# A Unified Theory of Human Judgements and Decision-Making under Uncertainty

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## Abstract

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## 1. Introduction

- (I)
- $\mathcal{S}\in \mathcal{A}$;
- (II)
- for every $E\in \mathcal{A}$, ${E}^{\prime}=\mathcal{S}\setminus E\in \mathcal{A}$;
- (III)
- for every countable family ${\{{E}_{i}\in \mathcal{A}\}}_{i\in \mathbb{N}}$, ${\cup}_{i\in \mathbb{N}}{E}_{i}\in \mathcal{A}$.

- (1)
- $p(\mathcal{S})=1$ (normalization);
- (2)
- for every countable family ${\{{E}_{i}\in \mathcal{A}\}}_{i\in \mathbb{N}}$ of pairwise disjoint sets, that is, ${E}_{i}\cap {E}_{j}=\varnothing ,\phantom{\rule{0.166667em}{0ex}}i\ne j$, $p({\cup}_{i\in \mathbb{N}}{E}_{i})={\sum}_{i\in \mathbb{N}}p({E}_{i})$ (additivity).

- (i)
- Probability judgement errors. Given two events E and F, people judge the conjunction event ‘$E\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}F$’ (disjunction event ‘$E\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}F$’) as more (less) probable than the event E or/and F separately. This entails a “violation” of the of the law of monotonicity (cfr., Equation (4)).
- (ii)
- DM errors. Given two actions f and g, people prefer f over g if they know that an event E occurs, and also if they know that E does not occur, but they prefer g over f if they do not know whether E occurs or not. This entails a “violation” of the law of total probability (cfr., Equation (3)).

- the most natural set-based models of cognition cannot in general be used to represent human judgements and decisions;
- the usual interpretation of human behaviour in terms of classical (Boolean) logic and classical (Kolmogorovian) probability theory is problematical when human judgements and decisions are at stake.

- (a)
- the conceptual entity under study is prepared by the test in a defined state and each participant is confronted with this uniquely prepared state;
- (b)
- an interaction occurs on a cognitive level between the conceptual entity and the participant, which acts as a (measurement) context for the entity;
- (c)
- the interaction is in general non-deterministic and the state of the conceptual entity is transformed into a new state;
- (d)
- the state change makes actual (potential) some properties of the conceptual entity which were potential (actual) in the initial state;
- (e)
- when the responses of all participants are collected, a statistics of outcomes is obtained.

## 2. Probability Judgement Errors: Over- and Under-Extension Effects

- (i)
- Concepts are “vague,” “fuzzy,” or “graded,” notions. For example, people judge an item like Robin as more typical than Stork when typicality is defined with respect to the concept Bird.
- (ii)
- The meaning of a concept depends on the “context” in which the concept is used. For example, an item like Snake, or Spider would score a low typicality with respect to the concept Pet. But, if typicality is defined with respect to the concept Weird Pet, or Pet of a Weird Person, then Snake, or Spider, would score a high typicality.

- (1)
- “Pet-Fish problem”, or “Guppy effect”. People judge an item like Guppy to be a very typical example of the conjunction Pet-Fish, without judging Guppy to be a typical example of either Pet or Fish [46].
- (2)
- “Overextension effects in the conjunction”. For several items, people judge the membership weight of the item with respect to the conjunction to be higher than the membership weight of the item with respect to one or both the component concepts [7].
- (3)
- “Under-extension effects”. For several items, people judge the membership weight of the item with respect to the disjunction to be lower than the membership weight of the item with respect to one or both the component concepts [8].
- (4)
- “Borderline contradictions”. A significant number of people judge the proposition “John is tall and John is not tall” to be true, in particular, for some borderline cases of “John” [47].

- “overextended with respect to the conjunction” if $\mu (A\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}B)>\mu (A)$ or $\mu (A\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}B)>\mu (B)$;
- “underextended with respect to the disjunction” if $\mu (A\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}B)<\mu (A)$ or $\mu (A\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}B)<\mu (B)$;
- “double overextended with respect to the conjunction” if $\mu (A\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}B)>\mu (A)$ and $\mu (A\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}B)>\mu (B)$;
- “double underextended with respect to the disjunction” if $\mu (A\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}B)<\mu (A)$ and $\mu (A\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}B)<\mu (B)$.

## 3. Probability Judgement Errors: Conjunctive and Disjunctive Fallacies

“Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.”

- (i)
- Linda is a bank teller.
- (ii)
- Linda is a bank teller and is active in the feminist movement.

