2 Related work

The task of inducing honest reporting of private information has been extensively studied in the fields of mechanism design and decision theory. My focus in this paper is on the elicitation of private information as subjective probabilities (beliefs) over uncertain outcomes.

Proper scoring rules provide a prominent technique to induce honest reporting of subjective probabilities. Proper scoring rules have been used in a variety of domains, e.g., when sharing rewards amongst a set of agents based on peer evaluations (Carvalho and Larson [2010],Carvalho and Larson [2011],Carvalho and Larson [2012]), when incentivizing agents to accurately estimate their own efforts to accomplish a task (Bacon et al. [2012]), to elicit opinions from policy makers regarding the occurrence of political and economic events (Tetlock [2005]), etc.

A standard assumption when using proper scoring rules is that agents are risk neutral. Focusing on the quadratic scoring rule, Winkler and Murphy [1970] investigated the effects of nonlinear utilities on how agents report their beliefs. More precisely, for some specific utility functions, Winkler and Murphy [1970] showed that a risk-seeking agent reports a very sharp probability distribution, whereas a risk-averse agent reports a probability distribution close to the uniform distribution. Winkler [1969] discussed how any proper scoring rule can be adjusted to an agent’s nonlinear utility function, resulting in what I refer to in this paper as tailored proper scoring rules.

The aforementioned works are still within the expected utility theory framework. Modern theories of choice under uncertainty based on rank-dependent utilities assert that, aside from nonlinear utilities, probability sensitivity also plays a role in defining an agent’s attitude towards uncertainty (Quiggin [1982],Schmeidler [1989]). Focusing on binary outcomes, Offerman et al. [2009] discussed how to calibrate a posteriori beliefs reported under the quadratic scoring rule by agents who take decisions based on rank-dependent utilities. Kothiyal et al. [2011] extended the work by Offerman et al. [2009] to all positive proper scoring rules. Moreover, Kothiyal et al. [2011] briefly mentioned that agents with rank-dependent utilities report vectors of decision weights instead of their true beliefs for the specific case when their utility functions are linear.

I generalize the results of Kothiyal et al. [2011] to any proper scoring rule, any finite number of outcomes, and any strictly increasing utility function. More specifically, I show that, when the utility function of an agent who makes decisions based on a rank-dependent utility is known and incorporated into a proper scoring rule, the agent still misreports his belief by reporting a vector of decision weights. Such reporting behavior happens because probability sensitivity, which is defined in terms of weighting functions, plays a crucial role when an agent reports his belief under a proper scoring rule.

I also show how to elicit weighting functions using tailored proper scoring rules. A popular method for eliciting weighting functions was proposed by Abdellaoui [2000]. Abdellaoui’s method implicitly assumes that agents are honest when reporting indifferences between lotteries. My approach, on the other hand, is based on the reports of beliefs for events with known objective probabilities (decision under risk), and honest reporting maximizes an agent’s rank-dependent utility, thus resulting in a more reliable elicitation process.

3 Proper scoring rules

Consider a set of exhaustive and mutually exclusive outcomesθ₁,θ₂,…,θ_n, for n≥ 2. I assume that agents have beliefs (subjective probabilities) regarding the occurrence of the outcomes. Formally, an agent’s belief is the probability vector p = ( p ₁,…, p _n), where p _k is his subjective probability regarding the occurrence of outcomeθ_k. Agents are self-interested and, consequently, they are not necessarily honest when reporting their beliefs. Therefore, I distinguish between an agent’s true belief p, and his reported belief q = ( q ₁,…, q _n).

Proper scoring rules are traditional devices used to promote honest reporting of subjective probabilities (Winkler and Murphy [1968]). Formally, a scoring rule R( q,θ_x) is a function that provides a score for the reported belief q upon observing the outcomeθ_x. Scores are somehow coupled with relevant incentives, be they social-psychological, such as praise or visibility, or material rewards through prizes or money. A scoring rule is called proper when an agent maximizes his expected score (according to his own beliefs) by reporting a belief q that corresponds to his true belief p (Winkler and Murphy [1968]). A strictly proper scoring rule means that an agent maximizes his expected score if and only if he reports q = p. The expected score of an agent for a real-valued scoring rule R( q,θ_x) is:

(1)

The best known strictly proper scoring rules, together with their scoring ranges, are:

