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7. Choice Under Uncertainty: Problems Solved and Unsolved
7. Choice Under Uncertainty: Problems Solved and Unsolved
Pages 134-188
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From page 134... ...
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.
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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)
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, 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 NeumannMorgenstern utility function is cardinal in that it can only be subjected to transformations that change the origin point or the scale (or both)
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,"'/ ~ '~' / 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.
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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)
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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.
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+ U(xn~pn. Although this preference function generalizes the expected value form HIPS + ...
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~ 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.
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B: Relatively flat indifference curves of a risk lover.
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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.
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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)
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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.
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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 .
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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.
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have found a tendency for higher values of p to lead to the "recovery" of higher valued utility functions (Figure 6a)
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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.
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* = 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 probabilities have led several researchers to generalize the expected utility model by positing nonlinear functional forms for the individual preference function.
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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.
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· Risk Aversion: Vie ) will exhibit global risk aversion if and only if the coefficients {U(xi)
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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.
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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.
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, 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)
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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)
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Excellent discussions 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 Schoemaker (1985~; and MacCrimmon and Wehrung (1986~. ~ report how the response mode study of Slovic and Lichtenstein (1968)
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. Unlike indifference curves for transitive preferences, however, these curves 4iAlgebraically, this expected value is given by the double summation Hi Hi r(~i,rj)
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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.
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.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.
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, we should expect most of the testable implications of this approach to appear as cross-institutional predictions, such as systematic violations 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 representing or "framing" probabilistically equivalent choice problems will lead to systematic differences in choice.
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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~.
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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 reference point. Theoretically, the variable that enters an individuals von Neumann-Morgenstern utility functions should be total (i.e., final)
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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.
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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.
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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)
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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.
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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")
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, the unweighted average of their outcomes ("principle of insufficient reason") , or similar criteria.57 Finally generalizations of expected utility theory that drop the standard additivity or compounding laws of probability theory (or both)
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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)
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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 individuals who are exhibiting each of these phenomena; should there not be a single model capable of gmerat~ng them all?
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Yet the types of systematic biases in the formulation and manipulation of subjective probabilities presented in the preceding section ('`Other Issues: Is Probability Theory Relevant?
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Although experimental subjects and real-world decision makers sometimes 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 departures from the strictures of probability theory should be corrected but that (systematic) departures from the strictures of expected utility theory should not?
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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)
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From page 176... ...
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.
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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.
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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.
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American Economic Review 75:597613. Becker, G., M
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1986 An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom. Joumal of Economic Theory 40:3~318.
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Grether, D., and C Plott 1979 Economic theory of choice and the preference reversal phenomenon.
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Organizational Behavior and Human Performance 21:61-72. 1979 Subjectively weighted utility and the Allais paradox.
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American Economic Review 73:428~32. MacCrimmon, K 1965 An Experimental Study of the Decision Making Behavior of Business Executives.
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443. 1983b Generalized expected utility analysis and the nature of observed violations of the independence axiom.
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Schneider, and P Zweifel 1982 Economic theory of choice and the preference reversal phenomenon: A reexamination.
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Unpublished man uscript, University of California, Los Angeles. 1987 The Ellsberg paradox and risk aversion: An anticipated utility approach.
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American Economic Review 73: 59 605. Slovic, P., and A
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American Economic Review Papers and Proceedings 75:381-385. 1985b A Bayesian perspective on biases in risk perception.
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