A revealed reference point for prospect theory
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Abstract
Without an instrument to identify the reference point, prospect theory includes a degree of freedom that makes the model difficult to falsify. To address this issue, we propose a foundation for prospect theory that advances existing approaches with three innovations. First, the reference point is not known a priori; if preferences are referencedependent, the reference point is revealed from behavior. Second, the key preference axiom is formulated as a consistency property for attitudes toward probabilities; it entails both a revealed preference test for referencedependence and a tool suitable for empirical measurement. Third, minimal assumptions are imposed for outcomes, thereby extending the model to general settings. By incorporating these three features we deliver general foundations for prospect theory that show how reference points can be identified and how the model can be falsified.
Keywords
Probability midpoint Prospect theory Rankdependence Reference point SigndependenceJEL Classification
D03 D811 Introduction
Prospect theory (PT; Tversky and Kahneman 1992) is regarded as one of the most successful descriptive theories for risk and ambiguity (Starmer 2000; Kahneman and Tversky 2000; Wakker 2010; Barberis 2013). Relative to classical expected utility (EU), PT incorporates nonlinear treatment of probabilities (Preston and Baratta 1948; Allais 1953; Quiggin 1982), or nonadditive event uncertainty resulting from ambiguity (Ellsberg 1961; Schmeidler 1989) and referencedependence. The latter requires the existence of a reference point, relative to which outcomes are seen as gains or losses, and constitutes a distinctive feature and a key assumption of PT. What exactly determines the reference point has been left unspecified, and not offering a plausible explanation for how the reference point is derived from primitives, i.e., from preferences over prospects, is regarded as a major shortcoming of PT (Fudenberg 2006, p. 696 footnote 2; Pesendorfer 2006, pp. 713–716).
We develop a revealed preference technique based on probability midpoints to show that PT can be obtained from preferences without assuming the reference point as exogenously given. Starting from an indifference between two prospects, probability midpoints are obtained by shifting probability mass across outcomes, such that a new indifference results (Kuilen and Wakker 2011). By keeping the outcomes of these prospects ordered in terms of preference and commonly fixed, and repeatedly shifting probability mass across adjacent outcomes, one can elicit a sequence of probabilities that are perceived equally far apart in terms of preferences.^{1} If the reference point is meaningful for preferences, the elicited probability midpoints will be affected when probability mass is shifted from losses to gains. This can be used to reveal the location of the reference point. Indeed, this feature of behavior is exploited to show how the reference point in PT is revealed from choices. It is in this sense that we obtain the reference point endogenous to the model.
To position our contribution, let us recall that the von Neumann and Morgenstern (1944) foundations for EU imply a linear treatment of probabilities. This means that elicited probability midpoints are arithmetic midpoints and independent of the type or magnitude of outcomes. The EUaxioms do not impose a specific interpretation for outcomes or restrictions on utility, except for the latter being a cardinal measure. For instance, one can interpret realvalued outcomes as final wealth positions (a frequent assumption made in theoretical applications) or as changes relative to a reference point (usually an implicit assumption made in experimental studies).^{2} In particular, imposing a utility value of 0 at the reference point, and treating outcomes with negative utility values as losses and those with positive utility as gains, is compatible with the EUaxioms as long as probabilities are treated linearly. The requirement that utility is 0 at the reference point then appears as an arbitrary restriction of the class of admissible cardinal utilities to a smaller ratioscale subclass. By contrast, in PT such restrictions on admissible utility functions follow from the asymmetric treatment of the probabilities attached to gains and losses and, thus, properties that capture deviations from linearity in probabilities. It is precisely this asymmetric nonlinear treatment of probabilities, revealed as an inconsistency in elicited probability midpoints, that we exploit in order to provide general foundations for PT.
A nonlinear treatment of probabilities has also been incorporated into the rankdependent utility (RDU) model (Quiggin 1981, 1982; Segal 1987; Wakker 1994). RDU can be seen as a special case of PT where the presence of a reference point is immaterial for attitudes toward probabilities. General foundations for RDU were provided by Nakamura (1995), Abdellaoui (2002), Abdellaoui and Wakker (2005), and Zank (2010), and they can readily be used to derive PT if the reference point is known in advance. Without knowing that a reference point exists, deriving PT from RDU becomes a challenge. To achieve foundation for PT, we employ a consistency test for specifically elicited probability midpoints. Consistency means that the treatment of probabilities is insensitive to replacements of the stimuli used to elicit midpoints. As our consistency property does not impose restrictions on the admissible probability weighting functions under PT, probability midpoint consistency can accommodate a wide range of behaviors (e.g., risk behavior captured through the popular inverseSshaped probability weighting functions; Prelec 1998).
To give further intuition for our main preference tool, suppose we have identified the reference point. Then our condition requires that probability midpoints elicited from preferences are independent of the outcomes (i.e., the stimuli used in the elicitations), whenever the latter are of the same type (i.e., either they are all gains or they are all losses). This is a natural requirement for the treatment of probabilities under PT, where a distinct nonlinear treatment for probabilities of gains as compared to probabilities of losses is explicitly allowed for; it can be inferred, for example, from the reflection examples in Kahneman and Tversky (1979, p. 268). This feature of referencedependent behavior, which we call signdependence, has been widely documented.^{3} Signdependence can be inferred from the presence of distinct probability midpoints for gains than for losses, as implied by empirically elicited parametric forms (e.g., Tversky and Kahneman 1992; Abdellaoui 2000). Conversely, sign independence of probability midpoints suggests that the reference point is immaterial for the treatment of probabilities (as in EU or RDU). Hence, if we do not know the location of the reference point, we can employ probability midpoints elicited for different outcomes to test for sign independence. This leads to a revealed preference technique, where replacing a gain by a different gain does not affect elicited probability midpoints and neither should the replacement of a loss by a different loss affect such midpoints; inconsistent midpoints are revealed only if a gain is replaced by a loss or vice versa. In a nutshell, our main preference tool requires that sign independence of revealed probability midpoints is violated once at the most, in which case we identify the location of the reference point.