For the sake of illustration, consider a coin toss experiment with two outcomes ( n=2):θ₁ =‶heads" andθ₂ =‶tails". Consider that an agent i has a true belief p = (0.4, 0.6). Assume that agent i reports the belief q = ( q ₁, q ₂), which is rewarded according to the logarithmic scoring rule. Then, agent i’s expected score is E_p [ R( q,·) ] = p ₁log q ₁ + p ₂log q ₂ = 0.4log q ₁ + 0.6log q ₂. In the future, if outcomeθ₁ is the observed outcome, then the score agent i receives is equal to log q ₁. Since the logarithmic scoring rule is a strictly proper scoring rule, agent i’s expected score is strictly maximized when he is honest, i.e., when q = p = (0.4, 0.6). To show this, note that E_p [ R( q,·) ] = p ₁log q ₁ + p ₂log q ₂ = p ₁log q ₁ + (1− p ₁)log(1− q ₁). Since the resulting expected score is concave in q ₁, the value of q ₁ that maximizes agent i’s expected score can be found by taking the first-order derivative of E_p [ R( q,·) ] with respect to q ₁, and equating the result to zero, i.e.:

Selten [1998] and Jose [2009] provided axiomatic characterizations of, respectively, the quadratic scoring rule and the spherical scoring rule in terms of desirable properties, e.g., sensitivity to small probability values, symmetry, etc. In a seminal work, Savage [1971] showed that any differentiable strictly convex function J( q) that is well-behaved at the endpoints of the scoring range can be used to generate a proper scoring rule. Formally:

For example, the logarithmic scoring rule can be derived from J( q) =∑_k=1 ⁿ q _klog q _k:

I say that a scoring rule is positive when all the returned scores are nonnegative, i.e., R( q,θ_x)≥ 0 for all x∈ {1,…, n}. The spherical scoring rule is an example of a positive scoring rule. A negative scoring rule, on the other hand, returns only nonpositive scores, i.e., R( q,θ_x)≤ 0 for all x∈ {1,…, n}. The logarithmic scoring rule is an example of a negative scoring rule. Finally, a mixed scoring rule might return both positive and negative scores. The quadratic scoring rule is an example of a mixed scoring rule.

On a side note, I observe that proper scoring rules not only induce honest reporting of subjective probabilities, but they also measure the accuracy of reported beliefs, a task often called forecast verification. In particular, the more an agent moves probability mass to the observed outcome, the higher the agent’s score will be.

3.1 Tailored proper scoring rules

An implicit assumption in the definition of proper scoring rules is that agents are risk neutral, i.e., they report their beliefs so as to maximize their expected scores. Since _q E_p [ R( q,·) ] = p, a risk-neutral agent has to honestly report his belief under a proper scoring rule R in order to maximize his expected score. Regarding risk neutrality, Savage [1971] said the following in his seminal work about the theoretical foundations of proper scoring rules:

“This assumption is not altogether unobjectionable; for it may imply that the person’s utility function is linear in money. But such linearity assumptions are made almost throughout the present paper and are presumably tolerable if only moderate sums of money are involved.” (Savage [1971], page 791)

In other words, the function that represents the value that an agent derives from a score, called the utility function, is linear with respect to the range of the score used in conjunction with the scoring rule. Theoretically, an agent’s utility function is approximately linear when the stakes are low (Arrow [1971], page 100). In practice, however, human beings’ utility functions tend to become nonlinear when the stakes are high (Wakker [2010],§2).

Expected utility theory tackles some of the problems concerning risk neutrality by assuming that utility functions might be nonlinear. More specifically, the curvature of the utility function determines an agent’s attitude towards uncertainty, e.g., a convex utility function implies that the agent is risk seeking, whereas a concave utility function indicates that the agent is risk averse. Risk-neutral behavior arises only when the utility function is linear. Naturally, agents are assumed to behave so as to maximize their expected utilities.

