FRAMING

Australian National University

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< empirical critique of EU theory. Kahneman and Tversky pointed to numerous phenomenae predictions of EU theory. However, the alternative 'prospect theory' model proposed by KT dealt with only a subset of these phenomena, including common consequence and common ratio effects. In addition, the original formulation of prospect theory involved pervasive violations of dominance for which there was no clear basis.

Much of the subsequent literature on generalizations of EU was motivated by the desire to model the phenomena analyzed by prospect theory while preserving normatively desirable properties such as transitivity and first-order stochastic dominance. This task was successfully accomplished with the development of a number of models in the early 1980s (Chew 1983, Machina 1982, Quiggin 1982). Subsequent analysis has generally focused on variants of these models. Fishburn (1988) offers a survey and Chew and Epstein (1989) present a unifying perspective.

In the process, however, some of the central ideas The most important are coding and framing. Coding refers to the process of transforming probability information into a form which may be used in evaluating prospects. Framing refers to the way in which the coding process may be affected by extraneous inputs, notably the way in which information is presented.

By adopting very general representation of coding and framing, it is possible to develop a framework which includes, as special cases, a number of models of choice under uncertainty. Quiggin (1992) presents a general model of probability transformations, which may be interpreted as a theory of coding. Quiggin derives conditions under which tractable results on preferences and comparative statics of chouce under uncertainty may be derived for very general preference structures. The central idea is that the model may be interpreted as EU with respect to a transformed probability distribution. As long as the transformation is 'well-behaved', EU results will carry across.

The main concern of the paper is to extend this framework to incorporate framing effects. Framing effects modify the coding process and may be interpreted as shifts from one transformation to another. For a given probability information input, a change in frame produces a change in the probability distribution yielded by the coding process. The effects of this change may be analyzed using the tools of EU comparative static analysis.

Ordinal and Cardinal Utility Functions - An Aside

By an interesting historical irony, the development by von Neumann and Morgenstern of a theory of choice under uncertainty based on cardinal utility functions occurred just after the completion of the ordinalist revolution which banished cardinal utility from both positive and normative economic theories of choice under certainty. A number of the pioneers of EU theory were at pains to argue that the cardinal utility function employed in the model had no implications in terms of intensity of preferences or interpersonal comparisons. The most notable dissenter from this view was Allais, whose critique of EU theory was grounded in his belief in a meaningful cardinal utility function and the view that risk preferences could not be explained solely in terms of concave utilities.

The development of generalized utility theories has had implications for cardianl utility which have received surprisingly little attention. Allais' (1986) formulation of the rank-dependent model in terms of cardinal utility (given a rigorous axiomatisation in recent unpublished work by Wakker) may be regarded as the culmination of his critique of the EU model. On the other hand, the Machina (1982) local utility function model may be regarded as ordinalism triumphant. Because the utility function is a purely local representation of preferences, it has no inherent cardinal properties.

The approach represented in this paper is unabashedly cardinalist. The existence of a fixed cardinal utility function over wealth is assumed. In the discussion of Machina's local utility function, a canonical utility function is introduced and all of the local utility functions are treated as transformations of that function.

The cardinalist approach has important implications for the relationship between choice under uncertainty, choice over time and social choice. It suggests that, beyond merely formal similarities, the utility function which is maximized in choice under uncertainty should be, in some sense the same one which is maximised in temporal and social choice.

The Model

The objects of choice in the model are probability distributions over a totally ordered metric space X. The existence of a cardinal utility function U over X is assumed. For simplicity, attention is confined to the case where there are n equi-probable states of the world. A prospect may then be defined as a vector x = (x₁, ... x_n). Note that the x_i are not necessarily distinct. For many purposes, it may be useful to arrange the x_i so that x₁ ¾ x₂ ... x_n. In order to distinguish the case when the outcomes have been rearranged in order we will use the notation x* = (x¹, ... xⁿ).

