An alternative approach to quadratic scoring rules using continuous-valued logic

Following tfhe seminal paper of Offerman et al. (2009), in this study, adaptations of constructions of continuous-valued logic to prospect theory are presented. Here, we demonstrate that the so-called kappa function and its special cases are viable alternatives to some elements of quadratic scoring rule prospects theory. After, we present the tau-eta scoring rule prospect and show that it may be treated as a generalization of the quadratic scoring rule prospect. Furthermore, we prove that if this new prospect for an uncertain event is evaluated using specific kappa functions as utility functions, then (1) the weighting measure of the event is a function of the optimal value of its reported probability, (2) the inverse of the latter function, and (3) the (risk-) corrected reported probability of the event, also as a function of the optimal value of its reported probability, all have a common formula. The parameters of the common formula are unambiguously determined by four tuning parameters. Lastly, we show that with our approach, by fitting one of the abovementioned functions to corresponding empirical data, we can immediately obtain the other two functions as well.


Introduction
In many situations, people's private opinions about the probability of an uncertain event contain subjective beliefs that may greatly influence their decisions and the confidence in their judgments. In cases where appropriate incentives are missing, these subjective beliefs are not sufficiently updated and this may make people more confident about their judgments than they in fact are (Li 2007). Manski (2004) presented a comprehensive overview of belief measurement. The incentive-compatible mechanisms in the form of proper scoring rules are well-known efficient techniques for belief measurement (Brier 1950;Good 1952). There are many areas of science where proper scoring rules have been successfully applied. These include accounting (Wright 1988 (Schotter and Trevino 2014;Andersen et al. 2014;Arieli and Mueller-Frank 2017;Murad et al. 2016;Harrison et al. 2017), education (Echternacht 1972), finance (Shiller et al. 1996;Johnstone 2007b, a), forecasting (Witkowski et al. 2017), mathematical statistics (Savage 1971), medical science (Spiegelhalter 1986), politics (Tetlock 2017), prospect theory (Offerman et al. 2009;Wakker 2010;Armantier and Treich 2013;Hossain and Okui 2013;Schlag et al. 2015) and psychology (McClelland and Bolger 1994).
Traditionally, when applying proper scoring rules, maximization of expected value is assumed, but in practice, many exceptions to this assumption have been identified. Also numerous deviations from the expected utility model have been presented (see, e.g., Allais (1953); Kahneman and Tversky (2013); Ellsberg (1961); Keynes (1921); Knight (1921)). Offerman et al. (2009) extended the proper scoring rules from the expected value model and demonstrated that subjective belief and ambiguity attitudes can be separated from risk attitude by using proper scoring rules.
Our own study may be viewed as a further development to the theoretical results presented in Offerman et al. (2009). The main contribution of our paper is the adaptation of constructions of continuous-valued logic to prospect theory. Our main findings can be summarized as follows. We will demonstrate that some important elements of prospect theory can be derived from probabilistic unary operators of continuousvalued logic. Namely, we will show that: (1) The functions in the quadratic scoring rule prospect can be generated by the probabilistic unary operators of continuous-valued logic; (2) The Prelec probability weighting function is a special case of the probabilistic unary operator; and (3) The widely applied power utility functions can also be generated by the probabilistic unary operators.
Next, we will utilize the kappa function, which is a unary operator of continuous-valued logic that can be induced by the generator functions of the Dombi operators. Here, we will show that the so-called extended kappa function, the kappa function and its special cases, the tau and the eta functions, can be applied in prospect theory as viable alternatives to: (1) The functions in the quadratic scoring rule prospect (2) The power utility functions (3) The Prelec probability weighting function (4) The Ostaszewski, Green and Myerson probability weighting function and (5) The (risk-)corrected reported probability function (inverse of Prelec's probability weighting function).