- “Joint test”. The same group of participants were asked to judge the likelihood of both the individual event and the conjunction event.
- “Separate test”. One group of participants were asked to judge the likelihood of the individual event and another group of participants were asked to judge the likelihood of the conjunction event.

- (i)
- Linda is a feminist;
- (ii)
- Linda is a bank teller;
- (iii)
- Linda is a feminist and a bank teller;
- (iv)
- Linda is a feminist or a bank teller.

- (1)
- Human judgement and DM rest on a preliminary subjective assessment of the probabilities of uncertain events.
- (2)
- In these processes, people rely on a limited number of heuristics principles, for example, representativeness, availability, adjustment and anchoring, and so forth.
- (3)
- Judgement heuristics are simple strategies/mental processes that people use to find quick solutions to a complex problem by focusing on the most relevant aspects of it.
- (4)
- These heuristics are generally very useful, but they may also lead to severe and systematic errors, or “biases”, because the ensuing judgements are all based on data of limited validity.
- (5)
- Examples of biases are given by the already mentioned over-/under-estimation of probabilities and other fallacies.

- Kahneman-Tversky explanation. The heuristics of representativeness is introduced to explain the conjunction fallacy. The event “being a feminist and a bank teller” is judged as more representative than the event “being a bank teller” for Linda’s story [5,9]. This explanation can be criticised, as the heuristics hypothesis works well at an intuitive level, but a general theory of heuristics is missing and different heuristics have to be assumed each time to accommodate different types of fallacies [6,48].
- Misunderstanding explanation. People misunderstand the Linda problem, in the sense that they misinterpret the terms “and” and/or “probability” in the story. This explanation can be criticised, as the fallacy also occurs when the terms “and” and “probability” are not mentioned in the story [48].
- Quantum probability explanation. People judge the sentence “Linda is a feminist and a bank teller” as an ordered sequence, namely, “Linda is a feminist”, then “Linda is a bank teller”. The two questions are incompatible in the standard quantum sense, that is, they produce different statistical distributions when asked in a different order [4,6]. This explanation can be criticised, as new tests seem to confirm that the conjunction fallacy is not directly related to question order effects [49].

People were asked in Reference [50] to rank likelihood of the following events.“Bill is 34 years old. He is intelligent, but unimaginative, compulsive and generally lifeless. In college, he was strong in mathematics.”

- (U)
- Bill is a reporter;
- (LU)
- Bill is an accountant and plays jazz for a hobby;
- (UU)
- Bill is a reporter and surfs for a hobby;
- (L)
- Bill works for the Inland Revenue;
- (U)
- Bill plays jazz for a hobby;
- (LL)
- Bill is an accountant and works for the Inland Revenue;
- (U)
- Bill surfs for a hobby;
- (L)
- Bill is an accountant.

## 4. Decision-Making Errors: The Disjunction Effect

“A businessman contemplates buying a certain piece of property. He considers the outcome of the next presidential election relevant. So, […], he asks whether he would buy if he knew that the Democratic candidate were going to win, and decides that he would. Similarly, he considers whether he would buy if he knew that the Republican candidate were going to win, and again finds that he would. Seeing that he would buy in either event, he decides that he should buy, even though he does not know which event obtains […].”

- (i)
- Respondents knew they had won the first gamble.
- (ii)
- Respondents knew they had lost the first gamble.
- (iii)
- Respondents did not know the outcome of the first gamble.

- (1)
- 69% of the respondents who knew they had won the first gamble decided to play again.
- (2)
- 59% of the respondents who knew they had lost the first gamble decided to play again.
- (3)
- 36% of the respondents who did not know the outcome of the first gamble decided to play again.

## 5. Decision-Making Errors: Ellsberg-Type Paradoxes

## 6. Elaboration of a SCoP Formalism for Cognitive Domains

- (i)
- Judgements and decisions are intrinsically and unavoidably probabilistic processes.
- (ii)
- Judgements and decisions involve constructive processes which create rather than record.
- (iii)
- In these processes, context plays a fundamental role in determining the final response among a range of possible alternatives.
- (iv)
- Different and mutually exclusive alternatives generally disturb each other.
- (v)
- The unrestricted validity classical (Boolean) logic and (Kolmogorovian) probability cannot be assumed in these cases.

