# A Boundedly Rational Decision-Making Model Based on Weakly Consistent Preference Relations

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## Abstract

**:**

## 1. Introduction

## 2. Strongly and Weakly Consistent Preferences

**Definition**

**1.**

**strongly consistent preference**if $\forall x,y\in A\cap B\ne \varphi $, $\left|A\right|\le \left|B\right|$, and we have $x{\mathrm{R}}_{B}y$ if and only if $x{\mathrm{R}}_{A}y$. A binary relation R is said to be a

**weakly**

**consistent**

**preference**if $\forall x,y\in A\cap B\ne \varphi $, $\left|A\right|\le \left|B\right|$, and we have $x{\mathrm{R}}_{B}y$ only if $x{\mathrm{R}}_{A}y$.

## 3. Boundedly Rational Choices Based on Strongly and Weakly Consistent Preferences

- (1)
- The DM is unable to perceive alternatives.
- (2)
- The DM is unable to rank all alternatives.
- (3)
- The DM chooses the alternative still according to the “optimization” principle within bounds of perceptibility and decidability.

#### 3.1. Choice Function and Conditions of Rationality

**nonempty finite**set $X$ and a domain $\mathsf{\Omega}=\left\{A\subset X:A\ne \varphi \right\}$. $X$ denotes all of the available alternatives of the finite set and $\mathsf{\Omega}$ denotes the set of nonempty subsets of $X$. A choice function on $X$ is a mapping $C:\mathsf{\Omega}\to \mathsf{\Omega}$. $C(A)$ of each $A\in \mathsf{\Omega}$ denotes the set of alternatives that in A cannot be ruled out. A reasonable choice function $C$ should satisfy the following postulates [19]:

- (1)
**Availability**: $C(A)\subset A$.- (2)
**Decisiveness**: $A\ne \phi \Rightarrow \left|C(A)\right|\ge 1$.

**Condition**

**1.**

**Condition**

**2.**

**Condition**

**3.**

**Condition**

**4.**

**Proposition**

**1.**

**Proof.**

#### 3.2. The Relationship between Choice Function and Conditions of Rationality

**Definition**

**2.**

**relation**

**dominant**in $A$ in terms of R if and only if $y{\overline{\mathrm{R}}}_{A}x$, $\forall y\in A\cap \overline{B}$.

**Proposition**

**2**

**Proof.**

**Proposition**

**3**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

## 4. An Example of the Choices of Chocolates Combined with Interval Ordinal Numbers

**an interval ordinal number**is defined as follows: For $\forall x\in A$, the utility or value of x is denoted as ${V}_{A}\left(x\right)\in \left[\underset{\_}{{V}_{A}\left(x\right)},\overline{{V}_{A}\left(x\right)}\right]\subseteq \left[0,1\right]$, where $\underset{\_}{{V}_{A}\left(x\right)}$ and $\overline{{V}_{A}\left(x\right)}$ represent the lower bound and the upper bound of ${V}_{A}\left(x\right)$, respectively. Meanwhile, the corresponding relations between perceived preferences and the intervals of ${V}_{A}\left(x\right)$ are as follows [32]: for $\forall x,y\in A$, $x{\mathrm{R}}_{A}y\Rightarrow \underset{\_}{{V}_{A}\left(x\right)}\ge \overline{{V}_{A}\left(y\right)}$ and $x{\mathrm{I}}_{A}y\Rightarrow \underset{\_}{{V}_{A}\left(x\right)}=\overline{{V}_{A}\left(x\right)}=\underset{\_}{{V}_{A}\left(y\right)}=\overline{{V}_{A}\left(y\right)}$ or the alternatives x and y are incomparable. In addition, the intervals of utilities or values are varied about the same alternatives, which the DM perceives in different sets of alternatives. So, it is assumed that $\left[\underset{\_}{{V}_{A}\left(x\right)},\overline{{V}_{A}\left(x\right)}\right]\subseteq \left[\underset{\_}{{V}_{B}\left(x\right)},\overline{{V}_{B}\left(x\right)}\right]$ for $\forall x\in A\cap B,\left|A\right|\le \left|B\right|$.

_{{wxyz}}x is perceived in the set of chocolates {wxyz} and hence also in the subset of chocolates {wxy}. A DM perceives preferences in the big set only if also in the small set, rather than vice versa; e.g., the DM can perceive the preference xRy in the set of chocolates {wxy}, but not in the set of chocolates {wxyz}. Moreover, the DM’s choices satisfy Condition 2 when the DM’s preferences are transitive; e.g., the DM’s perceived preferences in the set of chocolates {wxz} are wR

_{{wxz}}x, xR

_{{wxz}}z, wR

_{{wxz}}z, and the chocolate w is chosen in the small set of chocolates {wz} and hence also in the big set of chocolates {wxz}. Finally, the choices should attain the given utility threshold; e.g., the chosen chocolates in the set of chocolates {wxy} are those with utility values no smaller than $\mu \left(\left\{wxy\right\}\right)=0.4$.

