âchoose the lottery that has the highest expected utility.â This decision rule must be justified on its own terms as a valid rule of rationality by demonstration that ...
Forum Is Expected Utility Theory Normative for Medical Decision Making? BRIAN J. COHEN, MD Expected utility theory is felt by its proponents to be a normative theory of decision making under uncertainty. The theory starts with some simple axioms that are held to be rules that any rational person would follow. It can be shown that if one adheres to these axioms, a numerical quantity, generally referred to as utility, can be assigned to each possible outcome, with the preferred course of action being that which has the highest expected utility. One of these axioms, the independence principle, is controversial, and is frequently violated in experimental situations. Proponents of the theory hold that these violations are irrational. The independence principle is simply an axiom dictating consistency among preferences, in that it dictates that a rational agent should hold a specified preference given another stated preference. When applied to preferences between lotteries, the independence principle can be demonstrated to be a rule that is followed only when preferences are formed in a particular way. The logic of expected utility theory is that this demonstration proves that preferences should be formed in this way. An alternative interpretation is that this demonstrates that the independence principle is not a valid general rule of consistency, but in particular, is a rule that must be followed if one is to consistently apply the decision rule “choose the lottery that has the highest expected utility.” This decision rule must be justified on its own terms as a valid rule of rationality by demonstration that violation would lead to decisions that conflict with the decision maker’s goals. This rule does not appear to be suitable for medical decisions because often these are one-time decisions in which expectation, a long-run property of a random variable, would not seem to be applicable. This is particularly true for those decisions involving a non-trivial risk of death. Key words: expected utility theory; von Neumann-Morgenstern utility; strength of preference; decision analysis; independence principle. (Med Decis Making 1996;16:1-6)
Expected utility theory has been the dominant theory of decision making under uncertainty of the last half of the twentieth century. Though it is often violated in practice, its proponents hold it to be a normative theory of decision making. It forms the basis of decision analysis, which has been widely applied as an aid to clinical decision making. The normative nature of the theory is controversial, and the techniques derived from the theory can be used appropriately only with a thorough understanding of the basis of the theory. This article reviews the theoretical basis of expected utility theory and questions its status as a normative theory for medical decision making.
Expected Utility Theory: Historical Perspective The intellectual origins of expected utility theory date back to the 18th century in proposed solutions to the Saint Petersburg paradox.’ A game is proposed in which a fair coin is tossed until the first heads. turns up. If this first head occurs on the nth toss, the player wins 2” dollars. Since the probability of the first heads occurring on the nth toss is I/Z”, the expected value of this game is
which is infinite. The paradox lies in the fact that it is unlikely that anyone would pay any substantial sum to play this game. Daniel Bernoulli suggested that expectation is relevant, but that the appropriate expectation involves a measure of the personal worth, or utility, of the money at stake, rather than the expectation of the monetary amounts themselves. The paradox could be explained by assuming that the gain in
Received April 26, 1994, from the Division of Clinical Decision Making, Department of Medicine, New England Medical Center, Tufts University School of Medicine, Boston, Massachusetts. Accepted for publication May 23, 1995. Supported in part by Grant TGLM-7044 from the National Library of Medicine, Bethesda, Maryland. Address correspondence and reprint requests to Dr. Cohen: New England Medical Center, 750 Washington Street, Box 302, Boston, MA 02111.
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a
P
2
b
2. Independence: If a 2 (is preferred to or is equivalent to) b, then the lottery offering a with probability p and c with probability 1 - p 2 the lottery offering b with probability p and c with probability 1 - p, for any c (see figure 1).
a >
1-P (is preferred to) b, and b > c, then a > c.