In the context of proper scoring rules, an agent who behaves according to expected utility theory reports a belief q so that q = _z E_p [ U( R( z,·)) ], where U(·) is the agent’s utility function. Often in this setting, proper scoring rules are no longer proper, i.e., there are cases where (Winkler and Murphy [1970]). Winkler [1969] discussed how the composite function S = U ⁻¹∘ R is a proper scoring rule under a strictly increasing utility function U. That is, the scoring rule S( q,θ_x) is tailored to the agent’s utility function. For example, consider the logarithmic scoring rule R( q,θ_x) = log q _x , and a concave utility function U( y) = log y. Then, the tailored proper scoring rule Footnote ¹ is:

Clearly, tailored proper scoring rules subsume traditional proper scoring rules since the latter assume that utility functions are linear. In the following sections, I study the reporting behavior of agents under tailored proper scoring rules. Thus, an implicit assumption in my analysis is that an agent’s utility function is known a priori, for example, it was previously elicited using an approach such as the tradeoff method (Wakker and Deneffe [1996]). However, I make no assumptions on U, except that it is a strictly increasing function, which implies that there exists an inverse function U ⁻¹ defined over the range of the utility function U.

4 Rank-dependent utility

When selecting and reporting a probability vector q under a tailored proper scoring rule, an agent is essentially taking a decision under uncertainty, where the potential payoffs resulting from his choice are defined by S( q,θ_x), for x∈ {1,…, n}. Consequently, an agent’s reporting behavior can be analyzed from the perspective of different decision theories under uncertainty.

Unarguably, expected utility theory represents a crucial advancement in decision theory under uncertainty. Expected utility theory suggests an elegant and simple way of combining subjective probabilities and payoffs into a single measure of value, which has a number of appealing theoretical properties. However, several violations of the premises of expected utility theory have been widely reported. Many of these violations, such as the common consequence effect and the common ratio effect, can be explained by models that take subjective attitudes to probability into account, such as rank-dependent models (Quiggin [1982],Schmeidler [1989]).

Rank-dependent models assert that both sensitivity to payoffs and sensitivity to probabilities generate deviations from risk neutrality. In particular, these models convert subjective probabilities into decision weights, and agents are assumed to take decisions so as to maximize their rank-dependent utilities (RDU). A possible interpretation of decision weights is that they represent a cognitive bias concerning how human beings deal with probability values when making choices under risk and uncertainty.

Rank-dependent models are amongst the most satisfactory decision theories under uncertainty (but, as discussed later, other models may be better still). Starmer [2000] and Camerer [2004] documented the superior predictive performance of rank-dependent models over expected utility theory for a range of phenomena, including the disposition effect, the equity premium puzzle, asymmetric price elasticities, the excess sensitivity of consumption to income, elasticities of labour supply and asset pricing, etc.

By construction, rank-dependent models can explain everything that expected utility theory can, but the converse is false. Under expected utility theory, an agent reports his true belief under a tailored proper scoring rule. In the next sections, I show that this is no longer the case under a rank-dependent model. In order to build intuition, I first introduce RDU in terms of lotteries, which are event-contingent payoffs. Thereafter, I extend the initial definition of RDU to tailored proper scoring rules and characterize how an underlying agent reports his belief.

4.1 RDU and lotteries

Let l = [ y ₁∶θ₁,…, y _n∶θ_n] denote a lottery which yields a payoff of y _x∈ℜ if outcomeθ_x occurs. Since one can always rearrange the outcomes, I assume without loss of generality that y _n≥ y _n−1≥…≥ y ₁. Given that agents have beliefs over the occurrence of the outcomes, I can then represent a lottery as l = [ y ₁∶ p ₁,…, y _n∶ p _n], which yields a payoff of y _x∈ℜ with probability p _x.

A lottery is called positive when all payoffs are nonnegative, i.e., y _n≥ y _n−1≥…≥ y ₁≥ 0. I denote a positive lottery by l ⁺. A lottery is called negative when all payoffs are nonpositive, i.e., 0≥ y _n≥ y _n−1≥…≥ y ₁. I denote a negative lottery by l ⁻. Finally, a mixed lottery l ^± contains both positive and negative payoffs, i.e., y _n≥ y _n−1≥…≥ y _i≥ 0≥ y _i−1≥…≥ y ₁.