In what follows, we present preliminary notation and formal expressions for the models of EU, RDU and PT (Sect. 2), informally introduce probability midpoints, and look at the distinct predictions for midpoints resulting from these models. In Sect. 3 we proceed by recalling the preference conditions shared by all three models. We highlight potential difficulties in deriving PTfoundations by giving examples of referencedependent preferences that are similar to PTpreferences, which do not, in general, allow for the identification of both probability weighting functions. In the literature such preferences have hitherto been circumvented. In Sect. 4 we formally generalize the notion of probability midpoints and present our main preference condition and theorem for the case where the set of outcomes is finite. Extensions are discussed in Sect. 5. In particular, we allude to a procedure that shows how our probability midpoint tool can be employed to identify the location of reference points. The literature on reference points is growing, and different models and approaches have emerged; for instance, there are choice situations in which the reference point may not be a unique degenerate outcome as in PT; thus, a brief summary of this literature is in order. This is done in Sect. 6, where we also discuss issues related to midpoints, the central tool for our PTfoundation. Concluding remarks are in Sect. 7. The Appendix contains further elaborations and proofs.
2 Preliminaries
This section recalls the standard framework for decision under risk and the decision models of expected utility, rankdependent utility and prospect theory, explaining how the second model extends the first through deviating from the linear treatment of probabilities and how the latter model extends the second through the reference point impacting the nonlinear probability treatment.
2.1 Notation
Let X denote the nonempty set of outcomes. A prospect is a finite probability distribution over X. Prospects are labeled as \( P=(p_{1}:x_{1},\ldots ,p_{n}:x_{n})\) with the usual interpretation that outcome \(x_{j}\in X\) is obtained with probability \(p_{j}\), for \(j=1,\ldots ,n \). Naturally, \(p_{j}\ge 0\) for each \(j=1,\ldots ,n\) and \( \sum _{i=1}^{n}p_{i}=1\). Let \({\mathcal {L}}\) denote the set of all prospects.
A preference relation, denoted \(\succcurlyeq ,\) is assumed over \( {\mathcal {L}}\). Its restriction to subsets of \({\mathcal {L}}\) (e.g., all degenerate prospects where one of the outcomes is received for sure) is also denoted by \(\succcurlyeq \). The symbol \(\succcurlyeq \) means “weak preference” from which \(\succ \) (strict preference) and \(\sim \) (indifference) are defined as usual. The function Vrepresents (or is a representation of) the preference \(\succcurlyeq \) on \({\mathcal {L}}\), if V assigns a real value to each prospect, such that for all \(P,Q\in {\mathcal {L}}\) we have \(P\succcurlyeq Q\Leftrightarrow V(P)\ge V(Q)\). This general representation V will be required to satisfy several properties including those that reflect the behavior corresponding to the treatment of probabilities.
Next, we recall the functional expressions of expected utility, rankdependent utility and prospect theory, which are specific representations of the preference \(\succcurlyeq \) on \({\mathcal {L}}\). In all these models a utility function, u, for outcomes exists that is strictly monotonic (that is, \(u:X\rightarrow {\mathbb {R}}\) satisfies \( u(x_{i})\ge u(x_{j})\Leftrightarrow x_{i}\succcurlyeq x_{j}\)). As a result, outcomes that are indifferent receive the same utility value. To simplify the exposition we henceforth assume, without loss of generality, that no two distinct outcomes in X are indifferent. This allows us to strictly rank outcomes from best to worst within a prospect; this particular ordering of outcomes is meaningful in rankdependent models.
2.1.1 Expected utility
2.1.2 Rankdependent utility
2.1.3 Prospect theory
 if all outcomes in P are (weakly) preferred to the reference point (i.e., we have no losses), then P is evaluated by$$\begin{aligned} \mathrm{PT}(P)=\mathrm{RDU}^{+}(P)\text { using the weighting function }w^{+}\text {;} \end{aligned}$$
 if all outcomes in P are (weakly) dispreferred to the reference point (i.e., we have no gains), then P is evaluated by$$\begin{aligned} \mathrm{PT}(P)=\mathrm{RDU}^{}(P)\text { using the weighting function }w^{}\text {;} \end{aligned}$$
 if P assigns positive probability to both gains and losses, then the PTvalue of P is the sum of P’s gain and loss parts. That is, with \( P^{+} \), the gain part of P, being the prospect “P with all losses replaced by r” and the loss part, \(P^{}\), being the prospect “P with all gains replaced by r ,” the PTvalue of P is given bywhere the qualification \(u(r)=0\) applies. It is custom for PT to express the treatment of probabilities for losses using the dual weighting function, \({\hat{w}}^{}\). For instance, if \(r=x_{k}\) for some \(2\le k\le n1 \), the PTvalue of P is$$\begin{aligned} \mathrm{PT}(P)=\mathrm{PT}(P^{+})+\mathrm{PT}(P^{}), \end{aligned}$$$$\begin{aligned} \mathrm{PT}(P)= & {} \sum _{j=1}^{k1}w^{+}\left( p_{j}^{d}\right) \left[ u(x_{j})u(x_{j+1})\right] \nonumber \\&\quad +\sum _{j=k+1}^{n}{\hat{w}}^{}\left( p_{j}^{c}\right) \left[ u(x_{j})u(x_{j1})\right] . \end{aligned}$$(3)
Except for Schmidt and Zank (2012), all existing foundations for PT assume the reference point as given from the outset; hence, also the uniqueness of the reference point is assumed. Here we drop the assumption of knowing the reference point in advance and we also dispense of structural assumptions for outcomes (e.g., requiring outcomes to be real valued) that are usually imposed for obtaining PTfoundations. Therefore, in our uniqueness results we explicitly state that for PT it is the signdependence that implies the uniqueness of the reference point. This is indeed the feature that distinguishes PTpreferences from RDUpreferences and, hence, from EUpreferences. The corollary of this observation is that, in principle, signdependence can be used to reveal the reference point from behavior. To do this, we employ preference conditions that build on revealed or elicited probability midpoints. Next, we present this tool, and we look at the implication of midpoints for the just presented models.
2.2 Probability midpoints
The representations introduced in the preceding subsection make specific predictions for probability midpoints. In particular, they all imply that probability midpoints are, with modelspecific qualifications, independent of the gauges used to reveal those midpoints. We explore these implications next.
2.2.1 Modelspecific midpoints
3 Additive representation
The decision models presented in the preceding section share several properties that, when combined, imply an additive representation over prospects. For our specific framework these properties of preferences have been presented before (e.g., in Zank 2010); corresponding properties for general rankordered sets were provided in Wakker (1993). For completeness we recall these properties here and summarize their implications for preferences in a lemma. This also allows us to highlight some potential difficulties that are particular to PTpreferences. Since it simplifies the exposition, we assume that the set of outcomes is finite, i.e., \( X=\{x_{1},\ldots ,x_{n}\}\) for some natural number n. In Sect. 5 we indicate how our results can be extended to infinite sets of outcomes where, in contrast to the finite outcome case, preferences may be signdependent, but the reference point need not be included in the outcome set. Corresponding examples are then provided.