It should be added that each parameter of the kappa function has a semantic meaning related to the shape of the function curve. This property makes the kappa function suitable for modeling and curve fitting tasks. Also, by making use of the tau and the eta functions, for the uncertain event E, we will introduce the tau-eta scoring rule prospect as an alternative to the quadratic scoring rule prospect where ν s ∈ (1/2, 1) and r ∈ (0, 1). As the main results of this study, in Theorem 4, we will show that if a tau-eta scoring rule prospect in Eq. (1) is evaluated with a tau utility function that has a parameter value ν u ∈ (0, 1/2] and the probability weighting function w is the kappa probability weighting function w (λ w ) ν w , then (F1) The weighting measure W (E) of the uncertain event E being a function of the optimal value of the reported probability r = r E (F2) The inverse function of W (E) and The (risk-)corrected reported probability measure B, which is given by B(E) = w −1 (W (E)), being a function of the optimal value of the reported probability r = r E can all be written in the following common form: where and the variable and parameter values in Eq.
In Table 1, the parameters a and b are given by Note that in our approach, four tuning parameters are utilized to model the abovementioned three functions; namely, ν u , ν s , λ w and ν w . Each of these parameters has a semantic meaning. These are: The value of the parameter ν u ∈ (0, 1) determines the shape of the tau function (a special case of the kappa function), when it is used as a utility function. If ν u = 1/2, then the tau function coincides with the identity utility function. If ν u = 1/4, then the tau function approximates the square root utility function quite well. By changing the value of the parameter ν u , the tau function can be used to model various utility functions. ν s : ν s ∈ (0, 1) is the scoring parameter of the taueta scoring rule prospect. If ν s = 3/4, then the tau-eta scoring rule is approximately the quadratic scoring rule. λ w and ν w : These are the parameters of the kappa probability weighting function w (λ w ) ν w , where λ w > 0 and ν w ∈ (0, 1). The value of the parameter λ w determines the slope of w (λ w ) ν w (x) at x = ν w , while ν w is the unique fixed point of the function, i.e., w (λ w ) ν w (ν w ) = ν w . If λ w = 2/3 and ν w = 2/5, then the kappa probability weighting function approximates the Prelec probability weighting function quite well.
It is worth adding that the values of these four tuning parameters unambiguously determine the values of all the parameters in the common formula in Eq. (2) and Eq. (3). As findings in our study, we will propose particular values for the four tuning parameters. The functions W (E), r (W (E)) and B(E) all have an S-shape or an inverse S-shape (see Fig. 1).
Also, there is an important aspect of our approach that should be mentioned here. Suppose that we have empirical data for one of the abovementioned three functions. Then, the four tuning parameters can be determined by fitting the function to the empirical data, which means that we immediately have the other two functions determined as well.
Regarding the connections between fuzzy logic and the quadratic scoring rules, it is worth mentioning that in fuzzy control, the fuzzy affine models utilize quadratic Lyapunov functions (see, e.g., Ji et al. (2021Ji et al. ( , 2022). In our approach, we approximate two particular quadratic functions, q 1 (r ) = 1 − (1 − r ) 2 and q 2 (r ) = (1 − r 2 ), r ∈ [0, 1], using the tau and eta functions, which are unary operators in fuzzy logic (see Dombi and Jónás (2021)). That is, we apply unary operators of fuzzy logic to approximate quadratic functions. The question of whether the constructions of fuzzy affine models can be adapted to quadratic scoring rules is left for future study.
We should mention that with compared with some prevalent results (see, e.g., Kahneman and Tversky (2013); Knight (1921); Offerman et al. (2009)), the proposed approach has two big advantages. On the one hand, using Eq. (2), we can provide a unified theoretical approach to the three important functions of prospect theory that are described in (F1)-(F3). On the other hand, our approach simplifies the empirical studies. Namely, by fitting one of the abovementioned functions to empirical data, we get the other two functions as well.
This paper is organized as follows. In Sect. 2, the main elements of proper scoring rules are summarized. In Sect. 3, we briefly show how the operators of continuous-valued logic are related to elements of prospect theory. Next, Dombi operators of continuous-valued logic and the kappa function are discussed in Sect. 4. Fitting kappa functions to wellknown functions of prospect theory is introduced in Sect. 5. Here, we present the tau-eta scoring rule as an alternative to the quadratic scoring rule. In Sect. 6, we will present our results for the evaluation and analysis of tau-eta scoring rule prospects. Lastly, in Sect. 7, we summarize the key findings of this study and mention some future research plans.
The proofs of lemmas, propositions and theorems presented in this paper can be found in Appendix A.

