# Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance

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## Abstract

**:**

## 1. Introduction

## 2. Integral-Order Stochastic Dominance

#### 2.1. First- and Second-Order Stochastic Dominance

**Definition**

**1**(First-Order SD)

**.**

**Definition 2**(First-Order SD (Utility))

**.**

**Definition 3**(Second-Order SD)

**.**

**Definition 4**(Second-Order SD (Utility))

**.**

#### 2.2. Higher-Order Stochastic Dominance

**Definition**

**5**(Fishburn’s nth-Order SD)

**.**

**Definition**

**6**(nth-Order SD (Utility))

**.**

**Definition**

**7**(Conventional nth-Order SD)

**.**

- 1.
- ${F}_{n}\left(x\right)\le {G}_{n}\left(x\right)$ for all $x\in [a,b]$; and
- 2.
- ${F}_{j}\left(b\right)\le {G}_{j}\left(b\right)$ for all $j\in \{1,2,\dots ,n-1\}$.

#### 2.3. Discrete Utilities

**Theorem**

**1.**

**Proof.**

## 3. Discrete Fishburn’s Fractional-Order Stochastic Dominance

- 1.
- The domains of the probability mass functions are $\{0,1,\dots ,b\}$, unless otherwise specified.
- 2.
- The utility functions are discrete.

#### 3.1. Definition

**Theorem**

**2.**

**Proof.**

- The preservation of the SD hierarchy: for any $\nu >0$ and $\alpha \in [1,\infty )$, $F{\u2ab0}_{\alpha}G$ implies $F{\u2ab0}_{\alpha +\nu}G$;
- An equivalent definition by utilities: find the utility classes that are congruent with the $\alpha $th-order SD;
- The monotonicity of utility classes: for integral nth-order SD, show that every u in the utility class that we found satisfies ${(-1)}^{p+1}{\mathsf{\Delta}}^{p}u\left(i\right)\ge 0$ for all $p=1,2,\dots ,n$.

#### 3.2. Stochastic Dominance Hierarchy

**Theorem**

**3.**

- $F{\u2ab0}_{\alpha}G$ implies $F{\u2ab0}_{\alpha +\nu}G$;
- $F{\succ}_{\alpha}G$ implies $F{\succ}_{\alpha +\nu}G$.

**Proof.**

#### 3.3. Utility Classes

- $F{\u2ab0}_{u}G$ means that ${\sum}_{x=0}^{b}u\left(x\right)f\left(x\right)\ge {\sum}_{x=0}^{b}u\left(x\right)g\left(x\right)$ for all $u\in {U}_{\alpha}$;
- $F{\succ}_{u}G$ means that ${\sum}_{x=0}^{b}u\left(x\right)f\left(x\right)>{\sum}_{x=0}^{b}u\left(x\right)g\left(x\right)$ for all $u\in {U}_{\alpha}^{\prime}$.

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SD | Stochastic dominance |

## Appendix A. Proof of Theorem 1

## Appendix B. Proof of Theorem 5

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i | 0 | 1 | 2 | 3 |
---|---|---|---|---|

${p}_{i}$ | $0.1$ | $0.2$ | $0.3$ | $0.4$ |

${F}_{1}\left(i\right)={\displaystyle \sum _{t=0}^{x}}{p}_{t}$ | $0.1$ | $0.3$ | $0.6$ | 1 |

${F}_{2}\left(i\right)={\displaystyle \sum _{t=0}^{x}}{F}_{1}\left(t\right)$ | $0.1$ | $0.4$ | 1 | 2 |

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Yin, H.H.F.; Wang, X.; Mak, H.W.L.; Au Yong, C.S.; Chan, I.Y.Y.
Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance. *Axioms* **2023**, *12*, 564.
https://doi.org/10.3390/axioms12060564

**AMA Style**

Yin HHF, Wang X, Mak HWL, Au Yong CS, Chan IYY.
Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance. *Axioms*. 2023; 12(6):564.
https://doi.org/10.3390/axioms12060564

**Chicago/Turabian Style**

Yin, Hoover H. F., Xishi Wang, Hugo Wai Leung Mak, Chun Sang Au Yong, and Ian Y. Y. Chan.
2023. "Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance" *Axioms* 12, no. 6: 564.
https://doi.org/10.3390/axioms12060564