# Prospect theory for continuous distributions: A preference foundation

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## Abstract

Preference foundations give necessary and sufficient conditions for a decision model, stated directly in terms of the empirical primitive: the preference relation. For the most popular descriptive model for decision making under risk and uncertainty today, prospect theory, preference foundations have as yet been provided only for prospects taking finitely many values. In applications, however, prospects often are complex and involve infinitely many values, as in normal and lognormal distributions. This paper provides a preference foundation of prospect theory for such complex prospects. We allow for unbounded utility and only require finite additivity of the underlying probability distributions, leaving the restriction to countably additive distributions optional. As corollaries, we generalize previously obtained preference foundations for special cases of prospect theory (rank-dependent utility and Choquet expected utility) that all required countable additivity. We now obtain genuine generalizations of de Finetti’s and Savage’s finitely additive setups to unbounded utility.

## Keywords

Prospect theory Preference foundations Continuous distributions Unbounded utility## JEL Classification

D81 C60The most popular descriptive theory of decision making under risk and uncertainty today is prospect theory. Its original version was introduced by Kahneman and Tversky (1979), a paper that constituted a breakthrough in decision theory. Up to that point, the common thinking had been that irrational behavior was too chaotic to be modeled, and that models of rational choice were the best descriptive approximation of irrational behavior (Arrow 1951, p. 406). Prospect theory was the first convincing model that at the same time was tractable enough to allow for theoretical analyses and predictive applications, and was also able to model irrationalities commonly found in empirical choices.

Original prospect theory has some drawbacks. First, there are some theoretical problems in the way it implemented nonadditive probabilities. Second, it deals only with risk (known probabilities). Third, within risk it deals only with a limited set of prospects (only two nonzero outcomes).^{1} Quiggin (1982) discovered the solution to the first problem for risk. Schmeidler (1989, first version 1982) independently discovered that same solution and, importantly, extended it to uncertainty (unknown probabilities), jointly with Gilboa (1987), which resolves the second problem. Quiggin could handle any finite set of outcomes, not just the case of two nonzero outcomes, and Schmeidler could handle all bounded prospects, which attenuates the third problem. A limitation of their solution was, however, that they could not handle reference dependence and loss aversion, important empirical components of prospect theory, so that their solution applies only to gains or losses separately. It cannot handle mixed cases with both gains and losses. Tversky and Kahneman (1992) incorporated reference dependence and loss aversion in the models of Quiggin and Gilboa–Schmeidler, leading to new prospect theory, which will be called prospect theory henceforth.^{2} Thus they obtained the first theory that combines theoretical soundness with empirical realism.

The absence of any preference foundation for original prospect theory could have served as a signal of something being wrong.^{3} Preference foundations state conditions, directly in terms of preferences, that are necessary and sufficient for some decision model, such as prospect theory, to hold. Because preferences are directly observable, preference foundations identify the empirical meaning of a model. They show how to directly verify or falsify a model empirically, and how to defend or criticize it normatively. If the conditions in the preference foundation are natural then the theoretical soundness of the model has been established.

Historical examples illustrate the importance of preference foundations. The Bayesian approach to economics and statistics became popular through famous preference foundations by de Finetti (1931), von Neumann and Morgenstern (1944), and Savage (1954). The multiple priors model, introduced by Wald (1950) and extensively discussed as early as in Luce and Raiffa (1957, Ch. 13), only became popular when Gilboa and Schmeidler (1989) provided a preference foundation.

In retrospect it is clear why a preference foundation (with natural conditions) could never have been found for original prospect theory: the functional of this theory is not theoretically sound. The other theories discussed above did receive a preference foundation. Most of those preference foundations only concerned simple prospects, being prospects that take only finitely many values; Schmeidler (1989) could handle all bounded prospects. The preference foundations of Quiggin (1982) and Schmeidler (1989) were extended to continuous and possibly unbounded prospects by Wakker (1993).

Tversky and Kahneman (1992) did not define or analyze their new theory for continuous distributions. Such a definition can readily be given (Wakker and Tversky 1993, Eq. 6.1 and Section 8.4; Davies and Satchell 2003; Wakker 2010, Eqs. 9.7.1 and 12.8.1). This paper will present a preference foundation of the new version of prospect theory for continuous distributions. It is desirable to establish such a foundation because continuous distributions are widely used in applications. For example, stock prices are usually assumed to be lognormally distributed in finance (Hull 2006). Investment decisions deal with continua of investment options and, then, with continua of possible returns. Economic growth models consider continuous paths over time with continuous risks (Becker and Boyd 1997). Health economics typically deals with continuous outcome scales such as quality of life, considering continuous probability distributions over those scales and, further, profiles that are continuous over time (Drummond et al. 1987). References applying prospect theory with nonlinear probability weighting to continuous distributions, often lognormal and, hence, unbounded, include Barberis and Huang (2008), Barberis and Xiong (2009), Bernard and Ghossoub (2009), Carlier and Dana (2011), He and Zhou (2010), Henderson (2010), Kanbur et al. (2008), Spalt (2010), and Sumalee et al. (2009).