Main elements of proper scoring rules in prospect theory
Here, we will briefly review the most important notions, constructions and results related to proper scoring rule prospects, focusing mainly on the quadratic scoring rule prospects. Following the study of Offerman et al. (2009), we shall assume that judgments related to uncertain events are made by specialized agents. In such cases, if there is a lack of proper incentives, agents may pretend to be more confident about their judgment than they really are. From now on, let X be an event space, and let E ⊆ X be an uncertain event. The uncertainty of the agent about the event E depends on the information that the agent possesses about E. In many cases, the objective probability of the event E is unknown, therefore the decisions and judgments related to E need to be made based on beliefs and subjective probabilities. A prospect is generally defined as follows.
Definition 1 (see Offerman et al. (2009)). The prospect x E y is an event-contingent payment such that This means that the prospect x E y yields the outcome x if event E occurs, and it yields the outcome y if the complement of event E occurs, and the outcomes are financial amounts. From here on, we will use the notation E for the complement of set (event) E. When the probability p of event E is known, the general notation x p y is used to express the fact that the prospect yields x with probability p and it yields y with probability 1 − p, where p ∈ [0, 1].
It is generally agreed that the quadratic scoring rule (QSR) is the most widely used proper scoring rule (see McKelvey and Page (1990); Nyarko and Schotter (2002); Offerman et al. (2009)). Now, we will present an alternative to the wellknown quadratic scoring rule prospect, which is given by the following definition.
Definition 2 (see Offerman et al. (2009)). A quadratic scoring rule prospect for event E is where r ∈ [0, 1] is a function of event E and it is called the (uncorrected) reported probability of E.
It should be added that the notation r E is also used to indicate that the reported probability r E is a function of E. Furthermore, it is also assumed that if event E has a probability p, then r = r E depends only on p; that is, It is worth noting that the QSR prospect in Eq. (5) has an important symmetry property, which is stated in the following lemma.
holds if and only if Notice follows from Lemma 1. Lemma 1 allows us to state our results only for r ≥ 1/2, and the results for r < 1/2 follow from Eq. (6) and Eq. (7). We will now use the concept of monotone measures that are also known as fuzzy measures.

Definition 3
Let be a σ -algebra on the set X . Then the function g : → [0, 1] is a monotone measure on the measurable space (X , ) iff g satisfies the following requirements: (1) g(∅) = 0, g(X ) = 1; (2) If B ⊆ A, then g(B) ≤ g(A) for any A, B ∈ (monotonicity); If X is a finite set, then the continuity requirement in Definition 3 can be disregarded.
In modern decision models, the QSR prospect given in Eq. (5) is evaluated by using the following formula (see, e.g., Wakker (2010)): is a utility function which is strictly increasing, continuous and U (0) = 0, and W is a monotone measure over the event space X . Offerman et al. (2009) presented the following cases for the measure W , where each case represents a decision model.
1. Expected value. The utility function U is the identity function, and the measure W is a probability measure P on X . 2. Expected utility. The measure W is a probability measure P on X . 3. Probabilistic sophistication (with non-expected utility).
Note that there are two sub-cases for the probability measure P. Namely, it is either an objective probability measure, which is based on statistical data that everyone agrees on, or a subjective probability measure, which can be derived from preferences (Offerman et al. 2009). The preference foundations for subjective probabilities can be found in the publications of De Finetti (1937), Savage (1954) and Machina and Schmeidler (1992). It should be added that the sequence of the presented cases reflects the historical development of decision models. The well-known St.
Petersburg paradox presented by Daniel Bernoulli in 1738 (Commentaries of the Imperial Academy of Science of Saint Petersburg, see its translation in Bernoulli (1954)) demonstrates that the expected value model (Case 1) is not suitable for describing people's decision making under uncertainty. Next, Maurice Allais in 1953 pointed out that the expected utility model (Case 2) is often violated (see the Allais paradox in Allais (1953)), and so the function w and the measure W are non-additive. The well-known Ellsberg paradox, which was published in 1961, shows that the probabilistic sophistication (Case 3) is also often violated (Ellsberg 1961). Therefore, the modern decision making models usually consider the general case (Case 4).
As regards the quadratic scoring rule prospect given in Eq. (5), Offerman et al. (2009) presented the following results for the optimal value of the reported probability r E .
Theorem 1 (Offerman et al. 2009). In the general model (Case 4), the optimal choice r in Eq. (5) satisfies the following requirement: Corollary 1 Under the expected value (Case 1), Eq. (9) holds for all r and r = r E = P(E).
This was first presented by author Brier (1950). In Offerman et al. (2009), the authors stated that if there is a sufficient amount of general decisions under risk without proper scoring rules observable, but with known probabilities, then the whole probability weighting function w can be discovered. Also, the measure W could be found by observing enough decisions under uncertainty. Based on these findings, they introduced the following important measure, which is called the (risk-)corrected reported probability.

