have the property that U(~) = 1/2.9 Finding the certainty equivalent of a gamble yielding a 1/2 chance of (, and a In chance of 0 yields the value (2 satm6mg U(62) = 1/4. 7 How risk attitudes actually differ over gains versus losses is itself an unsolved problem: evidence consistent with or contradictory to the Friedman-Savage observation of risk seeking over gains and risk aversion over losses can be found in Williams (1966), Kahneman and Tversky (1979), Fishburn and Kochenberger (1979), Grether and Plott (19 79), Hershey and Schoemaker (1980a), Payne, Laughhunn, and Crum (1980, 1981), Hershey, Kunreuther, and Schoemaker (1982), and the references cited in these articles. Finally, Feather (1959) and Slovic (1969a) found evidence that subjects' risk attitudes over gains and losses systematically changed when hypothetical situ- ations were replaced by situations involving real money. For example, if Uc( ) and Ua( ) satisfy -U"(x)/U'(x) > -U`''(~)/U<~(~) for all x [i.e., if Uc( ) is at least as risk averse as U`, ( A, an individual with utility function UC ( ) would always be willing to pay at least as much as an individual with utility function Us ( ) for (complete) insurance against any risk. See also the related analyses of Ross (1981) and Kihlstrom, Romer, and Williams (1981~. 9Because the utility of it, will equal the expected utility of the gamble, it follows that UP ) = (1/2)U(M) + (1/2)U(0), which under the normalization U(o) = o and U(M) = 1 will equal l/2.

MARK ~ CHINA 139 Finding the certainty equivalent of a gamble yielding a 1/2 chance of M and a 1/2 chance of (1 yields the value <3 satiating U(63) = 3/4.10 By repeating this procedure (i.e., 1/8, 3/8, 5/8, 7/8, 1/16, 3/16, etc.), the utility function can (in the limit) be completely assessed. 1b see how the expected utility model can be applied to risk policy, let us consider a disastrous event that is expected to occur with probability p and involve a loss L (L can be measured in either dollars or lives). In many cases, there will be some scope for influencing the magnitudes of either p or L, often at the expense of the other. For example, replacing one large planned nuclear power plant with two smaller, geographically separated plants may (to a first approximation) double the possibility that a nuclear accident will occur. However, the same action may lower the magnitude of the loss (however it is measured) if an accident occurs. The key tool used in evaluating whether such adjustments should be undertaken is the individual's (or society's) marginal rate of substitution MRSp,L, which specifies the rate at which an individual (or society) would be willing to trade off a (small) change in p against an offsetting change in L. If the potential adjustment involves better terms than this minimum acceptable rate, it will obviously be preferred; if it involves worse terms, it will not be preferred. Although the exact value of this marginal rate of substitution will depend upon the individual's (or society's) utility function UP ), the expected utility model does offer some general guidance regardless of the shape of the utility function: namely, for a given loss magnitude L, a doubling (tripling, halving, etc.) of the loss probability p should double (triple, half, etc.) the rate at which one would be willing to trade reductions in p against increases in L.~i The discussion so far has paralleled the economic literature of the 1960s and 1970s by emphasizing the flexibility of the expected utility model in comparison with the Pascal-Fermat expected value approach. The need to analyze and respond to growing empirical challenges, however, has led economists in the l980s to concentrate on the behavioral restrictions implied by the expected utility hypothesis. These restrictions are the subject of the next section. iOAs in the previous note, U(~2) = (1/2)U(~) + (1/2)U(0) and U(63) = (~/2)U(M) + (~/2)U((, ), which from the normalization U(o) = o, U(M) = ~ and the fact that Up) = i/2 will equal t/4 and 3/4, respectively. ~ iBecause expected utility in this example is given by u = (I - p)U(W) + pU(W - L) (where W is initial wealth or lives), an application of the standard economic formula for the marginal rate of substitution (e.g., see Henderson and Quandt [1980:10-113) yields MRSp,~ = -(Bu/BL)/ (Bu/0p) = -pU'(W - L)/[U(W) - U(W - L)] which, for fixed L, varies proportionately with P.

140 CHOICE UNDER UNCERTAINTY A Modern Perspective: Linearity in the Probabilities as a Testable Hypothesis As a theory of Individual behavior, the expected utility model shares many of the underlying assumptions of standard economic consumer theory. In each case, it is assumed that the objects of choice, either commodity bundles or lotteries, can be unambiguously and objectively described and that situations that ultimately imply the same set of availabilities (e.g., the same budget set) will lead to the same choice. In each case, it is also assumed that the individual is able to perform the mathematical operations necessary to actually determine the set of availabilities for example, to add up the quantities in different sized containers or to calculate the probabilities of compound or conditional events. Finally, in each case, it is assumed that preferences are transitive, so that if an individual prefers one object (either a commodity bundle or a risky prospect) to a second, and prefers this second object to a third, he or she will prefer the first object to the third. The validity of these assumptions for choice under uncertainty is examined in later sections. The strongest and most specific implication of the expected utility hypothesis stems from the form of the expected utility maximand or pr~er- encefilnction U(x~)p1 + . .. + U(xn~pn. Although this preference function generalizes the expected value form HIPS + ... + caps by dropping the property of linearity in the payoff levels (i.e., the xi's), it retains the other key property of this form, namely, linearly in the probabilities. Graphically, the property of linearity in the probabilities may be illus- trated by considering the set of all lotteries or prospects over some set of fixed outcome levels ~1 < x2 < ~3, which can be represented by the set of all probability triples of the form P = (\Pi,P2,P3) where Pi = probed) and pi + P2 + p3 = 1.~2 Making the substitution P2 = 1 - pi-p3, this set of lotteries can be represented by the points in the unit triangle in the (P1,P3) plane, as in Figure 2.~3 Because upward movements in the triangle increase p3 at the expense Of P2 (i.e., shift probability from the outcome x2 up to ~3) and leftward movements reduce P1 to the benefit of P2 (i.e., shift probability from ~1 up to Aid these movements (and, more generally, all northwest movements) lead to stochastically dominating lot- teries and would accordingly be preferred. For the purposes of illustrating many of the following discussions it will be useful to plot the ~ndiv~dual's indifference curves in this diagram; that Is, the curves in the diagram that 12Thus, if x' = $20, ~2 = $30, and X3 = $100, the three prospects in footnote 6 would be repre sented by the points (p! ,p3 ) = (1/3,2/3), (p,, ,p3 ) = (O. ~ /2) and ~l ,p3) = (1/2, 1/2), respectively. 13Although it is fair to describe the renewal of interest in this approach as "modern," modified versions of this triangle diagram can be found as far back as Marschak (1950) and Markowitz (1959:Chap 11~.

11~4RK ~ CHINA 11 1 / pi = prod (x,) FIGURE 2 Expected utility indifference curares in the triangle diagram. 141 connect points of equal expected utility.~4 Because each such curve will consist of the set of all ~i, p3) points that solve an equation of the form u = U(~)p~ ~ Up-pi-p3) ~ U(X3)p3 = k for some constant k, and because the probabilities pi and p3 enter linearly (i.e., as multiplicative co- efficients) into this equation, the indifference curves will consist of parallel straight lines, with more preferred indifference curves lying to the north- west. This means that, to know an expected utility maximizer's preferences over the entire triangle, it suffices to know the slope of a single indifference curve. ~ see how this diagram can be used to illustrate attitudes toward risk, let us consider Figures 3a and 3b. The dashed lines in the figures are not indifference curves but rather iso-expected value lines; that is, lines connecting points with the same expected value that are hence given by the solutions to equations of the form ~ = Alps +x2~1-pi-p3~+~3p3 = k for some constant k. Because northeast movements along these lines do not change the expected value of the prospect but do increase the probabilities i4A useful analogy to the concept of indifference curves is the "constant-altitude" curves on a topographic map, each of which connect points of the same altitude. Just as these curares can be used to determine whether a given movement on the map will lead to a greater or lower altitude, indifference curves can be used to determine whether a given movement in the triangle will lead to greater or lower expected utility.

142 CHOICE UNDER UNCERTAINTY A P3 B P3 1  o 1 f / o lo P1 Non,. 1~ / '/ I'd  \ 2~ FIGURE 3 A: Relatively steep indifference curves of a risk averter. B: Relatively flat indifference curves of a risk lover.

M4RK J: AL4 CHINA 143 of the extreme outcomes x~ and x3 at the expense of the middle outcome x2, they are simple examples of mean preserving spreads or "pure" increases in risker When the utility function Up ~ is concave (i.e., risk averse), its indifference curves can be shown to be steeper than the iso-expected value lines (Figure 3a),~6 and such increases in risk will lead to less preferred indifference curves. When Up ~ is convex (risk loving), its indifference curves will be flatter than the iso-expected value lines (Figure 3b), and these increases in risk will lead to more preferred indifference curves. Finally, if one compares two different utility functions, the one that is more risk averse (in the above Arrow-Pratt sense) will possess the steeper indifference cu~ves.~7 Behaviorally, the property of linearity in the probabilities can be viewed as a restriction on the individual's preferences over probability mixtures of lotteries. If P* = (P~,...,Pn) and P = (P., APE) are two lotteries over a common outcome set {at, ..., anti the cat: (1-cry probability mixture of P* and P is the lottery clip* + (1-a)P = kept +(1-alps, . . ., rip* +(1-a)Pn). This may be thought of as that prospect that yields the same ultimate probabilities over fx~...~3cn} as the two-stage lottery that offers an c': (1- a) chance of winning P* or P. respectively. It can be shown that expected utility maximizers will exhibit the following property, known as the independence Axiom: If the lottery Pa is preferred (respectively indifferent) to the lottery P. then the mixture cop* + (1-~)P** will be preferred (respectively indifferent) to the mixture c'P + (1-c'jP** for all c' > 0 and P**. This property, which is in fact equivalent to linearity in the probabilities, can be interpreted as follows: In terms of the ultimate probabilities over the outcomes {at, . . ., an), choosing between the mixtures c'P* + (1-c~)P** and cap + (1-~)P** is the same as being offered a coin with a probability 1-cat of landing tails, in which case you will obtain the lottery P**, and being asked before the flip whether you would rather have Pe or P in the event of a head. Now either the coin will land tails, in which case your choice won't have mattered, or else it will land heads, in which case your are "in effect" 5 See, for example, Rothschild and Stiglitz (1970, 1971~. 16This follows because the slope of the indifference curves can be calculated to be [U(~2)- U(xi)]/[U(x3) - U(~2)], the slope of the iso-expected value lines can be calculated to be [x2 - =~]/[~3 - 2, and a concave shape for U(~) implies [U(~2) - U(~)]/[~2-XI] > [U(X3) - U(x2)]/[X3 - 272] wherever r~ < r2 < X3. 17Setting his v, w, x, and y equal to an, ~2, =2, and :1:3, respectively, this follows directly from theorem 1 of Pratt (1964~. 18See, for example, Marschak (1950) and Samuelson (1952~.

144 CHOICE UNDER UNCERTAINI'Y back to a choice between P. or P. and it is only "rational" to make the same choice as you would before. Although this is a prescriptive argument, it has played a key role in economists' adoption of expected utility as a descriptive theory of choice under uncertainty. The mounting evidence against the model has led to a growing tension between those who view economic analysis as the description and prediction of what they consider to be rational behavior and those who view it as the description and prediction of observed behavior. Let us turn now to this evidence. VIOLATIONS OF LINEARITY IN THE PROBABILITIES The Allais Paradox and "Fanning Out" One of the earliest and best-known examples of systematic violation of linearity in the probabilities (or, equivalently, of the independence axiom) is the well-known Allais paradox.l9 This problem involves obtaining the individual's preferred option from each of the following two pairs of gambles (readers who have never seen this Problem may want to circle their own choices before proceeding): a~:{l.OO chance of $1,000,000 versus and ~10 chance of $5,000,000 versus .10 chance of $5,000,000 a2: .89 chance of $1,000,000 .01 chance of $0 . J .11 chance of $1,000,000 a4 ~ .89 chance of $0 Defining Act, x2, x3) = {$0;$1 miDion;$5 million), these four gambles are seen to form to a parallelogram in the (pi,p3) triangle (Figures 4a and 4b). Under the expected utility hypothesis, therefore, a preference for al in the first pair would indicate that the individual's indifference curves were relatively steep (as in Figure 4a), which would imply a preference for a4 in the second pair. In the alternative case of relatively flat indifference curves, the gambles as and as would be preferred.20 Yet, such researchers as Allais (1953, 1979a), Morrison (1967), Raiffa (1968), and Slovic and Tversky (1974) have found that the most common choice has been for al in the first pair and as in the second, which implies that indifference curves are not parallel but rather fan out, as in Figure 4b.  19 See, for example, Allais (1952, 1953, 1979a). 20Algebraically, these two cases are equivalent to the expression [.1o U(5,000,000) - U(1,ooo,ooo) + .01 U(0)], being respectively negative or positive.

ARK J. MA CHINA A 1 P3 O ~ at B P3 145  .,. silo," i__ _ I'\ P. a4 a' ' /// at 1 p, a4 FIGURE 4 A: Expected utility indifference curves and the Allais Paradox. B: Indifference curves that "fan out" and the Allais Paradox.