- (1)
- $\Sigma $ is the set of all states of $\Omega $. A state $p\in \Sigma $ is the result of a preparation procedure of $\Omega $ at the end of which $\Omega $ is in the state p.
- (2)
- $\mathcal{M}$ is the set of all contexts of $\Omega $. A context $e\in \mathcal{M}$ has typically a cognitive nature and interacts with $\Omega $ changing in general its state.
- (3)
- $\mathcal{L}$ is the set of all properties of $\Omega $. A property $a\in \mathcal{L}$ can be either actual or potential for $\Omega $, depending on its state.
- (4)
- $\mu $ is the “state-transition probability function”, that is, an application $\mu :\Sigma \times \mathcal{M}\times \Sigma \u27f6[0,1]$, such that, for every $p,q\in \Sigma $, $e\in \mathcal{M}$, $\mu (q,e,p)$ is the probability that the state p of $\Omega $ is changed to the state q by the context e.
- (5)
- $\nu $ is the “property-applicability probability function”, that is, an application $\nu :\Sigma \times \mathcal{L}\u27f6[0,1]$, such that, for every $p,\in \Sigma $, $a\in \mathcal{L}$, $\nu (p,a)$ is the probability that the property a is actual in the state p of $\Omega $.

#### 6.1. Application of the SCoP Formalism to Human Probability Judgements

#### 6.2. Application of the SCoP Formalism to Human Decisions

## 7. A Quantum Framework to Represent Concepts and Their Combinations

- (i)
- A concept is an entity in a defined state, rather than a container of items.
- (ii)
- A context is a factor that influences the concept and generally changes its state.
- (iii)
- Quantities as typicality, membership, and so forth can be measured on concepts and have different probabilistic values in different states of the concept.

- $Re(\langle A|M|B\rangle )>0$ indicates to constructive interference;
- $Re(\langle A|M|B\rangle )=0$ indicates to absence of interference;
- $Re(\langle A|M|B\rangle )<0$ indicates to destructive interference.

#### 7.1. An Application to Conjunctive and Disjunctive Fallacies

#### 7.2. An Application to the Disjunction Effect

## 8. A Quantum Framework for Ellsberg-Type Paradoxes

- (i)
- (ii)
- (iii)
- successfully models shifts of attitudes towards uncertainty, from ambiguity averse to ambiguity seeking, and viceversa, as due to hope and fear effects [79];
- (iv)

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Table 1.**Test by Fisk and Pidgeon on the conjunction fallacy [50]. Values obtained from the quantum model in Section 7.1 are also reported.

Case | $\mathit{\mu}(\mathit{A})$ | $\mathit{\mu}(\mathit{B})$ | $\mathit{\mu}(\mathit{A}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\mathit{B})$ | ${\mathit{\theta}}_{\mathit{c}}$ | $|\mathit{A}\rangle $ | ${\mathit{e}}^{-\mathit{i}{\mathit{\theta}}_{\mathit{c}}}|\mathit{B}\rangle $ |
---|---|---|---|---|---|---|

LL | 0.84 | 0.62 | 0.71 | 94.65 | (0.92, 0, 0.4) | (0.27, 0.74, −0.62) |

LL | 0.59 | 0.85 | 0.63 | 111.28 | (0.77, 0, 0.64) | (0.32, 0.86, −0.39) |

LU | 0.76 | 0.28 | 0.37 | 111.15 | (0.87, 0, 0.49) | (0.48, 0.23, −0.85) |

LU | 0.85 | 0.31 | 0.42 | 119.82 | (0.92, 0, 0.39) | (0.35, 0.43, −0.83) |

UU | 0.33 | 0.11 | 0.13 | 118.19 | (0.82, 0, 0.57) | (−0.23, −0.91, 0.33) |

**Table 2.**Test by Fisk on the disjunction fallacy [51]. Values obtained from the quantum in Section 7.1 are also reported.

Case | $\mathit{\mu}(\mathit{A})$ | $\mathit{\mu}(\mathit{B})$ | $\mathit{\mu}(\mathit{A}\phantom{\rule{4pt}{0ex}}or\phantom{\rule{4pt}{0ex}}\mathit{B})$ | ${\mathit{\theta}}_{\mathit{d}}$ | $|\mathit{A}\rangle $ | ${\mathit{e}}^{-\mathit{i}{\mathit{\theta}}_{\mathit{d}}}|\mathit{B}\rangle $ |
---|---|---|---|---|---|---|

