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 (x₁ ..., x_n) with probabilities (p₁, ..., p_n) 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 ). 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 ; 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.

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 yon Neumann-Morgenstern utility of wealth function U(·). Rather than evaluating gambles on the basis of their expected value individuals will evaluate them on the basis of their expected utility 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 (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 U(·) 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(0) = 0 and U(M) = 1 for some large value M.

To see how this shape determines risk attitudes, let us consider Figures la and lb. The monotonicity of the curves in each figure reflects the property of stochastic dominance preference, by which one lottery is said to stochastically 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 x' and a 1/3 chance of a wealth levels of x". The amount in the figures gives the expected value of this gamble; and give its expected utility

for the utility functions U_a(·) and U_b(·). For the concave (i.e., bowed upward) utility function U_a(·), we have , which implies that this individual would prefer a sure gain of [which would yield utility ] 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 U_b(·), we have . Because this preference for bearing the risk rather than receiving the expected value will also extend to all gambles, U_b(·) 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 U(·) would allow the prediction of preferences (and hence behavior) in any risky 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 fractile 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" , . The next step involves obtaining the individual's certainty equivalent ξ₁ of a gamble yielding a 1/2 chance of M and a 1/2 chance of 0, which will have the property that U(ξ₁) = 1/2.⁹ Finding the certainty equivalent of a gamble yielding a 1/2 chance of ξ₁ and a 1/2 chance of 0 yields the value ξ₂ satisfying U (ξ₂) = 1/4.

Finding the certainty equivalent of a gamble yielding a 1/2 chance of M and a 1/2 chance of ξ₁ yields the value ξ₃ satisfying U(ξ₃) = 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.

To 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 M RS_p,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 U(·), 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.¹¹

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 1980s to concentrate on the behavioral restrictions implied by the expected utility hypothesis. These restrictions are the subject of the next section.

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 descried 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 preference function U(x₁)p₁ +... + U(x_n)p_n. Although this preference function generalizes the expected value form x₁p₁ + ... + x_np_n by dropping the property of linearity in the payoff levels (i.e., the x_i's), it retains the other key property of this form, namely, linearity in the probabilities.

Graphically, the property of linearity in the probabilities may be illustrated by considering the set of all lotteries or prospects over some set of fixed outcome levels x₁ << x₂ << x₃, which can be represented by the set of all probability triples of the form P = (p₁, p₂, p₃) where p_i = prob(x_i) and p₁ + p₂ + p₃ = 1.¹² Making the substitution p₂⁼ 1- p₁- p₃, this set of lotteries can be represented by the points in the unit triangle in the (p₁, p₃) plane, as in Figure 2.¹³ Because upward movements in the triangle increase p₃ at the expense of p₂ (i.e., shift probability from the outcome x₂ up to x₃) and leftward movements reduce p₁ to the benefit of p₂ (i.e., shift probability from x₁ up to x₂), these movements (and, more generally, all northwest movements) lead to stochastically dominating lotteries and would accordingly be preferred. For the purposes of illustrating many of the following discussions it will be useful to plot the individual's indifference curves in this diagram; that is, the curves in the diagram that connect points of equal expected utility.¹⁴ Because each such curve will consist of the set of all (p₁, p₃) points that solve an equation of the form for some constant k, and because the probabilities p₁ and p1, p₃ enter linearly (i.e., as multiplicative coefficients) into this equation, the indifference curves will consist of parallel straight lines, with more preferred indifference curves lying to the northwest. 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.

Figure 2

Expected utility indifference curves in the triangle diagram.

To 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 for some constant k. Because northeast movements along these lines do not change the expected value of the prospect but do increase the probabilities of the extreme outcomes x₁ and x₃ at the expense of the middle outcome x₂, they are simple examples of mean preserving spreads or ''pure" increases in risk.¹⁵ When the utility function U(·) is concave (i.e., risk averse), its indifference curves can be shown to be steeper than the iso-expected value lines (Figure 3a),¹⁶ and such increases in risk will lead to less preferred indifference curves. When U(·) 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 curves.¹⁷

Figure 3

A: Relatively steep indifference curves of a risk averter. B: Relatively fiat indifference curves of a risk lover.

