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Although the expected utility model has long been the standard theory of individual choice under objective and subjective uncertainty, experimental work by both psychologists and economists has uncovered systematic departures from the expected utility hypothesis, which has led to the development of alternative models of preferences over uncertain prospects.

The Expected Utility Model

In one of the simplest settings of choice under economic uncertainty, the objects of choice consist of finite-outcome objective lotteries of the form P = (x1, p1;...; xn, pn), yielding a monetary payoff of xi with probability pi, where p1 + ... + pn = 1. In such a case, the expected utility model of risk preferences assumes (or posits axioms sufficient to imply) that the individual ranks these prospects on the basis of an expected utility preference function of the form

$$ {V}_{EU}\left(\mathbf{P}\right)\equiv {V}_{EU}\left({x}_1,{p}_1,...,{x}_n,{p}_n\right)\equiv U\left({x}_1\right)\cdotp {p}_1+\dots +U\left({x}_n\right)\cdotp {p}_n $$

in the standard economic sense that the individual prefers lottery \( {\mathbf{P}}^{\ast }=\left({x}_1^{\ast },{p}_1^{\ast };...;{x}_n^{\ast },{p}_{n^{\ast}}^{\ast}\right) \) over lottery P = (x1, p1;...;;xn, pn) if and only if VEU (P*) > VEU (P), and is indifferent between them if and only if VEU (P*) = VEU (P). U( · ) is termed the individual’s von Neumann–Morgenstern utility function (von Neumann and Morgenstern 1944, 1947, 1953) and its various mathematical properties serve to characterize various features of the individual’s attitudes toward risk, for example:

  • VEU( · ) exhibits first-order stochastic dominance preference (a preference for shifting probability from lower to higher outcome values) if and only if U(x) is an increasing function of x.

  • VEU( · ) exhibits risk aversion (an aversion to all mean-preserving increases in risk) if and only if U(x) is a concave function of x.

  • \( {V}_{EU}^{\ast}\left(\cdotp \right) \) is at least as risk averse as VEU( · ) (in several equivalent senses) if and only if its utility function U*( · ) is a concave transformation of U( · ) (that is, if and only if U*(x) ≡ ρ(U(x)) for some increasing concave function ρ( · )).

As shown by Bernoulli (1738), Arrow (1965), Pratt (1964), Friedman and Savage (1948), Markowitz (1952) and others, this model admits of a tremendous flexibility in representing attitudes towards risk, and can be applied to many types of economic decisions and markets.

But in spite of its flexibility, the expected utility model has testable implications which hold regardless of the shape of the utility function U( · ), since they follow from the linearity in the probabilities property of the preference function VEU( · ). These implications can be best expressed by the concept of an α : (1 − α) probability mixture of two lotteries P = (x1, p1;...; xn, pn) and \( {\mathbf{P}}^{\ast }=\left({x}_1^{\ast },{p}_1^{\ast };...;{x}_n^{\ast },{p}_{n^{\ast}}^{\ast}\right) \), which is defined as the single-stage lottery α · P + (1 − α)· P* = (x1,α · p1;...; xn, α · pn;\( {x}_1^{\ast } \), (1 − α)· \( {p}_1^{\ast } \);...;\( {x}_{n^{\ast}}^{\ast } \), (1 − α)· \( {p}_{n^{\ast}}^{\ast } \) ). The mixture α · P + (1 − α)· P* can be thought of as a coin flip yielding lotteries P and P* with probabilities α : (1 − α), where the uncertainty in the coin and in the subsequent lottery is resolved simultaneously. Linearity in the probabilities is equivalent to the following property, which serves as the key foundational axiom of the expected utility model (Marschak 1950):

Independence Axiom If lottery P* is preferred (indifferent) to lottery P, then the probability mixture α · P* + (1 − α)· P** is preferred (indifferent) to α · P + (1 − α)· P** for every lottery P** and every mixture probability α ∈ (0, 1].

This axiom can be interpreted as saying ‘given an α : (1 − α) coin, the individual’s preferences for receiving P* versus P in the event of a head should not depend upon the prize P** that would be received in the event of a tail, nor upon the probability α of landing heads (so long as this probability is positive)’. The strong normative appeal of this axiom has contributed to the widespread adoption of the expected utility model.