3.1 Traditional preference conditions

Weak Order: The preference relation satisfies completeness and transitivity.

Dominance: The preference relation satisfies dominance (or monotonicity in decumulative probabilities) if \( P\succ Q\) whenever P firstorder stochastically dominates Q.

Continuity: The preference relation \(\succcurlyeq \) satisfies Jensencontinuity on the set of prospects \({\mathcal {L}}\) if for all prospects \(P\succ Q\) and R there exist \(\rho ,\mu \in (0,1)\) such that \( \rho P+(1\rho )R\succ Q\) and \(P\succ \mu R+(1\mu )Q\).
3.2 Additive separability properties
This subsection presents an independence property that is shared by EU, RDU and PT. It is formulated as a preference condition involving common shifts of probability mass between outcomes. We have informally used shifts in probabilities when defining elicited probability midpoints. Probability shifts can be regarded as substitutions of an outcome with probability \( \varepsilon \) by a different outcome with that very same probability. Given prospect \(P=(p_{1}:x_{1},\ldots ,p_{n}:x_{n})\) we denote the prospect resulting from a shift of probability \(\varepsilon \) from outcome \(x_{i}\) to outcome \(x_{j}\) in P as the prospect \(\varepsilon _{i,j}P:=(p_{1}^{\prime }:x_{1},\ldots ,p_{n}^{\prime }:x_{n})\), with \(p_{i}^{\prime }=p_{i}\varepsilon ,p_{j}^{\prime }=p_{j}+\varepsilon ,\) and \(p_{m}^{\prime }=p_{m}\) for \(m\notin \{i,j\}\). Whenever we use this notation, it is implicitly assumed that \(p_{i}\ge \varepsilon >0\) to ensure that \( \varepsilon _{i,j}P\) is a welldefined prospect in \({\mathcal {L}}\).
 Sure Thing Principle for Risk: The preference relation \( \succcurlyeq \) satisfies the sure thing principle (STP) for risk ifwhenever \(P,Q,\varepsilon _{i,i+1}P,\varepsilon _{i,i+1}Q\in {\mathcal {L}}\).$$\begin{aligned} P\succcurlyeq Q\Leftrightarrow \varepsilon _{i,i+1}P\succcurlyeq \varepsilon _{i,i+1}Q, \end{aligned}$$
 Comonotonic STP: The preference relation \(\succcurlyeq \) satisfies the comonotonic sure thing principle (CSTP) for risk ifwhenever \(P,Q,\varepsilon _{i,i+1}P,\varepsilon _{i,i+1}Q\in {\mathcal {L}}\) are such that \(p_{i}^{d}=q_{i}^{d}\).$$\begin{aligned} P\succcurlyeq Q\Leftrightarrow \varepsilon _{i,i+1}P\succcurlyeq \varepsilon _{i,i+1}Q, \end{aligned}$$
When CSTP is combined with the preference conditions in the preceding subsection, it implies an additive separability property across outcomes for the representing function V. The result, formally stated here, generally applies to the preference restricted to all prospects except perhaps the best prospect (\(x_{1}\)) and the worst prospect (\(x_{n}\)); the proof follows from Wakker’s (1993, Theorem 3.2) result for additive representations on comonotonic sets.
Lemma 1
 (i)The preference relation \(\succcurlyeq \) on \({\mathcal {L}}\backslash \{x_{1},x_{n}\}\) is represented by an additive functionwith strictly increasing functions \(V_{1},\ldots ,V_{n1}:\) [0, 1] \( \rightarrow \)\({\mathbb {R}}\) which are continuous and bounded with the exception of \(V_{1}\) and \(V_{n1}\), which could be unbounded at 1 and at 0, respectively.^{6}$$\begin{aligned} V(P)=\overset{n1}{\sum _{j=1}}V_{j}(p_{j}^{d}), \end{aligned}$$(4)
 (ii)
The preference relation \(\succcurlyeq \) is a Jensencontinuous weak order that satisfies dominance and the comonotonic sure thing principle for risk.
As shown in Wakker (1993, Proposition 3.5), by adding further preference conditions that imply a separation of the treatment of probabilities from the utility value assigned to outcomes, one can generally obtain boundedness of all functions in Lemma 1. This has been exploited, for instance, in the RDUderivations in Diecidue et al. (2009) and in Webb and Zank (2011). For PT this also applies, provided that there are at least two gains and two losses, whence proportionality arguments can be exploited (see our Theorem 2). As we do not make assumptions about which outcome is the reference point, we cannot, in general, invoke information about the number of gains and losses from the outset. This means that the potential unboundedness reported in Lemma cannot a priori be excluded. Consequently, this leads to difficulties for deriving a standard PTrepresentation, as we discuss next.
3.3 Extended prospect theory
Based on the additive representation of Lemma 1, in this subsection we further explore the consequences for obtaining PTfoundations when one of \(V_{1}\) or \(V_{n1}\) is unbounded. In general, such unboundedness precludes the identification of probability weighting functions as is required for PT. Our main result shows that we are able to obtain representations for “extended PTpreferences” where a reference point can nonetheless be identified and, further, a utility for losses and a corresponding probability weighting function (if \(V_{1}\) is unbounded), or a utility for gains with the associated probability weighting function (if \(V_{n1}\) is unbounded), can still be derived. In particular, at most one of \(V_{1}\) or \( V_{n1}\) can be unbounded. We provide specific examples to illustrate these extreme cases as they highlight specific types of preferences that are plausible when general outcome sets and referencedependence are jointly considered. In the foundational literature on PT such behavior has hitherto been excluded due to the structural requirements on the set outcomes and the properties reflected in the corresponding utility functions.
Example 1
Assume that the representation of Lemma 1 holds for \(X=\{x_{1},x_{2},x_{3}\}\) with \(x_{1}\succ x_{2}\succ x_{3}\) and that \(V_{1}\) and \(V_{2}\) are bounded. Then, the representation is a PTfunctional with \(r=x_{2}\) as the reference point, unless \(\succcurlyeq \) is represented by RDU.
If \(w^{+}(p)=w^{}(p)\) for all \(p\in [0,1]\), one can show that Eq. (5) reduces to an RDUrepresentation with the corresponding uniqueness results. For the case that \(w^{+}\ne w^{} \), one can show that the values \(V_{1}(0)\) and \(V_{2}(1)\) are constants that are immaterial for the ranking of prospects, such that the representation in Eq. (5) is indeed PT with the corresponding uniqueness results; this aspect is further elaborated on in the Appendix.