Focusing first on positive lotteries, rank-dependent models state that the value that a human being assigns to l ⁺ is described according to his rank-dependent utility (RDU) (Quiggin [1982]):

(2)

where:

(3)

,

for k∈ {1,…, n−1}. The function W ⁺∶[0, 1]→ [0,1], also known as the weighting function, is striclty increasing, and it satisfies W ⁺(0) = 0 and W ⁺(1) = 1. Henceforth, I drop the superscript whenever talking about weighting functions in general, and not only in the domain of gains. As suggested by Gonzalez and Wu [1999], the weighting functions model the“psychophysics of chance”, i.e., the way human beings subjectively distort probability values. Common findings suggest that the weighting function is a nonlinear transformation of the probability scale that overweights small probabilities and underweights moderate and high probabilities (Tversky and Kahneman [1992],Abdellaoui [2000]). In other words, the weighting function displays an inverse-S shape: it is concave near 0 and convex near 1. The weighting function proposed by Tversky and Kahneman [1992] is:

(4)

whereγ≥ 0.28 in order for W to be strictly increasing. Forγ = 1, the weighting function in (4) becomes the identity function. Decreasingγ results in a more pronounced inverse-S shape. Figure 1 illustrates the weighting function in (4) for different values ofγ.

There are two crucial points regarding the rank-dependent utility in (2). First, as in the expected utility theory, the value that an agent derives from a payoff in a lottery is given by a strictly increasing utility function U . Second, instead of an individual probability value p _k as in the expected utility theory, the weight of a utility U( y _k) in (2) is the difference between two transformed ranks, W( p _k+…+ p _n)− W( p _k+1+…+ p _n), also called a decision weight. For a lottery l ⁺, the rank of a payoff y _k is the probability of l ⁺ yielding a payoff better than y _k, i.e., the rank of y _k is equal to p _k+1+ p _k+2+…+ p _n. The weight of U( y _k) is then the transformed marginal contribution of the individual probability p _k to the total probability of receiving payoffs better than y _k.

Under rank-dependent models, positive and negative lotteries might be evaluated differently. For a negative lottery l ⁻, the rank-depend utility in (2) is now defined as:

where:

(5)

,

Figure 1: The weighting function in (4) for different parameter valuesγ.

for k∈{2,…, n}. While a decision weightπ_k ⁺ denotes the marginal contribution of an individual probability value p _k to the total probability of receiving better payoffs, a decision weightπ_k ⁻ denotes the marginal contribution of an individual probability value p _k to the total probability of receiving worse payoffs, measured in terms of a weighting function W ⁻∶[0, 1]→ [0,1].

Finally, for a mixed lottery l ^±, where y _n≥ y _n−1≥…≥ y _i≥ 0≥ y _i−1≥…≥ y ₁, the rank-depend utility is now defined as:

4.2 RDU and tailored proper scoring rules

Without loss of generality due to a possible rearrangement of outcomes, assume that the scores from a tailored proper scoring rule S are ordered, i.e., S( q,θ_n)≥ S( q,θ_n−1)≥…≥ S( q,θ₁). I note that the scores from a tailored proper scoring rule can be stated in terms of a lottery: [ S( q,θ₁)∶ p ₁,…, S( q,θ_n)∶ p _n]. Consequently, when reporting a belief q, an agent is essentially defining the payoffs of a lottery, where the associated probabilities are subjective probabilities. In other words, reporting a belief q is equivalent to choosing a lottery amongst a potentially infinite number of lotteries. This implies that an agent’s reporting behavior can be analyzed from the perspective of decision models such as rank-dependent models. For a positive, tailored proper scoring rule S( q,θ_x), the rank-dependent utility in (2) becomes:

(6)

Similarly, the RDU for a negative, tailored proper scoring rule S( q,θ_x) is:

(7)

Finally, the RDU for a mixed, tailored proper scoring rule S( q,θ_x) is:

(8)

From the above equations, one might expect that an agent who maximizes a rank-dependent utility will behave differently than an expected-utility maximizer and, consequently, will report a belief other than his true belief under a tailored proper scoring rule. I discuss this point in the following section.

5 Characterizing reporting behavior under tailored proper scoring rules and RDU

The following propositions characterize how an agent who behaves to maximize a rank-dependent utility reports his belief under a tailored proper scoring rule. In short, my results indicate that such an agent reports a vector of decision weights, instead of his true belief.