Definition 4
The (risk-)corrected reported probability B is given by for any E ⊆ X , where w is a probability weighting function and W is a monotone measure on the event space X .
It immediately follows from Definition 4 that B is a monotone measure on the event space X . Furthermore, we can see that under the probabilistic sophistication; that is, when where P is a probability measure on X . Based on these results, Offerman et al. concluded that B is a better measure of subjective beliefs than the reported probability r . Utilizing the measure B, they deduced the following results from Theorem 1.
Corollary 2 ( Offerman et al. (2009)). For the optimal choice r = r E , if r > 1/2, then Corollary 3 (Offerman et al. (2009)). Under probabilistic sophistication, we have for the optimal choice r = R( p): if r > 1/2, then Corollary 4 ( Offerman et al. (2009) In our study, by using techniques of continuous-valued logic, we will give viable alternatives to all the main elements of prospect theory that were briefly discussed in this section.

Connection of operators of continuous-valued logic with elements of prospect theory
In a study of continuous-valued logic it is necessary to discuss t-norms and t-conorms (Klement et al. 2013), which here correspond to the the conjunctive and disjunctive operators. From an application point of view, the strictly monotonously increasing (or decreasing) operators play an important role. It is well known that any strict conjunction operator c : have the following form (Aczél 1966):  (Dombi 1982). Note that the generator functions f c and f d are determined up to a constant multiplicative factor.
For these conjunctive and disjunctive operators, we can order the unary operators κ where ν, ν 0 ∈ (0, 1) and λ ∈ R are the parameters of the unary operators (Dombi 2012).
Here, we will utilize two operator families, namely the probabilistic operators (also known as product operators) and the Dombi operators.
The generator functions of the probabilistic operators c and d are given by Using Eq. (11) and Eq. (12), one can easily see that these generator functions yield the following operators: Later, we will show that the unary operators in Eq. (13) and in Eq. (14) Now, we will demonstrate that some important elements of prospect theory can be derived from probabilistic unary operators of continuous-valued logic. Namely, we will show that: -The functions in the quadratic scoring rule prospect can be generated by the probabilistic unary operators of continuous-valued logic; -Prelec's probability weighting function is a special case of the probabilistic unary operator; -The widely applied power utility functions can be generated by the probabilistic unary operators.

A new interpretation of the quadratic scoring rule prospect using probabilistic unary operators
Here, we will demonstrate that the functions in the quadratic scoring rule prospect can be generated by the unary operators given in Eq. (13) and in Eq. (14).

Proposition 1 If the generator function f d of the unary operator
can be written in the form where r ∈ (0, 1).
for any x ∈ (0, 1). That is, the QSR prospect in Eq. (15) can also be written in the following form: where r ∈ (0, 1).
Proposition 1 tells us that the functions in the quadratic scoring rule prospect can be generated by the unary operators given in Eq. (13) and in Eq. (14).

A new interpretation of Prelec's probability weighting function using the probabilistic unary operator
The Perlec's probability weighting function family (Prelec 1998) w P is given by where 0 < a < 1, b > 0 and x ∈ (0, 1]. The following proposition shows how the Perlec's probability weighting function family can be generated by the unary operator and

A new interpretation of the power utility functions using the probabilistic unary operator
In prospect theory, a utility function U : [0, 1] → [0, 1] is commonly represented by a power function (Offerman et al. 2009); that is, U (x) = x ρ . The following proposition shows how the power utility functions can be derived from the unary operators given in Eqs. (13) and (14).

Proposition 3
Let the generator function f c of the unary operator κ Notice that by making use of Proposition 3 with ρ = 1/2, we get the utility function U :

Dombi operators and the kappa functions
The structure of the continuous-valued logic operators when the fix point ν * of the negation operation can have any value in the interval (0, 1) is discussed in Dombi (2011). In these operator systems, we have infinitely many negation operators. In the following, we will use a system in which the generator function is a parametric function.

Definition 5
The generator function f D : (0, 1) → (0, ∞] of the Dombi operators in continuous-valued logic is given by where α ∈ R, α = 0. Remark 2 It should be noted that (a) if α > 0, then f D is a strictly decreasing generator function and so applying Eq. (11) with f c = f D results in a conjunctive operator, which is called the Dombi conjunction operator; (b) if α < 0, then f D is a strictly increasing generator function and so applying Eq. (12) with f d = f D results in a disjunctive operator, which is called the Dombi disjunction operator Dombi (2011). ν,ν 0 : (0, 1) → (0, 1), which is given by
Notice that the unary operators κ (λ) c,ν,ν 0 and κ (λ) d,ν,ν 0 induced from the generator function f D with α > 0 and α < 0, respectively, are both independent of the parameter α. This is why these two operators coincide if they are both induced from the generator function of the Dombi operator in Eq. (18).
Later, we will use the following extended kappa function.