In any given choice situation, each prospect is accompanied by a set of framing information F. A coding process associates with each pair (x, F) a vector w of weights such that w_i >= 0 " i and w_i = 1. For a given frame F andielding weights w, the evaluation of a prospect is given by

(1) V (x, F) = w_i U(x_i)

The RHS of (1) differs from the standard EU model only in that the weights vector w is noo the 'true' probability vector (1/n, ... 1/n). It will be useful to consider the random variable  = {x, w} = {(x₁, w₁), ... (x_n, w_n)} Œ D_n(X), where D_n(X) denotes the space of probability distributions over X with at most n distinct outcomes. The representation of preferences proposed here is one in which expected utility is calculated with respect to the transformed prSIZE=4>D_n(X), where .

A number of well-known models of choice under uncertainty fit easily into this framework. First, as has just been noted, the EU model corresponds to the case when w = (1/n, ... 1/n), so that x =  and:

(2) V (x, F) = 1/n U(x_i) E[U(x_i)]

Second, the RDEU case applies when w is independent of x and F. The RDEU probability weightiented by a mapping q:[0, 1] Æ [0, 1] which is composed with the cumulative distribution function to yield a transformed probability distribution. In the discrete case discussed here, this is represented by the relationship

(3) wⁱ((i+1)/n) - q(i/n)

j = h(x_j) / h(x_i)

A fourth example is the reference prospect theory of Viscusi (1989), which depends on the number of distinct outcomes. Supposing there are m distinct outcomes with the j-th such outcome occurring in r_j states (that is, having probability r_j/n). Then the subject is assumed to assign to each distinct outcome a weight of the form a(1/m) + (1-a)r_j/n, derived from a Bayesian updating process with a prior derived from the Laplace principle of insufficient reason. For any state i yielding outcome j, the associated value w_i is

(6) w_i = a(1/mr_j) + (1-a)1/n

None of the models discussed thus far incorporates framing effects, and indeed, there has been little formal analysis of framing. The classic example of framing is a problem posed by Kahneman and Tversky, concerning a choice of medical treatment for 600 patients. Treatment A will result (with probability 1) in 400 patients living and 200 dying. Treatment B is uncertain. With 1/3 probability all 600 patients will die. With 2/3 probability all 600 patients will live. Kahneman and Tversky present this information in two different ways. One group of respondents is told

Under treatment A, 400 patients will live with probability 1

Under treatment B, there is a 2/3 probability that 600 patients will live and a 1/3 probability that none will live.

A second group of respondents is told

Under treatment A, 200 patients will die with probability 1

Under treatment B, there is a 2/3 probability that no patients will die and a 1/3 probability that 600 will die.

With the first presentation, the majority of subjects choose treatment A, but with the second presentation a majority choose treatment B. This result may be explained in a variety of ways. In order to present the problem in the framework proposed here, begin with three equiprobable states of nature, and denote outcomes by pairs (x₁, x₂) where x₁ is the number dying and x₂ is the number living. Treatment A yields the same outcome (200,400) in all states. Treatment B yields (0,600) in states 1 and 2 and (600,0) in state 3. Because zero is an origin for the natural numbers, there is a tendency to focus more on zero outcomes than other outcomes. The problem above illustrates this phenomenon. In this case, there is no natural zero. The first presentation leads respondents to regard the outcome when no patients survive as the zero outcome, leading to a high weight on state 3, and a consequent preference for treatment A. The second presentation leads respondents to regard the outcome when no patients die as the zero outcome, leading to a high weight on states 1 and 2, and preference for treatment B. A broadly similar, though more complex, account may be given of the more general phenomenon of risk-preference in the domain of losses.

The phenomenon of preference reversal may also be discussed in terms of farming. In the typical case of preference reversal, individuals are asked to choose between two prospects and also how much they would accept to give up each prospect. Surprisingly, in certain cases the preferred prospect is given a lower price than the less preferred prospect. One way of interpreting the evidence is to suggest that framing the problem in terms of valuation encourages a high weight on the mean outcome. Framing the problem in terms of choice encourages a higher weight on the extreme outcomes.