146 CHOICE UNDER UNCERTAINTY One of the criticisms of this evidence has been that individuals whose choices violated the independence axiom would "correct" themselves once the nature of their violations were revealed by an application of the above coin-flip argument.21 Thus, although even Savage chose al and as when he was first presented with this problem, upon reflection, he concluded that these preferences were in error.22 Although his own reaction was undoubt- edly sincere, the prediction that individuals would invanably react in such a manner has not been sustained in direct empirical testing. In experiments in which subjects were asked to respond to Allais-type problems and then presented with written arguments both for and against the expected utilibr position, neither MacCrimmon (1968), Mosko~tz (1974), nor Slovic and Tversky (1974) found predominant net swings toward the expected utility choicest 1 Additional Evidence of Fanning Out Although the Allais paradox was originally dismissed as an isolated example, it is now known to be a special case of a general empirical pattern that is called the common consequence effect. This effect involves pairs of probability mixtures of the form b . ~ ~ chance of x b . ~ ~ chance of P ~ 1 1- cat chance of p** verSus 2~] 1- a chance of P* and ~ ~ chance of x b3: ~ 1- ~ chance of P* , c' chance of P versus b4:i 1-~> chance of P* 21Let P be a gain of $1 million, let Pa be a (10/11~:~1/11) chance of $5 million or $0, and let ~ = .11. The choice between al and as is then equivalent to a choice between c'P + (1 - Up** and APE + (1 - UP** when P** is a gain of $1 million; the choice between ad and as is a choice between aP + (1 - UP** and aP* + (1 - UP** when P*$ is a gain of $0. Thus one should choose a 1 and a. if P is preferred to P* or a ~ and as if Pa is preferred to P. 22Reports of this incident can be found in Savage (1954:101-103) and Allais (1979b:533-535~. In that instance the payoffs of {$0;$1 million;$5 million} in the displayed example were replaced by {$0;$500,000;$2.5 million} (1952 dollars). 23In each of MacCrimmon's experiments, for example, he obtained approximately 60 percent conformity with the independence axiom (1968:7-11~. However, when presented with opposing written arguments, the pro-expected utility argument was chosen by only 20 percent of the sub- jeets in the first experiment and 50 percent of the subjects in the second experiment (subjects in the third experiment were not presented with written arguments). In subsequent interviews with the experimenter, the percentage of subjects conforming to the independence axiom did rise to 75 percent. Although MaeCrimmon did not apply pressure to induce the subjects to adopt expected utility and "repeatedly emphasized that there was no right or wrong answer," he per- sonally believed in "the desirability of using the [expected utility] postulates in training decision  makers" (1968:21-22), a fact that Slovic and Tversly felt "may have influenced the subjects to conform to the axioms" (1974:369~.

AL4RK ~ CHINA 147 where P involves outcomes both greater and less than a, and P** stochas- tically dominates p*.24 Although the independence axiom clearly implies choices of either bi and b3 (if x is preferred to P) or b2 and b4 (d P is preferred to x), researchers have again found a tendency for subjects to choose be in the first pair and b4 in the second.25 When the distributions P. P*, and P** are each over a common outcome set ~ At, ~2, ~3) that includes x, the prospects bi, b2, be, and b4 will again form a parallelogram in the (pi, pa) triangle, and a choice of bi and b4 again implies indifference curves that fan out as in Figure 4b. The intuition behind this phenomenon can be described in terms of the coin-flip scenario noted earlier. According to the independence axiom, preferences over what would occur in the event of heads should not depend upon what would occur in the event of tails. In fact, however, they may well depend on what would otherwise happen.26 The common consequence effect states that the better off individuals would be in the event of tails (in the sense of stochastic dominance), the more risk averse they become over what they would receive in the event of heads. Intuitively, if the distribution P** in the pair Abe, b2} involves very high outcomes, an individual may prefer not to bear further risk in the unlucly event that he or she does not receive it, and prefer instead the sure outcome x over the distribution P in this event (i.e., choose be over by. If P* in {b3, b4) involves very low outcomes, however, an individual may be more willing to bear risk in the (lucly) event that he or she doesn't receive it, and prefer the lottery P to the outcome ~ in this case (i.e., choose b4 over bay. Note that it is not the individual's beliefs regarding the probabilities in P that are affected here, merely his or her willingness to bear them.27 A second class of systematic violations, stemming from another early example of Allais (1953), is known as the common ratio erect. This ohe- nomenon involves pairs of prospects of the form art . ~ p chance of $X . J q chance of $Y Hi ~ 1-p chance of $ 0 versusC2 ~ 1-q chance of $ 0 24The Allais Paradox choices al, an, as, and ad correspond to 5,, be, be, and be, where a = .11, x = $1 million, P is a (10/11~:~1/11) chance of $5 million or $0, P. = is a sure gain of $0, and P** is a sure gain of $1 million. 25See MacCrimmon (1968), MacCnmmon and Lesson (1979), Kahneman and Tversky (1979) and Chew and Walter (1986~. 26As Bell (1985) notes, "winning the top prize of $10,000 in a lottery may leave one much happier than receiving $10,000 as the lowest prize in a lottery." 27In a conversation with the author, Kenneth Arrow has offered an alternative phrasing of this argument: The widely maintained hypothesis of decreasing absolute risk aversion asserts that individuals will display more risk aversion in the event of a loss and less risk aversion in the event of a gain. In the common consequence effect, individuals display more risk aversion in the event  of an opportunity loss, and less risk aversion in the event of an opportunity gain.

148 and CHOICE UNDER UNCERTAINTY c { ap chance of $X versus C4:( 1_qq chance of $0 where p > q, O < X ~ Y and O < ~ < 1; it includes the 'Certainty effect" of Kahneman and Tversky (1979) and the ingenious "Bergen paradox" of Hagen (1979) as special cases.28 Setting {~t,~2,~3) = {O,X,Y) and plotting these prospects in the (pt,p3) triangle, the segments cock and C3C4 are seen to be parallel (as in Figure 5a), so that the expected utility model again predicts choices of cat and C3 (if the individual's indifference curves are steep) or c2 and C4 (if they are flat). Yet, experimental studies have found a systematic tendency for choices to depart from these predictions in the direction of preferring cat and c4,29 which again suggests that indifference curves fan out, as in the figure. In a variation on this approach, Kahneman and Tversky (1979) replaced the gains of $X and $Y in the above gambles with losses of these magnitudes and found a tendency to depart from expected utility in the direction of c2 and C3. Defining {x~,x2,~3) as ~-Y.-X, 0) (to maintain the ordering xl ~ x2 ~ X3) and plotting these gambles in Figure Sb, a choice of c2 and C3 is again seen to imply that indifference curves fan out. Finally, Battalio, Kagel, and MacDonald (1985) found that laboratory rats choosing among gambles that involved substantial variations in their actual daily food intake also exhibited this pattern of choices. A third class of evidence stems from the elicitation method described in the previous section. In particular, the reader should note that there is no reason why the mixture probability p must be l/2, as in the earlier example. Picking any other value say p* = Demand obtaining the individual's certainty equivalent (~ of the gamble offering a 1/4 chance of M and a 3/4 chance of O will lead to the property that U(41 ~ = 1/4; in addition, just as in the previous case of p = 1/2, the procedure using p* = 1/4 (or any other fixed value) can also be continued to (in the limit) completely recover UP ). Although this procedure should recover the same (normalized) utility function for any value of the mixture probability p, such researchers as Karmarkar (1974, 1978) and McCord and de Neufville (1983, 1984) have found a tendency for higher values of p to lead to the "recovery" of higher valued utility functions (Figure 6a). By illustrating the gambles used to 28The former involves setting p = 1, and the latter consists of a twofftep choice problem in which individuals exhibit the effect with Y = 2Xandp = 2q. Kahneman and Tversly (1979), for example, found that 80 percent of their subjects preferred a sure gain of 3,000 Israeli pounds to a .80 chance of winning 4,000; 65 percent, however, preferred a .20 chance of winning 4,000 to a .25 chance of winning 3,000. The name "common ratio effect" comes from the common value of prob(X)/prob(Y) in the pairs {c, ,c2} and {C3,C4}. 29See Tversly (1975), MacCrimmon and Larsson (1979), and Chew and Waller (1986~.

AWRY J. MA CHINA 149 A P3 o B 1 1 C3  P3 o ,,~ . \ :2 1~,~: pit C3 A ~` - \-C2 , - .~ V~' Lift FIGURE 5 A: Indifference curves that fan out and the common ratio effect. B: Indifference curves that fan out and the common ratio effect with negative payoffs.

150 CHOICE UNDER UNCERTAINTY obtain the certainty equivalents 61, (2, and 63 for the mixture probability p = 1/2, (1 for p* = 1/4, and <1* for p * * = 3/4, Figure 6b shows that, as with the common consequence and common ratio effects, this utility evaluation effect is precisely what would be expected from an individual whose indifference curves departed from expected utility by fanning out.30 Non-Expected Utility Models of Preferences The systematic nature of these departures from linearity in the prob- abilities have led several researchers to generalize the expected utility model by positing nonlinear functional forms for the individual prefer- ence function. Some examples of such forms and researchers who have studied them are given in Table 1. Many (though not all) of these forms are flexible enough to exhibit the properties of stochastic domi- nance preference, risk aversion/risk preference, and fanning out, and the Chew/MacCrimmon/Fishburn and Quiggin forms have proven to be par- ticularly useful both theoretically and empirically. Additional analyses of the above forms can be found in Chew, Karni, and Safra (1987~; Fishburn (1982, 1984a,b); Rdell (1987~; Segal (1984, 1987~; and Yaari (19874. For general surveys of these models, see Machina (1983a), Sugden (1986), and Weber and Camerer (1987~. Although such forms allow for the modeling of preferences that are more general than those allowed by the expected utility hypothesis, each requires a different set of conditions on its component functions z,( ), try ), rt ~ or 9~) for the properties of stochastic dominance preference, risk aversion/risk preference, comparative risk aversion, and so forth. In particular, the standard expected utility results that link properties of the function U(~) to such aspects of behavior generally will not extend to the corresponding properties of the function u(~) in the above forms. Does this imply that the study of non-expected utility preferences requires one to abandon the vast body of theoretical results and intuition that have been developed within the expected utility framework? Fortunately, the answer is no. An alternative approach to the analysis of non-expected utility preferences proceeds not by adopting a specific nonlinear function but by considering nonlinear functions in general, and using calculus to extend results from expected utility theory in the same manner in which it is typically used to extend results involving linear 30Having found the value (1 that solves U(61) = (~/2)U(M) + (1/2)U(0), let us now choose {X~,X2,X3} = {O,(l,M}, so that the indifference curve through the point (0,0) (i.e., a sure gain of (1) also passes through the point (1/2, 1/2) (a 1/2:1/2 chance of M or 0~. The ordering of the values At, (2, (3, (; and 6**1 in Figure 6a is derived from the individual's preference ordering over the five distributions in Figure 6b for which they are the respective certainty equivalents.

MARK J. MA CHINA A U 1 ~*() (p= 3/4) 3/4~ ~ ~7~ , Pi' I/ 1/2 ~ I__ /_* 1/4 B 1 1/2:1/2 Chance of Mork, (~431N Pa l Sure chance of 4, o 151 I) (p= 1/2) O 'A ~ · O ~ 42 (1 t1 41 43 () (If- 1/4)  M x \ 3/4:1/4 Chance of M or O (~41 7 i,~ it_ _ P. ~1 $ ~ /2:1/2 Chance \ of Mor O (~t,,) Aft- 114:314 Chance ,' ~ of M or O (at) 1/2:1/2 Chance of (' or O (~42) FIGURE 6 A: "Recovered" utility functions for mixture probabilities 1/4, 1t2, and 3/4. B: Fanning out indifference curves that generate the responses of Figure 6a. Note: ~ denotes indifference.

152 TABLE 1 Examples of Non-Expected Utility Preference Functions CHOICE UNDER UNCERTAINTY Function Researcher u(X l ) (Pi ) + + t/(Xn ) ~ (pit ) u(Xl)1r(pl) + · · - + ''(Xn)~(Pn) ~(P1)+ +1r(Pn) 7J(X1)P1 ~ + [~(Xn)Prl T(X 1 )p1 + + T(Xn )Pn Text )9(P1 ) + V(X2)[9(P2 + P1 )-9(P1 )]+ ~(X3)[g(p3 + P2 + P1)-§(P2 + P1)] + [~(Xl)Pl + + ~(Xn)Pn]+ [:T(Xi)P1 + + ~(xn)pn]2 Edwards (195S, 1962): Kahneman and Overly (1979) Kannarkar (1978, 1979) Chew and MacCnn~mon ( 1979~.1 ); Chew (1983); Fishhur'~ (1'383) ·. Quiggin (1982) Machina (1982) functions. (Readers who are not interested in the details of this approach may wish to skip ahead to the next section.3~) Specifically, let us consider the set of all probability distributions P = (P~ ~ . . . ~ Pn) over a fixed outcome set {at, . . ., an), so that the expected utility preference function can be written as V(P) = V(pl' APE) = U(x:)p~ + . . . + U(xn~pn. Let us also think of U(x') not as a "utility level" but rather as the coefficient of Pi = probed) in this linear function. If these coefficients are plotted against xi as in Figure 7, the expected utility results of the previous section can be stated as: . Stochastic Dominance Preference: Vt ) will exhibit global stochastic  dominance preference if and only if the coefficients {U(xi)) are increasing in xi, as in Figure 7. · Risk Aversion: Vie ) will exhibit global risk aversion if and only if the coefficients {U(xi)) are concave in xi,32 as in Figure 7. · Comparative RiskAversion: The expected utility preference function V*(P) = U*(x~)p1 + .. . + U*(xn~p~ will be at least as risk averse as Vt ) if and only if the coefficients {Uteri)} are at least as concave in xi as {U(xi)~.33 31 More complete developments of this approach may be found in Machina (1982, 1983b). 32As in footnote 16, this is equivalent to the condition that [U(=i+l) - U(=i)]/[xi+~ - ~i] < [U(xi) - U(=i_~)]/[=i - ri-~] for all i. 33This is equivalent to the condition that U* (hi ) _ p(U(xi)) for some increasing concave func- tion pig).