_{{vwxy}}x, wR

_{{vwxy}}x, so v and w are incomparable in the set of chocolates {vwxy} for her. This implies that if vR

_{{vwxy}}x, then wR

_{{vwxy}}x, and hence the DM’s preferences are negatively transitive. Moreover, the preference vR

_{{vwxy}}x is perceived in the set of chocolates {vwxy} and hence also in the set of chocolates {vwxz}. The chocolates v and w are chosen in the set of chocolates {vwxy} and hence also in the set of chocolates {vwxz}. This implies that the DM’s choices satisfy Condition 3.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Properties | Expressions | Properties | Expressions |
---|---|---|---|

Reflexivity | xRx | Transitivity | If xRy and yRz, then xRz |

Irreflexivity | x$\overline{\mathrm{R}}$x | Residual transitivity | If xR^{0}y and yR^{0}z,then xR^{0}z |

Symmetry | If xRy, then yRx | Cross transitivity | If xRy and y R^{0}z (or x R^{0}y and yRz), then xRz |

Asymmetry | If xRy, then y$\overline{\mathrm{R}}$x | Negative transitivity | If x$\overline{\mathrm{R}}$y and y$\overline{\mathrm{R}}$z, then x$\overline{\mathrm{R}}$z |

Acyclicity | If x_{1} R x_{2} R x_{3}…R x_{n}, then x_{n}$\overline{\mathrm{R}}$x_{1} | Completeness | If x$\overline{\mathrm{R}}$y, then yRx |

The Set of Chocolates | v | w | x | y | z | C(∙) | $\mathit{\mu}\left(\mathit{A}\right)$ |
---|---|---|---|---|---|---|---|

{wx} | → | [0.4, 0.45] | [0.35, 0.4] | → | → | {w} | 0.4 |

{vx} | [0.25, 0.3] | → | [0.35, 0.4] | → | → | {x} | 0.35 |

{xy} | → | → | [0.35, 0.4] | [0.25, 0.3] | → | {x} | 0.35 |

{xz} | → | → | [0.35, 0.4] | → | [0.25, 0.3] | {x} | 0.35 |

{wxy} | → | [0. 4, 0.55] | [0.3, 0.4] | [0.2, 0.3] | → | {w} | 0.4 |

{wxz} | → | [0. 4, 0.55] | [0.3, 0.4] | → | [0.2, 0. 3] | {w} | 0.4 |

{xyz} | → | → | [0.3, 0.4] | [0.2, 0.35] | [0.2, 0.35] | {xyz} | 0.3 |

{wxyz} | → | [0. 4, 0.6] | [0.25, 0.4] | [0.2, 0.4] | [0.2, 0.35] | {w} | 0.4 |

{vwxy} | [0.2, 0.3] | [0. 4, 0.6] | [0.25, 0.4] | [0.2, 0.4] | → | {w} | 0.4 |

{vwxz} | [0.2, 0.3] | [0. 4, 0.6] | [0.25, 0.4] | → | [0.2, 0.35] | {w} | 0.4 |

{vxyz} | [0.2, 0.3] | → | [0.25, 0.4] | [0.2, 0.4] | [0.2, 0.35] | {vxyz} | 0.25 |

**Table 3.**Perceived intervals and choices of the DM with acyclic and negatively

**transitive preferences**.

The Set of Chocolates | v | w | x | y | z | C(∙) | $\mathit{\mu}\left(\mathit{A}\right)$ |
---|---|---|---|---|---|---|---|

{wx} | → | [0.4, 0.45] | [0.2, 0.3] | → | → | {w} | 0.4 |

{vx} | [0.4, 0.45] | → | [0.2, 0.3] | → | → | {v} | 0.4 |

{xy} | → | → | [0.2, 0.3] | [0.45, 0.5] | → | {y} | 0.45 |

{xz} | → | → | [0.2, 0.3] | → | [0.35, 0.5] | {z} | 0.35 |

{wxy} | → | [0. 4, 0.55] | [0.2, 0.35] | [0.35, 0.5] | → | {wy} | 0.4 |

{wxz} | → | [0. 4, 0.55] | [0.2, 0.35] | → | [0.35, 0.55] | {wz} | 0.4 |

{xyz} | → | → | [0.2, 0.4] | [0.35, 0.5] | [0.35, 0.55] | {xyz} | 0.35 |

{wxyz} | → | [0. 4, 0.6] | [0.15, 0.4] | [0.35, 0.55] | [0.25, 0.55] | {wyz} | 0.4 |

{vwxy} | [0.4, 0.55] | [0. 4, 0.6] | [0.15, 0.4] | [0.35, 0.55] | → | {vwy} | 0.4 |

{vwxz} | [0.4, 0.55] | [0. 4, 0.6] | [0.15, 0.4] | → | [0.25, 0.55] | {vwz} | 0.4 |

{vxyz} | [0.4, 0.6] | → | [0.15, 0.4] | [0.35, 0.55] | [0.25, 0.55] | {vyz} | 0.4 |

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Wu, X.; Xiao, H.
A Boundedly Rational Decision-Making Model Based on Weakly Consistent Preference Relations. *Symmetry* **2022**, *14*, 918.
https://doi.org/10.3390/sym14050918

**AMA Style**

Wu X, Xiao H.
A Boundedly Rational Decision-Making Model Based on Weakly Consistent Preference Relations. *Symmetry*. 2022; 14(5):918.
https://doi.org/10.3390/sym14050918

**Chicago/Turabian Style**

Wu, Xinlin, and Haiyan Xiao.
2022. "A Boundedly Rational Decision-Making Model Based on Weakly Consistent Preference Relations" *Symmetry* 14, no. 5: 918.
https://doi.org/10.3390/sym14050918