Another difference between the modern version of prospect theory and its original version is that until recently no one could even define the latter for continuous distributions in any plausible manner. Only recently Rieger and Wang (2008) succeeded in proposing such an extension. They showed that, basically, only the derivative *w* ^{′}(0) of the weighting function *w* at *p* = 0 is empirically relevant for continuous distributions. In our interpretation, this is unsatisfactory. Thus Rieger and Wang revealed further limitations of original prospect theory. Their approach can be extended to all theories using transformations of probabilities of separate outcomes (Allais 1979; Edwards 1962; Viscusi 1989).

In what follows, Section 1 deals with decision under risk. It defines prospect theory for continuous distributions and presents the preference foundation for this case. Our result generalizes Wakker (1993, Corollary 4.5) by incorporating reference dependence and loss aversion. It is also structurally more general. As a corollary, we generalize Wakker’s extension of Quiggin’s (1982) rank-dependent utility from countably additive to the more general finitely additive prospects.

Section 2 provides extensions to uncertainty, which is modeled through states of nature and events for which no probabilities are available (ambiguity). This result, in combination with the aforementioned extension to finite additivity, implies that we obtain genuine generalizations of de Finetti (1931, 1972) and Savage (1954), allowing for unbounded utility. An additional feature of our approach is that the only richness required is that all simple (finitely valued) prospects are available, plus truncations of the prospects considered. Other than that, our results can be applied to any domain of prospects. It thus serves as justification of using prospect theory in many applications, including applications that consider *only* normal, or lognormal, distributions and their truncations. Section 3 concludes. Appendix A shows how measure-theoretic structure can be introduced, and Appendices B and C give proofs. Appendix C introduces a method for embedding decision under risk into decision under uncertainty that generalizes Wakker’s (1993, Section 4.2) technique.

## 1 Risk

### Structural definitions

*X* is an *outcome* set. It is often taken to be the set ℝ of reals, designating monetary outcomes. Everything that follows remains valid for more general outcome sets, such as commodity bundles or health states. We will assume for the utility function defined later that its range is a nonpoint interval, which will imply that the outcome set is a continuum and cannot be finite. One outcome, the *reference outcome*, denoted *r*, will play a special role in what follows. A common case is *r* = 0 for monetary outcomes, designating the status quo.

*probability distribution*

*P*on

*X*maps the subsets of

*X*to [0, 1], with

*P*( ∅ ) = 0,

*P*(

*X*) = 1, and

*countable additivity*, is useful:

If a countable sequence of events *A* _{ j } are mutually disjoint then \(P(\cup _{j=1}^{\infty}A_{j}) = \sum_{j=1}^{\infty}P(A_{j}).\)

\(\mathcal{P}\) is a subset of the set of all probability distributions over *X* . Its elements are called *prospects*. Our analysis will be flexible regarding the domain (\(\mathcal{P}\)) of prospects. Our analysis holds with no modification if we restrict attention to countably additive prospects. It also holds if we add measure theoretic structure (Appendix A).

*simple prospects*, i.e. the prospects that assign probability 1 to a finite set of outcomes. A simple prospect can be denoted (

*p*

_{1}:

*x*

_{1}, ...,

*p*

_{ n }:

*x*

_{ n }) with the obvious meaning. \(\succcurlyeq\) is a preference relation over \(\mathcal{P}\). ≻, \(\preccurlyeq\), ≺, and ∼ are defined as usual. Any

*degenerate*prospect (1:

*α*), assigning probability 1 to an outcome

*α*, is identified with that outcome. Preferences over prospects thus generate preferences over outcomes. Outcomes strictly preferred to

*r*are

*gains*, and outcomes strictly less preferred than

*r*are

*losses*. Outcomes indifferent to

*r*are

*neutral*. For prospect

*P*, outcome

*α*is a

*certainty equivalent*(

*CE*) if

*α*∼

*P*. A function

*V*

*represents*\(\succcurlyeq\) if \(V:\mathcal{P} \rightarrow\mathbb{R}\) and

### Defining prospect theory

A function *w* is a *(probability) weighting function* if it maps [0, 1] to itself, satisfies *w*(0) = 0 and *w*(1) = 1, and is nondecreasing. Under prospect theory, we use a different weighting function for gains than for losses.