1. The independence principle.
utility of a given increment in wealth decreases as the starting wealth increases (decreasing marginal utility), so that the expected utility of the game is finite, and perhaps small. An alternative explanation is that since the probability of winning a very large amount is so small, the player would consider this probability essentially zero.” Anyone playing the game once would be virtually guaranteed of losing the bulk of a large entrance fee. A very large number of games must be played before the odds of winning a substantial sum become considerable, and all but the wealthiest would bankrupt themselves in the process. The St. Petersburg paradox does not prove that decisions are (or should be) made on the basis of expected utility, only that they are not, in some circumstances at least, made on the basis of expected value. Bernoulli’s solution represents the earliest suggestion that numbers can be attached to monetary amounts to represent the personal worth of these amounts, and that decisions under risk are made based on the expectation of this personal worth. The concept of decreasing marginal utility played a key role in the development of economic theory, which dealt with preferences under certainty. The concept of the maximization of expected utility as the basis for decision making under risk lay dormant until revived by Ramsey in the 1920s, 3 and then fully developed by von Neumann and Morgenstern” and others.5
Expected Utility Theory The approach of expected utility theorists is to start with axioms that they claim are basic rules that any rational person would adhere to when considering their preferences. Two of these axioms deal strictly with consistency among preferences, in that they dictate that a rational agent must hold a specified preference given other stated preferences. a, b, and c referred to in these axioms could be certain prizes (i.e., those obtained with probability 1) or lotteries.
Won Neumann and Morgenstern did not explicitly specify an independence axiom, but others have shown that it was implicit in their assumptions. 6) If one accepts the full set of axioms, a numerical quantity, generally referred to as utility, can be assigned to each outcome, with the “preferred” course of action being that which has the highest expected utility.* It is the “preferred” course of action in the sense that choosing any other course of action would be inconsistent with the decision maker’s preferences, as expressed in the utilities assigned to the individual possible outcomes. These utilities can be assessed through elicitation of preferences between pairs of hypothetical lotteries. The inconsistency specifically refers to a violation of one of the axioms discussed above.
Is the Independence Principle a Valid Rule of Rationality? The claim of normative status for expected utility theory is based on the assumption that the axioms are self-evident rules of rational choice. The independence axiom holds that given Lottery A offering prize a at probability p, and Lottery B offering prize b at probability p, with both lotteries offering prize c at probability 1 -p, the preference order for Lotteries A and B should be the same as the preference order for prizes a and b. The basic argument for it can be summarized as follows’: When choosing between a Lottery A, offering a p chance of winning a and a 1 - p chance of winning c, and Lottery B, offering a p chance of winning b and a 1 - p of winning c, the prize c is irrelevant since it is common to both lotteries, and we are operating in the probabilistic realm where in each lottery, only one of the two prizes can be won. Therefore, the preference ordering of Lotteries A and B should be the same as the preference ordering for a and b. A reversal of preference induced by the possibility of winning c would imply some perceived complementarity between the two prizes in a given lottery, as if the presence of one prize altered feelings about the other. This type of complementarity is commonly observed for goods that can be enjoyed together, but should not be a factor with mutually exclusive outcomes.
*Expected utility is the weighted sum of the utilities of all possible outcomes of the given course of action, with the probability of occurrence of each outcome serving as its weight.
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The defense of expected utility theory as a normative theory of decision making is based on this type of intuitive defense of the axioms, along with the implicit claim that this constitutes an adequate basis for defining what is rational. Any discussion of the normative nature of expected utility theory must start with a broader discussion of rationality, and the criteria that will be used for judging whether a given rule is a valid rule of rationality. Hume’s view of rationality was that it is purely instrumental, i.e., it can serve as an aid to reaching one’s goals, but cannot serve to justify those goals.8 Some modern writers hold this view, 9 but others find this definition too narrow, claiming that decisions about which goals to adopt can be judged by standards of rationality. 10 Central to either view is the idea that the value of rationality is its ability to help people further their own interests, so that intuitive rules of consistency alone cannot serve as a sufficient basis for rationality. 10-12 They may form a necessary basis for rationality, though, if it can be demonstrated that a specific type of inconsistency would potentially interfere with the decision maker’s ability to achieve his or her goals, or at least in some way lead to a less desirable state of affairs. For example, violations of transitivity would potentially lead to someone’s spending money to remain at the status quo. Assume I prefer a to b, and b to c. If I start out with c, I would presumably be willing to part with c, and pay at least a small sum of money to receive b. I would then be willing to part with b, and pay an additional sum to receive a. But if I violate transitivity by also preferring c to a, I would be willing to part with a, and pay additional money to receive c. I would have returned to the starting point, only with decreased wealth. Even this type of argument has been questioned as an adequate defense as it refers to a hypothetical sequence of trades rather than the actual choice problem. 13 Thus, the logic of those who defend expected utility theory based on the axioms would appear to be backward.‘” Consistency axioms that can be shown to be equivalent to a specific prescription for forming preferences cannot be viewed as general rules of rationality unless that prescription for forming preferences can be justified through external considerations. The independence axiom has intuitive appeal when applied to certain outcomes. If I am indifferent between a car and $20,000, it seems reasonable that I should be indifferent between one lottery offering a 50% chance of winning a car and a 50% chance of winning a motorboat, and a second lottery offering a 50% chance of winning $20,000 and a 50% chance of winning a motorboat. One feels comfortable substituting items felt to be equivalent under certainty in one limb of a lottery and not altering preference rankings. Willingness to make this substitution is independent of how one compared the car and the $20,000 and what went into the judgement that they were equivalent. This is a rule
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Lottery I A
l
3
Lottery II .80 ,$4000
1
$3000
Y .2h$O .20,/$4000 *2> $3000 ,
B YiLr ‘,, Lottery IV
Lottery III
FIGURE 2. Lotteries used in an empirical test of expected utility theory (see text).