Some widely discussed models of choice may be interpreted in terms of framing effects. First prospect theory itself incorporates one element of framing in the determination of the reference point against which prospects are evaluated. Given a reference point x*, the coding process may be axiomatized as in the cumulative prospect theory model of TK, WT.

Another notable example is the regret theory model of Loomes and Sugden (1982). In their original presentation, Loomes and Sugden examined the case of pairwise choices. As in the present approach they assumed a known utility function. Loomes and Sugden modelled regret/rejoicing as a convex function of the utility difference between the outcome obtained in state i as a result of a given choice and the outcome which would have been received had an alternative choice been made. Quiggin (1990) examined the problem of extending regret theory to arbitrary choice sets and showed, that the natural extension is one in which regret depends only on the best attainable outcome in a given state. This result depends upon the assumptions that the state-independence property of regret theory is maintained and that the addition of statewise-dominated prospects to the choice set does not affect regret. Given a prospect x and a set of alternatives A (of which x is an element) let a* be defined by a*_i = max{a_i: a Œ A}. Following Loomes and Sugden, Quiggin defines regret as a function of utility differences and take the expectation of that function. The generalized regret associated with the choice of x is given by

(7) E, A)] = (1/n) f(U(a*_ix_i))

where f is a convex monotone function.

An equivalent procedure would be to replace the probabilities with decision weights depending on the relative importance of regret effects in each state of the world. For any given prospect, regret will depend on both the outcomes available from the prospect and on the alternatives in the choice set. In terms of the approach used here, the set of alternatives constitutes the frame F and the weights are determined as a function w(x, F). Thus,

(8) h_i = f(U(a*_i) - U(x_i))/U(x_i)

w_i = h_i/

As well as the formal models discussed above, the model proposed here incorporates a widapproximate and heuristic evaluation procedures. A number of these are worth discussing. First, there are procedures based on the choice of some representative outcome. Important examples are the maximin rule, based on selection of the worst outcome, modified maximin rules in which the bottom 5 or 10 per cent of the distribution is disregarded and the median rule. Since all of these procedures select a representative outcome based solely on its rank in the distribution, they are all special cases of the RDEU model. For example, the maximin model sets w = (1, 0, ... 0).

Another important group of representative value models are based on modal outcomes. There are some technical difficulties with the definition of a modal value for discrete probability distributions of the kind discussed here. In general, there will be n distinct values, each occurring with probability 1/n. However, for large n, an approximation to a density function can be defined and the modal value will be that at which the density is maximized. For the modal value rule, as with other representative value rules, the weights vector w will have a single unit entry and the rest zero. Unlike the rules considered above, w will depend on x.

A more complex example of a modal value rule arises in contexts such as project planning and appraisal in which a number of relevant variables are uncertain and the object is to select a representative value for some variable of ultimate interest, such as the present value of the discounted stream of returns. A natural extension of the modal value approach is to derive an estimate of present value on the assumption that all relevant uncertain variables take their modal values. However, as is discussed by Quiggin and Anderson (1990) this approach does not necessarily yield a modal value for present value. In particular, if most surprises are unpleasant so that distributions are negatively skewed, the estimated present value will lie above the mode. As the number of uncertain variables becomes large, this procedure will yield an estimate in the upper tail of the distribution. This more complex problem cannot be fitted exactly into the framework outlined here, and suggests the need for a more general approach.

The first element of this approach is derived from the observation that, outside laboratory settings, we are rarely confronted with probability distributions over outcomes. A more typical, though still highly formalized representation arises if we suppose that instead of a probability distribution for x, we are presented with a sample of n observations. The notation of (1) may be maintained, although the interpretation is now different. In a Bayesian decision procedure, the frame F is the information used to derive the prior distribution. The coding procedure is the Bayesian updating rule which incorporates the sample information x to yield the posterior distribution w, and the evaluation of the prospect is given by (1).