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 and P = (p₁, ..., p_n) are two lotteries over a common outcome set {x₁, ..., x_n}, the α: (1 -α) probability mixture of P* and P is the lottery . This may be thought of as that prospect that yields the same ultimate probabilities over {x₁,..., x_n} as the two-stage lottery that offers an α: (1- α) 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 :¹⁸

If the lottery P* is preferred (respectively indifferent) to the lottery P, then the mixture αP* + (1-α)P** will be preferred (respectively indifferent) to the mixture αP + (1-α)P** for all α > 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 {x₁,..., x_n}, choosing between the mixtures αP* + (1-α)P** and αP + (1-α)P** is the same as being offered a coin with a probability 1 - α of landing tails, in which case you will obtain the lottery P**, and being asked before the flip whether you would rather have P* 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" 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.

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.¹⁹ 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):

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	versus

Defining {x₁, x₂, x₃} = {$0;$1 million;$5 million}, these four gambles are seen to form to a parallelogram in the (p₁, p₃) triangle (Figures 4a and 4b). Under the expected utility hypothesis, therefore, a preference for a₁ 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 a₂ 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 a₁ 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.

Figure 4

A: Expected utility indifference curves and the Allais Paradox. B: Indifference curves that "fan out" and the Allais Paradox.

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 a₁ and a₃ when he was first presented with this problem, upon reflection, he concluded that these preferences were in error.²² Although his own reaction was undoubtedly sincere, the prediction that individuals would invarialy, 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 utility position, neither MacCrimmon (1968), Moskowitz (1974), nor Slovic and Tversky (1974) found predominant net swings toward the expected utility choices.²³

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 where P involves outcomes both greater and less than x, and P** stochastically dominates P*.²⁴ Although the independence axiom clearly implies choices of either b₁ and b₃ (if x is preferred to P) or b₂ and b₄ (if P is preferred to x), researchers have again found a tendency for subjects to choose b₁ in the first pair and b₄ in the second.²⁵ When the distributions P, P*, and P** are each over a common outcome set {x₁, x₂, x₃} that includes x, the prospects b₁, b₂, b₃, and b₄ will again form a parallelogram in the

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and
	versus

(p₁, p₃) triangle, and a choice of b₁ 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 mils (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 {b₁, b₂} involves very high outcomes, an individual may prefer not to bear further risk in the unlucky 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 b₁ over b₂). If P* in {b₃, b₄} involves very low outcomes, however, an individual may be more willing to bear risk in the (lucky) event that he or she doesn't receive it, and prefer the lottery P to the outcome x in this case (i.e., choose b₄ over b₃). 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 effect. This phenomenon involves pairs of prospects of the form where p > q, 0 < X < Y and 0 < α < 1; it includes the "certainty effect" of Kahneman and Tversky (1979) and the ingenious "Bergen paradox" of Hagen (1979) as special cases.²⁸ Setting {x₁, x₂, x₃} = {0, X, Y} and plotting these prospects in the (p₁, p₃) triangle, the segments and are seen to be parallel (as in Figure 5a), so that the expected utility model again predicts choices of c₁ 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 c₁ 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 {x₁, x₂, x₃} as {-Y, -X, 0} (to maintain the ordering x₁ < x₂ < x₃) and plotting these gambles in Figure 5b, 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.

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.

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	versus
and
	versus

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 must be 1/2, as in the earlier example. Picking any other value—say obtaining the individual's certainty equivalent of the gamble offering a 1/4 chance of M and a 3/4 chance of 0 will lead to the property that ; in addition, just as in the previous case of , the procedure using (or any other fixed value) can also be continued to (in the limit) completely recover U(·).

Although this procedure should recover the same (normalized) utility function for any value of the mixture probability , such researchers as Karmarkar (1974, 1978) and McCord and de Neufville (1983, 1984) have found a tendency for higher values of to lead to the "recovery" of higher valued utility functions (Figure 6a). By illustrating the gambles used to obtain the certainty equivalents ξ₁, ξ₂, and ξ₃ for the mixture probability , for and , for 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

Figure 6

A: "Recovered" utility functions for mixture probabilities 1/4, 1/2, and 3/4. B: Fanning out indifference curves that generate the responses of Figure 6a. Note: ~ denotes indifference.

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. 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 dominance preference, risk aversion/risk preference, and fanning out, and the Chew/MacCrimmon/Fishburn and Quiggin forms have proven to be particularly useful both theoretically and empirically. Additional analyses of the above forms can be found in Chew, Karni, and Safra (1987); Fishburn (1982, 1984a,b); Röell (1987); Segal (1984, 1987); and Yaari (1987). For general surveys of these models, see Machina (1983a), Sugden (1986), and Weber and Camerer (1987).

TABLE 1

Examples of Non-Expected Utility Preference Functions.