AL4RK J. MA CHINA U(xn) a' U(Xn l) In c o C U(X2) 3 u(xl) in c o 153 . _ ~ Xl X2 . · · · Xn 1 Xn Wealth FIGURE 7 van Neumann-Morgenstern utilities as coefficients of the expected utility preference function V(pt,...,pn) = U(x~)p~ + .. + U(x")p". Now, let us consider the case in which the individual's preference function V(P) = V(p~,. . . ,Pn) is not linear (i.e., not expected utility) but at least differentiable, and let us consider its partial derivatives U(~i;P) = ~V(P)/0pi = dV(P)/0prob(~i). Some probability distribution PO can be chosen and these U(~i;Po) values plotted against xi. If they are increasing in ~i, it is clear that any infinitesimal stochastically dominating shift from PO, such as a decrease in some Pi and matching increase in Pi+, will be preferred. If they are concave in xi, any infinitesimal mean preserving spread, such as a drop in Pi and (mean preserving) rise in Pi- ~ and Pi+ I, ~1 make the individual worse off. In light of this correspondence between the coefficients {U(~i)) of the expected utility preference function Vie ~ and the partial derivatives {U(z`;Po)} of the non-expected utility preference function V(~), {U(xi;PO)) as the individual's local utility indices at PO. Of course, the above results will ondy hold exactly for infinitesimal shifts from the distribution PO. However, another result from standard calculus can be exploited to show how "expected utility" results may be applied to the exact global analysis of non-expected utility preferences. The reader should recall that, in many cases, a differentiable function will exhibit a specific global property if and only if that property is exhibited by its linear approximations at each point. For example, a differentiable function will be globally nondecreasing if and only if its linear approximation at each  point is nonnegative. In fact, most of the fundamental properties of risk

154 CHOICE UNDER UNCERTAINTY attitudes and their expected utility characterizations are precisely of this type. In particular, the following can be shown: . Stochastic Dominance Preference: A non-expected utility preference function Vie ~ will exhibit global stochastic dominance preference if and only if its local utility indices {U(xi;P)) are increasing in hi at each distribution P. · Risk Aversion: V(< ~ win exhibit global risk aversion if and only if its local utility indices {U(xi;P)) are concave in ~` at each distribution P. · Comparative Risk Aversion: The preference function V*( ~ will be globally at least as risk averse34 as Vie ~ if and only if its local utility indices {U*(xi;P)) are at least as concave in xi as {U(~i:P)) at each P. Figures 8a and 8b are a graphic illustration of this approach for the outcome set {x~,~2,~3~. Here, the solid curves denote the indifference curves of the non-expected utility preference function V(P). The parallel lines near the lottery PO denote the tangent "expected utility" indifference curves that correspond to the local utility indices {U(xi;PO)) at PO. As always with differentiable functions, an infinitesimal change in the proba- bilities at PO will be preferred if and only if it would be preferred by this tangent linear (i.e., expected utility) approximation. Figure 8b illustrates the above "risk aversion" result. It is clear that these indifference curves will be globally risk averse (averse to mean preserving spreads) if and only if these are everywhere steeper than the dashed iso-expected value lines. However, this is equivalent to all of their tangents being steeper than these lines, which in turn is equivalent to all of their local expected utility ap- proximations being steeper-or, in other words, to the local utility indices {U(xi;P)) being concave in hi at each distribution P. My fellow researchers and I have shown how this and similar techniques can be applied to further extend the results of expected utility theory to the case of non-expected utility preferences, to characterize and explore the implications of preferences that "fan out," and to conduct new and more general analyses of economic behavior under uncertainty.35 Still, although I feel that they constitute a useful and promising response to the phenomenon of nonlinearities in the probabilities, these models do not provide solutions to the more problematic empirical phenomena described in the following sections. 34For the appropriate generalizations of the expected utility concepts of "at least as risk averse" in this result, see Machina (1982, 1984~. 35See, for example, Machina (1982, 1983b, 1984~; Chew (1983~; Fishburn (1984a); Epstein (1985~; Dekel (1986~; Allen (1987~; Chew, Karni, and Safra (1987~; Karni and Safra (1987), and Machina and Neilson (1987~.

AL4RK J. MA CHINA A 1 P3 o B P3 1  o 155 , / /\ Po it, / / \ P1 1~. my/ , \ ,.~ P. FIGURE 8 A: Tangent "expected utility" approximation to non-expected utility indifference curves. Note: Solid lines are local expected utility approximation to non-expected utility indifference curves at PO. B: Risk aversion of every local expected utility approximation is equivalent to global risk aversion. Note: Dashed lines are iso-expected value lines.

156 ~., .^ ~_ CHOICE UNDER UNCERTAINTY THE PREFERENCE REVERSAL PHENOMENON The Evidence The finding now known as the preference reversal phenomenon was initially reported by psychologists Lichtenstein and Slovic (1971~. In this study, subjects were first presented with a number of pairs of bets and asked to choose one bet out of each pair. Each of these pairs took the following form: p chance of $X P-bet: 1-p chance of $ x versus $-bet: ~ q chance of $Y l 1-q chance of $y where X and Y are respectively greater than x and y, p is greater than q, and Y is greater than X (the names "P-bet" and "$-bet" come from the greater probability of winning in the first bet and greater possible gain in the second). In some cases, x and y took on small negative values. The subjects were next asked to "value" (state certainty equivalents for) each of these bets. The different valuation methods that were used consisted of (a) asking subjects to state their minimum selling price for each bet if they were to own it, (b) asking them to state their maximum bid price for each bet if they were to buy it, and (c) the elicitation procedure of Becker, DeGroot, and Marschak (1964), in which it is in a subject's best interest to reveal his or her true certainty equivalent.36 In the latter case, real money was in fact used. The expected utility model, as well as each of the non-expected utility models of the previous section, clearly implies that the bet that is actually chosen out of each pair will also be the one that is assigned the higher certainty equivalent.37 However, Lichtenstein and Slovic (1971) found a systematic tendency to violate this prediction In the direction of choosing the P-bet In a direct choice but assigning a higher value to the $-bet. In one experiment, for example, 127 out of 173 subjects assigned a higher sell price to the $-bet in every pair In which the P-bet was chosen. Similar findings were obtained by Lindman (1971) and, in an interesting variation on the usual experimental setting, by Lichtenstein and Slavic (1973) in a 36Roughly speaking, the subject states a value for the item and then the experimenter draws a random price. If the price is above the stated value, the subject forgoes the item and receives the price. If the drawn price is below the stated value, the subject keeps the item. The reader  can verify that under such a scheme, it will never be in a subject's best interest to report anything other than his or her true value. 37Economic theory tells us that income effects may well lead an individual to assign a lower bid price to the object that, if both were free, would actually be preferred. However, such an effect will not apply to either selling prices or the Becker, DeGroot, and Marschak procedure. For discussions of the empirical evidence on sell price/bid price disparities, see Knetsch and Sinden (1984) and the references cited there.

WOK ~ CHINA 157 Las Vegas casino where customers actually staked (and hence sometimes lost) their own money. In another real-money experiment, Mowen and Gently (1980) found that groups who could discuss their Joint) decisions were if anything more likely than individuals to exhibit the phenomenon. Although these above studies involved deliberate variations in design in order to check for the robustness of this phenomenon, they were nev- ertheless received skeptically by economists, who perhaps not unnaturally felt they had more at stake than psychologists in this type of finding. In an admitted attempt to "discredit" this work, economists Grether and Plott (1979) designed a pair of experiments in which they corrected for issues of incentives, income effects,38 strategic considerations, the ability to indicate indifference, and so forth. They expected that the experiments would not generate this phenomenon, but they nonetheless found it in both. Fur- ther design modifications by Pommerehne, Schneider, and Zweifel (1982) and Reilly (1982) yielded the same results. Finally, the phenomenon has been found to persist (although in mitigated form) even when subjects are allowed to engage in experimental market transactions involving the gambles (Knez and Smith, 1987), or when the experimenter is able to act as an arbitrager and make money from such reversals (Berg, Dickhaut, and O'Brien, 1983~. I we Interpretations of This Phenomenon How one interprets these findings depends on whether one adopts the world view of an economist or a psychologist. An economist would reason as follows: Filch individual possesses a unique underlying preference ordering over objects (in this case lotteries), and information about this preference ordering can be gleaned from either direct choice questions or (properly designed) valuation questions.39 Someone exhibiting the preference reversal phenomenon is therefore indicating that (a) they are indifferent regarding the choice between the P-bet and some sure amount lip, (b) they strictly prefer the P-bet to the $-bet, and (c) they are indifferent regarding the choice between the $-bet and an amount [,$ greater than lip. Assuming that they in fact prefer t$ to the lesser amount lip, this implies that their preferences over these four objects are cyclic or intransitive. 38In addition to the problem with bid prices discussed in the previous note, Grether and Plott (1979) noted that subjects' changing wealth (as a result of the actual play of these gambles during the course of the experiment), or the changing of their expected wealth (in those experiments in which chosen gambles would be played at the end), could be a source of income effects. 39Formally, this ordering is represented by the individual's weak preference relation >, where "A > B " is read "A is at least as preferred as B." From this one may in turn derive the individual's strictpreference relation > and indifference relation a, where "A> B" denotes that A > B but not B >A, and "A ~ B" denotes that both A > B and B > A.

158 CHOICE UNDER UNCERTAINTY Psychologists, on the other hand, would deny the premise of an com- mon underlying mechanism generating both choice and valuation behavior. Rather, they view choice and valuation (even different forms of valuation) as distinct processes, subject possibly to different influences. In other words, individuals exhibit what are termed response mode effects. Excellent discus- sions and empirical examinations of this phenomenon and its implications for the elicitation of both probabilistic beliefs and utility functions can be found in Hogarth (1975, 1980~; Hershey, Kunreuther, and Schoemaker (1982~; Slovic, Fischhoff, and Lichtenstein (1982~; Hershey and Schoe- maker (1985~; and MacCrimmon and Wehrung (1986~. ~ report how the response mode study of Slovic and Lichtenstein (1968) actually led them to predict the preference reversal phenomenon, I can do no better than to quote the authors themselves: ~ ,, The impetus for this study [Lichtenstein and Slovic (1971~] was our observation in our earlier 1968 article that choices among pairs of gambles appeared to be influenced primarily by probabilities of winning and losing, whereas buying and selling prices were primarily determined by the dollar amounts that could be won or lost.... In our 1971 article, we argued that, if the information in a gamble is processed differently when making choices and setting prices, it should be possible to construct pairs of gambles such that people would choose one member of the pair but set a higher price on the other. [Slovic and Lichtenstein (1983:597~] Implications of the Economic World View The issue of intransitivity is new neither to economics nor to choice under uncertainty. May (1954), for example, observed intransitivities in pairwise rankings of three alternative marriage partners, in which each candidate was rated highly in two of three attributes (intelligence, looks, wealth) and low in the third. In an uncertain context, Blyth (1972) has adapted this approach to construct a set of random variables (x,y,z) such that prober ~ y) = probity > z) = prober > x) = 2/3, so that individuals making panwise choices on the basis of these probabilities would also be intransitive. In addition to the preference reversal phenomenon, Edwards (1954a)4° and Tversly (1969) have also observed intransitivities in preferences over risky prospects. On the other hand, researchers have also shown that many aspects of economic theory, in particular the existence of demand functions and of general equilibrium, are surprisingly robust to the phenomenon of intransitivity (Sonnenschein, 1971; Mas-Colell (1974~; Shafer, 1974, 1976; Kim and Richter, 1986; Epstein, 1987~. 40See also the discussions of these findings by Edwards (1954b:4(~05), Davis (1958:28), and Weinstein (1968:337).