The property of linearity in the probabilities, as well as the senses in which it has been found to be empirically violated, can be illustrated in the special case of preferences over all lotteries \( \mathbf{P}=\left({\overline{x}}_1,\kern0.5em {p}_1;\kern0.5em {\overline{x}}_2,\kern0.5em {p}_2;\kern0.5em {\overline{x}}_3,\kern0.5em {p}_3\right) \) over a fixed set of outcome values \( {\overline{x}}_1<{\overline{x}}_2<{\overline{x}}_3 \). Since we must have p2 = 1 – p1 – p3, each such lottery can be completely summarized by its pair of probabilities (p1, p3), as plotted in the ‘probability triangle’ of Fig. 1. Since upward movements in the diagram (increasing p3 for fixed p1) represent shifting probability from outcome x2 up to x3, and leftward movements represent shifting probability from x1 up to x2, such movements constitute first-order stochastically dominating shifts and will thus always be preferred. Expected utility indifference curves (loci of constant expected utility) are given by the formula

$$ U\left({\overline{x}}_1\right)\cdotp {p}_1+U\left({\overline{x}}_2\right)\cdotp \left[1-{p}_1-{p}_3\right]+U\left({\overline{x}}_3\right)\cdotp {p}_3=\mathrm{constant} $$

and are thus seen to be parallel straight lines of slope \( \left[U\left({\overline{x}}_2\right)-U\left({\overline{x}}_1\right)\right]/\left[U\left({\overline{x}}_3\right)-U\left({\overline{x}}_2\right)\right] \), as indicated by the solid lines in the figure. The dotted lines in Fig. 1 are loci of constant expected value, given by the formula \( {\overline{x}}_1\cdotp {p}_1+{\overline{x}}_2\cdotp \left[1-{p}_1-{p}_3\right]+{\overline{x}}_3\cdotp {p}_3=\mathrm{constant} \), with slope \( \left[{\overline{x}}_2-{\overline{x}}_1\right]/\left[{\overline{x}}_3-{\overline{x}}_2\right] \). Since north-east movements along the constant expected value lines shift probability from x2 down to x1 and up to x3 in a manner that preserves the mean of the distribution, they represent simple increases in risk (Rothschild and Stiglitz 1970, 1971). When U( · ) is concave (that is, risk averse), its indifference curves will have a steeper slope than these constant expected value lines, and such increases in risk move the individual from more to less preferred indifference curves, as illustrated in the figure. It is straightforward to show that the indifference curves of any expected utility maximizer with a more risk-averse (that is, more concave) utility function U*( · ) will be steeper than those generated by U( · ).

Non-expected Utility Theory, Fig. 1
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Expected utility indifference curves in the probability triangle

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Systematic Violations of the Expected Utility Hypothesis

In spite of its normative appeal, researchers have uncovered several types of widespread systematic violations of the expected utility model and its underlying assumptions. These can be categorized into (a) violations of the Independence Axiom (such as the common consequence and common ratio effects), (b) violations of the hypothesis of probabilistic beliefs (such as the Ellsberg Paradox) and (c) violations of the model’s underlying assumptions of descriptive and procedural invariance (such as reference-point and response-mode effects).

Violations of the Independence Axiom

The best-known violation of the Independence Axiom is the so-called Allais Paradox, in which individuals are asked to rank the lotteries in each of the following pairs, where $1 M = $1; 000, 000:

$$ {a}_1:\left\{\begin{array}{l}\hfill \\ {}1.00\ \mathrm{chance}\ \mathrm{of}\ \$1\mathrm{M}\kern1em \mathrm{versus}\kern1em {a}_2:\hfill \\ {}\hfill \end{array}\right.\left\{\begin{array}{l}.10\ \mathrm{chance}\ \mathrm{of}\ \$5\mathrm{M}\hfill \\ {}.89\ \mathrm{chance}\ \mathrm{of}\ \$1\mathrm{M}\hfill \\ {}.01\ \mathrm{chance}\ \mathrm{of}\ \$0\hfill \end{array}\right. $$
$$ {a}_3:\left\{\begin{array}{l}.10\ \mathrm{chance}\ \mathrm{of}\ \$5\mathrm{M}\hfill \\ {}.90\ \mathrm{chance}\ \mathrm{of}\ \$0\hfill \end{array}\kern1em \mathrm{versus}\kern1em {a}_4\right.:\left\{\begin{array}{l}.11\ \mathrm{chance}\ \mathrm{of}\ \$1\mathrm{M}\hfill \\ {}.89\ \mathrm{chance}\ \mathrm{of}\ \$0\hfill \end{array}\right. $$