Definition 6 [extended kappa function] The extended kappa function
Remark 3 Notice that the extended kappa function given in Definition 6 can be obtained by a linear transformation of the kappa function κ (λ) ν,ν 0 from the domain (0, 1) to the domain (a, b). Therefore, the main properties of the extended kappa function κ (λ) (a,b),x ν ,ν 0 can be readily derived from the properties of the kappa function κ (λ) ν,ν 0 .

Properties of the kappa functions
We will use the kappa function, its inverse function, the extended kappa function and the compositions of kappa functions to create an alternative to the quadratic scoring rule prospect and derive new techniques to evaluate it. The elementary properties of the kappa function κ (λ) x ν ,ν 0 , namely, continuity, monotonicity, limits, role of the parameters and shape are summarized in Appendix B. Figure 2 shows some example plots of the kappa function κ (λ) ν,ν 0 (x) with various parameter values.

Compositions of kappa functions
Now, we will show that the composition of a kappa function with an extended kappa function is a kappa function as well.

Inverse of the kappa function
Now, we will show that the inverse of a kappa function is a kappa function as well.

The tau and eta functions as special cases of the kappa function
Now, we will define the tau and eta function as a special case of the kappa function.

Some properties of the tau and eta functions
Here, we will describe some important properties of the tau and eta functions that we will use later on.
The following two propositions concern the inverse of tau and eta functions.

Proposition 8 If η ν is an eta function with the parameter
The following two propositions are about the compositions of tau and eta functions.

Kappa functions as alternatives to well-known functions of prospect theory
Here, we will discuss how the kappa function and its special cases, the tau and the eta functions, can be applied in prospect theory as viable alternatives to -Functions in the quadratic scoring rule prospect (Sect. 5.1); -Power utility functions (Sect. 5.2); -Prelec's probability weighting function (Sect. 5.3); -The Ostaszewski, Green and Myerson probability weighting function (Sect. 5.4); -The (risk-)corrected reported probability function (inverse of Prelec's probability weighting function) (Sect. 5.5).

Tau and eta functions in the quadratic scoring rule prospect
Here, we will show that the functions in a quadratic scoring rule prospect (1 − (1 − r ) 2 ) E (1 − r 2 ) can be approximated by tau and eta functions. Let the functions f and g be given by f (r ) = 1 − (1 − r ) 2 and g(r ) = 1 − r 2 , where r ∈ (0, 1). Notice that function f has a similar shape to that of the tau function with a parameter value of ν < 1/2. Also, function g has a similar shape to that of the eta function with a parameter value of ν > 1/2. Let the scoring parameter ν s be ν s ∈ (1/2, 1). Then, 1 − ν s < 1/2 and so the function τ 1−ν s has a similar shape to that of function f and the function η ν s has a similar shape to that of function g. That is, we may use the following approximations: where r ∈ (0, 1). It can also be seen that f (r ) = g(r ) = 3/4 if and only if r = 1/2. Based on these observations, if we require that both the tau function τ 1−ν s and the eta function η ν s take the value of 3/4 at r = 1/2, then τ 1−ν s and η ν s need to fulfill the following requirements: By solving Eq. (30) and Eq. (31) for ν s , we get ν s = 3/4 from both equations, which means that 3/4 is an appropriate value for the scoring parameter ν s . Hence, we may use the following approximations: where r ∈ (0, 1). It can be shown numerically that (r ) may be viewed as an alternative to the quadratic scoring rule prospect (1 − (1 − r ) 2 ) E (1−r 2 ). Based on these findings, we define the tau-eta scoring rule prospect as follows.

Definition 9 A tau-eta scoring rule prospect for event E is
where r ∈ (0, 1) is a function of event E and ν s ∈ (1/2, 1).
Note that this definition is general in the sense that the value of the scoring parameter ν s may be arbitrary in the interval (1/2, 1). However, we will use it with the setting ν s = 3/4. Notice that for the tau-eta scoring rule prospect the value of the reported probability r is in the open interval (0, 1). It should be added that this does not mean any restriction in practice because the following limit properties of the tau and eta functions hold: It is worth noting that, similar to the QSR prospect, the tau-eta scoring rule prospect in Eq. (32) has the following important symmetry property.
be a tau-eta scoring rule prospect for the event E and let τ 1−ν s (r E ) E η ν s (r E ) be a tau-eta scoring rule prospect for the event E, where r E , r E ∈ (0, 1). Then, holds if and only if Remark 4 Notice that the tau function τ 1−ν s is symmetric to the line given by l(r ) = 1 − r and the eta function η ν s is symmetric to the diagonal line l(r ) = r , while the functions f (r ) = 1 − (1 − r ) 2 and g(r ) = 1 − r 2 do not have these symmetry properties.