MARK J. MA CHINA 159 In any event, economists have begun to develop and analyze models of nontransitive preferences over lotteries. The leading example of this is the C`regret theory" model developed independently by Bell (1982, 1983) (see also Bell and Raiffa 11980~), Fishburn [1981, 1982, 1984a,b], and Loo mes and Sugden [1982, 1983a,b]~. In this model of pairwise choice the von Neumann-Morgenstern utility function U(x) is replaced by a regret/rejoice function r~x,y) that represents the level of satisfaction (or, if negative, dissatisfaction) the individual would experience if he or she were to receive the outcome x when the alternative choice would have yielded the outcome y (this function is assumed to satisfy r~x,y) = -rgy,x) for all values of x and y). In choosing between statistically independent gambles P = (`Pi ~ ~ Pn`) and P* = (Pi, . . ., Pn`) over a common outcome set {ad, , an I, the individual will choose P* if the expected value of the function Rex, y) is positive and P if it is negatived (Once again, readers who wish to skip the mathematical details of this approach may proceed to the following subsection.) It is interesting to note that when the regret/rejoice function takes the special form rid, y) = U(~)-U(y) this model reduces to the expected utility model.42 In general, however, such an individual will neither be an expected utility maximizer nor have transitive preferences. Yet, this intransitivity does not prevent the graphing of such preferences or even the application of the "expected utility" analysis to them. ~ see the former, let us consider the case in which the individual is facing alternative independent lotteries over a common outcome set T~,~2,x3T, so that the triangle olagram may agam be used to Illustrate their ~muluerence curves," which will appear as in Figure 9. In such a case, it is important to understand what is and is not still true of these indifference curves. The curve through P will still correspond to the points (i.e., lotteries) that are indifferent to P. and it will still divide the points that are strictly preferred to P (the points in the direction of the arrow) from the ones to which P is strictly preferred. Furthermore, if (as in the figure) P* lies above the indifference curve through P. then P will lie below the indifference curve through P* (i.e., the individual's ranking of P and P* will be unambiguous). Unlike indifference curves for transitive preferences, however, these curves 4iAlgebraically, this expected value is given by the double summation Hi Hi r(~i,rj)P, Pj. 42This follows because Hi Hi r(=i,~j)p' pi = Hi ~i[U(=i) ( j)]p' p; Zi U(~i)p, - Ej U(=j)pj, so that theindividualwillpreferP*toPifandonlyifEi U(xi)pi* > Hi U(~j)pj. When r( ,.) takes the form r(x,y) = v(~)~r(y) - u(y)T(x), the expected regret model reduces to the (transitive) Chew/MacCrimmon/Fishburn form of Table 1. This is the most general form of the model compatible with transitivity.

160 1 Pa o P. FIGURE 9 "Indifference curves" for the expected regret model. CHOICE UNDER UNCERTAINTY \ it, p** \ - Ace `/P \ 1 will cross,43 and preferences over the lotteries P. P*, and P** are seen to form an intransitive cycle. In regions in which the indifference curves do not cross (such as near the origin), however, the individual will be indistinguishable from- someone with transitive (albeit non-expected utility) preferences. 1b see how elected utility results can be extended to this nontransitive framework, let us fix a lottery P = Opt, . . . ,Pn) and consider the question of when an (independent) lottery P* = Opt, . . ., p* ~ will be preferred or not  preferred to P. Defining the "utilipr function" ¢~;P) = r('c,x~)pl +...+ Rex, ~n)Pn, it is possible to show that P* will be preferred to P if and only if ¢(x~;P)p: + · · .+~n;P)pn > ¢(x~;P)p: + . . .+~=n;P)pn- in other words, if and only if P* implies a higher expectation of the function ¢(x;P) than does P.44 Thus, if ¢(x;P) is increasing in x for all lotteries P. the individual will exhibit global stochastic dominance preference; if +~;P) is concave in ~ for all P. the individual will exhibit global risk aversion, even though he or she is not necessarily transitive (these conditions will clearly be satisfied 43In this model, the indifference curves will necessarily all cross at the same point. This (unique) point will accordingly be ranked indifferent to all lotteries in the triangle. 44Because r(=,y) = -r(y,~c) forallxandyimplies~i Hi r(~i,~i )PiPj = 0,P* willbe preferred to P if and only if ° < ~` Hi r(xi,~j)p,Pj = Ski ~,j r(~i,~j)P Pj Hi ~'j = Ei[Ej r(=i'=i)Pi]P' -Zi[Ej r(=i,~j)Pj]Pi = Zi q)(xi;p~pi -Ei ¢(=i;P)Pi.

MARK J. MA CHINA 161 if r(x, y) is increasing and concave in X).45 The analytics of expected utility theory are robust, indeed. Bell, Raiffa, Loomes, Sugden, and Fishburn have also shown how specific assumptions about the form of the regret/rejoice function will generate the common consequence effect, the common ratio effect, the preference reversal phenomenon, and other observed properties of choice over lotteries.46 The theoretical and empirical prospects for this approach seem quite impressive. Implications of the Psychological World View On the other hand, how should economists respond if it turns out that the psychologists are right and that the preference reversal phenomenon really Is generated by some form of response mode effect (or effects)? In that case, the first thing to do would be to try to determine if there were analogues of such effects in real-world economic situations.47 Will individuals behave differently when they are determining their valuation of an object (e.g., reservation bid on a used car) than they will when reacting to a fixed and nonnegotiable price for the same object? Because a proper test of this question would require correcting for any possible strategic or information-theoretic (e.g., signaling) issues, it would not be a simple undertaking. However, in light of the experimental evidence, I feel it is crucial that it be attempted. Let us say that it was found that response mode effects did not occur outside of the laboratory. In that case, we scientists could rest more easily, although we could not forget about such issues completely: experimenters testing other economic theories and models (e.g., auctions) would have to be forever mindful of the possible influence of the particular response mode used in their experimental design. On the other hand, what if response mode effects were found out In the field? In such circumstances, we would want to determine, perhaps by going back to the laboratory, whether the rest of economic theory re- ma~ned valid provided the response mode were held constant. If this 45It is important to note that although the function ¢(~i;P) plays a role very similar to the local utility index U(=i;P), it is a different concept. Unlike the linear approximation to a nonlinear preference function, the previous inequality is both exact and global. 46Loomes and Sugden (1982), for example, have shown that the many of these effects follow if one assumes that r(x,y ) = Q(= - y ) where Q is convex for positive values and concave for negative values. 47Although this point in the discussion has been reached by an examination of the preference revemal phenomenon over risky prospects, it is important to note that neither the evidence of re- sponse mode effects (e.g., Slovic, 1975) nor their implications for economic analysis are confined to the case of choice under uncertainty.

162 CHOICE UNDER UNCERTAINTY were true, then with further evidence on exactly how the response mode mattered, we could presumably incorporate it into existing theories as a new independent variable. Because response modes tend to be constant within specific economic models (e.g., quantity responses to fixed prices in competitive markets, valuation announcements-truthful or otherwis~in auctions, etc.), we should expect most of the testable implications of this approach to appear as cross-institutional predictions, such as systematic vi- olations of the various equivalency results involving prices versus quantities, or second price/sealed bid versus oral English auctions. I feel that the new results and implications for our theories of institutions and mechanisms would be exciting indeed.48 FRAMING EFFECTS Evidence In addition to response mode effects, psychologists have uncovered an even more disturbing phenomenon: namely, that alternative means of rep- resenting or "framing" probabilistically equivalent choice problems will lead to systematic differences in choice. An early example of this phenomenon is reported by Slovic (1969b), who found, for example, that offering a gain or loss contingent on the joint occurrence of four independent events with probability p elicited responses different from offering it on the occurrence of a single event with probability p4 (all underlying probabilities were stated explicitly). In comparison with the single-event case, making a gain contin- gent on the joint occurrence of events was found to make it more attractive; making a loss contingent on the joint occurrence of events made it more unattractive.49 In another study, Payne and Braunstein (1971) used pairs of gambles of the type illustrated in Figure 10. Each of the gambles in the figure, known as a duplex gamble, involves spinning the pointers on both its 'Gain wheel" (on the left) and its "loss wheel" (on the right), with the individual 48A final "twist" on the preference reversal phenomenon: Karni and Safra (1987) and Holt (1986) have shown how the procedures used in most of these studies, namely, the Becker, De- Groot, and Marschak elicitation technique (see footnote 363 and the practice of only selecting a few questions to actually play, will lead to truthful revelation of preferences only under the addi- tional assumption that the individual satisfies the independence axiom. Accordingly, it is possible to construct (and these researchers have done so) examples of non-expected utility individuals with transitive underlying preferences and no response mode effects, whose optimal responses in such experiments consist of precisely the typical "preference revemal" responses. How (and whether) experimenters will be able to address this issue remains to be seen. 49Even though all underlying probabilities were stated explicitly, Slavic found that individuals tended to overestimate the probabilities of these compound events.

MARK J. MA CHINA $.40 - - \ 163 -$.4o - .6 $0 $.40 \ GAMBLE A - .5 \  .5 $0 $~4o / - so FIGURE 10 Duplex gambles with identical underlying distributions. - .6 - GAMBLE B - - $0 - / face of wealth variations has also been observed in several experimental studies.5i Markowitz (1952:155) also suggested that certain circumstances may cause the individual's reference point to deviate temporarily from current wealth. If these circumstances include the manner in which a given problem is verbally described, then differing risk attitudes over gains and losses can  51See the discussion and references in Machina (1982:285-286~.

164 CHOICE UNDER uNcERTAINry receiving the sum of the resulting amounts. Thus, an individual choosing gamble A would win $.40 with probability .3 (i.e., if the pointer in the gain wheel landed up and the pointer in the loss wheel landed down), would lose $.40 with probability .2 (if the pointers landed in the reverse positions), and would break even with probability .5 (if the pointers landed either both up or both down). An examination of gamble B reveals that it has an identical underlying distribution; thus, subjects should be indifferent regarding a choice between the two gambles, regardless of their risk preferences. Payne and Braunstein, however, found that individuals in fact chose between such pairs (and indicated nontrivial strengths of preference) in manners that were systematically affected by the attributes of the component wheels. When the probability of winning in the gain wheel was greater than the probability of losing in the loss wheel for each gamble (as in the figure), subjects tended to choose the gamble whose gain wheel yielded the greater probability of a gain (gamble A). In cases in which the probabilities of losing in the loss wheels were greater than the probabilities of winning in the gain wheels, subjects tended to choose the gamble with the lower probability of losing in the loss wheel. Finally, although the gambles in Figure 10 possess identical underly- ing distributions, continuity suggests that worsening of the terms of the preferred gamble could result in a pair of nonequivalent duplex gambles in which the individual will actually choose the one with the stochastical) dominated underlying distribution. In an experiment in which subjects were allowed to construct their own duplex gambles by choosing one from a pair of prospects involving gains and one from a pair of prospects involving losses, stochastically dominated combinations were indeed, chosen (T\rersly and Kahneman, 1981; Kahneman and Tversly, 1984~.5° A second class of framing effects exploits the phenomenon of a ref- erence point. Theoretically, the variable that enters an individuals von Neumann-Morgenstern utility functions should be total (i.e., final) wealth, and gambles phrased In terms of gains and losses should be combined with current wealth and reexpressed as d~stnbutions over final wealth levels before being evaluated. However, economists since Markow~tz (1952) have observed that risk attitudes over gains and losses are more stable than can be explained by a fixed utility function over final wealth, and have suggested that the utility function might be best defined In terms of changes from the "reference point" of current wealth. The stability of risk attitudes In the 50Subjects were asked to choose either (A) a sure gain of $240 or (B) a 1/4:3/4 chance of $1,000 or $0, and to choose either (C) a sure loss of $750 or (D) a 3/4:1/4 chance of - $1,000 or $0. Eighty-four percent of the subjects chose A over B and 87 percent chose D over C, even though B + C dominates A + D, and choices over the combined distributions were unanimous when they were presented explicitly.

AL4RK J. MA CHINA 165 lead to different choices, depending on the exact description. A simple example of this, from Kahneman and Tversly (1979:273), involves the following two questions: In addition to whatever you own, you have been given 1,000 (Israeli pounds). You are now asked to choose between a 1/2:1/2 chance of a gain of 1,000 or O or a sure chance of a gain of 500. and In addition to whatever you own, you have been given 2,000. You are now asked to choose between a 1/2:1/2 chance of a loss of 1,000 or O or a sure loss of 500. These two problems involve identical distributions over final wealth. When put to two different groups of subjects, however, 84 percent chose the sure gain in the first problem, but 69 percent chose the 1/2:1/2 gamble in the second. A nonmonetary version of this type of example, from Tversly and Kahneman (1981:453), posits the following scenario: Imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimate of the consequences of the programs are as follows: If Program A is adopted, 200 people will be saved. If Program B is adopted, there is 1/3 probability that 600 people will be saved, and 2t3 probability that no people will be saved. Seventy-two percent of the subjects who were presented with this form of the question chose program ~ A second group was given the same initial information, but the descriptions of the programs were changed to read (p. 453~: If Program C is adopted 400 people will die. If Program D is adopted there is 1/3 probability that nobody will die, and 2/3 probability that 600 people will die. Although this statement once again implies a problem that is identical to the former one, 78 percent of the respondents chose program D. In other studies, Schoemaker and Kunreuther (1979~; Hershey and Schoemaker (l9SOb); Kahneman and Tversky (1982, 1984~; Hershey, Kun- reuther, and Schoemaker (1982~; McNeil et al. (1982~; and Slovic, Fis- chhoff, and Lichtenstein (1982) have found that subjects' choices in oth-  erwise identical problems will depend on whether the choices are phrased as decisions about whether to gamble or to insure, whether the statisti- cal information for different therapies is presented in terms of cumulative survival probabilities over time or cumulative mortality probabilities over