Researchers such as Allais (1953), Morrison (1967), Raiffa (1968), Slovic and Tversky (1974) and others have found that the modal if not majority preference of subjects is for a1 over a2 in the first pair of choices and for a3 over a4 in the second pair. However, such preferences violate expected utility, since the first ranking implies the inequality U ($1 M) > .10 · U ($5 M)+ .89 · U ($1 M)+ .01 · U ($0) whereas the second implies the inconsistent inequality ..10 · U ($5 M)+ .90 · U ($0) > .11 · U ($1 M)+ .89 · U ($0). By setting \( {\overline{x}}_1=\$0 \),\( {\overline{x}}_2=\$1\mathrm{M} \) and \( {\overline{x}}_3=\$5\mathrm{M} \), the lotteries a1, a2, a3 and a4 are seen to form a parallelogram when plotted in the probability triangle (Fig. 2), which explains why the parallel straight line indifference curves of an expected utility maximizer must either prefer a1 and a4 (as illustrated for the relatively steep indifference curves of the figure) or else prefer a2 and a3 (for relatively flat indifference curves). Fig. 3 illustrates non-expected utility indifference curves which fan out, and are seen to exhibit the typical Allais Paradox rankings of a1 over a2 and a3 over a4.

Non-expected Utility Theory, Fig. 2
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Expected utility indifference curves and the Allais Paradox choices

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Non-expected Utility Theory, Fig. 3
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Allais Paradox choices and indifference curves which ‘fan out’

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Although the Allais Paradox was originally dismissed as an isolated example, subsequent experimental work by psychologists, economists and others have uncovered a similar pattern of violations over a range of probability and payoff values, and the Allais Paradox is now seen to be a special case of a widely observed phenomenon known as the common consequence effect. This effect involves pairs of prospects (probability mixtures) of the form:

$$ {b}_1:\left\{\begin{array}{ccc}\hfill \alpha \hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill x\hfill \\ {}\hfill 1-\alpha \hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill {\mathbf{P}}^{\ast \ast}\hfill \end{array}\right.\kern1em \mathrm{versus}{b}_2:\left\{\begin{array}{ccc}\hfill \alpha \hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \mathbf{P}\hfill \\ {}\hfill 1-\alpha \hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill {\mathbf{P}}^{\ast \ast}\hfill \end{array}\right. $$
$$ {b}_3:\left\{\begin{array}{ccc}\hfill \alpha \hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill x\hfill \\ {}\hfill 1-\alpha \hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill {\mathbf{P}}^{\ast}\hfill \end{array}\right.\kern1em \mathrm{versus}{b}_4:\left\{\begin{array}{ccc}\hfill \alpha \hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \mathbf{P}\hfill \\ {}\hfill 1-\alpha \hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill {\mathbf{P}}^{\ast}\hfill \end{array}\right. $$

where the lottery P involves outcomes both greater and less than the amount x, and P** first order stochastically dominates P* (in Allais’s example, x = $1M, P = ($5M, 10 = 11, $0, 1 = 11), P* = ($0, 1), P** = ($1M, 1) and α = .11). Although the Independence Axiom clearly implies choices of either b1 and b3 (if x is preferred to P) or else b2 and b4 (if P is preferred to x), researchers have found a tendency for subjects to choose b1 in the first pair and b4 in the second. When the distributions P, P* and P** are each over a common outcome set \( \left\{{\overline{x}}_1,\kern0.5em {\overline{x}}_2,\kern0.5em {\overline{x}}_3\right\} \) with \( {\overline{x}}_2=x \), the prospects {b1, b2, b3, b4} again form a parallelogram in the (p1, p3) triangle, and a choice of b1 and b4 again implies indifference curves which fan out.

The intuition behind this phenomenon can be described in terms of the above ‘coin-flip’ scenario. According to the Independence Axiom, one’s preferences over what would occur in the event of a head ought not depend upon what would occur in the event of a tail. However, they may well depend upon what would otherwise happen (as Bell 1985, p. 1, notes, ‘winning the top prize of $10,000 in a lottery may leave one much happier than receiving $10,000 as the lowest prize in a lottery’). The common consequence effect states that the better off individuals would be in the event of a tail (in the sense of stochastic dominance), the more risk averse their preferences over what they would receive in the event of a head. That is, if the distribution P** in the pair {b1, b2} involves very high outcomes, one may prefer not to bear further risk in the unlucky event that one doesn’t receive it, and hence opt for the sure outcome x over the risky distribution P (that is, choose b1 over b2). But, if P* in {b3, b4} involves very low outcomes, one might be more willing to bear risk in the lucky event that one doesn’t receive it, and prefer going for the lottery P rather than the sure outcome x (choose b4 over b3).