Tau functions as alternatives to power utility functions
As we pointed out previously in Sect. 3.3, a utility function U : [0, 1] → [0, 1] is commonly given by a power function (Offerman et al. 2009); that is, U (x) = x ρ . Here, we will demonstrate that tau functions with appropriate parameter values may be viewed as alternatives to the most widely used power utility functions. Identity utility function (ρ = 1) It immediately follows from the definition for a tau function given in Definition 7 that where x ∈ (0, 1). Square root utility function (ρ = 1/2) Notice that the utility function U : x has a similar shape to that of the tau function τ ν u with a parameter value of ν u < 1/2, where x ∈ (0, 1). If ν u = 1 4 , then τ 1

4
(1/4) = 1 2 ; that is, Furthermore, it can be shown numerically that max x∈(0,1) Hence, the tau function τ 1 4 (x) may be viewed as an alternative to the square root utility function One can also see that the utility function U : [0, 1] → [0, 1], U (x) = x 2 has a similar shape to that of the tau function τ ν u with a parameter value of ν u > 1/2, where x ∈ (0, 1). It can be shown numerically that if ν = 3 4 , then max x∈(0,1) Therefore, the tau function τ 3 4 (x) approximates the square utility function U (x) = x 2 quite well, where x ∈ (0, 1).  Table 2 summarizes how the tau function τ ν u can be used to approximate the most commonly used power utility functions.
Based on the above results, we conclude that the tau function may be treated as an alternative to the most commonly used power utility functions of prospect theory.

The Ostaszewski, Green and Myerson probability weighting function as a special case of the kappa function
The Ostaszewski, Green and Myerson probability weighting function family (Ostaszewski et al. 1998) w OG M is given by where 0 < a < 1, b > 0 and p ∈ [0, 1]. Note that this probability weighting function family was introduced independently by Lattimore, Baker and Witte as well in 1992 (see Lattimore et al. (1992)). The following lemma demonstrates that the Ostaszewski, Green and Myerson probability weighting function is a special case of the kappa function κ (λ) ν,ν 0 in (35). then holds for any p ∈ (0, 1]. Lemma 3 tells us that the Ostaszewski, Green and Myerson probability weighting function is a special case of the kappa function κ (λ) ν,ν 0 .

The kappa function as an alternative to the (risk-)corrected reported probability function
According to Eq. (10), the (risk-)corrected reported probability B is given by where E is an uncertain event of an event space X , W is a monotone measure on X and w −1 is the inverse of a probability weighting function. As pointed out previously, the most commonly used probability weighting function is Prelec's probability weighting function given in Eq. (34) with the parameter values a = 0.65 and b = 1. Next, based on the results of Sect. 5.3, this function can be replaced by the probability weighting function w (λ w ) ν w in Eq. (36) with the parameter values ν w = 2/5 and λ = 2/3. Therefore, the inverse of the function w (λ) ν w is a suitable (risk-)corrected reported probability function. Noting that w (λ) ν w is a kappa function and making the use of Proposition 6, we get that is a kappa function as well, where p ∈ (0, 1), ν w ∈ (0, 1), λ w ∈ R and λ w = 0.

Evaluating and analyzing tau-eta scoring rule prospects
Here, we will discuss the evaluation and analysis of taueta prospects. Following the approach presented by Offerman et al. (2009), for the tau-eta scoring rule prospect τ 1−ν s (r ) E η ν s (r ), we will identify the relationship between the optimal value of the reported probability r = r E and the measure W (E). Next, we will show that this relationship is given by an extended kappa function when the utility function is the function τ ν u with a parameter value ν u ∈ (0, 1/2]. Furthermore, we will demonstrate that if the probability weighting function w is the kappa probability weighting function w (λ w ) ν w given in Eq. (36) and the utility function is the tau function τ ν u with a parameter value ν u ∈ (0, 1/2], then the (risk-)corrected reported probability B, which is given by B(E) = w −1 (W (E)), is also an extended kappa function of the optimal value of the reported probability r .
Later, we will use the following lemma.
Lemma 4 can be evaluated by where ν s ∈ (1/2, 1] and r ∈ (0, 1). Recall that according to the general model (Case 4 in Sect. 2), E is an uncertain event of an event space X , W is a monotone measure on X and U : [0, 1] → [0, 1] is a utility function such that U (0) = 0. Now, we will investigate the optimal value of the reported probability r = r E of the uncertain event E.
From now on, we will only discuss the case r ≥ 1/2, and the results for r < 1/2 can be derived from Eq. (44).