As Tversky and Kahneman (1974) have shown, even people with a relatively sophisticated knowledge of statistics do not, in general, follow Bayesian updating rules. Among the patterns observed by Tversky and Kahneman are a tendeht to sample information than would be suggested by Bayes' theorem. This 'conservative' tendency might be modelled by deriving w as a weighted average of the prior distribution and the Bayesian posterior.

Preferences

The framework presented here is capable of incorporating a very wide range of preferences. Quiggin (1992) considers the case when framing effects are absent. For a given frame F, the following conditions are imposed on preferences.

(A.1) State-independence: If y₁ and y₂ have the same cumulative distribution function F, then y₁ I y₂.

(A.2) Completeness and Transitivity: P is a total order on Y

(A.3) Strict dominance: y₁ FSD y₂, y₁ != y₂ fi y₁ P* y₂

(A.4) Continuity: y₁ P y₂ P y₃ fi $ l Œ [0, 1], y₂ I ly₁ ~ (1-l) y₃,

F, then existence is guaranteed.

The discussion above is sufficient to indicate that there will not, in general, be a unique representation of the form (1) for given preferences. As will be argued below, this abundance of representations is not a handicap in the present context.

The absence of uniqueness, and the fact that only very mild restrictions are imposed upon preferences mean that it is difficult to formulate (1) as a testable hypothesis about preferences. This is not the purpose of the approach proposed here.osed is an extension of the approach of Machina (1982). The key idea is to show that, under appropriate conditions, central results of EU theory can be extended to very general classes of preference functionals. Empirical evidence on preferences may then be used to impose appropriate restrictions on preference functionals..

A powerful approach to the analysis of preferences in the present model arises from the observation that it may be interpreted as deriving Expected Utility with respect to the transformed cumulative distribution function  = {x, w(x, F)}. Let R denote a partial ordering relation on the space D_n of probability distributions, such as first or second stochastic dominance. The coding function w will be said to preserve the partial ordering R, (for a given frame F) if x₁ R x₂ fi ₁ R ₂. If w preserves R, any EU results conceI>R will carry over to the present context. In addition to first and second stochastic dominance it is useful to consider the issue of risk-aversion. An important feature of generalized models, including RDEU, is that d stochastic dominance is not equivalent to risk-aversion in the standard sense of preferring a certain outcome d_c to a risky prospect x such that E[x] ¾ d_c. This standard definition yields a partial order R_d. It is well-known in the RDEU model that the latter condition is satisfied if and only if q(p) >= p " p. From (3), the corresponding condition on w is that w^j(x) >= i/n" i, x. If this condition is satisfied, w will preserve(y+ the relation R_d. Since the condition derived here amount to placing more weight on the bad outcomes, it is often referred to as pessimism.

Other useful relations are those of a monotone spread in the outcomes and a monotone adverse shift in the probabilities. We say that x₂ is derived from x₁ by a monotone spread in the outcomes if x_2i - x_1i is increasing in i. If, in addition, E[x₂] ¾ E[x₁], it is easy to show that x₁ second stochastically dominates x₂. Monotone adverse shifts in the probabilities are defined on weights over a given vector of outcomes x. We say that w₂ is derived from w₁ by a monotone adverse shift in the probabilities if w_2i - w_1i is decreasing in i. It follows that {x, w₁} first stochastically dominates {x, w₂}.

 

PROPOSITION 1:

(a) Preferences preserve first stochastic dominance if there exists a representation V such that U is monotone increasing and w preserves first stochastic dominance

(b) Preferences preserve second stochastic dominance if there exists a representation V such that U is concave and w preserves second stochastic dominance

(c) Preferences display risk-aversion if there exists a representation V such that U is concave and w is pessimistic.

Proof: (a) By hypothesis, xx₂ fi ₁ FSD ₂

fi E[U(₁)] >= E[U(₂)]

fi V(x₁) >= V(x₂)

The proof of (b) and (c) is similar. n

Proposition 1 applies trivially in the case of the EU model. For the RDEU model, it is straightforward to show that a necessary and sufficient condition for part (a) is that q be monotone increasing and for part (b) that q be concave. As stated above, for part (c) it is necessary and sufficient that q(p) >= p " p. More generally, the problem is to obtain a representation of preferences such that the conditions of the proposition are met.