A second type of systematic violation of linearity in the probabilities, also noted by Allais and subsequently termed the common ratio effect, involves prospects of the form:

$$ {c}_1:\left\{\begin{array}{ccc}\hfill p\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$\mathrm{X}\hfill \\ {}\hfill 1-p\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$0\hfill \end{array}\right.\kern1em \mathrm{versus}{c}_2:\left\{\begin{array}{ccc}\hfill q\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$\mathrm{Y}\hfill \\ {}\hfill 1-q\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$0\hfill \end{array}\right. $$
$$ {c}_3:\left\{\begin{array}{ccc}\hfill \alpha \cdot p\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$\mathrm{X}\hfill \\ {}\hfill 1-\alpha \cdot p\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$0\hfill \end{array}\right.\kern1em \mathrm{versus}{c}_4:\left\{\begin{array}{ccc}\hfill \alpha \cdot q\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$\mathrm{Y}\hfill \\ {}\hfill 1-\alpha \cdot q\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$0\hfill \end{array}\right. $$

where p > q, 0 < X < Y and α ∈ (0, 1). (The term ‘common ratio effect’ comes from the common value of prob($X)/prob($Y) in the upper and lower pairs.) Setting \( \left\{{\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3\right\}=\left\{\$0,\$\mathrm{X},\$\mathrm{Y}\right\} \) and plotting these prospects in the probability triangle as in Fig. 4, the line segments \( \overline{c_1{c}_2} \) and \( \overline{c_3{c}_4} \) are seen to be parallel, so that the expected utility model again predicts choices of c1 and c3 (if the indifference curves are relatively steep) or else c2 and c4 (if they are flat). However, experimental studies by MacCrimmon (1968), Tversky (1975), MacCrimmon and Larsson (1979), Kahneman and Tversky (1979), Hagen (1979), Chew and Waller (1986) and others have found a systematic tendency for choices to depart from these predictions in the direction of preferring c1 over c2 and c4 over c3, which again suggests that indifference curves fan out, as in the figure. For example, Kahneman and Tversky (1979) found that, while 86 per cent of their subjects preferred a .90 chance of winning $3,000 to a .45 chance of $6,000, 73 per cent preferred a .001 chance of $6,000 to a .002 chance of $3,000. Kahneman and Tversky (1979) observed that, when the positive outcomes $3000 and $6000 in the above gambles are replaced by losses of these magnitudes, to obtain the lotteries \( {c}_1^{\prime },{c}_2^{\prime },{c}_3^{\prime } \) and \( {c}_4^{\prime } \), preferences typically ‘reflect,’ to prefer \( {c}_2^{\prime } \)over \( {c}_1^{\prime } \) and \( {c}_3^{\prime } \) over \( {c}_4^{\prime } \). Setting \( {\overline{x}}_1=-\$6000 \), \( {\overline{x}}_2=-\$3000 \) and \( {\overline{x}}_3=-\$0 \) (to preserve the ordering \( {\overline{x}}_1<{\overline{x}}_2<{\overline{x}}_3 \)) and plotting as in Fig. 5, such preferences again suggest that indifference curves in the probability triangle fan out. Battalio et al. (1985) found that laboratory rats choosing among gambles involving substantial variations in their daily food intake also exhibited this pattern of choices.

Non-expected Utility Theory, Fig. 4
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Common ratio effect and fanning out indifference curves

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Non-expected Utility Theory, Fig. 5
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Common ratio effect for losses and fanning out indifference curves

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One criticism of this evidence has been that individuals whose initial choices violated the Independence Axiom in the above manners would typically ‘correct’ themselves once the nature of their violations was revealed by an application of the above type of coin-flip argument. Thus, while even Leonard Savage chose a1 and a3 when first presented with such choices by Allais, he concluded upon reflection that these preferences were in error (Savage 1954, pp. 101–3). Although Moskowitz found that allowing subjects to discuss opposing written arguments led to a decrease in the proportion of violations, 73 per cent of the initial fanning-out type choices remained unchanged after the discussions (1974, pp. 232–7, Table 6). When written arguments were presented but no discussion was allowed, there was a 93 per cent persistency rate of such choices (1974, p. 234, Tables 4 and 6). In experiments where subjects who responded to Allais-type problems were 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.

Further descriptions of these and other violations of the Independence Axiom can be found in Camerer (1989), Machina (1983, 1987), Starmer (2000), Sugden (1986) and Weber and Camerer (1987).

