# Cumulative Prospect Theory Version with Fuzzy Values of Outcome Estimates

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## Abstract

**:**

## 1. Introduction

- Decision-making problems under certainty. The outcomes of alternative decisions coincide with the decisions themselves.
- Decision-making problems under risk conditions. The outcomes of alternative decisions depend on the decisions themselves and uncertain factors (random events). The probabilities of occurrence of relevant random events are given.
- Decision-making problems under conditions of uncertainty. The outcomes of alternative decisions depend on the decisions themselves and uncertain factors (random events). The probabilities of random events occurrence are not specified.

## 2. Probability Weighting

## 3. Rank Dependent Utility

- the probability of getting a win greater than 40 is equal 0, since prospect A has no gain with such an estimate;
- the probability of getting a win greater than 30 is equal to $p(1)=0.40$, since such a gain is ensured by outcome (1);
- the probability of getting a win greater than 15 is equal to $p(1)+p(2)=0.40+0.40=0.80$ since any of the outcomes, (1) or (2), provides such a gain;
- the probability of getting a win greater than 0 equals $p(1)+p(2)=p(3)=0.40+0.40+0.20=1.00$, since the occurrence of any from prospect outcomes leads to a positive gain.

- the probability of getting a win less than 40 is $p(2)+p(3)=0.40+0.20=0.60$;
- the probability of getting a win less than 30 is $p(3)=0.20.$

## 4. Cumulative Prospect Theory

## 5. Cumulative Prospect Theory’s Version with Fuzzy Outcome Estimates

- A group of methods that use the distance values from the centroids of fuzzy numbers to certain original points (Wang et al. 2006; Cheng 1998).
- A group of methods that use the specific areas as an evaluation function (Rao and Shankar 2012; Wang and Lee 2008).
- A group of methods that use the concepts of maximum and minimum values (Chou et al. 2011).

## 6. Illustrative Example

Prospect A: | Prospect B: |

$\tilde{v}(1)=(1.232,1.237,1.241)$ | $\tilde{v}\left(1\right)=\left(1.222,1.227,1.232\right)$ |

$\tilde{v}\left(2\right)=\left(1.210,1.216,1.222\right)$ | $\tilde{v}\left(2\right)=\left(1.195,1.203,1.210\right)$ |

$\tilde{v}\left(3\right)=\left(1.194,1.203,1.210\right)$ | $\tilde{v}\left(3\right)=\left(1.175,1.185,1.195\right)$ |

$v\left(4\right)=\left(-2.644,-2.614,-2.576\right)$ | $\tilde{v}\left(4\right)=\left(-2.576,-2.524,-2.439\right)$ |

$\tilde{v}\left(5\right)=\left(-2.689,-2.666,-2.644\right)$ | $\tilde{v}\left(5\right)=\left(-2.643,-2.614,-2.576\right)$ |

$\tilde{v}\left(6\right)=\left(-2.722,-2.704,-2.686\right)$ | $\tilde{v}\left(6\right)=\left(-2.772,-2.706,-2.688\right)$ |

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Graphical representation of good and bad news probabilities for prospect $A=\left({k}_{1},{p}_{1},\dots ,{k}_{n},{p}_{n}\right)$ with ${k}_{1}>{k}_{2}>\dots >{k}_{n}$.

**Figure 3.**Schematic representation of the concept of decision weight in rank-dependent utility theory.

**Figure 4.**Graphs of probability weighting functions ${w}^{+}(p)$, ${w}^{-}(p)$, by expressions (12), (13).

**Figure 8.**Graphs of the membership functions of the generalized fuzzy values ${\tilde{V}}_{PT}\left(A\right)$, ${\tilde{V}}_{PT}\left(B\right)$.

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Uzhga-Rebrov, O.; Grabusts, P.
Cumulative Prospect Theory Version with Fuzzy Values of Outcome Estimates. *Risks* **2021**, *9*, 72.
https://doi.org/10.3390/risks9040072

**AMA Style**

Uzhga-Rebrov O, Grabusts P.
Cumulative Prospect Theory Version with Fuzzy Values of Outcome Estimates. *Risks*. 2021; 9(4):72.
https://doi.org/10.3390/risks9040072

**Chicago/Turabian Style**

Uzhga-Rebrov, Oleg, and Peter Grabusts.
2021. "Cumulative Prospect Theory Version with Fuzzy Values of Outcome Estimates" *Risks* 9, no. 4: 72.
https://doi.org/10.3390/risks9040072