166 CHOICE UNDER UNCERTAINTY time, and so forth (see also the additional references in Tversly and Kah- neman [1981] as well as the examples of this phenomenon in nonstochastic situations given in Thaler [1980, 19853~. In a final class of examples, not based on reference point effects, Moskow~tz (1974) and Keller (1985) found that the proportion of subjects that choose in conformance with or in violation of the independence axiom in examples like the Allais paradox was significantly affected by whether the problems were described in the standard matrix form (e.g., Raiffa, 1968:7), in a decision tree form, or as minimally structured written statements. Interestingly enough, the form that was judged to be the "clearest representation" by the majority of Moskowitz's subjects (the tree form) led to the lowest degree of consistency with the independence axiom, the highest proportion of Allais-type (i.e., fanning out) choices, and the highest persistency rate of these choices (1974:234, 237-38~. Iwo Issues Regarding Framing The replicability and pervasiveness of the above group of examples is indisputable. Their implications for economic modeling involve two issues (at least). The first is whether these experimental observations possess any analogue outside of the laboratory. Real-world decision problems are never as neatly packaged as those that appear on experimental questionnaires; thus, monitoring such effects would not be as straightforward. This diffi- culty in monitoring does not mean that such efforts do not exist, however, or that they cannot be objectively observed or quantitatively measured. The real-world example that comes most quickly to mind, and is presumably of no small importance to the involved parties, is whether gasoline price dif- ferentials should be represented as "cash discounts" or "credit surcharges." Similarly, Russo, Krieser, and Miyashita (1975) and Russo (1977) found that the practice and even the method of displaying unit price information in supermarkets (information that allowed consumers to calculate for them- selves) affected both the level and distribution of consumer expenditures. The empirical marketing literature is no doubt replete with findings that could legitimately be interpreted as real-world framing effects. The second, more difficult issue is that of the independent observability of the particular frame that an individual will adopt in a given problem. In the duplex gamble and matrix/decision tree/written statement examples of the previous section, the different frames seem unambiguously determined by the form of presentation. In instances in which framing involves the choice of a reference point, however, instances that presumably include the majority of real-world cases, this point might not be objectively determined by the form of presentation. Rather, it might be chosen differently and, what

1~4RK ~ CHINA 167 is worse, unobservably, by each individual.52 In a particularly thorough and insightful study, Fischhoff (1983) presented subjects with a written decision problem that allowed for different choices of a reference point. The study went on to explore different ways of predicting which frame individuals would adopt in order to be able to predict their actual choices. Although the majority choice of subjects was consistent with what would appear to be the most appropriate frame, Fischhoff noted "the absence of any relation within those studies between {separately elicited] frame preference and option preference." Indeed, to the extent that frame preferences varied across his experiments, they did so inversely to the incidence of the predicted choice (Fischhoff, 1983:115-116~.53 If such problems can occur in predicting responses to specific written questions in the laboratory, imagine how they could plague the modeling of real-world choice behavior. Framing Effects and Economic Analysis: Has This Problem Already Been Solved? What response is appropriate if it turns out that framing actually is a real-world phenomenon of economic relevance and, in particular, if individuals' frames cannot always be observed? I would argue that the means of responding to this issue can already be found in the "tool box" of existing economic analysis. Let us consider first the case in which the frame of a particular economic decision problem (even though it should not matter from the point of view of standard theory), can at least be independently and objectively observed. I believe that, in fact, economists have already solved such a problem in their treatment of the phenomenon of "uninformative advertising." Although it is hard to give a formal definition of this term, it is widely felt that economic theory is hard put to explain a large portion of current advertising in terms of traditional informational considerations.54 This constraint, however, has hardly led economists to abandon classical consumer theory. Rather, models of uninformative advertising proceed by quantifying this variable (e.g., air time) and treating it as an additional independent variable in the utilibr function, the demand function, or both. Standard results like the Slutsky equation need not be abandoned but rather reinterpreted as properties of demand functions holding this new variable 52This is not to say that well-defined reference points never exist. The reference points involved in credit surcharges versus cash discounts, for example, seem unambiguous. 53Fischhoff notes that "tiff one can only infer frames from preferences after assuming the truth of the theory, one nuns the risk of making the theory itself untestable" (p. 116~. 54A wonderful example, offered by my colleague Joel Sobel, is that of milk advertisements that make no reference either to price or to a particular daily. What commodity could be more wet 1 known than milk?

168 CHOICE UNDER UNCERTAINTY constant. The degree of advertising itself is determined as a maximizing variable on the part of the firm (given some cost curve) and is thus subject to standard comparative static analysis. In cases in which decision frames can be observed, framing effects presumably can be modeled in an analogous manner. ~ do so, one would begin by adopting a method of quantifying-or at least of categorizing- frames. The activity of the second step, some of which has of course already been done, would be to study both the effect of this new independent variable holding the standard economic variables constant, and, conversely, to retest standard economic theories in conditions in which the frame was carefully held in a fixed position. With any luck, one would find that, holding the frame constant, the Slutsky equation still held. The next step in any given modeling situation would be to discover `'who determines the frame." If (as with advertising) it is the firm, then the effect of the frame on consumer demand, and hence on the firm's profits, can be incorporated into the firm's maximization problem. The choice of the frame, as well as the other relevant variables (e.g., prices and quantities), can be simultaneously determined and subjected to comparative static analysis just as in the case of uninformative advertising. A seemingly more difficult case is when the individual chooses the frame (for example, a reference point), and this choice cannot be observed. Although findings of Fischhoff (1983) should be kept in mind, let us assume that this choice is at least systematic in the sense that the consumer will jointly choose the frame and make the subsequent decision in a way that maximizes a "utility function" that depends both on the decision and on the choice of frame. In other words, individuals make their choices as part of a joint maximization problem, the other component of which (the choice of frame or reference point) cannot be observed. Such models are hardly new to economic analysis. Indeed, most economic models presuppose that the agent is simultaneously maximizing his or her choices with respect to variables other than the ones being studied. When assumptions are made on the individual's joint preferences over the unobserved and observed variables, the well-known theory of induced preferences can be used to derive testable implications on choice behavior over the observables.55 With a little more knowledge on exactly how frames are chosen, such an approach could presumably be applied here as well. The above remarks should not be taken as implying that the problems of framing in economic analysis have already been solved or that there Is no need to adapt and, if necessary, abandon standard economic models in 55See, for example, Milne (1981~. For an application of the theory of induced preferences to choice under uncertainty, see Machina (1984~.

II~RK J. MA CHINA 169 light of this phenomenon. Rather, the remarks reflect the view that when psychologists are able to present enough systematic evidence on how these effects operate, economists will be able to respond appropriately. OTHER ISSUES: IS PROBABILITY THEORY RELEVANT? The Manipulation of Subjective Probabilities The evidence discussed so far has consisted primarily of cases in which subjects were presented with explicit (i.e., "objective") probabilities as part of their decision problems and the models that addressed these phenomena possessed the corresponding property of being defined over objective probability distributions. There is extensive evidence, however, that when individuals have to estimate or revise probabilities for themselves, they will make systematic mistakes in doing so. The psychological literature on the processing of probabilistic infor- mation is much too large even to summarize here. Yet, it is worth noting that experimenters have uncovered several "heuristics" used by subjects that can lead to predictable errors in the formation and manipulation of subjective probabilities. Kahneman and Tversky (1973), Bar-Hillel (1974), and Grether (1980), for example, all found that probability updating sys- tematically departs from Bayes' law in the direction of underweighting prior information and overweighting the `'representativeness" of the current sam- ple. In a related phenomenon termed the "law of small numbers," Tversly and Kahneman (1971) found that individuals overestimated the probabil- ity of drawing a perfectly representative sample out of a heterogeneous population. Finally, Bar-Hillel (1973), Tversky and Kahneman (1983), and others have found systematic biases in the formation of the probabilities of conjunctions of both independent and nonindependent events. For surveys, discussions, and examples of the psychological literature on the formation and handling of probabilities, see Edwards, Lindman, and Savage (1963~; Edwards (1971~; Slovic and Lichtenstein (1971~; 1,versky and Kahneman (1974~; and Grether (1978), as well as the collections in Acta Psycholog- ica (December 1970~; Kahneman, Slovic, and l~rersly (1982~; and Arkes and Hammond (19863. For examples of how economists have responded to some of these issues, see Arrow (1982), Viscusi (1985a,b) and the references cited there. The Existence of Subjective Probabilities The evidence referred to above indicates that when individuals are asked to formulate probabilities they seldom do it correctly. These findings may be rendered moot, however, by evidence that suggests that when

170 CHOICE UNDER UNCERTAINTY individuals making decisions under uncertainty are not explicitly asked to form subjective probabilities they might not do it at all. In one of a class of examples developed by Ellsberg (1961), subjects were presented with a pair of urns: the first contained 50 red balls and 50 black balls, and the second also contained 100 red and black balls but in an unknown proportion. When faced with the choice of staking a prize on (R~) drawing a red ball from the first urn, (R2) drawing a red ball from the second urn, (B~) drawing a black ball from the first urn, or (B2) drawing a black ball from the second urn, a majority of subjects strictly preferred (R~) over (R2) and strictly preferred (B~) over (Bay. It is clear that there can exist no subjectively assigned of probabilities p: (1-p) of drawing a red versus a black ball from the second urn not even 1/2:1/2, that can simultaneously generate both of these strict preferences. Similar behavior in this and related problems has been observed by Raiffa (1961), Becker and Brownson (1964), MacCrimmon (1965), Slovic and I versly (1974), and MacCrimmon and Larsson (1979~.56 Life (and Economic Analysis) Without Probability Theory One response to this type of phenomenon as been to suppose that individuals "slant" whatever subjective probabilities they might otherwise form in a manner that reflects the amount of confidence or ambiguity asso- ciated with them (Fellner, 1961, 1963; Becker and Brownson, 1964; Brewer and Fellner, 1965; Fishburn, 1985, 1986; Hogarth and Kunreuther, 1985, 1986; and Einhorn and Hogarth, 1986~. In the case of complete ignorance regarding probabilities, Arrow and HuIwicz (1972), Maskin (1979), and others have presented axioms that imply such principles as ranking options solely on the basis of their best or worst possible outcomes or (both ~ (e.g., manumit, maximax), the unweighted average of their outcomes ("principle of insufficient reason"), or similar criteria.57 Finally generalizations of ex- pected utility theory that drop the standard additivity or compounding laws of probability theory (or both) have been developed by Schmeidler (1989) and Segal (1987~. Although the above models may well capture aspects of actual decision processes, analytically the most useful approach to choice in the presence of uncertainty but the absence of probabilities is the so-called state-preference model of Arrow (1953/1964), Debreu (1959), and Hirshleifer (1965, 1966~.58 56See also the discussions of Fellner (1961, 1963), Brewer (1963), Ellsberg (1963), Roberts (1963), Brewer and Fellner (1965), MacCrimmon (1968), Smith (1969), Sherman (1974), and Sinn (1980~. 57For an excellent discussion of the history, nature, and limitations of such approaches, see Arrow (1951~. 58For a comprehensive overview of this model and its analytics, see Karni (1985~.

llL4RK ~ CHINA 171 In this model, uncertainty is represented by a set of mutually exclusive and exhaustive states of nature S = {si). This partition of all possible unfoldings of the future could be either coarse, such as the pair of states {it rains here tomorrow, it does not rain here tomorrow), or else very fine (so that the definition of a state might read "it rains here tomorrow and the temperature at Gibraltar is 7S degrees at noon and the price of gold in London is below $700 per ounce"~. Note that it is neither feasible nor desirable to capture all conceivable sources of uncertainty when specifying the set of states for a given problem. It is not feasible because no matter how finely the states are defined, there will always be some other random criterion on which to further divide them; it is not desirable because such criteria may affect neither individuals' preferences nor their opportunities. Rather, the key requirements are that the states be mutually exclusive and exhaustive so that exactly one will be realized, and that the extent to which the individual is able to influence their probabilities (if at all) be explicitly specified. Given a fixed (and, let us say, finite) set of states, the objects of choice in this framework consist of alternative state-payoff bundles, each of which specifies the outcome the individual will receive in every possible state. When, for example, the outcomes are monetary payoffs, state- payoff bundles take the form act, . . ., en), where ci denotes the payoff the individual will receive should state i occur. In the case of exactly two states of nature, this set can be represented set by the points in the act, c2) plane. Because bundles of the form (c, c) represent prospects that yield the same payoff in each state of nature, the 45-degree line in this plane is known as the certain line. Now, if the individual happens to assign some set of probabilities {Pi) to the states {si), each bundle (c,, . . ., cn) will imply a specific probability distribution over the payoffs, and his or her preferences could be inferred (i.e., indifference curves) over state-payoff bundles.S9 Yet because these bundles are defined directly over the respective states and without reference to any probabilities, it is possible to speak of preferences over such bundles without making any assumptions regarding the coherency, or even the existence, of probabilistic beliefs. Researchers such as those listed above, as well as Yaari (1969), Diamond and Yaari (1972), and Mishan (1976), have used this indifference cu~ve-based approach to derive results from individual demand behavior through general equilibrium in a context that requires win generating these indifference cumes from individuals' preferences over probability distri- butions, one implicitly assumes that their level of satisfaction from a given amount of money does not depend on the particular state of nature which that (i.e., that their preferences are state independent. Beginning with the next sentence, this assumption will no longer be required.