For the case of Example 1, we are not aware of a preference condition, that explicitly identifies \(x_{2}\) as a reference point in a manner that pins down PT with a ratioscale utility, beyond the properties that imply additive separability as required in Lemma 1.^{7} More generally, we observe that it is only in the case of \(n=3\) that a reference point may exist, but that both \(V_{1}\) and \( V_{n1}(=V_{2})\) in Lemma 1 are unbounded. Such unboundedness precludes a separation of utility and signdependent probability weighting functions. Including such representations under a general notion of PTpreferences means that Lemma 1 already gives necessary and sufficient conditions for a corresponding representation. Given this observation, from here onward we require that X has at least four strictly rankordered outcomes. The next two examples show that the assumption of more than three outcomes restricts, but does not eliminate unboundedness for a representation in the presence of referencedependence.
Example 2
One can regard the preference in Example 2 as that of a patient who has been diagnosed with a severe disease, such as cancer. Suppose some potential medical interventions can lead to a range of outcomes, the best being \(x_{1}=\)“fully cured from cancer,” while other interventions may prolong life duration but do not offer positive probability for \(x_{1}\). It is conceivable that the latter interventions are all perceived as leading to losses and, hence, they are perceived unattractive relative to an intervention with a positive probability for \(x_{1}\). A related example is documented in Thaler and Johnson (1990) and analyzed in Barberis et al. (2001). After having faced a series of losses, many investors attempt to break even by engaging in very risky trades, despite the chances of breaking even being relatively small. Such investors appear to perceive the event of breaking even as leading to an extremely attractive outcome, while the complementary event is viewed as leading to losses of a tolerable magnitude in utility terms. The counterpart of Example 2 is the following one.
Example 3
The functional in Example 3 can be thought of as a representation for a preference with an extreme form of aversion or pessimism, where the possible loss \(x_{n}\) is extremely unattractive (e.g., ruin) and any prospect with a positive probability for \(x_{n}\) will be regarded inferior to a prospect with zero probability for \(x_{n}\). Individuals exhibiting this form of pessimism are willing to buy insurance at prices far above the actuarially fair value to completely avoid the loss \( x_{n}\). Such behavior is reported in relation to substantially increased demand for flood and earthquake insurance after a corresponding event has occurred (Kunreuther et al. 1978; Palm 1995)^{8} and in the willingness to pay for the complete elimination of risk associated with a hazardous product (Viscusi et al. 1987).
Having elaborated on potential issues for deriving PT when the set of outcomes includes a single gain or a single loss, we proceed by keeping in mind that in such special cases extreme sensitivity in probabilities of best or worst outcomes may preclude a derivation of PT in which both probability weighting functions are uniquely specified. Instead, we may obtain what we term extended prospect theory: Preferences are either represented by PT on \({\mathcal {L}}\), or they are represented on \({\mathcal {L}}\backslash \{x_{1}\}\) by the functional in Eq. (6) with \( V_{1}:[0,1)\rightarrow [0,\infty )\) unbounded at \(p=1\), or they are represented on \({\mathcal {L}}\backslash \{x_{n}\}\) by Eq. (7) with \(V_{n1}:(0,1]\rightarrow (\infty ,0]\) unbounded at \( p=0 \). As the preference conditions presented in the next section show, in such cases we can still identify the reference point from behavior if preferences are referencedependent.
4 Consistent probability midpoints
We have already established that for EUpreferences probability midpoints are algebraic midpoints, that they are independent of outcomes and, hence, reference point independent. Similarly, RDUpreferences imply reference independence although, due to flexibility in the treatment of probabilities, RDU allows probability midpoints to differ from algebraic midpoints. PTpreferences are different. They allow for elicited probability midpoints to depend on the type (i.e., gain or loss), but not the magnitude of outcomes.
The preceding analysis shows that revealed probability midpoints elicited through shifts of probability mass to \(x_{1}\) are meaningful for \(V_{1}\), i.e., they are consistent and independent of stimuli other than \(x_{1}\). Similarly, midpoints for each function \(V_{j},j=2,\ldots ,n1\) can be elicited and are meaningful concepts given the additive separable representation of Lemma 1. Next we present properties that demand further consistency for elicited probability midpoints.
4.1 Goodnews and badnews midpoint consistency
 GoodNews Midpoint Consistency: For \(m\in \{2,\ldots ,n1\}\), the preference relation \(\succcurlyeq \) satisfies goodnews midpoint consistency (GMC) above \(x_{m}\), if for \(P=(\alpha :x_{1},p_{m}\alpha :x_{m},p_{m+1}:x_{m+1},\ldots ,p_{n}:x_{n})\) and \(Q=(\beta :x_{1},q_{m}\beta :x_{m},q_{m+1}:x_{m+1},\ldots ,q_{n}:x_{n})\) we havefor all \({\tilde{m}}\in \{1,\ldots ,m\}\) whenever \(\alpha \le \beta \le \gamma \) are probabilities such that \(P,Q,(\beta \alpha )_{m,1}P,\) and \( (\gamma \beta )_{m,1}Q\) are from \({\mathcal {L}}\).^{9}$$ \begin{aligned} P\sim Q \& (\beta \alpha )_{m,1}P\sim (\gamma \beta )_{m,1}Q\Rightarrow (\beta \alpha )_{m,{\tilde{m}}}P\sim (\gamma \beta )_{m, {\tilde{m}}}Q, \end{aligned}$$
It can be verified that RDU satisfies GMC above \(x_{m}\) for all \(m=2,\ldots ,n1\). This has been shown in Zank (2010). Similarly, this holds for the preference with the representation of Example 3. The next lemma shows that PT satisfies GMC above \(x_{m}\) if all outcomes above \(x_{m}\) are gains, hence, in particular if \(x_{m}\) is the reference point.
Lemma 2
Assume that the preference \(\succcurlyeq \) on \( {\mathcal {L}}\) is represented by PT (or by extended PT on \({\mathcal {L}} \backslash \{x_{n}\})\). Then \(\succcurlyeq \) satisfies goodnews midpoint consistency above x whenever all outcomes ranked above x are gains.