### **Definition 1**

*Prospect theory*(

*PT*) holds if there exist weighting functions

*w*

^{+}and

*w*

^{−}, and a

*utility function U*:

*X*→ ℝ with

*U*(

*r*) = 0, such that the evaluation

*PT*(

*P*) is the

*prospect theory value*of

*P*.

Tversky and Kahneman (1992) only considered simple prospects, for which the integral in our Eq. 1 coincides with their definition of PT. Equation 1 agrees with Quiggin’s (1982) rank-dependent utility for gains separately, and also for losses separately. It may deviate when both gains and losses are present. We have maintained the classical symbol *U* and the classical term utility, instead of the symbol *v* and the term value function that Kahneman and Tversky used. For losses, our function *U* implicitly includes loss aversion; i.e., it is the loss aversion parameter times the value function of prospect theory. This is why we do not explicitly write the loss aversion parameter in this paper. Wakker (2010) called *U* the overall utility function.

### Preference conditions for prospect theory

*weak order*. That is, then \(\succcurlyeq\) is

*complete*(\(P \succcurlyeq Q\) or \(Q \succcurlyeq P\) for all \(P,Q \in\mathcal{P}\)) and transitive. For the extension of PT from simple to bounded prospects, we use the following two preference conditions.

- (i)
*Certainty equivalence*, or*CE equivalence*, holds if for each prospect \(P \in\mathcal{P}\) there exists a CE. - (ii)
*(Weak) stochastic dominance*holds if, for all \(P,Q \in\mathcal{P}\), we have \(P \succcurlyeq Q\) whenever: \(P\{\beta\in X: \beta\preccurlyeq\alpha\} \leq Q\{\beta\in X: \beta\preccurlyeq\alpha\} \}\) for all*α*∈*X*.

We next consider the extension to prospects with unbounded utility (of their outcomes) that, however, still have a finite and well-defined PT value. For this purpose we use a truncation-continuity condition, introduced by Wakker (1993). The condition imposes restrictions on the preference relation in combination with the preference domain \(\mathcal{P}\). It will rule out all unbounded prospects that have infinite or undefined PT values, and will do so entirely in terms of preferences.

For prospect *P* and outcome *μ*, *P* ^{ ∧ μ }, the *above truncation of P* * at μ* *,* is equal to *P* on \(\{\alpha\in X: \alpha \preccurlyeq\mu\} \backslash\{\mu\}\), and assigns all remaining probability *P*{*μ*} + *P*{*α* ∈ *X*: *α* ≻ *μ*} to outcome *μ*. For outcome *ν*, *P* _{ ∨ ν }, the *below truncation of P* * at ν* *,* is equal to *P* on \(\{\alpha\in X: \alpha\succcurlyeq\nu\} \backslash\{\nu\}\), and assigns all remaining probability *P*{*ν*} + *P*{*α* ∈ *X*: *α* ≺ *ν*} to outcome *ν*.

*P*is

*truncation-robust*if for all outcomes

*α*the following holds:

*Truncation-continuity*holds if all prospects \(P\in\mathcal{P}\) are truncation-robust. We will assume the following richness on \(\mathcal{F}\):

*Truncation-closedness*holds if for every \(P \in\mathcal{P}\) and

*α*∈

*X*, we have \(P^{\wedge\alpha} \in\mathcal{P}\) and \(P_{\vee\alpha} \in\mathcal{P}\).

### Our main theorem for risk

We will not relate the following theorem to one particular preference foundation of PT. We will state it in general, so that it can be used as an add-on to every preference foundation of PT for simple prospects that has a continuum of utility. This includes the common cases with real-valued outcomes or commodity bundles as outcomes and with continuous utility. Preference foundations for PT have so far mostly been studied for uncertainty. Chateauneuf and Wakker (1999) gave a preference foundation for risk with a continuum of utility, but only for simple prospects. Theorem 2 can be combined with their result to include continuous distributions.