Analyzing the tau-eta scoring rule prospects when the utility function is a tau function
Now, we will discuss Eq. (43) in cases where the utility function U is the tau function τ ν u with a parameter value ν u ∈ (0, 1/2]. Here, we will demonstrate that in these cases -W (E) is an extended kappa function of the optimal value of the reported probability r ; -The (risk-)corrected reported probability B, which is given by B(E) = w −1 (W (E)), is also an extended kappa function of the optimal value of the reported probability r if the probability weighting function w is the kappa probability weighting function w (λ w ) ν w given in Eq. (36).
Theorem 3 Let τ 1−ν s (r ) E η ν s (r ) be a tau-eta scoring rule prospect with a scoring parameter value ν s ∈ (1/2, 1), r ∈ (0, 1) and let the utility function U be the tau function τ ν u with a parameter value ν u ∈ (0, 1/2]. Then, in the general model, if r ≥ 1/2 and τ 1−ν s (r ) E η ν s (r ) is evaluated according to Eq. (42), then the following hold: (1) The measure W (E) is an extended kappa function of the optimal value of the reported probability r ; namely, where λ = 2, a = 1 (2) The optimal value of the reported probability r can be expressed in terms of W (E) by where the values of parameters λ, a, b, r ν and ν 0 are given by Eqs. (46). where Remark 6 Note that r = r E ∈ (0, 1); that is, we interpret functions W (E) and B(E) in the domain (0, 1) despite the fact that the corresponding expressions would allow r ∈ (a, b). Accordingly, the inverse function of W (E); that is, Therefore, the equation a + b = 1 holds for each of the three statements of Theorem 3.
It is worth mentioning here that the parameters ν u , ν s , λ w and ν w unambiguously determine all the parameters of the functions W (E), r (W (E)) and B(E) in Eqs. (46) and Eqs. (49).
Notice that, based on Theorem 3, the measure W (E) as a function of the optimal value of the reported probability r in Eq. (45), the inverse function of W (E) in Eq. (47) and the (risk-)corrected reported probability B(E) as a function of the optimal value of the reported probability r in Eq. (48) can be written in a common form. Therefore, the main results of Theorem 3 can be summarized by using a common form for these three functions.
Theorem 4 Let τ 1−ν s (r ) E η ν s (r ) be a tau-eta scoring rule prospect with a scoring parameter value ν s ∈ (1/2, 1), r ∈ (0, 1) and let the utility function U be the tau function τ ν u with a parameter value ν u ∈ (0, 1/2]. Then, in the general model, if r ≥ 1/2 and τ 1−ν s (r ) E η ν s (r ) is evaluated according to Eq. (42), then the following functions all have the same form: (1) The weighting measure W (E) of the uncertain event E as a function of the optimal value of the reported probability r = r E ; (2) The optimal value of the reported probability r = r E as a function of the weighting measure W (E); (3) In the case where the weighting function w is the kappa probability weighting function w (λ w ) ν w , the (risk-) corrected reported probability B(E) = w −1 (W (E)) as a function of the optimal value of the reported probability r = r E .
The common formula is where and the variable and parameter values in Eq. (50) and Eq. (51) for the above-listed three functions are given in Table 3. In Table 3, the parameters a and b are given by and Remark 7 Note that in our approach, there are four tuning parameters used to model the abovementioned three functions; namely, ν u , ν s , λ w and ν w . It should be added that once the values of these four tuning parameters have been determined, then all the parameters of the common formula in Eq. (50) and Eq. (51) are unambiguously given as well. Also, if α = 1, then the second term in Eq. (50) is zero f is an extended kappa function. Recall that a + b = 1 (see Remark 6), which implies that if α = b, then 1 − α = a.
Making use of Theorem 4, we now present a step-by-step method that can be used to compute the following functions: 1. (F1) The weighting measure W (E) of the uncertain event E being a function of the optimal value of the reported probability r = r E ; 2. (F2) The inverse function of W (E) (denoted by r (W (E))); 3. (F3) The (risk-)corrected reported probability measure B, which is given by B(E) = w −1 (W (E)), being a function of the optimal value of the reported probability r = r E .
-s for the utility function U (x) = x, x ∈ [0, 1], (i.e., the utility function is the identity function) set the value of parameter ν u to ν u = 1/2.