Proposition 1  Let S( q,θ_x)be a positive, tailored proper scoring rule where S( q,θ_n)≥ S( q,θ_n−1)≥…≥ S( q,θ₁)≥ 0. Assume that an agent reports his belief q so as to maximize his RDU shown in (6). Then,

Proof. I start by noting that U( S( q,θ_x)) = U( U ⁻¹( R( q,θ_x))) = R( q,θ_x), for some proper scoring rule R. Ifπ⁺ = (π₁ ⁺,π₂ ⁺,…,π_n ⁺) is a probability vector, then∑_k=1 ⁿπ_k ⁺ R( q,θ_k) = E_π⁺ [ R( q,·) ], as in equation (1), and, consequently, _q∑_k=1 ⁿπ_k ⁺ R( q,θ_k) =π⁺. Thus, I just need to prove thatπ⁺ = (π₁ ⁺,…,π_n ⁺) is indeed a probability vector. From (3), I deduce that∑_k=1 ⁿπ_k ⁺ = W ⁺(∑_k=1 ⁿ p _k) = 1. Since W ⁺ is a strictly increasing function and its image is equal to [0, 1], then 0≤π_k ⁺≤ 1, for all k∈ {1,…, n}, thus completing the proof.

A similar result holds for negative, tailored proper scoring rules, as shown in Proposition 2.

Proposition 2  Let S( q,θ_x)be a negative, tailored proper scoring rule where0≥ S( q,θ_n)≥ S( q,θ_n−1)≥…≥ S( q,θ₁). Assume that an agent reports his belief q so as to maximize his RDU shown in (7). Then,

Proof. Given that U( S( q,θ_x)) = R( q,θ_x), for some proper scoring rule R, ifπ⁻ = (π₁ ⁻,π₂ ⁻,…,π_n ⁻) is a probability vector, then , as in equation (1). Consequently, _q∑_k=1 ⁿπ_k ⁻ R( q,θ_k) =π⁻. Thus, I just need to prove thatπ⁻ = (π₁ ⁻,…,π_n ⁻) is indeed a probability vector. From (5), I deduce that∑_k=1 ⁿπ_k ⁻ = W ⁻(∑_k=1 ⁿ p _k) = 1. Since W ⁻ is a strictly increasing function and its image is equal to [0, 1], then 0≤π_k ⁻≤ 1, for all k∈ {1,…, n}, thus completing the proof.

Propositions 1 and 2 imply that positive and negative tailored proper scoring rules induce different reporting behavior whenever the weighting functions W ⁺ and W ⁻ are different. In other words, a simple positive affine transformation of a proper scoring rule might induce different reporting behavior. I illustrate this point in Section 5.1. I show in the following proposition that mixed, tailored proper scoring rules induce agents to report decision weights as well when W ⁺(ρ) + W ⁻(1−ρ) = 1, for allρ∈ [0, 1].

Proposition 3  Let S( q,θ_x)be a mixed, tailored proper scoring rule where S( q,θ_n)≥ S( q,θ_n−1)≥…≥ S( q,θ_i)≥ 0≥ S( q,θ_i−1)≥…≥ S( q,θ₁). Assume that an agent reports his belief q so as to maximize his RDU shown in ( 8 ). If W ⁺(ρ) + W ⁻(1−ρ) = 1, for anyρ∈ [0, 1], then

Proof. Ifπ^±= (π₁ ⁻,…,π_i−1 ⁻,π_i ⁺,…,π_n ⁺) is a probability vector, then the result follows naturally because U( S( q,θ_x)) = R( q,θ_x), for some proper scoring rule R. Consequently, I just need to prove thatπ^± is indeed a probability vector. From (3) and (5), I have that∑_k=1 ⁱ⁻¹π_k ⁻ +∑_{k= i} ⁿπ_k ⁺ = W ⁻(∑_k=1 ⁱ⁻¹ p _k) + W ⁺(∑_{k= i} ⁿ p _k) = 1, where the last equality follows from the assumption that W ⁺(ρ) + W ⁻(1−ρ) = 1 , for allρ∈ [0, 1]. Since both W ⁺ and W ⁻ are strictly increasing functions and their images are equal to [0, 1], then 0≤π_j ⁻,π_k ⁺≤ 1, for all j∈ {1,…, i−1} and k∈ { i,…, n}, thus completing the proof.

5.1 Numerical example

In this subsection, I illustrate the theoretical results proved in Propositions 1 and 2 by using the weighting function proposed by Tversky and Kahneman [1992] shown in (4). Tversky and Kahneman [1992] found that the best fit for their data happened when using W ⁺ and W ⁻ as defined in (4) with parameter values equal to, respectively,γ = 0.61 andγ = 0.69.