The first stochastic dominance result in Proposition 1(a) extends fairly straightforwardly to the case when x is sample data and the coding process is a Bayesian derivation of a posterior distribution. If one sample first stochastically dominates another, the same relationship will holds for the posterior distributions. It follows that the result will also hold for conservative updating procedures in which is a weighted average of the Bayesian posterior and the prior. Thus, this heuristic does not lead to violations of first stochastic dominance.

For the weighted utility model, Proposition 1(c) holds whenever h(x) is positive and decreasing. On the other hand, although the weighted utility model preserves first stochastic dominance, the conditions of 1(a) and 1(b) do not apply in general. For example if h(x) is decreasing and positive, an increase in the value of the best outcome will lead to a reduction in the weight on that outcome. By the averaging process used to obtain weights, the weights on all the other outcomes (including the worst one) will increase so that the resulting shift will not satisfy first stochastic dominance. A similar counter-example can be constructed when h(x) in increasing.

An alternative approach to preferences follows that developed by Machina (1982). The frame F will be held constant, and it will be assumed that U is differentiable and that w is a continuously differentiable function of x. Instead of a local utility function, we will consider the case where V can be locally approximated in a neighborhood of x by an RDEU functional with utility function U and a constant weighting vector w = w(x, F).

That is, for any Dx

(9) V(x + Dx, x, F) + w_i(x, F)U'(x_i)(Dx)_i + o(||Dx||)

where ||…|| is the usual Euclidean norm on Xⁿ.

It should be noted at this point that the notion of smoothness used in the present weaker than that used by Machina. The evaluation mapping V is treated as a mapping from Xⁿ to ¬. Machina's functionals are defined on the infinite-dimensiSIZE=4>F) = V(x, n becomes large, the topology associated with the approach used here approximates that of the supremum norm.

resentation of preferences for some given U. In a neighborhood of any x, the preference functional may be locally approximated by an RDEU functional with w held constant at w(x). In the RDEU literature it has been shown that first stochastic dominance will be preserved whenever U and q are monotone increasing and that second stochastic dominance will be preserved whenever U and q are concave. From (2) q will be monotone increasing if and only if the elements of w are all positive and q will be concave if and only if the elements of w (in the rank-ordered setting) are decreasing. With these preliminaries, we are ready to state:

 

PROPOSITION 2:

(a) V preserves first stochastic dominance if and only if there exists a local RDEU representation of preferences such that U is monotone increasing and w_i(x) >= 0 " i, x.

(b) V preserves second stochastic dominance if and only if there exists a local RDEU representation of preferences such that U is concave and 0 ¾ w_i+1(x) ¾ w_i(x) " i, x.

(c) V displays risk-aversion if and only if there exists a local RDEU representation of preferences such that U is concave and w_j(x) >= i/n" i, x.

 

Proof: (a) Let x₁ FSD x₂ and define Dx = k(x₂ - x₁) for suitably small k. By the first stochastic dominance condition Dx_i ¾ 0 " i. Hence from (9)

V(x₁ + Dx, F) - V(x, F) = w_i(x, F)U'(x_i)(Dx)_i + o(||Dx||) ¾ 0

This is true if we replace x₁ with lx₂ + (1-l)x₁. Hence

/ lV( lx₂ + (1-l)x₁, F) ¾ 0 " l .

Integrating over [0, 1] yields the desired result. The proof of parts (b) and (c) is similar.n

 

SIZE=4>F) - V(x,

The approach proposed here yields a straightforward way of dealing with framing effects. Preferences are treated as Expected Utility with respect to the transformed distribution . The effects of change in frame from F₁ to F₂ are captured as a change from ₁ to ₂. Hence the usual analysis of preferences under EU is applicable. For example, if ₁ first stochastically dominates ₂, the prospect x will be more attractive (have a higher certainty equivalent) under frame F₁ than under F₂.