Non-existence of Probabilistic Beliefs

Although the expected utility model was first formulated in terms of preferences over objective lotteriesP = (x1, p1;... ; xn, pn) with pre-specified probabilities, it has also been applied to preferences over subjective acts f (·) = [x1 on E1;... ; xn on En], where the uncertainty is represented by a set {E1;.. .; En} of mutually exclusive and exhaustive events (such as the alternative outcomes of a horse race) (Savage 1954). As long as an individual possesses well-defined subjective probabilities μ(E1);.. .;μ(En) over these events, their subjective expected utility preference function takes the form

$$ {\displaystyle \begin{array}{c}{W}_{SEU}\left(f\left(\cdotp \right)\right)\equiv {W}_{SEU}\left({x}_1\mathrm{on}\ {E}_1;...;{x}_n\mathrm{on}\ {E}_n\right)\\ {}\equiv \mathrm{U}\left({x}_1\right)\cdotp \upmu \left({E}_1\right)+\dots +U\left({x}_n\right)\cdotp \upmu \left({E}_n\right).\end{array}} $$

However, researchers have found that individuals may not possess such well-defined subjective probabilities, in even the simplest of cases. The best-known example of this is the Ellsberg Paradox (Ellsberg 1961), in which the individual must draw a ball from an urn that contains 30 red balls, and 60 black or yellow balls in an unknown proportion, and is offered the following bets based on the colour of the drawn ball:

 

30 balls

60 balls

 

Red

Black

Yellow

f1( · )

$100

$0

$0

f2( · )

$0

$100

$0

f3( · )

$100

$0

$100

f4( · )

$0

$100

$100

Most individuals exhibit a preference for f1( · ) over f2( · ) and f4( · ) over f3( · ). When asked, they explain that the chance of winning under f2( · ) could be anywhere from 0 to 2/3 whereas under f1( · ) it is known to be exactly 1/3, and they prefer the bet that offers the known probability. Similarly, the chance of winning under f3( · ) could be anywhere from 1/3 to 1 whereas under f4( · ) it is known to be exactly 2/3, so the latter is preferred. However, such preferences are inconsistent with any assignment of subjective probabilities μ(red), μ(black), μ(yellow) to the three events. If the individual were to be choosing on the basis of such probabilistic beliefs, the choice of f1( · ) over f2( · ) would ‘reveal’ that μ(red) > μ(black), but the choice of f4( · ) over f3( · ) would reveal that μ(red) < μ(black). A preference for gambles based on probabilistic partitions such as {red, black ∪ yellow} over gambles based on subjective partitions such as {black, red ∪ yellow} is termed ambiguity aversion.

In an even more basic example, Ellsberg presented subjects with a pair of urns, the first containing 50 red balls and 50 black balls, and the second with 100 red and black balls in an unknown proportion. When asked, a majority of subjects strictly preferred to stake a prize on drawing red from the first urn over drawing red from the second urn, and strictly preferred staking the prize on drawing black from the first urn over drawing black from the second. It is clear that there can exist no subjective probabilities p:(1 − p) of red:black in the second urn, including 1/2:1/2, which can simultaneously generate both of these strict preferences. Similar behaviour 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).

Violations of Descriptive and Procedural Invariance

Researchers have also uncovered several systematic violations of the standard economic assumptions of stability of preferences and invariance with respect to problem description in choices over risky prospects. In particular, psychologists have found that alternative means of representing or framing probabilistically equivalent choice problems lead to systematic differences in choice. Early examples of this were reported by Slovic (1969), who found that offering a gain or loss contingent on the joint occurrence of four independent events with probability p elicited different responses than offering it on the occurrence of a single event with probability p4 (all 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, and making a loss contingent on the joint occurrence of events made it more unattractive.

One class of framing effects exploits the phenomenon of a reference point. According to economic theory, the variable which enters an individual’s von Neumann–Morgenstern utility function should be total (that is, final) wealth, and gambles phrased in terms of gains and losses should be combined with current wealth and re-expressed as distributions over final wealth levels before being evaluated. However, risk attitudes towards gains and losses tend to be more stable than can be explained by a fixed utility function over final wealth, and utility functions might be best defined in terms of changes from the reference point of current wealth. In his discussion of this phenomenon, Markowitz (1952, p. 155) suggested that certain circumstances may cause the individual’s reference point to temporarily deviate from current wealth. If these circumstances include the manner in which a problem is verbally described, then differing risk attitudes towards gains and losses from the reference point can lead to different choices, depending upon the exact description of an otherwise identical problem. A simple example of this, from Kahneman and Tversky (1979, p. 273), involves the following two choices:

  • 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. But, when put to two different groups of subjects, 84 per cent chose the sure gain in the first problem but 69 per cent chose the 1/2:1/2 gamble in the second.