Consider an agent with belief p = (0.2, 0.8) who behaves so as to maximize his rank-dependent utility. Under a positive, tailored proper scoring rule, Proposition 1 implies that the agent reports:

Proposition 2 implies that the same agent reports:

under a negative, tailored proper scoring rule. The deviation of the agent’s reported belief q from his true belief p according to the mean absolute error is equal to: 0.5×|1− W ⁺(0.8)− 0.2| + 0.5×| W ⁺(0.8)− 0.8| = 0.193, for a positive, tailored proper scoring rule, and 0.5×| W ⁻(0.2)− 0.2| + 0.5×|1− W ⁻(0.2)− 0.8| = 0.057 for a negative, tailored proper scoring rule.

The above example illustrates that positive and negative tailored proper scoring rules might induce different reporting behavior whenever the weighting functions W ⁺ and W ⁻ are not equal to each other. In particular, tailored proper scoring rules with positive scores seems to result in stronger deviations from honest reporting and, consequently, risk neutrality than with negative scores, a fact that is empirically plausible (Wakker [2010], page 264). Furthermore, the above example illustrates that agents overweight low probabilities by reporting probability values greater than their true beliefs, and they underweight high probabilities by reporting probability values less than their true beliefs.

6 Using tailored proper scoring rules to elicit an agent’s weighting functions

The results from the previous section are negative in nature because they mean that RDU agents report biased beliefs under tailored proper scoring rules. On the positive side, I discuss in this section how tailored proper scoring rules can elicit weighting functions in a parameter-free manner.

My approach assumes that there are two exhaustive and mutually exclusive outcomes,σ₁ andσ₂, with known, objective probability valuesφ and 1−φ. For example,σ₁ andσ₂ can be the outcomes“heads” and“tails” in an experiment where a biased coin with known Bernoulli distribution is tossed. An agent is then asked to report his belief µ = (µ, 1−µ), forµ∈ [0, 1].

Consider a proper scoring rule R(µ,σ_x), for x∈ {1, 2}, defined as follows:

where R ^′ is a bounded proper scoring rule, i.e., a proper scoring rule where all the returned scores are real numbers, sgn is the sign function, and m is the maximum score returned by R ^′. Then, by construction, R( µ,σ₂)≥ R( µ,σ₁) for anyµ, which means that R is comonotonic (Kothiyal et al. [2011]). I now construct a tailored proper scoring rule to elicitµ, i.e., S(µ,σ_x) = U ⁻¹( R(µ,σ_x)). Since U ⁻¹ is strictly increasing, I then obtain S(µ,σ₂)≥ S(µ,σ₁) for anyµ, which implies that S also satisfies comonotonicity.

In previous sections, for ease of exposition and mathematical notation, I assumed that S( q,θ_n)≥ S( q,θ_n−1)≥…≥ S( q,θ₁). I claimed that such an assumption is without loss of generality because the outcomes could always be rearranged a posteriori. In this section, however, I do not allow the outcomes to be rearranged and, by construction, S(µ,σ₂)≥ S(µ,σ₁) for any beliefµ. For example, in the aforementioned coin experiment, one agent will always receive higher scores if outcomeσ₂ =“tails” occurs than if outcomeσ₁ =“heads” occurs, no matter what the agent reports.

First, consider the case where the resulting tailored proper scoring rule S is negative, i.e., 0≥ S(µ,σ₂)≥ S(µ,σ₁). Proposition 2 implies that an agent who maximizes a rank-dependent utility reports the probability vectorµ = (π₁ ⁻,π₂ ⁻) = ( W ⁻(φ), 1− W ⁻(φ)). In other words, I obtain the value of W ⁻ for the objective probability valueφ. For a sufficiently dense set of objective probabilities, e.g., taking all values in the set {0, 0.05, 0.1,…, 0.95, 1 }, I obtain a parameter-free estimate of the weighting function W ⁻.

Alternatively, if S is a positive tailored proper scoring rule, Proposition 1 says that an agent who maximizes a rank-dependent utility reports the probability vectorµ = (π₁ ⁺,π₂ ⁺) = (1− W ⁺(1−φ), W ⁺(1−φ)). Then, for a sufficiently dense set of objective probabilities, I obtain a parameter-free estimate of the weighting function W ⁺.