### **Theorem 2**

*Assume that PT holds on*\(\mathcal{P}^{s}\)

*, with utility function U*

*and weighting functions w*

^{+}

*and w*

^{−}

*. Assume the following richness:*

- (1)
*U*(*X*)*is an interval that contains 0 in its interior.* - (2)
*We have:*\(\mathcal{P}^{s} \subset\mathcal{P}\)*, certainty equivalence, and truncation-closedness.*

*Then PT holds on all of*\(\mathcal{P}\)

*with respect to the same U*

*, w*

^{+}

*, and w*

^{−}

*, with the PT value of all prospects in*\(\mathcal{P}\)

*well defined and finite, if and only if the following conditions hold on*\(\mathcal{P}\)

*:*

- (i)
*Weak ordering;* - (ii)
*Stochastic dominance;* - (iii)
*Truncation-continuity.*

Wakker (1993, Corollary 4.5) did not use the proof technique that we use in Appendix C (constructing the state space *S* as a high-dimensional product space) and, hence, provided his extension of rank-dependent utility to unbounded prospects only for countably additive prospects. We can now extend it to finitely additive prospects. Another difference between Wakker’s (1993) analysis and ours is that he used a step equivalence assumption instead of our simpler certainty equivalence assumption. If the utility range is a continuum, as in Wakker’s Corollary 4.5 and in our paper, then equivalence of the two conditions follows elementarily. A third difference is that Wakker (1993) assumed a condition called weak truncation-closedness, instead of our truncation-closedness, throughout his paper. Our results could be generalized to Wakker’s condition. That condition is, however, more complex, and we nowadays prefer accessibility to mathematical generality.

### **Corollary 3**

*Under truncation-closedness, Corollary *4.5* of *Wakker (1993)^{4} * also holds without the restriction of countable additivity.* □

## 2 Uncertainty

### Structural definitions

*X* is again an outcome set, with *r* the *reference outcome*. Now no probabilities are given for uncertainties, but uncertainties are modelled through states (of nature). *S* denotes the *state space*. Exactly one state is true, the others are not true, and the decision maker does not know for sure which state is true. *Events* are subsets of the state space. An event is *true* if it contains the true state of nature. ∅, the vacuous event, surely is not true, and *S*, the *universal event*, surely is. The results in this paper all hold true both for finite and for infinite state spaces *S*. For finite state spaces they, however, do not contribute anything new, with all extra conditions vacuously satisfied. Only for infinite state spaces do our results bring novelty.

We consider a subset \(\mathcal{F}\) of the set of all functions from *S* to *X* . Its elements are called *prospects*.^{5} We assume that \(\mathcal{F}\) contains the set \(\mathcal{F}^{s}\) of all *simple* functions from *S* to *X*, taking only finitely many outcomes. A simple prospect can be denoted (*E* _{1}:*x* _{1}, ..., *E* _{ n }:*x* _{ n }). Here it is implicitly understood that *E* _{1}, ..., *E* _{ n } are events that partition the state space. The prospect assigns outcome *x* _{ j } to every state in *E* _{ j }. Using the same letter s to indicate states of nature and, as a superscript, to indicate simple prospects will not raise confusion.

\(\succcurlyeq\) is a *preference relation* over \(\mathcal{F}\), with \(\succ, \preccurlyeq, \prec\), and ~ as usual. Any *degenerate* prospect (*S*:*α*), assigning outcome *α* to each *s* ∈ *S*, is identified with that outcome *α*. Preferences over prospects thus, again, generate preferences over outcomes. Gains, losses, neutral outcomes, certainty equivalents, and representing functions are defined as before.

### Defining prospect theory

A function *W* is a *weighting function* if it maps events to [0, 1], and satisfies *W*( ∅ ) = 0, *W*(*S*) = 1, and \(A \supset B \Rightarrow W(A) \geq W(B)\) (*set monotonicity*). Weighting functions need not satisfy additivity. Probability measures concern the special case of weighting functions that do satisfy additivity.

### **Definition 4**

*Prospect theory*(

*PT*) holds if there exist weighting functions

*W*

^{+}and

*W*

^{−}, and a

*utility function U*:

*X*→ ℝ with

*U*(

*r*) = 0, such that the evaluation

### Preference conditions for prospect theory

*Certainty equivalence* (*CE equivalence*) holds if for every prospect in \(\mathcal{F}\) there exists a CE. The following condition adapts stochastic dominance from risk to uncertainty. *Pointwise monotonicity* holds if, for all \(f,g \in\mathcal{F}\), we have \(f \succcurlyeq g\) whenever \(f(s) \succcurlyeq g(s)\) for all states *s*.