LL | 0.82 | 0.6 | 0.78 | 74.88 | (0.91, 0, 0.42) | (0.3, 0.72, −0.63) |

LL | 0.85 | 0.62 | 0.72 | 93.60 | (0.92, 0, 0.39) | (0.26, 0.74, −0.62) |

LU | 0.61 | 0.2 | 0.47 | 79.28 | (0.62, 0, 0.78) | (−0.56, −0.7, 0.45) |

LU | 0.82 | 0.31 | 0.62 | 81.02 | (0.91, 0, 0.42) | (0.39, 0.4, −0.83) |

UU | 0.36 | 0.11 | 0.26 | 82.78 | (0.8, 0, 0.6) | (−0.25, −0.91, 0.33) |

UU | 0.3 | 0.08 | 0.23 | 75.04 | (0.84, 0, 0.55) | (−0.19, −0.94, 0.28) |

**Table 3.**Average data on the two-stage gamble tests by Tversky and Shafir (TS 1992) [11], Kühberger, Kamunska and Perner (KKP 2001) [52], and Lambdin and Burdsal (LB 2007) [53] on the disjunction effect. Values obtained from the quantum in Section 7.2 are also reported.

Test | $\mathit{\mu}(\mathit{A})$ | $\mathit{\mu}(\mathit{B})$ | $\mathit{\mu}(\mathit{A}\phantom{\rule{4pt}{0ex}}\mathbf{or}\phantom{\rule{4pt}{0ex}}\mathit{B})$ | ${\mathit{\theta}}_{\mathit{d}}$ | $|\mathit{A}\rangle $ | ${\mathit{e}}^{-\mathit{i}{\mathit{\theta}}_{\mathit{d}}}|\mathit{B}\rangle $ |
---|---|---|---|---|---|---|

TS 1992 | 0.69 | 0.58 | 0.37 | 137.26 | (0.73, 0, 0.68) | (0.61, 0.45, −0.66) |

KKP 2001 | 0.72 | 0.47 | 0.48 | 107.37 | (0.85, 0, 0.53) | (0.45, 0.51, −0.73) |

LB 2007 | 0.63 | 0.45 | 0.41 | 106.75 | (0.79, 0, 0.61) | (0.57, 0.36, −0.74) |

$\mathit{p}({\mathit{E}}_{\mathit{R}})=1/3$ | $\mathit{p}({\mathit{E}}_{\mathit{Y}})+\mathit{p}({\mathit{E}}_{\mathit{B}})=2/3$ | ||
---|---|---|---|

Act | Red | Yellow | Black |

${f}_{1}$ | $100 | $0 | $0 |

${f}_{2}$ | $0 | $0 | $100 |

${f}_{3}$ | $100 | $100 | $0 |

${f}_{4}$ | $0 | $100 | $100 |

Urn I | Urn II | |||
---|---|---|---|---|

Acts | ${\mathit{E}}_{\mathit{R}}$: Red Ball | ${\mathit{E}}_{\mathit{B}}$: Black Ball | ${\mathit{E}}_{\mathit{R}}$: Red Ball | ${\mathit{E}}_{\mathit{B}}$: Black Ball |

$\mathit{p}({\mathit{E}}_{\mathit{R}})\in [\mathbf{0},\mathbf{1}]$ | $\mathit{p}({\mathit{E}}_{\mathit{B}})=\mathbf{1}-\mathit{p}({\mathit{E}}_{\mathit{R}})$ | $\mathit{p}({\mathit{E}}_{\mathit{R}})=\mathbf{1}/\mathbf{2}$ | $\mathit{p}({\mathit{E}}_{\mathit{B}})=\mathbf{1}/\mathbf{2}$ | |

${f}_{1}$ | $100 | $0 | ||

${f}_{2}$ | $100 | $0 | ||

${f}_{3}$ | $0 | $100 | ||

${f}_{4}$ | $0 | $100 |

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Pisano, R.; Sozzo, S.
A Unified Theory of Human Judgements and Decision-Making under Uncertainty. *Entropy* **2020**, *22*, 738.
https://doi.org/10.3390/e22070738

**AMA Style**

Pisano R, Sozzo S.
A Unified Theory of Human Judgements and Decision-Making under Uncertainty. *Entropy*. 2020; 22(7):738.
https://doi.org/10.3390/e22070738

**Chicago/Turabian Style**

Pisano, Raffaele, and Sandro Sozzo.
2020. "A Unified Theory of Human Judgements and Decision-Making under Uncertainty" *Entropy* 22, no. 7: 738.
https://doi.org/10.3390/e22070738