Finally, if S is a mixed tailored proper scoring rule, Proposition 3 says that an agent who behaves so as to maximize a rank-dependent utility reports the probability vectorµ = (π₁ ⁻,π₂ ⁺) = ( W ⁻(φ), W ⁺(1−φ)), under the assumption that W ⁻(φ) + W ⁺(1−φ) =1. Then, for a sufficiently dense set of objective probabilities, I obtain a parameter-free estimate of both the weighting function W ⁻ and the weighting function W ⁺.

It is noteworthy that without the comonotonicity property,π₁ ⁻ is always less than or equal to 0.5, andπ₂ ⁺ is always greater than or equal to 0.5 (Kothiyal et al. [2011]). Consequently, the weighting function W ⁻ could not be estimated for probability values greater than 0.5, whereas the weighting function W ⁺ could not be estimated for probability values less than 0.5.

On a final note, I observe that traditional methods for eliciting weighting functions assume that agents report indifferences between lotteries honestly (Abdellaoui [2000]). Under my approach, on the other hand, it is in the best interest of an agent to reportµ honestly since this maximizes his rank-dependent utility.

7 Obtaining true beliefs from vectors of decision weights

In Section 5, I showed how tailored proper scoring rules elicit vectors of decision weights from agents who behave so as to maximize a rank-dependent utility. In Section 6, I discussed how to use tailored proper scoring rules to elicit an agent’s weighting functions. A natural question that then arises regards how to combine these two results in order to obtain an agent’s true belief p when that agent reports a vector of decision weights. In the following subsections, I show how an agent’s true belief can be obtained by using simple recursive procedures. The proposed procedures are sound as long as S( q,θ_n)> S( q,θ_n−1)>…> S( q,θ₁), i.e., when there are only inequalities in the scores from the tailored proper scoring rule. Otherwise, the underlying proper scoring rule might have to satisfy comonotonicity (Kothiyal et al. [2011]).

7.1 Positive tailored proper scoring rule

If a positive, tailored proper scoring rule is used in the elicitation process, then Proposition 1 says that the belief q = ( q ₁,…, q _n) = (π₁ ⁺,…,π_n ⁺) is reported by a rank-dependent utility maximizer, which implies that:

Once W ⁺ is known, the above system of equations can be solved by using backward substitution, i.e., by first computing p _n, then substituting that into the next equation to find p _n−1, and so on. Starting with the base case p _n, I have p _n = W ⁺⁻¹( q _n). For p _n−1, I have p _n−1 = W ⁺⁻¹( q _n−1 + q _n)− p _n. More generally, for all k∈ {2,…, n−1}, I obtain p _k by solving the equation p _k = W ⁺⁻¹(∑_{x= k} ⁿ q _x)−∑_{x= k+1} ⁿ p _x. Finally, p ₁ = 1−∑_x=2 ⁿ p _x.

7.2 Negative tailored proper scoring rule

If a negative, tailored proper scoring rule is used in the elicitation process, then Proposition 2 says that the belief q = ( q ₁,…, q _n) = (π₁ ⁻,…,π_n ⁻) is reported by a rank-dependent utility maximizer, which implies that:

Once W ⁻ is known, the above system of equations can be solved by using forward substitution, i.e., by first computing p ₁, then substituting that into the next equation to find p ₂, and so on. Starting with the base case p ₁, I have p ₁ = W ⁻⁻¹( q ₁). For p ₂, I have p ₂ = W ⁻⁻¹( q ₁ + q ₂)− p ₁. More generally, for all k∈ {2,…, n−1}, I obtain p _k by solving the equation p _k = W ⁻⁻¹(∑_x=1 ^k q _x)−∑_x=1 ^k−1 p _x. Finally, p _n = 1−∑_x=1 ⁿ⁻¹ p _x.

7.3 Mixed tailored proper scoring rule

Finally, if a mixed, tailored proper scoring rule is used in the elicitation process, then Proposition 3 says that the belief q = ( q ₁,…, q _n) = (π₁ ⁻,…,π_i−1 ⁻,π_i ⁺,…,π_n ⁺) is reported under the assumption that W ⁺(ρ) + W ⁻(1−ρ) = 1, for allρ∈ [0, 1], which implies that:

Once the weighting functions W ⁺ and W ⁻ are known, the above system of equations can be solved by using forward and backward substitution, i.e., forward substitution can be used to obtain the values of p ₁,…, p _i−1 as discussed in Section 7.2, whereas backward substitution can be used to obtain the values of p _i,…, p _n as discussed in Section 7.1.