In another class of examples, not based on reference point effects, Moskowitz (1974), Keller (1985) and Carlin (1990) found that the proportion of subjects choosing in conformity with the Independence Axiom in examples like the Allais Paradox was significantly affected by whether the problems were described in the standard matrix form, decision tree form, roulette wheels, or as minimally structured written statements. Interestingly, the form judged 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 (fanning out) choices, and the highest persistency rate of these choices Moskowitz (1974, pp. 234, 237–8).

In other studies, Schoemaker and Kunreuther (1979), Hershey and Schoemaker (1980), Kahneman and Tversky (1982, 1984), and Slovic et al. (1977) found that subjects’ choices in otherwise identical problems depended upon whether they were phrased as decisions whether or not to gamble as opposed to whether or not to insure, whether statistical information for different therapies was presented in terms of cumulative survival probabilities or cumulative mortality probabilities, and so on (see the references in Tversky and Kahneman 1981).

Whereas framing effects involve alternative descriptions of an otherwise identical choice problem, alternative response formats have also been found to lead to different choices, leading to what have been termed response-mode effects. For example, under expected utility, an individual’s von Neumann–Morgenstern utility function can be assessed or elicited in a number of different manners, which typically involve a sequence of pre-specified lotteries P1, P2, P3, .. .; and ask for (a) the individual’s certainty equivalent CE(Pi) of each lottery Pi, (b) the gain equivalent Gi that would make the gamble (Gi,1/2, $0,1/2) indifferent to Pi, or (c) the probability equivalent ℘i that would make the gamble ($1000, ℘i, $0,1 − ℘i) indifferent to Pi. Although such procedures should generate equivalent assessed utility functions, they have been found to yield systematically different ones (for example, Hershey et al. 1982; Hershey and Schoemaker 1985).

In a separate finding now known as the preference reversal phenomenon, subjects were first presented with a number of pairs of lotteries and asked to make one choice out of each pair. Each pair of lotteries took the following form:

$$ p-\mathrm{bet}:\left\{\begin{array}{ccc}\hfill p\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$\mathrm{X}\hfill \\ {}\hfill 1-p\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$0\hfill \end{array}\right.\kern1em \mathrm{versus}\kern1em \$-\mathrm{bet}:\left\{\begin{array}{ccc}\hfill q\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$\mathrm{Y}\hfill \\ {}\hfill 1-q\hfill & \hfill \mathrm{chance}\ \mathrm{of}\hfill & \hfill \$0\hfill \end{array}\right. $$

where 0 < X < Y and p > q. The terms ‘p-bet’ and ‘$-bet’ derive from the greater probability of winning in the first bet, and greater possible gain in the second bet. Subjects were next asked for their certainty equivalents of each of these bets, via a number of standard elicitation techniques. Standard theory predicts that, for each such pair, the prospect selected in the direct choice problem would also be assigned the higher certainty equivalent. However, subjects exhibit a systematic departure from this prediction in the direction of choosing the p-bet in a direct choice, but assigning a higher certainty equivalent to the $-bet (Lichtenstein and Slovic 1971). Although this finding initially generated widespread scepticism, it has been replicated by both psychologists and economists in a variety of settings involving real-money gambles, patrons in a Las Vegas casino, group decisions and experimental market trading. By expressing the implied preferences as ‘$-bet ∼ CE ($-bet) ≻ CE(p-bet) ∼ p-bet ≻ $-bet’, some economists have categorized this phenomenon as a violation of transitivity and tried to model it as such (see the ‘regret theory’ model below). However, most psychologists and economists now view it as a response-mode effect: specifically, that the psychological processes of valuation (which generates certainty equivalents) and direct choice are differentially influenced by the probabilities and payoffs involved in a lottery, and that under certain conditions this can lead to choices and valuations which ‘reveal’ opposite preference rankings over a pair of gambles.

Non-expected Utility Models of Risk Preferences

Non-expected Utility Functional Forms

Table 2 Non-expected Utility Theory, Table 1
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Researchers have responded to departures from linearity in the probabilities in two manners. The first consists of replacing the expected utility form VEU (P) = U (x1)· p1 + ... + U (xn)· pn by some more general form for the preference function V (P)= V (x1, p1;...; xn, pn). Several such forms have been proposed (for the Rank Dependent, Dual and Ordinal Independence forms, the payoffs must be labelled so that x1 ≤ ... ≤ xn, and G( · ) must satisfy G(0)= 0 and G(1)= 1):

Most of these forms have been formally axiomatized, and, under the appropriate monotonicity and/or curvature assumptions on their constituent functions υ( · ), G( · ), and so on, most are capable of exhibiting first-order stochastic dominance preference, risk aversion, and the above types of systematic violations of the Independence Axiom. Researchers such as Konrad and Skaperdas (1993), Schlesinger (1997) and Gollier (2000) have used these forms to revisit many of the applications previously modelled by expected utility theory, such as asset and insurance demand, in order to determine which expected-utility-based results are, and which are not, robust to departures from linearity in the probabilities, and which additional properties of risk-taking behaviour can be modelled.