Remark 8 Note that
Step 3 in the above method gives the following values for parameters a, b and z: -When the utility function is the identity function (ν u = 1/2), using the parameter values given in Eq. (54) and the condition that ν u = 1/2, after direct calculation, we get -When the utility function is the tau function τ ν u with ν u = 1/4, taking into account the parameter values given in Eq. (54) and the condition that ν u = 1/4, after direct calculation, we find that Figures 4 and 5 show the plots of the functions W (E), r (W (E)) and B(E) for the parameter values ν u = 1/2 and ν u = 1/4, respectively, when ν s = 3/4, λ w = 2/3 and ν w = 2/5.
There is a nice aspect of our approach that should be mentioned here. Suppose that we have empirical data for one of the abovementioned three functions. Then, the four tuning parameters can be determined by fitting the function to the empirical data, which also means that we immediately determine the other two functions as well.

Conclusions and future work
The main findings of our study can be summarized as follows. We demonstrated that some important elements of prospect theory can be derived from probabilistic unary operators of continuous-valued logic. Namely, we showed that -The functions in the quadratic scoring rule prospect can be generated by the probabilistic unary operators of continuous-valued logic; -Prelec's probability weighting function is a special case of the probabilistic unary operator; -The widely applied power utility functions can be generated by the probabilistic unary operators.
Next, we pointed out that the extended kappa function, the kappa function and its special cases, the tau and the eta functions, can be applied in prospect theory as viable alternatives to -Functions in the quadratic scoring rule prospect; -Power utility functions; -Prelec's probability weighting function; -The Ostaszewski, Green and Myerson probability weighting function; -The (risk-)corrected reported probability function (inverse of Prelec's probability weighting function).
Also, by making use of the tau and the eta functions, we introduced the tau-eta scoring rule prospect τ 1−ν s (r ) E η ν s (r ), where ν s ∈ (1/2, 1), and said that if this is evaluated with a utility function U which is a tau function τ ν u with a parameter value ν u ∈ (0, 1/2], then -W (E) is an extended kappa function of the optimal value of the reported probability r ; -The (risk-)corrected reported probability measure B, which is given by B(E) = w −1 (W (E)), is also an extended kappa function of the optimal value of the reported probability r if the probability weighting function w is the kappa probability weighting function w (λ w ) ν w given in Eq. (36).
Furthermore, we demonstrated that if a tau-eta scoring rule prospect is evaluated with a utility function U which is a tau function τ ν u with a parameter value ν u ∈ (0, 1/2]) and the kappa probability weighting function w (λ w ) ν w is used in probabilistic sophistication, then -The measure W (E) as a function of the optimal value of the reported probability r ; -The inverse function of W (E); -The (risk-)corrected reported probability B(E) being a function of the optimal value of the reported probability r ; can all be written in a common form. In practice, this result can be utilized as follows. If we have empirical data for one of the abovementioned three functions, then the four tuning parameters can be determined by fitting the function to the empirical data, which means that we immediately obtain the other two functions at the same time.
As part of our future research plans, we would like to find how the kappa function and the common formula given in Eq. (50) can be fitted to empirical data. Also, we plan to investigate how the weighting measure W could be modeled by λ-additive or ν-additive measures (see Dombi and Jónás (2019)). And, we plan to see whether the constructions of fuzzy affine models (see, e.g., Ji et al. (2021Ji et al. ( , 2022) can be adapted to quadratic scoring rules.
Proof of Proposition 6. Let λ = 0. In this case, based on the properties of the kappa function, κ (λ) ν,ν 0 is a bijective function. By a direct calculation, we get that the inverse of function κ (λ) ν,ν 0 can be written as Now, taking into account the definition of the kappa function in Eq. (19), from Eq. (61), we get that κ ν 0 ,ν (x) holds for any x ∈ (0, 1).
Proof of Proposition 9. Since every tau function and every eta is a kappa function, the proposition immediately follows from Corollary 5.
Proof of Proposition 10. Noting that the tau function and the eta function are both kappa functions, the proposition immediately follows from Corollary 5.
Proof of Lemma 2. Noting the definition for a tau-eta scoring rule prospect given in Definition 9, the definitions for tau and eta functions given in Definition 7 and Definition 8, respectively, and the fact that r E , r E ∈ (0, 1), if where the values of parameters λ, a, b, r ν and ν 0 are given