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 v(·), p(·), or g(·) 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 v(·) 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 functions. (Readers who are not interested in the details of this approach may wish to skip ahead to the next section.³¹)

Specifically, let us consider the set of all probability distributions P = (p₁,... ,p_n) over a fixed outcome set {x₁, ..., x_n}, so that the expected utility preference function can be written as V(P) = V(p₁, ..., p_n) = U(x₁)p₁ +... + U(x_n)p_n. Let us also think of U(x_i) not as a "utility level" but rather as the coefficient of p_i = prob( x_i) in this linear function. If these coefficients are plotted against x_i as in Figure 7, the expected utility results of the previous section can be stated as:

Figure 7

von Neumann-Morgenstern utilities as coefficients of the expected utility preference function v(p₁,...,p_n) = U(x₁)p₁ + ... + U(x_n)p_n.

• Stochastic Dominance Preference: V(·) will exhibit global stochastic dominance preference if and only if the coefficients {U(x_i)} are increasing in x_i, as in Figure 7.
• Risk Aversion: V(·) will exhibit global risk aversion if and only if the coefficients {U(x_i)} are concave in x_i,³² as in Figure 7.
• Comparative Risk Aversion: The expected utility preference function V*(P) = U*(x₁)p₁ + ... + U*(x_n)p_n will be at least as risk averse as v(·) if and only if the coefficients {U*(x_i)} are at least as concave in x_i as {U(x_i)}.³³

Now, let us consider the case in which the individual's preference function V(P) = V(p₁, ... p_n) is not linear (i.e., not expected utility) but at least differentiable, and let us consider its partial derivatives Some probability distribution P0 can be chosen and these values plotted against x_i. If they are increasing in x_i, it is clear that any infinitesimal stochastically dominating shift from

P₀, such as a decrease in some p_i and matching increase in p_i+1, will be preferred. If they are concave in x_i, any infinitesimal mean preserving spread, such as a drop in p_i and (mean preserving) rise in P_i-1 and P_i+1, will make the individual worse off. In light of this correspondence between the coefficients {U(x_i)} of the expected utility preference function V(·) and the partial derivatives of the non-expected utility preference function V(·), as the individual's local utility indices at P₀.

Of course, the above results will only hold exactly for infinitesimal shifts from the distribution P₀. 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 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 V(·) will exhibit global stochastic dominance preference if and only if its local utility indices are increasing in x_i at each distribution P.
• Risk Aversion: V(·) will exhibit global risk aversion if and only if its local utility indices are concave in x_i at each distribution P.
• Comparative Risk Aversion: The preference function V*(·) will be globally at least as risk averse³⁴ as V(·) if and only if its local utility indices are at least as concave in x_i as at each P.

Figures 8a and 8b are a graphic illustration of this approach for the outcome set {x₁, x₂, x₃}. Here, the solid curves denote the indifference curves of the non-expected utility preference function V(P). The parallel lines near the lottery P₀ denote the tangent "expected utility" indifference curves that correspond to the local utility indices at P₀. As always with differentiable functions, an infinitesimal change in the probabilities at P₀ 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 approximations being steeper—or, in other words, to the local utility indices being concave in x_i at each distribution 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 P₀. B: Risk aversion of every local expected utility approximation (more...)

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.

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:

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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 S-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 Slovic (1973) in a Las Vegas casino where customers actually staked (and hence sometimes lost) their own money. In another real-money experiment, Mowen and Gentry (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 nevertheless 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. Further 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).

Two 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: Each 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 ξp (b) they strictly prefer the P-bet to the $-bet, and (c) they are indifferent regarding the choice between the $-bet and an amount ξ_$ greater than ξ_p. Assuming that they in fact prefer ξ_$ to the lesser amount ξp, this implies that their preferences over these four objects are cyclic or intransitive.

Psychologists, on the other hand, would deny the premise of an common 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 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). To 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 Slavic (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 () such that prob() = prob() = prob() = 2/3, so that individuals making pairwise choices on the basis of these probabilities would also be intransitive. In addition to the preference reversal phenomenon, Edwards (1954a)⁴⁰ and Tversky (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).

In any event, economists have begun to develop and analyze models of nontransitive preferences over lotteries. The leading example of this is the "regret theory" model developed independently by Bell (1982, 1983) (see also Bell and Raiffa [1980]), Fishburn [1981, 1982, 1984a,b], and Loomes 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) = -r(y,x) for all values of x and y). In choosing between statistically independent gambles P = (p₁,..., p_n) and over a common outcome set {x₁,...,x_n}, the individual will choose P* if the expected value of the function r(x,y) is positive and P if it is negative⁴¹ (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 r(x,y) = U(x) - 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. To see the former, let us consider the case in which the individual is facing alternative independent lotteries over a common outcome set {x₁,x₂,x₃}, so that the triangle diagram may again be used to illustrate their "indifference 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 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.