 BadNews Midpoint Consistency: For \(m\in \{2,\ldots ,n1\}\), the preference relation \(\succcurlyeq \) satisfies badnews midpoint consistency (BMC) below \(x_{m}\), if for \(P=(p_{1}:x_{1},\ldots ,p_{m1}:x_{m1},p_{m}\alpha :x_{m},\alpha :x_{n})\) and \( Q=(q_{1}:x_{1},\ldots ,q_{m1}:x_{m1},q_{m}\beta :x_{m},\beta :x_{n})\) we havefor all \({\tilde{m}}\in \{m,\ldots ,n\}\) whenever \(\alpha \le \beta \le \gamma \) are probabilities such that \(P,Q,(\beta \alpha )_{m,n}P,\) and \( (\gamma \beta )_{m,n}Q\) are from \({\mathcal {L}}\).$$ \begin{aligned} P\sim Q \& (\beta \alpha )_{m,n}P\sim (\gamma \beta )_{m,n}Q\Rightarrow (\beta \alpha )_{m,{\tilde{m}}}P\sim (\gamma \beta )_{m, {\tilde{m}}}Q, \end{aligned}$$
Lemma 3
Assume that the preference \(\succcurlyeq \) on \( {\mathcal {L}}\) is represented by PT (or by extended PT on \({\mathcal {L}} \backslash \{x_{1}\})\). Then \(\succcurlyeq \) satisfies badnews midpoint consistency below x whenever all outcomes ranked below x are losses.
If we have referencedependence and we know the location of the reference point, then it is easy to formulate a signdependent midpoint consistency condition that characterizes PT when combined with the properties in Lemma 1. If there are two or more gains and two or more losses, all we need is GMC above all outcomes that are not losses and BMC below all outcomes that are not gains. Although not stated formally, this result is new and provides PTfoundations for general sets of outcomes, thereby directly extending the RDU foundations of Nakamura (1995), Abdellaoui (2002), Abdellaoui and Wakker (2005), and Zank (2010). In the case that there is one gain or one loss, we further need to assume that the additive representation consists of bounded functions; alternatively, the domain of the preference is limited to all but the best and worst prospects, in which case we obtain foundations for extended PT. Next we proceed without assuming that we know the location of the reference point.
4.2 Reference point revealing midpoint consistency

Reference Point Revealing Midpoint Consistency: The preference relation \(\succcurlyeq \) satisfies reference point revealing midpoint consistency (RMC) if for each \(m\in \{2,\ldots ,n1\}\) the preference satisfies goodnews midpoint consistency above \(x_{m}\) or badnews midpoint consistency below \(x_{m1}\) (or both).
Theorem 1
 (i)
The preference relation \(\succcurlyeq \) is represented by extended PT.
 (ii)
The preference relation \(\succcurlyeq \) is a Jensencontinuous weak order that satisfies dominance, the comonotonic sure thing principle for risk and reference point revealing midpoint consistency.
The proof of the preceding theorem is in the Appendix. The next section looks at extensions of our main result.
5 Extensions
In the previous sections we have assumed that no outcomes are indifferent. This requirement can be relaxed if there are at least four strictly ordered outcomes in the finite set X. All results remain valid if we restrict the analysis to the set of representatives for each indifference set of outcomes; within an indifference set all outcomes have the same utility value. As pointed out in Sect. 3, our results also remain valid if we include the case that there are exactly three strictly ordered outcomes, however, then RMC trivially holds and, given Lemma 1 with boundedness conditions for the additive representation satisfied, PT follows as indicated in Example 1. If boundedness conditions do not hold one must allow for extended PTrepresentations as suggested in Examples 2 and 3 , where the utility of the single gain, respectively, the single loss and the corresponding probability weighting function cannot be identified separately as the representing functions exhibit asymptotic behavior when approaching extreme probabilities. Such behavior is excluded if preferences agree with RDU, the special case of PT with sign independence. Finally, we recall that, in the trivial case of having at most two strictly ordered outcomes, the dominance property ensures the existence of an ordinal representation; it is well known that there is insufficient structure on the set of prospects to obtain more meaningful results for the twooutcome case.
In our derivation of PT it has been essential that the weighting functions are continuous at 0 and at 1. Discontinuities at these extreme probabilities are, however, empirically meaningful. We could adopt a weaker version of continuity for probabilities that is restricted to prospects that have common best and worst outcomes each with a positive objective probability. Such conditions have been used in Cohen (1992) and more recently in Webb and Zank (2011) where probability weighting functions are derived that are linear and discontinuous at extreme probabilities. These weighting functions can then be described by two parameters. As Webb and Zank show, this relaxation of continuity in probabilities comes at a price. They require additional structural assumptions for the preference in order to obtain consistency of those parameters across sets of prospects with different worst and best outcomes. Also, specific consistency principles that imply the uniqueness of those parameters are required. We conjecture that in our framework such consistency principles can be formulated for nonlinear weighting functions that are discontinuous at 0 and at 1. A formal derivation of PT with such weighting functions is, however, beyond the scope of this paper.
5.1 Omitting asymptotic behavior

Bounded RMC: The preference relation \(\succcurlyeq \) satisfies bounded RMC (bRMC) if RMC holds and further GMC holds above \(x_{3}\) and BMC holds below \(x_{n2}\).
If the set of outcomes is too small, declaring some outcomes as gains and some as losses may in fact pin down the model. For example, it should be clear that if there are only four strictly rankordered outcomes in X, then bRMC implies that GMC holds above \(x_{3}\) and BMC holds below \(x_{2}\). For the additive representation in Lemma 1 this has the implication that probability midpoints for \(V_{1}\) are also midpoints for \(V_{2}\) (inferred from GMC above \(x_{3}\)) and probability midpoints for \(V_{3}\) are also midpoints for \(V_{2}\) (inferred from BMC below \(x_{2}\)). Hence, midpoints are consistent throughout and this excludes signdependence; hence, RDU is implied.
The next result explicitly assumes that there are at least five strictly ordered outcomes, to allow for referencedependence when bRMC is invoked.
Theorem 2
 (i)
The preference relation \(\succcurlyeq \) is represented by PT with either signdependence (in which case the reference point is \(r=x_{k}\) for some \(k\in \{3,\ldots ,n2\})\) or sign independence (RDU holds).
 (ii)
The preference relation \(\succcurlyeq \) is a Jensencontinuous weak order that satisfies dominance, the comonotonic sure thing principle for risk and bounded reference point revealing midpoint consistency.
The proof of the preceding result is in the Appendix. In Theorem we exploit the fact that the set of outcomes is finite. If X is infinite a property like bRMC cannot simply be extended because, for example, for uncountable sets identifying the two best outcomes or the two worst outcomes may not be possible. But RMC can be formulated to hold on specific subsets of prospects as our next subsection shows. We now proceed to the discussion of how to obtain extended PTfoundations for infinite outcome sets.