Although the form \( {\sum}_{i=1}^n\upsilon \left({x}_i\right)\cdotp \pi \left({p}_i\right) \) was the earliest non-expected utility model to be proposed, it was largely abandoned when it was realized that, whenever the weighting function π( · ) was nonlinear, the generic inequalities π(pi)+ π(pj) ≠ π(pi + pj) and π(p1)+ ... + π(pn) ≠ 1 implied discontinuities in the payoffs and inconsistency with first-order stochastic dominance preference. Both problems were corrected by adopting weights \( \left[G\left({\sum}_{j=1}^i{p}_j\right)-G\left({\sum}_{j=1}^{i-1}{p}_j\right)\right] \) based on the cumulative probability values p1, p1 + p2, p1 + p2 + p3, .. ., to obtain the Rank-Dependent form. Under the above-mentioned restrictions on this form, these weights necessarily sum to unity, and the Rank-Dependent form has emerged as the most widely adopted model in both theoretical and applied analyses. The Dual Expected Utility and Ordinal Independence forms are based on similar weighting formulas.

Unlike the other models, the regret theory form dispenses with the assumption of a preference function over lotteries, and instead derives choice from the psychological notions of rejoice and regret – that is, the reaction to receiving outcome x when an alternative decision would have led to outcome x*. The primitive of this model is a regret:rejoice function R(x, x*) which is positive if x is preferred to x*, negative if x* is preferred to x, zero if they are indifferent, and satisfies the skew-symmetry condition R(x, x*)≡ −R(x*, x). In a choice between lotteries P = (x1, p1;...; xn, pn) and \( {\mathbf{P}}^{\ast }=\left({x}_1^{\ast },{p}_1^{\ast };...;{x}_{n^{\ast}}^{\ast },{p}_{n^{\ast}}^{\ast}\right) \) which are realized independently, the individual’s expected rejoice from choosing P over P* is given by \( {\sum}_{i=1}^n{\sum}_{j=1}^{i^{\ast }}R\left({x}_i,{x}_j^{\ast}\right)\cdotp {p}_i\cdotp {p}_j^{\ast } \), and the individual is predicted to choose P if this value is positive, P* if it is negative, and be indifferent if it is zero (various proposals for extending this approach beyond pairwise choice have been offered). Since this model specifies choice in pairwise comparisons rather than preference levels of individual lotteries, it allows choice to be intransitive, so the individual might select P over P*, P* over P**, and P** over P. Though some have argued that such cycles allow for the phenomenon of ‘money pumps’, it has allowed the model to serve as a proposed solution to the Preference Reversal Phenomenon.

Generalized Expected Utility Analysis

An alternative approach to non-expected utility preferences does not rely upon any specific functional form, but links properties of attitudes toward risk directly to the probability derivatives of a general ‘smooth’ preference function V (P)= V (x1, p1;...; xn, pn). Such analysis reveals that the basic analytics of the expected utility model are in fact quite robust to general smooth departures from linearity in the probabilities. This approach is based on the observations that for the expected utility function VEU (x1, p1;...; xn, pn) ≡ U (x1)· p1 + ... + U (xn)· pn, the value U(xi) can be interpreted as the coefficient of pi, and that many theorems involving a linear function’s coefficients continue to hold when generalized to a nonlinear function’s derivatives. By adopting the notation U (x; P) ≡∂V (P)/∂prob(x) and the term ‘local utility function’ for the function U( · ;P), standard expected utility characterizations such as those listed at the beginning of this article can be generalized to any smooth non-expected utility preference function V(P) in the following manners (Machina 1982):

  • V( · ) exhibits global first order stochastic dominance preference if and only if, at each lottery P, its local utility function U(x; P) is an increasing function of x.

  • V( · ) exhibits global risk aversion (aversion to small or large mean-preserving increases in risk) if and only if, at each lottery P, its local utility function U(x; P) is a concave function of x.