172 CHOICE UNDER UNCERTAINTY neither the expected utility hypothesis nor the existence or commonality of subjective probabilities. In other words, life without probability theory does not imply life without economic analysis.60 IMPLICATIONS FOR: PRIVATE AND PUBLIC DECISION MAKING Fifteen years ago, a decision analyst who was advising an individ- ual, firm, or government agency in a choice under uncertainty might use something like the following stylized procedure. 1. Collect as much information as possible about the decision, and construct an explicit list of the currently (and potentially) available options. 2. Assess the decision maker's (or, alternatively, the "experts"') sub- jective probability distributions over consequences implied by each option. 3. Evaluate the decision maker's (or, alternatively, "society's') pref- erences regarding the alternative consequences, including their attitudes toward risk fin other words, assess their van Neumann-Morgenstern utility function). 4. Determine the option that would yield the highest (individual or social) expected utility.6i Of course, the consequences might involve several dimensions (requiring the assessment of a multivariate utility function),62 or the experts might disagree on the probabilities (requiring some form of consensus, aggregation, or pooling of beliefs).63 Nevertheless, researchers working on these aspects remained confident of the validity of this overall (expected utilit,r-based) approach. Should the developments surveyed In this paper change the way private decision analysts or public decision makers go about their jobs? Do they imply new or different business or governmental responsibilities in keeping customers or citizens informed of any voluntary (or involuntary) risks they 60A final issue is the lack of a unified model that is capable of simultaneously handling all of the phenomena described in this paper fanning out, the preference reversal phenomenon, framing effects, probability biases, and the Ellsberg paradox. After all, it is presumably the same indi- viduals who are exhibiting each of these phenomena; should there not be a single model capable of gmerat~ng them all? I doubt whether our current ability allows us to do this; I also doubt the need for a unified model as a prerequisite for further progress. The aspects of behavior consid- ered in this paper are quite diverse, and if (like the wave-versus-particle properties of light) they cannot be currently unified, this does not mean that we cannot continue to learn by studying and modeling them separately. 61The classic introductory expositions of the process of decision analysis are Raiffa (1968) and Schlaifer (1969~. 62 See, for example, Keeney and Raiffa (1976~. 63See, for example, GroEman and Owen (1986~.

MARK J. AL4 CHINA 173 may be facing? The following section discusses some of the issues that these new developments raise. Implications for Private-Sector Decision Analysis _ ~ How should private-sector decision analysts adapt their procedures in light of these new empirical findings and theoretical models? It is hard to see how step 1 (formulating the options) could or should change. Yet the types of systematic biases in the formulation and manipulation of sub- jective probabilities presented in the preceding section ('`Other Issues: Is Probability Theory Relevant?") should cause the analyst to be especially careful in obtaining mutually consistent estimates of the underlying event likelihoods used in constructing the probability distributions over conse- quences implied by each option in step 2. Note that this step has nothing to do with the client's attitudes toward bearing these risks (i.e., whether or not they do, or should, maximize expected utility). Rather, it consists of applying probability theory to establish the internal consistency and (once that has been established) the logical implications of the client's or experts' probabilistic beliefs. If the client assigns probability .3 to the occurrence of some event A, probability .2 to the occurrence of some mutually exclusive event B. and probability .6 to the occurrence of neither, then at least one of these numbers will have to change before the pieces will fit. This situation is no different from that of asking a client for the length, width, and area of his or her living room before offering advice on a choice of carpet: if the numbers do not multiply out correctly then something is wrong, and the advising process should stop short until corrections are made. Although I suspect practitioners in the field have been aware of such inconsistencies (and of how to "iron them out") for some time now, the type of system- atic and specific biases that psychologists have been uncovering now give decision analysts the opportunity, and, I feel, much more of an obligation, to search explicitly for and eliminate biases and inconsistencies in clients' probabilistic beliefs which might otherwise remain hidden. Although I feel the suggestions of the previous paragraph are impor- tant, they are more of a technical improvement than a basic change in how step 2 is carried out. On the other hand, I would argue that the developments reviewed in this paper do imply a fundamental change in the way modern decision analysts should proceed with steps 3 and 4 (expli- cating clients' risk preferences and determining their optimal action). The classical approach would be essentially to impose the property of linearity in the probabilities on the client by assessing his or her von Neumann- Morgenstern utility function and then using it to calculate the `'optimal" (i.e., expected utility-maximizing) choice. If clients made choices like those in the Allais paradox, the common consequence effect, or the common

174 CHOICE UNDER UNCERTAINTY ratio effect discussed earlier, or their responses to alternative assessment methods yielded different "recovered" utility functions, they would often be told that they had "inconsistent" (i.e., not expected utility) preferences that would have to be corrected before their optimal action could be determined. Although experimental subjects and real-world decision makers some- times do make mistakes in expressing their preferences, I feel that the widespread and systematic nature of "fanning out"-type departures from expected utility, and the growing number of models that can simultaneously accommodate this phenomenon, as well as the more traditional properties of stochastic dominance preference and risk aversion, increase both the analyst's ability and obligation to fit and represent clients' risk attitudes within a consistent non-expected utility framework when their expressed risk preferences are pointing in that direction.64 Why do I feel that depar- tures from the strictures of probability theory should be corrected but that (systematic) departures from the strictures of expected utility theory should not? Because the former involve the determination of the risks involved in an option, which is a matter of accurate representation, whereas the latter involve the client's willingness to bear these risks, which is a matter of preference. 1b continue my earlier analogy, reporting a length, width, and area of a room that are not commensurate implies an internally inconsistent description of the room and is simply wrong; preferring purple polka-dot carpeting, however, is a matter of clients' tastes, to which they have every right if it is their living room. In the case of health or environmental risks, this would correspond to the distinction between measuring the detrimental effects of a drug or a pollutant versus determining the individual patient's or society's attitudes toward bearing these consequences. Does this increased respect for clients' preferences mean that the decision analyst should not play any guiding role in steps 3 or 4? The answer is no: conscientious decision analysts will still try to elicit and explicitly represent the client's risk attitudes, their underlying properties (e.g., whether they are risk averse, linear in the probabilities, etc.), and their logical implications. Even more important, they will continue explicitly to separate their client's beliefs from their preferences. For example, let us say that, although option 1 offers a very high chance of an acceptable but not terrific outcome (e.g., amputation of a gangrenous limb), the client insists on `'optimism" or '~wishful thinking" in connection with option 2 (e.g., drug therapy), which is not as likely to succeed but does offer a small chance of obtaining the best possible outcome. In that case the decision analysis should take pains formally to represent the client's attitude as a 64The components of such models (e.g., the functions v( ), ~r( ), T(-~' and 9( ) in Table 1) can be assessed by procedures similar to the one described earlier for van Neumann-Morgenstern utility functions.

AtIRK J. MA CHINA 175 willingness to bear risk (either by a convex utility function, such as as in Figure lb, or by some non-expected utility counterpart) rather than as an exaggerated probability estimate of obtaining the best outcome under option 2. The job of the decision analyst has hardly become obsolete. Implications for Public Decision Making Although private-sector decision analysts typically act on behalf of an individual client or firm, the decision maker in federal, state, or local government is faced with the obligation of acting on the behalf of citizens whose preferences and interests will generally differ from one another. In the case of decisions under certainty, economists have developed a large body of techniques, collectively termed welfare economics or welfare analysis, with which to analyze such situations.65 Not surprisingly, economic theorists have also used the expected utility model as a framework for extending such analyses to a world of uncertainty (e.g., Arrow, 1953/1964, 1974; Diamond, 1967~. Let us say, however, that we wish to respect what the recent evidence implies about individuals' actual attitudes toward risk. Can classical welfare analysis, the economist's most important tool for formal policy evaluation, be undertaken with these newer models of preferences? The answer to this question depends on the model. Fanning-out be- havior and the non-expected utility models used to characterize it, as well as the state-navoff anoroach discussed earlier, are completely consistent r ~1 ~- ~- w~th the assumption or well-uennec~, transitive ~na~v~aua~ pram orucr- ings and hence with traditional welfare analysis along the lines of Pareto (1909), Bergson (1938) and Samuelson (1947/1983:Chap. 8~. For exam- ple, the proof of the general efficiency ("Pareto efficiency") of a system of complete contingent-commodity markets (Arrow, 1953/1964; Debreu, l959:Chap. 7) requires neither the expected utility hypothesis nor the as- sumption of well-defined probabilistic beliefs. On the other hand, it is clear that the preference reversal phenomenon and framing effects, and at least some of the nontransitive or noneconomic models used to address them, will prove much more difficult to reconcile with welfare analysis, or at least with welfare analysis as currently practiced. ~ see how some of these (transitive) non-expected utility models can be applied to policy questions, the reader may recall the earlier expected utility-based analysis of the trade-off between the probability p and mag- nitude L of a disastrous event (see the section entitled "The Expected Utility Model"~. Under the expected utility hypothesis, it was seen that an individual's marginal rate of substitution (i.e., acceptable rate of trade-off) 65The standard policy techniques of "benefit-cost analysis," "benefit-risk analysis," and so forth fall into this category.

176 CHOICE UNDER uNcERTAINry between these variables would vary exactly proportionally to the loss prob- ability p (as seen in footnote 11~. Although it requires a bit of algebra to do so, it is possible to demonstrate that if preferences depart from ex- pected utility by "fanning out" (Figures 4b, Sa, 5b, and 6), then individuals' marginal rates of substitution between p and L will always vary less than proportional) to the loss probability p (Mach~na, 1983b:282-289~. Although this is not as strong a prediction as expected utility theory's prediction of exact proportionality, it can be used to place at least a one-sided bound on how individuals' acceptable rates of trade-off behave. In any event, it is at least more closely tied to what has actually been observed about preferences over risky prospects.66 Public and Corporate Obligations in the Presentation of Information The final issue concerns the public policy implications of framing effects. If individuals' choices actually depend on the manner in which pub- licly or privately supplied probabilistic information (e.g., cancer incidences or flood probabilities) is presented, then the manner of presentation itself becomes a public policy issue over which interest groups may well contend. Should "freedom of information" imply that a government or manufacturer has an obligation to present a broad range of "legitimate" frames when dis- closing required information, or would this practice lead to confusion and waste? Should legal rights of recourse for failures to provide information (e.g., job or product hazards) extend to failures to frame it "properly"? The general issue of public perception of risk is of growing concern to a number of government agencies In particular, the Environmental Protec- tion Agency.67 ~ the extent that new products, medical techniques, and environmental hazards continue to appear and the government takes a role their regulation, these issues will become more and more pressing. Although the issue of the public and private framing of probabilistic information Is a comparatively new one, I feel that there are several 66In the nuclear power plant example the earlier section noted above, let us say that there was a probability p of an accident involving a loss of L, that some expected utility maximizer was just willing to accept an increase of AL' in this potential loss in return for a reduction of Hip in its probability, and that some individual with fanning-out preferences was just willing to ac- cept an increase of ILL in return for this same reduction in p (^L2 could be greater or less than ~L, ). Should new technology reduce the initial loss probability ~ half to p/2, the extra loss the expected utility maximizer would be willing to accept for a (further) reduction of up would also drop by half (i.e., to AL, /2. The extra loss that the individual with fanning-out pref- erences would be willing to accept, however, would drop by less than half (i.e., to some amount greater than AL2/2. In otherwords, as the probability of the accident drops, the individual with fanning-out preferences will exhibit a comparatively greater willingness to trade off increases in loss magnitude in exchange for further reductions in loss probability. 67See Russell (in this volume).

I\L4RK ~ TRICHINA 177 analogous issues (not all of them fully resolved) from which useful insights may be derived. Previous examples have included the cash discount/credit surcharge issue mentioned earlier, rotating warning labels on cigarette packages, financial disclosure regulations, bans on certain forms of alcohol advertising, publicity requirements for product recall announcements, and current debate cover such issues as requiring special labels on irradiated produce or on products imported from countries that engage in human or animal rights violations, or both. If these issues do not provide ready-made answers for the case of probabilistic information, they at least allow glimpse of how policy makers, interest groups, and the public feel and act toward the general issue of the presentation of information. ACKNOWLEDGMENT This paper is an adapted and expanded version of Machina (1987~. I am grateful to Brian ginger, John Conlisk, Rob Coppock, Jim Cox, Vince Crawford, Gong Jin-Dong, Elizabeth Hoffman, Brett Hammond, Paul Portney, Howard Raiffa, Michael Rothschild, Carl Shapiro, Vernon Smith, Joseph Stiglitz, Timothy Taylor, Richard Thaler, and especially Joel Sobel for helpful discussions on this material. EDITORS' NOTE In the preceding paper, Mark Machina addresses an important set of questions relevant to the use of benefit-cost analysis in regulatory decision making. These include: How do (and should) decision makers and analysts cope with uncertainties in the science underlying decisions? · How does the way in which information is presented- or framed- affect analysis and regulatory decisions? · Is it possible to get better estimates of uncertain for purposes of practical decision making? Machina examines recent theoretical and empirical findings on how people actually evaluate risks and assign probabilities in arriving at policy preferences. Much of this literature challenges the traditional economic approach to preferences, in that it points out that individuals are able to sustain nonlinearity in their subjective assessments of probabilities, reversal of preferences over time or between different situations, and differences in preferences and probability assessments depending on the way in which a problem (or analysis) is framed or presented. At the heart of his paper is the conclusion that predicted policy out- comes should be differentiated from individual and collective policy prefer- ences for purposes of analysis and decision making. As he puts it, observed