$0 Lottery II
Lottery IV Lottery IV (figure 2) rewritten to highlight the relationship between Lottery II and Lottery IV.
FIGURE 3.
governing the extension of preferences formed under certainty into the world of uncertainty. Things change when we apply the principle to preferences formed in the face of uncertainty, as expected utility theory requires.515 If I am indifferent between Lottery A and Lottery B, whether or not I would also be indifferent between Lottery X offering a 50% chance of winning the opportunity to play Lottery A and a 50% chance of winning the opportunity to play Lottery C, and Lottery Y offering a 50% chance of winning the opportunity to play Lottery B and a 50% chance of winning the opportunity to play Lottery C (see figure 1), turns out to be dependent on how I compared the initial lotteries and formed my preference between them. I will be willing to make the substitution now only if a specific decision rule (“choose the lottery with the highest expected utility”) was used to form the preferences. Rather than being a consistency rule whose intuitive appeal proves the expected utility decision rule, the independence principle, when applied to preferences for lotteries, can be viewed as a rule that is followed only when those preferences are formed in a particular way, and is therefore not a valid general rule of consistency. What has been demonstrated is that the independence axiom is a rule that should be followed if one wants to consistently choose the lottery
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with maximum expected utility, not that decisions should be made in this way. This can be made more concrete by focusing on preferences between lotteries offering $A at probability p* (and $0 at probability 1 - p,) and $B at probability pB (and $0 at probability 1 - p,). Expected utility theory dictates that only the ratio pdpS matters in choosing the preferred lottery. The indifference point should be w h e n pJps equals U,/U,, where Us are von Neumann-Morgenstern utilities. The requirement that preferences for this type of lottery be based on probability ratios can be illustrated with reference to the lotteries of figure 2: If $3,000 for certain (Lottery I) were felt to be equivalent
to a lottery offering a 0.80 chance of winning $4,000, and a 0.20 chance of winning $0 (Lottery II), then, according to the independence principle, a lottery offering a 0.25 chance of winning $3,000 and a 0.75 chance of winning $0 (Lottery III) should be equivalent to a lottery offering a 0.20 chance of winning $4,000 and a 0.80 chance of winning $0 (Lottery IV). This can be demonstrated by rewriting Lottery IV as shown in figure 3. The relationship between Lottery II and Lottery IV becomes clear. Lotteries III and IV represent a 0.25 chance of winning the opportunity to play Lotteries I and II, respectively (with a 0.75 chance of winning nothing in both cases), so by the independence principle, preferring Lotteries I and III and preferring Lotteries II and IV are the only permissible combinations.
In both sets of lotteries the ratio of the probability of winning $3,000 to the probability of winning $4,000 is 1.25 to 1 when the lotteries are equivalent. In the first pair these probabilities are 1 and 0.80, while in the second pair they are 0.25 and 0.20. The ratio is the same in both pairs, but the probability difference is 0.2 in the first pair, and 0.05 in the second pair,: Kg cording to the independence principle, then, only the ratio, and not the difference, should be taken into account when determining preference among lotteries of this type. Empirically, though, a large percentage of experimental subjects preferred Lottery I to Lottery II, while also preferring Lottery IV to Lottery III, thus violating the independence principle.16 In another study using similar lotteries, experimental subjects reported that their choices are often influenced by probability differences.17 When the excess probability of winning the smaller amount is large enough, the lottery offering the smaller amount as, a prize may be preferred, but when the probability difference is small enough (even if the ratio is kept constant), the lottery offering the larger payoff may be preferred. This type of tradeoff violates the independence principle. Thus, the independence principle, when applied to outcomes that are themselves lotteries, is not just a principle of consistency among preferences, it is actually a specific psychological recipe for forming preferences. Conversely, the independence principle will
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be followed only when that recipe is used, and is therefore not a valid general consistency rule. In a sense, it is possible to say that certain preferences, not just pairs of preferences, violate the independence principle. Those who are influenced by probability differences have formed their preferences “incorrectly.” While the independence principle is based on the idea that the two limbs of a lottery should be independent since the outcomes are mutually exclusive, it places constraints on the way lotteries can be compared. To be accepted as a rule of rational choice, external justification is needed.