5.2 Infinite outcome sets

Comonotonic STP: The preference relation \(\succcurlyeq \) satisfies the comonotonic sure thing principle (CSTP) on \( {\mathcal {L}}\) if \(\succcurlyeq \) satisfies CSTP on \({\mathcal {L}}_{Y}\) for all finite sets \(Y\subset X\).
 Extended RMC: The preference relation \(\succcurlyeq \) satisfies extended reference point revealing midpoint consistency (eRMC) on \({\mathcal {L}}\) if for each outcome \(x\in X\) one (or both) of the following statements apply:
 (a)
For each finite set Y with all outcomes ranked above x the preference \(\succcurlyeq \) restricted to \({\mathcal {L}}_{Y\cup \{x\}}\) satisfies GMC above y and BMC below y for each \(y\in Y\).
 (b)
For each finite set Z with all outcomes ranked below x the preference \(\succcurlyeq \) restricted to \({\mathcal {L}}_{\{x\}\cup Z}\) satisfies BMC below z and GMC above z for each \(z\in Z\).
 (a)
Example 4
The preference in Example 4 satisfies all properties required for the existence of a general additive representation (weak order, Jensencontinuity, dominance, CSTP) and also eRMC. However, the value 0, which acts as a reference point, is not an outcome that is contained in the set X. Hence, the preference is signdependent, but no reference point within X exists; the reference point is “outside the model.”
Demanding convexity for the outcome set can circumvent the problems alluded to in Example 4. But even if the set of outcomes is a closed interval, we may have signdependent preferences but no reference point. This can be inferred from the following example that resembles features from Example 3.
Example 5
To exclude preferences that are signdependent, but where a reference point cannot be identified, such as the preferences in Examples 4 and 5 one needs to add further structural assumptions for the set of outcomes or exclude extreme outcomes (e.g., requiring that X is an open interval in \({\mathbb {R}}\)). Alternatively, one can consider preferences where the reference point is outside the model as in the extended PT of Example 5 with an “imaginary” reference point \(r^{*}\). We provide foundations for both cases. The first result provides a preference foundation for PT for the case that the set of outcomes is an open interval of the real numbers. As the set of outcomes is a connected separable topological space that does not contain a best or worst outcome, the reference point can be identified within the model.
Theorem 3
 (i)
The preference relation \(\succcurlyeq \) on \({\mathcal {L}}\) is represented by PT.
 (ii)
The preference relation \(\succcurlyeq \) is a Jensencontinuous weak order that satisfies dominance, the comonotonic sure thing principle for risk and extended reference point revealing midpoint consistency.
The proof of Theorem 3 is in the Appendix. From that proof one can infer that Theorem 3 also applies for the case that \(X={\mathbb {R}}\), which is the most frequent assumption considered in the literature. The next result applies to the most general case in which no best and worst outcomes are allowed.
Theorem 4
 (i)
The preference relation \(\succcurlyeq \) on \({\mathcal {L}}\) is represented by PT with a possibly imaginary reference point.
 (ii)
The preference relation \(\succcurlyeq \) is a Jensencontinuous weak order that satisfies dominance, the comonotonic sure thing principle for risk and extended reference point revealing midpoint consistency.
The proof of Theorem 4 is in the Appendix. Having explored how the probability midpoint tool can be used to obtain information about signdependence and reference points from a foundational perspective, we proceed to a practical application of midpoint consistency. The next subsection indicates how the midpoint tool can be applied to empirically test for signdependence.
5.3 Detecting reference points empirically
Our theoretical results, in particular the application of GMC and BMC as combined in RMC suggests that it is possible to test for signdependence using probability midpoints. Here we present a tool that can be used to experimentally implement such a test. Suppose, for simplicity of exposition, that we have the best outcome, labeled \(G\in X\) (which may objectively be seen as a gain), and the worst outcome, labeled \(L\in X\) (potentially regarded as a loss). Let there be finitely many outcomes ranked between G and L, say \(G\succ y_{1}\succ \cdots \succ y_{s}\succ L\) for some \(s\ge 2\). Assume that this list of intermediate outcomes is exhaustive and that we have PTpreferences, but do not know if one of the \(y_{j}\)’s is the reference point. An example of a procedure to identify the reference point involves repeated elicitations of probability midpoints and subsequent checks for consistency for those midpoints. This results in an algorithm searching for a reference point, as follows.
Step 1: Fix a small probability \(\alpha \) (e.g., \(\alpha =0.15\)) and probabilities p, q (e.g., \(p=0.1,q=0.2\)). Next consider the prospect \( P_{y_{1}}^{\alpha }=(\alpha :G,1p\alpha :y_{1},p:L)\) and elicit the probability \(\beta \) that makes a subject indifferent between \( P_{y_{1}}^{\alpha }\) and \(Q_{y_{1}}^{\beta }=(\beta :G,1q\beta :y_{1},q:L)\) . Such elicitations can be facilitated by using, for instance, choice lists in which outcomes are fixed and probabilities vary, as suggested in Holt and Laury (2002).
Step 2: Replace \(\alpha \) by \(\beta \) in \(P_{y_{1}}^{\alpha }\), hence obtaining \(P_{y_{1}}^{\beta }\), and elicit \(\gamma \) such that \( P_{y_{1}}^{\beta }\sim Q_{y_{1}}^{\gamma }\). This way the experimenter obtains a probability midpoint \(\beta \) between \(\alpha \) and \(\gamma \).
Step 3: Next, replace \(y_{1}\) by \(y_{2}\) in \(P_{y_{1}}^{\alpha }\) and obtain the prospect \(P_{y_{2}}^{\alpha }=(\alpha :G,1p\alpha :y_{2},p:L)\). Subsequently, elicit the probability \({\tilde{q}}\) that makes a subject indifferent between \(P_{y_{2}}^{\alpha }\) and \(Q_{y_{2}}^{\beta }=(\beta :G,1{\tilde{q}}\beta :y_{2},{\tilde{q}}:L)\).
Step 4: As in Step 2, replace \(\alpha \) by \(\beta \) in \( P_{y_{2}}^{\alpha }\), giving \(P_{y_{2}}^{\beta }\), and elicit \(\gamma ^{*}\) such that \(P_{y_{2}}^{\beta }\sim Q_{y_{2}}^{\gamma ^{*}}\). The experimenter obtains a probability midpoint \(\beta \) between \(\alpha \) and \( \gamma ^{*}\). If \(\gamma =\gamma ^{*}\) we have observed a consistency, from which we conclude that \(y_{1}\) and \(y_{2}\) are outcomes with utilities of the same sign; we proceed to the next step. Otherwise, if \( \gamma \ne \gamma ^{*}\) we have an inconsistency, which can occur only if \(y_{1}\) and \(y_{2}\) have utilities of a different sign. Therefore, \(y_{1}\) is identified as the reference point and the “search algorithm” stops.