178 CHOICE UNDER UNCERTAINTY "departures from the strictures of probability theory should be corrected [by the analyst or the decision maker] but that [systematic] departures from the structures of expected utility theory should not." This is because the former involved the determination of the risks associated with alternative actions or policies, which are in fact matters of accurate representation, while the latter involve the willingness of individuals, organizations, and society to bear these risks, which is a matter of preference. He concludes that analysis must be designed to account for actual preferences, even those that depart from the tenets of expected utility theory. Therefore, analysts and decision makers, in assigning values to policy alternatives, may need to consider departures from expected utility and weighting schemes to reflect those departures. REFERENCES Akerlof, G. 1970 The market for "Lemons": Quality uncertainty and the market mechanism. Quarter) Journal of Economics 84:488 500. 1984 An Economic Theonst's Booic of Tales. Cambridge: Cambridge University Press. Allais, M. 1952 Fondements d'une theorie positive des choix comportant un risque et critique des postulate et axiomes de ltecole Americaine. Econometrie, Colloques In ternationaux du Centre National de la Recherche Scientifique 40:Paris, 1953, 257-332. 1953 Le comportement de lthomme rationel devant le risque, Critique des postulates et amomes de l'ecole Amdncaine. Econometrica 21:503-546. Summarized version of Allais (1952~. 1979a The foundations of a positive theory of choice involving risk and a criticism of the postulates and axioms of the American school. Pp. 27-145 in M. Allais and 0. Hagen, eds. Expected Utility Hypotheses and the Allais Pardon. Dordrecht, Holland: D. Reidel Publishing Company. 1979b The so-called Allais paradox and rational decisions under uncertainty. Pp. 437 - i81 in M. Allais and O. Hagen, eds. Elected Utility Hypotheses and the Allais Pardon. Dordrecht, Holland: D. Reidel Publishing Company. Allen, B. 1987 Smooth preferences and the local expected utility hypothesis. Journal of Ecm nomic Theory 41:34~355. Arrow, K 1951 Alternative approaches to the theory of choice in risk-taking situations. Icon metrica 19404 437. 19S3 Le role des valeurs boursibres pour la repartition le meilleure des risques. Econam~tne, Collogues Internationaux du Centre National de la Recherche Scientifique 40, Paris, 1953, 41-47. 1963 Comment. Review of Economics and Statistics 45(Suppl.~:24-27. 1964 The role of securities in the optimal allocation of risk-bearing. Review of Economic Studies 31:91-96.  1974 Essays in the Theory of Risk-Bearing. Amsterdam: North-Holland Publishing Company.

AL4RK J. MA CHINA 1982 Risk perception in psychology and economics. Economic Inquiry 20:1-9. Arrow, K, and L. Hurwicz 179 1972 An optimality criterion for decision-making under ignorance. Pp. 1-11 in Carter, D., and F. Ford, eds., Uncertainty and Expectations in Economics. Oxford: Basil Blackwell. Arrow, K, and M. Intnligator, eds. 1981 Handbook of Mathematical Economics, vo! 1. Amsterdam: North-Holland Publishing Company. Bar-Hillel, M. 1973 On the subjective probability of compound events. Organizational Behavior Ed Human Performance 9:396~06. 1974 Similarity and probability. Organizational Behavior and Human Performance 11:277-282. Battalio, R., J. Kagel, and D. MacDonald 1985 Animals' choices over uncertain outcomes. American Economic Review 75:597- 613. Becker, G., M. DeGroot, and J. Marschak 1964 Measuring utility by a single-response sequential method. Behavioral Science 9:226-232. Becker, S., and F. Brownson 1964 What price ambiguity? Or the role of ambiguity in decision-making. Joumal of Political Economy 72:62-73. Bell, D. 1982 Regret in decision making under uncertainty. Operations Research 30:961-980. 1983 Risk premiums for decision regret. Management Science 29:1066 1076. 1985 Disappointment in decision making under uncertainty. Operations Research 33:1-27. Bell, D., and H. Raiffa 1980 Decision Regret: A Component of Risk Aversion. Unpublished manuscript, Harvard University. Berg, J., J. Dickhaut, and J. O'Brien 1983 Preference Reversal and Arbitrage. Unpublished manuscript, University of Minnesota. Bergson, A. 1938 A reformulation of certain aspects of welfare economics. Quarterly Joumal of Economics 52:31~334. Bernoulli, D. 1738 Specimen theoriae novae de mensura sortie. Commentary Academiae Scien tianan Imperialis Petropolitanae [Papers of the Imperial Academy of Sciences in Petersburg] 5:175-192. English translation: Exposition of a new theory on the measurement of risk. Econometnca Z2~1954~:2~. Blyth, C 1972 Some probability paradoxes in choice from among random alternatives. Joumal of He American Statistical Association 67:366 373.  Brewer, K 1963 Decisions under uncertainty: Comment. Quarterly Joumal of Economics 77:159- 161. Brewer, K, and W. Fellner 1965 The slanting of subjective probabilitie~Agreement on some essentials. Quarterly Joumal of Economics 79:657~63.

180 ClIOICE UNDER UNCERTAINTY Chew, S. 1983 A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais Paradox. Econometrica 51:1065-1092. Chew, S., and K MacCrimmon 1979a Alpha-Nu Choice Theory: A Generalization of Expected Utility Theory. Uni- versity of British Columbia Faculty of Commerce and Business Administration Working Paper No. 669. 1979b Alpha Utility Theory, Lottery Composition, and the Allais Paradox. University of British Columbia Faculty of Commerce and Business Administration Working Paper No. 686. Chew, S., and W. Walter 1986 Empirical tests of weighted utility theory. Joumal of Mathematical Psychology 30:55-72. Chew, S., E. Karni, and Z. Safra , , . 1987 Risk aversion in the theory of expected utility with rank dependent probabilities. Journal of Economic Theory 42:37~381. Davis, J. 1958 The transitivity of preferences. Behavioral Science 3:26-33. Debreu, G. 1959 Theory of Value: An Idiomatic Analysis of General Equilibrium. New Haven: Yale University Press. Dekel, E. 1986 An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom. Joumal of Economic Theory 40:3~318. Diamond, P. 1967 The role of a stock market in a general equilibrium model with technological uncertainty. American Economic Review 57:759-773. Diamond, P., and M. Rothschild, eds. 1978 Uncertainly in Economics. New York: Academic Press. Diamond, P., and M. Yaari 1972 Implications of the theory of rationing for consumer choice under uncertainty. American Economic Review 62:33~343. Edwards, W. 1954a Probability-preferences among bets with differing expected value. American Journal of Psychology 67:5~67. 1954b The theory of decision making. Psychological Bulletin 51:380 417. 1955 The prediction of decisions among bets. Journal of Experimental Psychology 50:201-214. Subjective probabilities inferred from decisions. Psychological Review 69:109-135. Bayesian and regression models of human information processing A myopic perspective. Organizational Behavior and Human Performance 6:639-648. Edwards, W., H. Lindman, and L. Savage  1963 Bayesian statistical inference for psychological research. Psychological Review 70:19~242. Einhorn, H., and R. Hoganh 1986 Decision making under ambiguity. Journal of Business 59(Suppl.~:S225-S250. Ellsberg, D. 1961 Risk, ambiguity, and the Savage axioms. Quarterp Journal of Economics 75:64 669. 1962 1971

MARK ~ MA CHINA 181 1963 Risk, ambiguity, and the Savage axioms: Reply. Quarters Joumal of Economics 77:336 342. Epstein, ~ 1985 Decreasing risk aversion and mean-vanance analysis. Econometn'ca 53:945-961. 1987 The unimportance of the intransitivity of separable preferences. Intemational Economic Review 28:315-322. Feather, N. 1959 Subjective probability and decision under uncertainty. 66:15~164. Psychological Review Fellner, W. 1961 Distortion of subjective probabilities as a reaction to uncertainty. Quarterly Journal of Economics 75:67~689. Slanted subjective probabilities and randomization: Reply to Howard Raiffa and K. R. W. Brewer. Quarterly Joumal of Economics 77:676-690. 1963 Fischhoff, B. 1983 Predicting frames. Journal of Experimental Psychology: Learned, Memory and Cognition 9:10~116. Fishburn, P. 1981 An axiomatic characterization of skew-symmetric bilinear functionals, with ap plications to utility theory. Economics Letters 8:311-313. 1982 Nontransitive measurable UtilitY. Joumal of Mathematical Psychology 26:31~7. 1 982 ~ 1984a SSB utility theory: An economic perspective. Mathematical Social Sciences 8:6~94. 1984b SSB utility theory and decision making under uncertainty. Mathematical Social Sciences 8:25~285. 1986 A new model for decisions under uncertainty. Economics Letters 21:127-130. 1988 Uncertainty aversion and separated effects in decision making under uncer tainty. In J. Kacprzyk and M. Fedrizzi, eds., Combining Fuzzy Impressions With Probabilistic Uncertainty in Decision Making. Berlin: Springer Verlag. Fishburn, P., and G. Kochenberger 1979 Two-piece Van Neumann-Morgenstern utility functions. 10:50~518. ~ , , Transitive measurable utility. Joumal of Economic Theory 31:29~317. Decision Sciences Friedman, M., and L. Savage 1948 The utility analysis of choices involving risk. Journal of Political Economy 56:27~304. Reprinted with revisions in Stigler and Boulding (1952~. Grether, D. 1978 Recent psychological studies of behavior under uncertainty. American Economic  Review Papers and Proceedings 68:7~74. 1980 Bayes rule as a descriptive model: The representativeness heuristic. Quarterly Joumal of Economics 95:537-557. Grether, D., and C. Plott 1979 Economic theory of choice and the preference reversal phenomenon. American Economic Rewew 69:6234638. Grofman, B., and G. Owen, eds. 1986 Information Pooling and Group Decision Making. Greenwich, Conn.: JAI Press. Hagen, O. 1979 Towards a positive theory of preferences under risk. Pp. 271-302 in M. Allais and O. Hagen, eds. Expected Utility Hypotheses and the Allais Paradom Dordrecht, Holland: D. Reidel Publishing Company.

182 CHOICE UNDER UNCERTAINTY Henderson, J., and R. Quandt 1980 Microeconomic Theory: A Mathematical Approach, ad ed. New York: McGraw- Hill. Hershey, J., and P. Schoemaker 1980a Prospect theo~y's reflection hypothesis: A critical examination. Organizational Behavior and Human Performance 25:395018. 1980b Risk-taking and problem context in the domain of losses-An expected utility analysis. Joumal of Risk and Insurance 47:111-132. 1985 Probability versus certainty equivalence methods in utility measurement: Are they equivalent? Management Science 31:121~1231. Hershey, J., H. Kunreuther, and P. Schoemaker 1982 Sources of bias in assessment procedures for utility functions. Management Science 28:936 954. Hey, J. 1979 Uncertain in Microeconomics. Oxford: Martin Robinson and Company, Ltd. Hirshleifer, J. 1965 Investment decision under uncertainty: Choice-theoretic approaches. Quarterly Journal of Economics 79:509 536. 1966 Investment decision under uncertainty Applications of the state-preference approach. Quarter) Joumal of Economics 80 25~277. Hogarth, R. 1975 Cognitive processes and the assessment of subjective probability distributions. Joumal of the American Statistical Association 70:271-289. Judgment and Choice: The Psychology of Decision. New York: John Wiley and Sons. Hogarth, R., and H. Kunreuther 1986 Risk, Ambiguity and Insurance. Unpublished manuscript, University of Chicago. 1989 Risk, ambiguity and insurance. Joumal of Risk and Uncertainty 2:5-35. Hogarth, R., and M. Reder, eds. 1987 Rational Choice: The Contrast Between Economics and Psychology. Chicago: University of Chicago Press. Holt, C. 1986 Preference reversals and the independence axiom. American Economic Review 76:50~515. Kahneman, D., and A. Tvemky 1973 On the psychology of prediction., Psychological Review 80:237-251. 1979 Prospect theory: An analysis of decision under risk. Econometrica 47:26~291. 1982 The psychology of preferences. Scientific American 246:16~173. 1984 Choices, values and frames. American Psychologist 39:341-350. Kahneman, D., P. Slavic, and A. Tversky, eds. 1982 Jud~nent Under Uncertainty: Heuristics and Biases. Cambridge: Cambridge University Press. Karmarkar, U. 1974 The Effect of Probabilities on the Subjective Evaluation of Lotteries. Mas  sachusetts Institute of Technology Sloan School of Business Working Paper No. 698-74. 1978 Subjectively weighted utility: A descriptive extension of the expected utility model. Organizational Behavior and Human Performance 21:61-72. 1979 Subjectively weighted utility and the Allais paradox. Organizational Behavior md Human Performance 24:67-72. 1980

MARK ~ MA CHINA 183 Karni, E. 1985 Decision Making Under Uncertainty: The Case of State Dependent Preferences. Cambridge, Mass.: Harvard University Press. Karni, E., and Z. Safra 1987 "Preference reversal" and the observability of preferences by experimental methods. Econometnca 55:675~85. Keeney, R., and H. Raiffa 1976 Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: John Wiley and Sons. Keller, L. 1985 The effects of decision problem representation on utility conformance. Manage ment Science 6:738-751. Kihlstrom, R., D. Romer, and S. Williams 1981 Risk aversion with random initial wealth. Econometnca 49:911-920. Kim, T., and M. Richter 1986 Nontransitive-nontotal consumer theory. Joumal of Economic Theory 38:32~363. Knetsch, J., and J. Sinden 1984 Willingness to pay and compensation demanded: Experimental evidence of an unexpected disparity in measures of value. Quarterly Journal of Economics 99:507-521. Knez, M., and V. Smith 1987 Hypothetical valuations and preference reversals in the context of asset trading. Pp. 131-154 in A. Roth, ea., Laboratory Embedments in Economics: Sur Pouts of Flew. Cambridge: Cambridge University Press. Lichtenstein, S., and P. Slovic 1971 Reversals of preferences between bids and choices in gambling decisions. Joumal of E~ruer~tal Psychology 89:4~55. 1973 Response-induced reversals of preference in gambling: An extended replication in Las Vegas. Journal of E~p~nmental Psycholofg, 101:1~20. Lindman, H. 1971 Inconsistent preferences among gambles. Journal of Experimental Psychology 89.39~397. Lippman, S., and J. McCall 1981 The economics of uncertainty: Selected topics and probabilistic methods. Pp. 211-284 in K Arrow and M. Intriligator, eds., Handbook of Mathematical Economics, vol. 1. Amsterdam: North-Holland Publishing Company. Loomes, G., and R. Sugden 1982 Regret theory: An alternative theory of rational choice under uncertainty. Economic Journal 92:805-824. 1983a Regret theory and measurable utility. Economics Letters 12: 1~22. 1983b A rationale for preference reversal. American Economic Review 73:428~32. MacCrimmon, K 1965 An Experimental Study of the Decision Making Behavior of Business Executives.  Doctoral dissertation, University of California, Los Angeles. 1968 Descriptive and normative implications of the decision-theo~y postulates. Pp. 3-23 in K Borch and J. Mossin, Risk and Uncertainty: Proceedings of a Conference lIeld by the International Economic Association. London: Macmillan and Company.