Figure 9

"Indifference curves" for the expected regret model.

To see how expected utility results can be extended to this nontransitive framework, let us fix a lottery P = (p₁,...,p_n) and consider the question of when an (independent) lottery will be preferred or not preferred to P. Defining the "utility function" , it is possible to show that P* will be preferred to P if and only if —in other words, if and only if P* implies a higher expectation of the function than does P.⁴⁴ Thus, if is increasing in x for all lotteries P, the individual will exhibit global stochastic dominance preference; if is concave in x 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 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 remained valid—provided the response mode were held constant. If this 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 otherwise—in auctions, etc.), 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

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. 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 contingent 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 face of wealth variations has also been observed in several experimental studies.⁵¹

Figure 10

Duplex gambles with identical underlying distributions.

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 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 underlying 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 stochastically 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 (Tversky and Kahneman, 1981; Kahneman and Tversky, 1984).⁵⁰

A second class of framing effects exploits the phenomenon of a reference point. Theoretically, the variable that enters an individual's yon 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 distributions over final wealth levels before being evaluated. However, economists since Markowitz (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 lead to different choices, depending on the exact description. A simple example of this, from Kahneman and Tversky (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 0 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 0 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 Tversky 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 2/3 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. 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 (1980b); Kahneman and Tversky (1982, 1984); Hershey, Kunreuther, and Schoemaker (1982); McNeil et al. (1982); and Slovic, Fischhoff, and Lichtenstein (1982) have found that subjects' choices in otherwise identical problems will depend on whether the choices are phrased as decisions about whether to gamble or to insure, whether the statistical information for different therapies is presented in terms of cumulative survival probabilities over time or cumulative mortality probabilities over time, and so forth (see also the additional references in Tversky and Kahneman [1981] as well as the examples of this phenomenon in nonstochastic situations given in Thaler [1980, 1985]).

In a final class of examples, not based on reference point effects, Moskowitz (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).

Two 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 difficulty 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 differentials 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 themselves) 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 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 flame 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 flaming 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 flame 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 utility 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 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. To 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 join fly 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 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.

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 information 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-s (1974), and Grether (1980), for example, all found that probability updating systematically departs from Bayes' law in the direction of underweighting prior information and overweighting the "representativeness" of the current sample. In a related phenomenon termed the "law of small numbers," Tversky and Kahneman (1971) found that individuals overestimated the probability 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); Tversky and Kahneman (1974); and Grether (1978), as well as the collections in Acta Psychologica (December 1970); Kahneman, Slovic, and Tversky (1982); and Arkes and Hammond (1986). 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 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 (R₁) drawing a red ball from the first urn, (R₂) drawing a red ball from the second urn, (B₁) drawing a black ball from the first urn, or (B₂) drawing a black ball from the second urn, a majority of subjects strictly preferred (R₁) over (R₂) and strictly preferred (B₁) over (B₂). 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 Tversky (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 associated 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 Hurwicz (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., maximin, maximax), 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) 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

In this model, uncertainty is represented by a set of mutually exclusive and exhaustive states of nature S = {s_i}. 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 75 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 (c₁,...,c_n), where c_i 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 (c₁,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 certainty line.

Now, if the individual happens to assign some set of probabilities {p_i} to the states {s_i}, each bundle (c₁,...,c_n) 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.⁵⁹ Yet bemuse 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 curve-based approach to derive results from individual demand behavior through general equilibrium in a context that requires 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 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 behavior and the non-expected utility models used to characterize it, as well as the state-payoff approach discussed earlier, are completely consistent with the assumption of well-defined, transitive individual preference orderings and hence with traditional welfare analysis along the lines of Pareto (1909), Bergson (1938) and Samuelson (1947/1983:Chap. 8). For example, the proof of the general efficiency ("Pareto efficiency") of a system of complete contingent-commodity markets (Arrow, 1953/1964; Debreu, 1959:Chap. 7) requires neither the expected utility hypothesis nor the assumption 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.

To 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 magnitude 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) between these variables would vary exactly proportionally to the loss probability 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 expected utility by "fanning out" (Figures 4b, 5a, 5b, and 6), then individuals' marginal rates of substitution between p and L will always vary less than proportionally to the loss probability p (Machina, 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 publicly 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 disclosing 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 Protection Agency.⁶⁷ To 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 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 a glimpse of how policy makers, interest groups, and the public feel and act toward the general issue of the presentation of information.