Step \((3+m)\) (\(m=2,\ldots ,s\)): If \(\gamma =\gamma ^{*}\) at Step \((2+m)\), repeat Steps \((1+m)\) and \( (2+m) \), with \(y_{m+1}\) and \(y_{m}\) replacing \(y_{m}\) and \(y_{m1}\), respectively.
If this procedure terminates after Step \((3+m^{*})\) for some index \( m^{*}\le s\), we conclude that \(y_{m^{*}}\) is the reference point; otherwise, there is no reference point that affects the treatment of probabilities.
A few comments on the above procedure are in order. Obviously, there are alternative ways of implementing the above search procedure. For instance, one can start the procedure in Steps 1 and 2 at any outcome \(y_{m},m\in \{1,\ldots ,s\}\) and adjust the subsequent elicitation steps, or one could elicit midpoints using outcome \(y_{s}\) in Steps 3 and 4 in the above procedure and then continue the elicitation of midpoints by alternation between the remaining best and worst ranked outcomes that have not yet been identified as gains or losses by the procedure. If X is an open interval in \({\mathbb {R}}\), one can use this procedure to narrow down the interval of outcomes in which the reference point is located by repeating the procedure on specific finite subsets Y of X.
While the above search procedure appears compelling and, from a theoretically perspective, feasible, implementing the procedure in experiments would need to account for some practicalities. First, the probability interval is narrow such that small changes in stimuli in the form of actual probabilities may hardly be noticed by subjects. This can be circumvented by “scaling up” the stimuli, e.g., by framing choices as events resulting from draws using urns containing 100 or even 1000 equally likely balls. Second, the midpoint procedure is based on eliciting indifferences. Irrespective of whether indifferences are elicited by varying outcomes or by varying probabilities, such elicitation tasks are cognitively demanding for many subjects and appropriate experimental procedures are needed. The aforementioned choice lists design (Holt and Laury 2002), which invokes an final interpolation step, has proven to be quite an efficient mechanism in dealing with the issue of eliciting indifferences.
Third, the decision criterion in how far to tolerate differences between \( \gamma \) and \(\gamma ^{*}\) in Step 4 of the above elicitation procedure usually needs to be specified as a rule where small differences can be regarded as a measurement error and large ones as a genuine inconsistency. What determines such bounds is essentially an empirical question, and setting appropriate thresholds can be based on existing data regarding empirically observed probability weighting functions. Fourth, chained measurements, as employed in our search procedure, have been criticized on the grounds of incentive compatibility and error propagation. Both aspects of the elicitation procedure are theoretically important, but empirically, these issues are not a serious concern as subjects treat choice tasks in isolation (Kahneman and Tversky 1979; Cubitt et al. 1998; see also Abdellaoui et al. 2005, on negligible error propagation).
Finally, in contrast to EU or RDUpreferences, the choice of outcome stimuli to detect signdependence is important. The experimenter needs to ensure that the range of outcomes chosen to implement the above procedure is not too narrow (such that reference points are excluded) and likewise that the number of stimuli is not too large as this raises the number of required elicitations. This calls for a tradeoff between precision of the method and the cognitive demands put on subjects that, in turn, can influence the precision in the obtained data. Finding the right balance is, however, a common challenge for all experimental studies.
6 Discussion
Here, we review some of the literature on endogenous reference points before we comment on the relation of our midpoint principle to midpoint notions developed elsewhere.
6.1 Models with reference points
The majority of existing PTderivations assume the reference point is exogenously given (e.g., Tversky and Kahneman 1992; Wakker and Tversky 1993; Chateauneuf and Wakker 1999; Köbberling and Wakker 2003; Neilson 2006). In alternative models, attention has been paid to endogenous reference points that are choicedependent (e.g., Gul 1991; Sugden 2003; Delquié and Cillo 2006; Bleichrodt 2007; Schmidt et al. 2008). Such multiple reference points are explicitly allowed for in the referencedependent theories of Munro and Sugden (2003) and Sagi (2006) where, motivated by the status quo effect, adjustments of preferences to new reference points are permitted. There the decision maker can be seen as having multiple preferences, each depending on a reference point. Those preferences are required to be consistent in the sense that they do not generate cyclical choices. Such consistency requirements for behavior are also appearing in theories that build on the classical revealed preference approach, however, by using choice functions that are referencedependent, such as in Apesteguia and Ballester (2009) and Bossert and Sprumont (2009).
Multiple reference points can also be found in the choice model of Ok et al. (2015). Those reference points are feasible alternatives in a choice set, but they are always dominated by some other alternatives and, hence, are never revealed preferred. By contrast, the endogenous reference points in Shalev (2000, 2002), Kőszegi and Rabin (2006, 2007), and Kőszegi (2010), correspond to a person’s rational expectations held in the recent past, which in turn are determined in the socalled personal equilibrium. Since there may be multiple equilibria, the decision maker is required to choose the most preferred one, e.g., a preferred personal equilibrium in Kőszegi and Rabin (2006) and Kőszegi (2010). Beyond the lack of uniqueness, the choice aspect is markedly different to the reference point concept in PT where, as pointed out by Shleifer (2012, p.1086), the reference point cannot be chosen deliberately.
Schmidt and Zank (2012) provide an alternative way to identify reference points from primitives by exploiting PT features, such as diminishing sensitivity of the utility (convexity for losses and concavity for gains) and consistent utility measurement paired with signdependence. The present approach is complementary to Schmidt and Zank (2012), and, unlike theirs, it does not impose structural richness on the set of outcomes. As a result, the present foundations for PT can be extended to more general settings like health and insurance where outcomes might be discrete, thus allowing for wider applications of PT.
6.2 Outcome and probability midpoints
Consistency requirements for outcomes are familiar in economics and finance and are commonly used for utility measurements or for comparative analyses. Specifically, the shape of utility functions can be inferred from preferencebased outcome midpoints (Baillon et al. 2012, Theorem 2.2), where a strictly concave utility requires that the utility midpoint of two outcomes is always below the corresponding algebraic midpoint and, further, independent of the probabilities of those outcomes. Similarly, consistently lower midpoints indicate more concavity of one utility relative to the other. The utility midpoint tool, which applies likewise to expected utility and nonexpected utility theories, has been advanced further in Baillon et al. (2012) to compare the utility for risk with that for ambiguity.