Can Maximization of Expected Utility be Defended Independently of the Axioms? Baron has offered a defense of this rule that is independent of the axioms.” He views utility as a measure of goal achievement, so that choosing based on expected utility allows one to best achieve one’s goals in the long run. This argument assumes the existence of a cardinal strength of preference utility function that can describe the relative value an individual attributes to various outcomes under certainty. Even if we accept this, the use of expectation, a long-run property of a random variable, in dealing with preferences under uncertainty would seem to be valid, by definition, only if one is considering the average results of many choices. Either the utility achieved from sequentially made decisions must be added over time, or we must consider the total utility achieved from the decisions of many people Many significant medical decisions are onetime decisions, not repetitive decisions. With a onetime decision, expectation, a long-run property, is meaningless. This is particularly true for those medical decisions involving more than a trivial risk of death. To be reasonably assured of achieving an average gain close to the expected value in any game of chance requires a large number of repetitions of the game. If there is a significant chance that the game will be cut short (by death or bankruptcy, for example) the chances that the long-run average will be observed becomes small. Basing decisions on expectation in this situation can then be justified only by reference to the results of a large number of people making the same decision.” Baron has argued18, pp295-296: . . . if many people choose to have a surgical operation that they know has a .000l probability of causing death, in order to achieve some great medical benefit, some of these people will die, but, if the utility of the medical benefit is high enough, the extent to which all the people together achieve their goals will still be greater than if none of them chooses the operation. When we add utilities across people, we are saying that the loss to some . . is more than compensated by the gain to others . . .
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This argument for maximizing the sum of strength of preference utility for decisions involving groups of people is that of the ethical theory of utilitarianism, not an argument for rationality in individual decision making. Even in the absence of a significant risk of death, a rule that dictates choosing the therapy that has the highest expected strength of preference utility runs counter to our intuitions about the types of tradeoffs that one might make in choosing under uncertainty. Assume someone is in Health State I, which that person assigns a strength of preference utility value of 0.6. Therapy A offers a certain increase from Health State I to Health State II assigned a value of 0.9, while Therapy B offers an increase from Health State I to Perfect health (utility of 1) with probability 0.8, and a 0.2 probability of remaining in Health State I. Without worrying about how strength of preference utilities might be measured, we will assume that the decision maker feels comfortable with the assertion that a change from Health State I to Perfect Health represents a 33% larger gain in utility than a change from Health State I to Health State II. The expected utility of Therapy B is 0.8-l + 0.2 - 0.6 = 0.92. The expected gain in strength of preference utility for Therapy A is 0.3, and that for Therapy B, 0.32. It is hard to see what would be irrational about choosing Therapy A, even though Therapy B has the higher expected strength of preference utility. If 100 different people are treated with Therapy A, all 100 people treated receive a utility gain of 0.3. When 100 people receive treatment with Therapy B, 80 experience a gain of 0.4, but 20 experience no gain.? Choosing Therapy A would be a reflection of pure risk aversion, i.e., the willingness to choose an option with lower expected strength of preference utility to avoid risk. In this example, it is a willingness to accept the smaller strength of preference utility gain if the therapy is successful (0.3 as compared with 0.4) in order to avoid the 20% chance of receiving no gain Any theory not allowing a willingness to make this type of trade when the risk of death is involved can be defended only by insisting that some people take risks in order to improve the collective outcome. Once we drop the requirement that a rational agent choose so as to maximize expected strength of preference utility, expected utility theory cannot be defended without reference to the axioms. We can always calculate a von Neumann-Morgenstern utility for an outcome based on expressed preferences between lotteries, but this number is merely a representation of those preferences, calculated by assuming that indifference between lotteries implies equal expected utility. von Neumann and Morgenstern recognized that TObviously in a probabilistic world, in a single trial of 100 gambles, the exact values predicted by the stated probabilities would
not be achieved.
