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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

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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.

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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.

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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.

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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.

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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.

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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~.

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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.

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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.

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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~.

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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.

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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.

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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.

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