Similarly to the comparison of utility functions based on outcome midpoints, attitudes toward probabilities can be inferred from comparisons of the probability weighting functions for risk and ambiguity (Abdellaoui et al. 2011) by adopting an analogous midpoint technique for probabilities. In Kuilen and Wakker (2011) it was demonstrated that probability midpoints are suitable for experimental studies and their elicitation is relatively easy. In empirical research, however, utility measurements were required prior to the elicitation of probability weighting functions (Abdellaoui 2000; Bleichrodt and Pinto 2000, and Kuilen and Wakker 2011). Our elicitation method and the derived results suggest that one can completely dispense of utilitybased measurements when employing the probability midpoint tool. Indeed, we advance the probability midpoint tool in three different ways: Our PTfoundation delivers in one stroke a new preference tool for experimental testing and empirical measurements, a tool for comparative analyses and, more fundamentally, a method to identify reference points endogenously from behavior.
7 Conclusion
A consequence of referencedependence is that risk behavior in PT is manifested through a combination of attitudes toward gains and losses, captured by a utility function (e.g., concave for gains and convex for losses), and attitudes toward probabilities of gains and losses, captured by corresponding probability weighting functions (e.g., inverseSshaped), which together imply a fourfold pattern of risk attitudes (Tversky and Kahneman 1992; Wu and Gonzalez 1996; Prelec 1998).^{12} Loss aversion, the component of risk attitude expressed as a stronger sensitivity toward losses as compared to equally sized gains (Wakker and Tversky 1993; Neilson 2002; Köbberling and Wakker 2005; Blavatskyy 2011) is a further cornerstone of PT that completes the picture on empirically observed choice behavior under risk. Yet, without knowledge of the location of the reference point, both loss aversion and the fourfold pattern become ambiguous concepts. While a model with an unspecified reference point gives additional flexibility for the analyst (e.g., it leads to an easier organization of experimental and field data, or it allows for framing effects to be incorporated), the absence of behavioral conditions that imply the existence of the reference point renders PT too general and it makes the model difficult to falsify. This is unwarranted both empirically and foundationally.
The importance of having sound preference foundations for decision models, in particular for PT, has recently been reiterated by Kothiyal et al. (2011, pp. 196–197). If a continuous utility is not available because outcomes are discrete (e.g., as in health or insurance), the relationship between the empirical primitive (i.e., the preference relation) and the assumptions of PT becomes unclear. In that case, one can no longer be sure that the PTpredictions are in line with the behavior underlying the preferences.
Our aim was to make PT falsifiable. We have addressed the foundational aspect of the reference point in PT under risk and showed that a specific consistency property for probability midpoints can be formulated in a way that allows for the identification of the reference point and, jointly, of PT. The conditions presented here are necessary and sufficient for PT; thus, they clarify which assumptions one makes by invoking the model. In particular, the new foundations highlight the difference between expected utility, rankdependent utility and prospect theory in a transparent way. Before, this was not possible as foundations for PT in the von Neumann and Morgenstern setting were not available. In the presence of standard preference conditions, the two principles of goodnews and badnews midpoint consistency, when combined as in our RMC properties, are sufficient to obtain either RDU, a special case of PT, or PT with signdependent probability weighting and a reference point endogenously revealed from behavior. This way, we have obtained a complete foundation for prospect theory.
Footnotes
 1.
This elicitation method is similar to the (dual analog) elicitation technique for utility measurement, where standard sequences of equally spaced outcomes on the utility scale are obtained (Wakker and Deneffe 1996). For PTpreferences, probability midpoints are equally spaced on the corresponding probability weighting scales.
 2.
That the EUaxioms were not tied to a specific interpretation of outcomes was also noted by Kahneman and Tversky (1979, p. 264). A similar argument holds for the rankdependent utility model of Abdellaoui (2002) who uses the von Neumann and Morgenstern framework as we do. Although Abdellaoui’s preference conditions allow for a nonlinear treatment of probabilities, they do not restrict the cardinal family of utility functions in any way nor do they require a specific interpretation for outcomes.
 3.
For monetary outcomes, the term “signdependence” is sometimes used to indicate that the utility for gains (i.e., positive outcomes) reveals a different shape than the utility for losses (negative outcomes), e.g., concave versus convex. Here we use the term “signdependence” to indicate that the weighting functions under PT generate different weights for negative utility (from losses) as compared to the weights for positive utility (from gains). There are plenty of studies providing empirical evidence of signdependence, including Edwards (1953, 1954), Tversky and Kahneman (1992), Abdellaoui (2000), or Abdellaoui, l’Haridon et al. (2010).
 4.
There is, however, empirical and theoretical interest in discontinuous weighting functions at 0 and at 1; see Kahneman and Tversky (1979), Birnbaum and Stegner (1981), Bell (1985), Cohen (1992), Wakker (1994, 2001), Chateauneuf et al. (2007), alNowaihi and Dhami (2010), Webb and Zank (2011), Andreoni and Sprenger (2010, 2012); we briefly discuss such potential extensions in Sect. 5. Webb (2017) suggests continuous extensions for a class of piecewise linear probability weighting functions that are empirically indistinguishable from those with discontinuities at 0 and 1 (see also Webb (2015) for event weighting under ambiguity).
 5.
This function may be unbounded when the probability of \(x_{n}\) approaches 0 or the probability of \(x_{1}\) approaches 1.
 6.
As \(p_{n}^{d}=1\) for all prospects, Eq. (4 ) does not need to include the term \(V_{n}(1)\).
 7.
 8.
Etner and Jeleva (2014) explain underinvestment in prevention schemes through the treatment of probabilities as captured in RDU.
 9.
In this definition we could have included the cases \(m=1\) and \(m=n\) as the property then trivially holds.
 10.
In the literature different formulations of the von Neumann and Morgenstern independence axiom have been used, e.g., \(P\succcurlyeq Q\Leftrightarrow \rho S+(1\rho )P\succcurlyeq \rho S+(1\rho )Q\) for all \(P,Q,S\in \mathcal {L }\) and all \(\rho \in [0,1]\). The definition presented here is equivalent given the employed weak order, continuity and dominance properties of \(\succcurlyeq \).
 11.
The idea of formulating preference conditions such that they imply the existence of an outcome where behavior changes suddenly has also been used in Schmidt and Zank (2012).
 12.
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