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they “defined numerical utility as being that thing for which the calculus of expectation is legitimate.“ 4,page28 Returning to the hypothetical clinical example, we know that Therapy A is preferred when Therapy B offers a 0.8 probability of success. Let us assume that this patient would be indifferent between the two when Therapy B offers a 0.9 probability of success. The von Neumann-Morgenstern utility for Health State II can now be calculated (using the same values for Health State I and Perfect Health as their strength of preference utilities) as 0.9 * 1 + 0.1’ 0.6 (the expected utility of Therapy B) = 0.96, as compared with its strength of preference utility of 0.9. This is a calculated number that can serve to represent the decision maker’s preference under risk, using this particular model of decision making. It reflects both the strength of preference for this health state and risk preference.” This view is well summarized in a passage from a standard treatise on decision theory2l: First, we may think of the theory as a description of
preference . . .; the preferences among lottery tickets logically preceded the introduction of a utility function.
Second, we may think of the theory as a guide to consistent action.. . . The point is that there is no need to assume, or to philosophize about, the existence of an underlying subjective utility function, for we are not attempting to account for the preferences or the rules of consistency. We only wish to devise a convenient way to represent them.
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There is no argument for trying to maximize this calculated value other than adherence to the axioms. Another approach to defending expected utility theory is through appeal to the principle that beliefs (probabilities) and preferences should be independent.” When the independence axiom is violated, unique utilities cannot be assigned to outcomes. Preferring $3,000 to a lottery offering a 0.8 chance of winning $4,000 means that the apparent von NeumannMorgenstern utility for $3,000 is greater than 0.8, given arbitrarily assigned utility values of 1 to $4,000 and 0 to $0. Preferring a 0.2 chance of winning $4,000 to a 0.25 chance of winning $3,000 means that the apparent von Neumann-Morgenstern utility of $3,000 is less than 0.8. The conclusion that this violation of the independence principle implies a change in preference with a change in belief follows only if we equate the calculated von Neumann-Morgenstern utilities with preference. This type of argument for expected utility theory is thus circular, in that this can be demonstrated only by using the theory to calculate the apparent preferences. Someone adhering to expected utility theory does maintain independence of beliefs and preferences, but to argue that the theory is normative because of this requires a demonstration that someone violating the theory necessarily violates this type of independence. Independence of preferences
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and beliefs can be demonstrated with theories of choice that do not require adherence to the independence axiom, in which preferences for lotteries are, modeled by a more complex function of the probability distribution of utilities than is assumed in expected utility theory.22
Implications for Clinical Decision Analysis Decision analysis is a tool that aims to provide decision makers with a method of gaining insight into complex decisions, in order to help them make “better” decisions. One way in which it attempts to do this is through utility assessment, which allows decomposition of a complex probability distribution of outcomes into smaller components. Utilities for all outcomes am assessed using simple lotteries, and these utilities are substituted for those outcomes anywhere in the more complex decision tree. Doubts about the normative force of expected utility theory as well as the frequent empirical violations limit the ability of a decision analyst to assess utilities, plug them into a model, and tell someone which strategy they “should” prefer. Decision analysis is best seen as an aid to decision making in which insights are gained during the process of building models and doing sensitivity analyses. 2 3 The structuring required may help avoid some of the biases that can creep into intuitive approaches, and facilitate a common understanding of the decision problem at hand for patient and physician. 24 It also forces the decision maker to look ahead at the downstream consequences of a decision, rather than focusing narrowly on the next step. 25 The potential tradeoffs among different attributes, such as length and quality of life, are made explicit and quantified. Sensitivity analysis allows exploration of the dependence of the apparent rankings on the change in one or more variables whose values may be considered ambiguous. Robustness of the ranking of strategies over a reasonable range of those variables may counter ambiguity avoidance. Expected utility, theory creates a framework to use in analyzing complex decisions, allowing that analysis to be based on relatively simple calculations. The utility concept can be used in a qualitative way, to explore how preferences might affect a particular decision. The dependence of the rankings on the degree of risk aversion, or disutility assigned to a particular health state, can be explored in a general way, aiding the development of preference-based guidelines. Decision analysis has much to offer, both to clinicians aiding an individual patient making a complex decision and to policy makers. The limits of the underlying theory must be recognized, however, and results of analyses interpreted accordingly.