*f*and outcome

*μ*,

*f*

^{ ∧ μ }, the

*above truncation of f*

*at μ*

*,*is defined by

*below truncation of f*

*at ν*

*,*

*f*

_{ ∨ ν }, is defined by

*truncation-robust*if for all outcomes

*α*the following holds:

*Truncation-continuity* holds if all prospects \(f\in\mathcal{F}\) are truncation-robust. We will assume the following richness on \(\mathcal{F}\): *truncation-closedness* holds if for every \(f \in\mathcal{F}\) and *α* ∈ *X*, we have \(f^{\wedge\alpha} \in\mathcal{F}\) and \(f_{\vee \alpha} \in\mathcal{F}\). As in the case of risk we will not relate the following theorem to one particular preference foundation of PT. We will state it in general, so that it can be used as an add-on to every preference foundation of PT existing today. Thus, we assume that a PT representation has been obtained for the simple prospects, and that the utility range is a continuum. These conditions are satisfied for all PT preference foundations that exist today. References are given after the following theorem.

### **Theorem 5**

*Assume that PT holds on*\(\mathcal{F}^{s}\)

*, with utility function U*

*and weighting functions W*

^{+}

*and W*

^{−}

*. Assume the following richness:*

- (1)
*U*(*X*)*is an interval that contains 0 in its interior.* - (2)
*We have:*\(\mathcal{F}^{s} \subset\mathcal{F}\)*, certainty equivalence, and truncation-closedness.*

*Then PT holds on all of*\(\mathcal{F}\)

*with respect to the same U*

*, W*

^{+}

*, and W*

^{−}

*, with the PT value of all prospect in*\(\mathcal{F}\)

*well defined and finite, if and only if the following conditions hold on*\(\mathcal{F}\)

*:*

- (i)
*Weak ordering;* - (ii)
*Pointwise monotonicity;* - (iii)
*Truncation-continuity.*

Theorem 5 can be combined with any presently existing preference foundation of PT (all obtained only for simple prospects) to provide a complete preference foundation of PT including nonsimple prospects. The presently existing preference foundations include Köbberling and Wakker (2003, Theorem 12), Luce and Fishburn (1991, Theorem 5 and its corollary), Tversky and Kahneman (1992, Theorem 2), Wakker and Tversky (1993, Theorem 6.3), and Wakker (2010, Theorem 12.3.5). It may seem that certainty equivalence is implied by the *PT* representation, given that *U*(*X*) is an interval, in which case the certainty equivalence condition could have been added to the necessary and sufficient conditions in the theorem. In virtually all cases of interest this is correct. There are, however, exceptional examples in which this is not so (Example 10 in Appendix B), because of which the condition has been stated as a richness restriction.

## 3 Conclusion

Prospect theory had as yet only received preference foundations for simple prospects (taking finitely many values). Applications in finance, health, and other domains often deal with complex situations with continua of possible outcomes. This paper has given a preference foundation for such cases, i.e. preference conditions that provide the critical demarcation for validity of the theory in such cases. Thus this paper supports applications of prospect theory to situations of complex uncertainties.

## Footnotes

- 1.
Extensions to more nonzero outcomes, proposed in the literature, all have serious problems (Wakker 2010, Appendix 9.8).

- 2.
The new version is, as we see it, an advance relative to the original version, as expressed in the title of the 1992 paper. Hence we use the convenient term prospect theory for the new theory rather than for the original one, in agreement with the terminology in Wakker, Thaler, and Tversky (1997) for instance.

- 3.
Kahneman and Tversky (1979, Appendix), aware of the importance of preference foundations, tried to provide one as good as possible, but they only obtained a partial foundation and could not identify the full empirical meaning of their model.

- 4.
This corollary is the analog of Theorem 2 for Quiggin’s rank-dependent utility instead of prospect theory.

- 5.
To distinguish the prospects defined in this section from the prospects defined before for decision under risk, the term

*event-contingent prospect*can be used here. We can then call the prospects for risk*probability-contingent*. - 6.
This can always be arranged by replacing all outcomes by their preference indifference classes. In view of pointwise dominance for uncertainty and stochastic dominance for risk, replacing outcomes in a prospect by other equivalent outcomes does not affect the preference value of the prospect, so that preferences over prospects are well defined if only indifference classes of outcomes are specified. Because indifference classes are preference intervals and, hence, are contained in \(\mathcal{D}\), we can modify \(\mathcal{D}\) accordingly.

- 7.
This can be seen most easily by viewing the left integral as a regular integral of the function

*W*^{+}{*s*∈*S*:*U*(*f*(*s*)) ≥*τ*}. (The integral over ℝ^{+}is per definition the limit of the integrals over [0,*μ*^{′}].) It can also be seen by viewing the left integral as a Choquet integral. - 8.
This condition is slightly more restrictive than the common condition of boundedness if

*U*(*X*) is bounded above but does not contain its supremum.

## Notes

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