The Kahneman and Tversky example may be analyzed inSIZE=4> F₁ to F₂ puts a high weight on the best states. Hence the transformed distribution for the risky option derived from F₂ first stochastically dominates that derived from F₁. Respondents are more likely to choose the risky option under F₂.

Similarly, advertising for a lottery seeks to induce framing effects which increase the weight placed upon the prize-winning states. Among the techniques used are direct statements about probability (e.g. 'nobody has a better chance than you'), graphic depictions of prize-winners, and appeals to the regret which might arise from faiSIZE=4>F₁ than under le of a first stochastically dominant shift in frame arises in regret theory. S₁ = (y₁, ... y_n) of maximum attainable outcomes, and that the vector y = (y₁, y₂ ... y_n) is comonotonic with x. Now consider a shift to a frame F₂ associated with y^# =  (y_n, y₁ ... y_n-1). From (8) this increases h₁ and reduces h_i i > 1. By the concavity of f, h_i is increased, so that w₁ is increased and all other w_i are reduced. Thus ₁ first stochasticaC="graphics/Framing32.jpeg" WIDTH="6" HEIGHT="10" NATURALSIZEFLAG="3" ALIGN=BOTTOM>₂ and the certainty equivalent of x is higher under F₁ than under F₂.

Comparative Statics

The majority of the literature on generalized theories of choice under uncertainty has been concerned with, if theories are to have any economic application, it is necessary to go beyond preferences to consider comparative static analysis. This has proved difficult for many generalized models. For example Machina's smooth preferences model has a straightforward extension for problems characterized by a functional separation between the control variable and the random variable of interest (Machina 1989), but this condition is not satisfied for many of the most important problems. Quiggin (1990) has shown that the RDEU model permits an extension of the main comparative static results of EU theory. The central tool in the argument is a correspondence principle which is an extension of the ideas used in previous sections.

This correspondence principle may be stated as follows. Let R be a relation on D_n, let U be a family of utility functions. Then a typical comparative static result states that if the movement from the initial to the final distribution of some relevant random variable satisfies the relation R, all Expected Utility maximizers with U ΠU will alter their optimal action in a particular way e.g. by increasing the value of some control variable.

Now let

(15) F*(R) = {f: x₁ R x₂ fi ₁ R ₂ }

Then the correspondence principle states that, provided the outcomes from every action are better, the higher is the realization of the relevant random variable, the conclusion of the comparative static result will be true for the general r U Œ U and f Œ F*(R) .

Quiggin (1990) deals with the following control problem, which is a slight generalization of that treated by Feder (1977):

(16) Max_a V = E[U(f(q, a, W₀))]

where U is a von Neumann-Morgenstern utility function, a is a control variable (assumed to take positive values), q is an economically relevant random variable with cumulative distribution function F, W₀ is initial wealth (here taken to be non-stochastic), and f is a function mapping actions and realizations of q into outcomes, normally taken to be wealth levels. Problems which may be cast in this form include portfolio choice (Diamond and Stiglitz 1974), savings decisions (Sandmo 1970), labor supply (Newbery and Stiglitz 1981) and the theory of the firm (Sandmo 1971, Coes 1977).

 

Changes in frame may also be considered in the comparative static context. As in the analysis preferences, a change in frame may be analyzed in the samin the relevant probability distribution for an EU control problem. Hence, whenever w(x, R w(x, F₂) an increase in a

Concluding Comments

The framework proposed here may be viewed as a variant on the approach suggested by Machina (1982). As with Machina's approach, the objective is to present a general and flexible framework, capable of accommodating a wide range of generalizations ofpose procedures for extending EU results on preferences and comparative statics. In order to make practical use of these procedures, it is necessary to show that specific testable models of choice under uncertainty may be incorporated into the framework in a manner which satisfies the conditions of the extension theorems proved here. A few examples of the way in which the framework may be applied to existing models have been offered here. The characterization of the decision process in terms of coding and framing suggests a wide range of alternative approaches.

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