References 1. Jorland G. The Saint Petersburg paradox 1713-1937. In: Kruger L, Daston LJ, Heidelberger M, eds. The Probabilistic Revolution, Volume 1: Ideas in History. Cambridge, MA: MIT Press, 1987, pp 157-90. 2. Gorovitz S. The St. Petersburg puzzle. In: Allais M, Hagen 0, eds. Expected Utility Hypothesis and the Allais Paradox. Dordrecht, The Netherlands: D. Reidel Publishing, 1979, pp 259-70. 3. Ramsey FP. Truth and probability. Reprinted in: Kyburg HE, Smokler HE, eds. Studies in Subjective Probability. Second edition. New York: Kreiger, 1980, pp 23-52. 4. Von Neumann J, Morgenstern 0. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press, 1944. 5 Savage LJ. The Foundations of Statistics. New York: Dover, 1954. 6 Malinvaud E. Note on Von Neumann-Morgenstern’s strong independence axiom. Econometrica. 1952;20:679. 7 Samuelson PA. Probability, utility and the independence axiom. Econometrica. 1952;20:670-8. 8 Hume D. A Treatise on Human Nature. Mossner EG, ed. London, England: Penguin Books, 1969. 9. Simon H. Reason in Human Affairs. Stanford, CA: Stanford University Press, 1983. 10. Rescher N. Rationality: A Philosophical Inquiry into the Nature and the Rationale of Reason. Oxford, England: Clarendon Press, 1988. 11. Kekes J.A Justification of Rationality. Albany, NY: State University of New York Press, 1976. 12. Mullen JD. Does the logic of preference rest on a mistake? Metaphilosophy. 1979;10:247-55. 13. Fishburn PC. Normative theories of decision making under risk and under uncertainty. In: Bell DE, Raiffa H, Tversky A, eds. Decision Making: Descriptive, Normative and Prescriptive Interactions. Cambridge, MA: Cambridge University Press, 1988, pp 78-98. 14. Allais M. The foundations of a positive theory of choice involving risk and a criticism of the postulates and axioms of the American school. In: Allais M, Hagen 0, eds. Expected Utility Hypothesis and the Allais Paradox. Dordrecht, The Netherlands: D. Reidel Publishing, 1979, pp 27-145. 15 McClennen EF. Rationality and Dynamic Choice: Foundational Explorations. Cambridge, MA: Cambridge University Press, 1990. 16 Kahneman D, Tversky A. Prospect theory: an analysis of decision under risk. Econometrica. 1979;47:263-91. 17 MacCrimmon KR, Larsson S. Utility theory: axioms versus paradoxes. In: Allais M, Hagen 0, eds. Expected Utility Hypothesis and the Ailais Paradox. Dordrecht, The Netherlands: D. Reidel Publishing, 1979, pp 333-409. 18. Bamn J, Thinking and Deciding. Cambridge, MA: Cambridge University Press, 1988. 19. Peirce CS. The doctrine of chances. Reprinted in: Cohen MR, ed. Chance, Love, Logic. New York: Harcourt Brace, 1923. 20. Bell DE, Raiffa H. Marginal value and intrinsic risk aversion. In: Bell DE, Raiffa H, Tversky A, eds. Decision Making: Descriptive, Normative, and Prescriptive Interactions. Cambridge, MA: Cambridge University Press, 1988, pp 384-97. 21. Luce RD, Raiffa H. Games and Decisions: Introduction and Critical Survey. New York: John Wiley & Sons, 1957. 22. Machina M, Schmeidler D. A more robust definition of subjective probability. Econometrica. 1992;60:745-80. 23. French S. Readings in Decision Analysis. London, England: Chapman and Hall, 1989. 24. Berkeley D, Humphreys P. Structuring decision problems and the ‘bias heuristic.’ Acta Psychologica. 1982;50:201-52. 25. Moskowitz AJ, Kuipers BJ, Kassirer JP. Dealing with uncertainty, risks, and tradeoffs in clinical decisions: a cognitive science approach. Ann Intern Med. 1988;108:435-49.