184 CHOICE UNDER UNCERTAINTY MacCrimmon, K, and S. Larsson 1979 Utility theory: Axioms versus "Paradoxes." Pp. 333-409 in M. Allais and 0. Hagen, eds., Expected Utility Hypotheses and the Allais Paradox. Dordrecht, Holland: D. Reidel Publishing Company. MacCrimmon, K, and D. Wehrung 1986 Taking Risks: The Management of Uncertainty. New York: The Free Press. Machina, M. 1982 "Expected utility" analysis without the independence axiom. Econometrica 50:277-323. 1983a The Economic Theory of Individual Behavior Toward Risk: Theory, Evidence and New Directions. Stanford University Institute for Mathematical Studies in the Social Sciences Technical Report No. 443. 1983b Generalized expected utility analysis and the nature of observed violations of the independence axiom. Pp. 26~293 in B. Stigum and F. Wenstop, Foundations of Utility and Risk Theory with Applications. Dordrecht, Holland: D. Reidel Publishing Company. 1984 Temporal risk and the nature of induced preferences. Journal of Economic Theory 33:199-231. 1987 Choice under uncertainty Problems solved and unsolved. Joumal of Economic Perspectives 1:121-154. Machina, M., and W. Neilson 1987 The Ross measure of risk aversion: Strengthening and extension. Econometrica 55:1139-1149. Markowitz, H. 1952 The utility of wealth,. Journal of Political Economy 60:151-158. 1959 Portfolio Selection: Efficient Diversification of Invesonents. New Haven, Conn.: Yale University Press. Marschak, J. 1950 Rational behavior, uncertain prospects, and measurable utility. Econometrica 18:111-141 (Errata. Econometrica 18:312~. 1974 Economic Information, Decision, and Prediction, 3 vole. Dordrecht, Holland: D. Reidel Publishing Company. Mas-Colell, ~ 1974 An equilibrium existence theorem without complete or transitive preferences. Joumal of Mathematical Economics 3:237-246. Maskin, E. 1979 Decision making under ignorance with implications for social choice. Theory and Decision 11:319-337. May, K. 1954 Intransitivity, utility, and the aggregation of preference patterns. Econometrica 22:1-13. McCord, M., and R. de Neufville 1983 Empirical demonstration that expected utility analysis is not operational. Pp. 181-199 in B. Stigum and F. Wenst0p, Foundations of Utility and Risk Theory with Applications. Dordrecht, Holland: D. Reidel Publishing Company.  McCord, M., and R. de Neufville 1984 Utility dependence on probability. An empirical demonstration. Large Scale Systems 6:91-103. McNeil, B., S. Pauker, H. Sox, and A. Iversky 1982 On the elicitation of preferences for alternative therapies. New England Journal of Medicine 306:1259-1262.

AL4RK ~ MA CHINA 185 Milne, F. 1981 Induced preferences and the theory of the consumer. Joumal of Economic Theory 24:205-217. Mishan, E. 1976 Choices involving risk: Simple steps toward an ordinalist analysis. Economic Journal 86:750777. Morrison, D. 1967 On the consistency of preferences in Allais' paradox. Behavioral Science 12:373- 383. Moskowitz, H. 1974 Effects of problem representation and feedback on rational behavior in Allais and Morlat-type problems. Decision Sciences 5:225-242. Mowen, J., and J. Gentry 1980 Investigation of the preference-reversal phenomenon in a new product intro- duction task. Journal of Applied P<ycholo~ 65:715-722. Pareto, V. 1909 Manuel d'Economie Politulue. Paris: V. Girard et E. Briere. Parkin, M., and A. Nobay, eds. 1975 Current Economic Problems. Cambridge: Cambridge University Press. Payne, J., and M. Braunstein 1971 Preferences among gambles with equal underlying distributions. Joumal of E~perunental Psychology 87:1~18. Payne, J., D. Laughhunn, and R. Crum 1980 Translation of gambles and aspiration level effects in nsly choice behavior. Management Science 26:1039-1060. 1981 Further tests of aspiration level effects in risky choice behavior. Management Science 27:953-958. Pommerehne, W., F. Schneider, and P. Zweifel 1982 Economic theory of choice and the preference reversal phenomenon: A reexamination. American Economic Review 72:569-574. Pratt, J. 1964 Risk aversion in the small and in the large. Econometrica 32:12~136. Quiggin, J. 1982 A theory of anticipated utility. Journal of Economic Behavior and Organization 3:323-343. Raiffa, H. 1961 Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics 75:690 694. 1968 Decision Analysis: Introductory Lectures on Choice- Under Uncertain~. Reading, Mass.: Addison-Wesley. Reilly, R. 1982 Preference reversal: Further evidence and some suggested modifications of experimental design. American Economic Review 72:57~584.  Roberts, H. 1963 Risk, ambiguity, and the Savage axioms: Economics 77:327-336. Comment. Quarters Joumal of Roell, A. 1987 Risk aversion in Quiggin and Yaari's rank-order model of choice under uncer tainty. Economic Joumal 97(Suppl.~:143-159.

186 CHOICE UNDER UNCERTAINTY Ross, S. 1981 Some stronger measures of risk aversion in the small and in the large with applications. Econometnca 49:621~. Rothschild, M., and J. Stiglitz 1970 Increasing risk: I. A definition. Joumal of Econamic Theory 2:225-243. 1971 Increasing risk: II. Its economic consequences. Joumal of Economic Theory 3:6~4. Russo, J. 1977 The value of unit price information. Joumal of Marketing Research 14:193 201. Russo, J. G. Krieser, and S. Miyashita 1975 An effective display of unit price information. Journal of Marketing 39:11-19. Samuelson, P. 1947 Foundations of Economic Analysis. Cambridge, Mass.: Harvard University Press. Enlarged edition, 1983. 1952 Probability, utility, and the independence axiom. Econometaca 20:670-678. 1977 St. Petersburg paradoxes: Deranged, dissected, and historically described. Joumal of Economic Literature 15:2A 55. Savage, ~ 1954 The Foundations of Statistics. New York: John Wiley and Sons. Revised and enlarged edition published lay Dover Publications (New York, 1972~. Schlaifer, R. 1969 Analysis of Decisions Under Uncertainty. New York: McGraw-Hill Publishing Co.. Schmeidler, D. 1989 Subjective probability and expected utility without additivity. Econometrica 57:571-587. Schoemaker, P. 1980 Experiments on Decisions Under Risk The Expected Utility Hypothesis. Boston: Martinus Nijhoff Publishing. Schoemaker, P., and H. Kunreuther 1979 An experimental study of insurance decisions. Joumal of Risk and Insurance 46:603~18. Segal, U. 1984 Nonlinear Decision Weights with the Independence Axiom. Unpublished man uscript, University of California, Los Angeles. 1987 The Ellsberg paradox and risk aversion: An anticipated utility approach. International Economic Review 28:175-202. Shafer. W. 1974 The nontransitive consumer. Econometrica 42:913-919. 1976 Equilibrium in economies without ordered preferences or free disposal. Joumal of Mathematical Economics 3:135-137. Sherman, R. 1974 The psychological difference between ambiguity and risk. Quarters Journal of Economics 88:166 169.  Sinn, H.-W. 1980 A rehabilitation of the principle of sufficient reason. Quarterp Joumal of Economics 94:493-506. Slovic, P. 1969a Differential effects of real versus hypothetical payoffs on choices among gambles. Joumal of E~perunental Psychology 80: 43~437.

AL4RK ~ AL4 CHINA 187 1969b Manipulating the attractiveness of a gamble without changing its expected value. Journal of Experimental Psychology 7913~145. 1975 Choice between equally valued alternatives. Joumal of Experimental Psychology: Human Perception and Performance 1:28~287. Slavic, P., and S. Lichtenstein 1968 Relative important of probabilities and payoffs in risk taking. Journal of Experimental Psychology 78~3, Part 20: 1-18. Slovic, P., and S. Lichtenstein 1971 Comparison of Bayesian and regression approaches to the study of information processing in judgment. Organizational Behavior and Human Performance 6:649- 744. Slovic, P., and S. Lichtenstein 1983 Preference reversals: A broader perspective. American Economic Review 73: 59 605. Slovic, P., and A. Tvemly 1974 Who accepts Savage's axiom? Behavioral Science 19:368 373. Slovic, P., B. Fischhoff. and S. Lichtenstein 1982 _, Response mode, framing, and information processing effects in risk assessment. Pp. 21-36 in R. Hogarth, ea., New Directions for Methodology of Social and Behavioral Science: Question Framing and Response Consistency. San Francisco: Jossey-Bass. Smith, V. 1969 Measuring nonmonetary utilities in uncertain choices: The Ellsberg urn. Quar ter1y Journal of Econamics 88:320329. Sonnenschein, H. 1971 Demand theory without transitive preferences, with applications to the theory of competitive equilibrium. Pp. 215-233 in J. Chipman, Lo Hurwicz, M. Richter, and H. Sonnenschein, eds., Preferences, Utility, and Demand. New York: Harcourt Brace Jovanovich, Inc. Spence, M., and R. Zeckhauser 1971 Insurance, information, and individual action. American Economic Review 61:38(}387. Stigler, G., and K Boulding, eds. 1952 Stiglitz, ]. 1975 A, Readings u' Pnce Theory. Chicago: Richard D. Irwin. Information and economic analysis. Pp. 27-52 in M. Parkin and A. Nobay, eds., Current Econamic Problems. Cambridge: Cambridge University Press. 1985 Information and economic analysis: A perspective. Economic Joumal 95(Suppl.): 21-41. Stiglitz, J., ed.  1966 Collected Scientific Papers of PaulA. Samuelson, vol. 1. Cambridge, Mass.: MIT Press. Sugden, R. 1986 New developments in the theory of choice under uncertainty. Bulletin of Economic Research 38:1-24. Thaler, R. 1980 1hward a positive theory of consumer choice. Joumal of Economic Behavior and Organization 1:39 60. 1985 Mental accounting and consumer choice. Marketing Science 4:19~214. Iversly, A. 1969 Intransitivity of preferences. Psychological Review 76:31~8.

188 CHOICE UNDER UNCERTAINTY 1975 A critique of expected utility theory: Descriptive and normative considerations. Erkenntnis 9:163-173. Iversly, As, and D. Kahneman 1971 Belief in the law of small numbers. Psychological Bulletin 2:105-110. 1974 Judgment under uncertainty: Heuristics and biases. Science 185:1124 1131. 1981 The framing of decisions and the psychology of choice. Science 211:453~58. 1983 Extensional vs. intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review 90:29~315. visCUSi, w. 1985a Are individuals Bayesian decision makers? American Economic Review Papers and Proceedings 75:381-385. 1985b A Bayesian perspective on biases in risk perception. Economics Letters 17:59~2. van Neumann, J., and O. Morgenstern 1947 Theory of Games and Economic Behavior, 2d ed. Princeton, N.J.: Princeton University Press. Weber, M., and C. Camerer 1987 Recent developments in modeling preferences under risk. OR Spekn~um 9:129- 151. Weinstein, A. 1968 Individual preference intransitivity. Southem Economic Joumal 34:335-343. Williams, C. 1966 Attitudes toward speculative risks as an indicator of attitudes toward pure risks. Joumal of Risk and Insurance 33:577-586. Yaari, M. 1969 Some remarks on measures of risk aversion and on their uses. Joumal of  Economic Theory 1:315-329. 1987 The dual theory of choice under risk. Economeoica 55:95-115.