  • V*( · ) is globally at least as risk averse as V( · ) if and only if, at each lottery P, V*( · )’s local utility function U*(x; P) is a concave transformation of V( · )’s local utility function U(x; P).

Similar generalizations of expected utility results and characterizations can be obtained for general comparative statics analysis, the theory of asset demand, and the demand for insurance. With regard to the Allais Paradox and other observed violations of the Independence Axiom, it can be shown that the indifference curves of a smooth preference function V( · ) will fan out in the probability triangle if and only if U(x; P*) is a concave transformation of U(x; P) whenever P* first-order stochastically dominates P. This analytical approach has been extended to larger classes of preference functionals and distributions by Chew et al. (1987), Karni (1987, 1989) and Wang (1993), formally axiomatized by Allen (1987), and applied to the analysis of choices under uncertainty by Chew et al. (1988), Chew and Nishimura (1992), Dekel (1989), Green and Jullien (1988), Machina (1984, 1989, 1995) and others.

Non-expected Utility Preferences Under Subjective Uncertainty

Recent years have seen a growing interest in models of choice under subjective uncertainty, with efforts to represent and analyse departures from both expected utility risk preferences and probabilistic beliefs. A non-expected utility preference function W (f (·)) ≡ W (x1 on E1;...; xn on En) over subjective acts f (·) = [x1 on E1;...; xn on En] is said to be probabilistically sophisticated if it takes the form W (f (·)) ≡ V (x1, μ(E1);...; xn, μ(En)) for some subjective probability measure μ( · ) over the space of events and some non-expected utility preference function V (P)= V (x1, p1;...; xn, pn). Such preferences have been axiomatized in a manner similar to Savage’s (1954) axiomatization of the subjective expected utility form WSEU (f (·)) ≡ U (x1)· μ(E1)+ ... + U (xn)· μ(En) (Machina and Schmeidler 1992). Although such preferences can be consistent with Allais-type departures from linearity in (subjective) probabilities, they are not consistent with Ellsberg-type departures from probabilistic beliefs.

Efforts to accommodate the Ellsberg Paradox and the general phenomenon of ambiguity aversion have led to the development of several non-probabilistically sophisticated models of preferences over subjective acts (see the analysis of Epstein 1999, as well as the surveys of Camerer and Weber 1992, and Kelsey and Quiggin 1992). One such model, the maximin expected utility form, replaces the unique probability measure μ( · ) of the subjective expected utility model by a finite or infinite family M of such measures, to obtain the preference function

Wmaximinx1onE1...xnonEn≡minμ·∈MUx1·μE1+...+Uxn·μEn

When applied to the Ellsberg Paradox, the family of subjective probability measures M = {(μ(red), μ(black), μ(yellow)} = (1/3, γ, 2/3 − γ)|γ ∈[0, 2/3]} will yield the typical Ellsberg-type choices of f1( · ) over f2( · ) and f4( · ) over f3( · ) (Gilboa and Schmeidler 1989).

Another important model for the representation and analysis of ambiguity averse preferences, based on the Rank Dependent form under objective uncertainty, is the Choquet expected utility form:

$$ {W}_{Choquet}\left({x}_1\;\mathrm{on}\ {E}_1;...;{x}_n\ \mathrm{on}\ {E}_n\right)\equiv \sum \limits_{i=1}^nU\left({x}_i\right)\cdotp \left[C\left({\cup}_{j=1}^i{E}_j\right)-C\left({\cup}_{j=1}^{i-1}{E}_j\right)\right] $$

where for each act f (·) = [x1 on E1;...; xn on En], the payoffs must be labelled so that x1 ≤ ... ≤ xn, and C( · ) is a nonadditive measure over the space of events which satisfies C(∅) = 0 and \( C\left({\cup}_{i=1}^n{E}_j\right)=1 \) (Gilboa 1987; Schmeidler 1989). This model has been axiomatized in a manner similar to the subjective expected utility model, and with proper assumptions on the shape of the utility function U( · ) and the nonadditive measure C( · ) it is capable of demonstrating ambiguity aversion as well as a wide variety of observed properties of risk preferences.

The technique of generalized expected utility analysis under objective uncertainty has also been adopted to the analysis of general non-expected utility/non-probabilistically sophisticated preference functions W (f (·))≡ W (x1 on E1;...; xn on En) over subjective acts. So long as such a function is ‘smooth in the events’ it will possess a ‘local expected utility function’ (which may be state-dependent) and a ‘local probability measure’ at each act f( · ), and classical results involving expected utility risk preferences and probabilistic beliefs can typically be generalized in the manner described above (Machina 2005).

See Also