#
Attaining Fairness in Communication for Omniscience^{ †}

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Summary of Main Results

#### 1.2. Organization

## 2. Communication for Omniscience

#### 2.1. Preliminaries

#### 2.2. Fairness

**Example**

**1.**

## 3. Decomposable Coalitional Game

#### 3.1. Coalition Game Model

**Example**

**2.**

#### 3.2. Core

#### 3.3. Decomposition

**Lemma**

**1**

**.**The game $\mathsf{\Omega}(V,{\widehat{f}}_{\alpha})$ can be decomposed by the fundamental partition ${\mathcal{P}}^{*}$ so that

- (a)
- the dimension of ${\mathcal{R}}^{*}\left(V\right)$ is $\left|V\right|-|{\mathcal{P}}^{*}|$ and$$\begin{array}{cc}\hfill {\mathcal{R}}^{*}\left(V\right)& \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\underset{C\in {\mathcal{P}}^{*}}{\u2a01}{\mathcal{R}}^{*}\left(C\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\left\{\underset{C\in {\mathcal{P}}^{*}}{\u2a01}{\mathbf{r}}_{C}:{\mathbf{r}}_{C}\in {\mathcal{R}}^{*}\left(C\right),C\in {\mathcal{P}}^{*}\right\}.\hfill \end{array}$$
- (b)
- The following holds for any ${\mathbf{r}}_{V}\in {\mathcal{R}}^{*}\left(V\right)$:
- (i)
- For any $C,{C}^{\prime}\in {\mathcal{P}}^{*}$ such that $C\ne {C}^{\prime}$, ${\mathbf{r}}_{V}+\u03f5({\chi}_{i}-{\chi}_{j})\notin {\mathcal{R}}^{*}\left(V\right)$, for all $\u03f5>0$, $i\in C$ and $j\in {C}^{\prime}$;
- (ii)
- For all $C\in {\mathcal{P}}^{*}$, ${\mathbf{r}}_{V}+\u03f5({\chi}_{i}-{\chi}_{j})\in {\mathcal{R}}^{*}\left(V\right)$ for some $\u03f5>0$ and $i,j\in C$.

#### Interpretation

**Example**

**3.**

## 4. Shapley Value

#### 4.1. Decomposition

**Theorem**

**1.**

**Proof.**

**Example**

**4.**

#### 4.2. Complexity and Approximation

**Example**

**5.**

## 5. Egalitarian Solution

**Example**

**6.**

#### 5.1. Steepest Descent Algorithm

**Lemma**

**2.**

**Proof.**

Algorithm 1: Steepest descent algorithm (SDA). |

input: a positive integer $K\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}|{\mathcal{P}}^{*}|-1$ and an initial point ${\mathbf{r}}_{V}^{\left(0\right)}\in {\mathcal{R}}^{*}\left(V\right)\cap {\mathbb{Q}}_{|{\mathcal{P}}^{*}|-1}^{\left|V\right|}$ output: ${\mathbf{r}}_{V}^{\left(n\right)}$, the minimizer of (12) |

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Example**

**7.**

#### 5.2. Dependence Function

#### 5.3. Complexity and Distributed Implementation

**Example**

**8.**

#### 5.4. Decomposition

**Theorem**

**3.**

**Remark**

**2.**

**Example**

**9.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Preliminaries

## Appendix B. Background on Coalitional Game

**Definition**

**A1**

**.**A convex game $\mathsf{\Omega}(V,f)$ with the characteristic cost function f is decomposable if

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**Figure 1.**The 5-user system with $V\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{1,\dots ,5\}$ in Example 1. The users encode and broadcast ${\mathsf{Z}}_{i}$s so as to attain omniscience of the source ${\mathsf{Z}}_{V}$. In the corresponding CCDE problem, each ${\mathsf{W}}_{j}$ denotes a packet that belongs to a field ${\mathbb{F}}_{q}$, and each user $i\in V$ broadcasts linear combinations of ${\mathsf{Z}}_{i}$ to help others recover all packets in ${\mathsf{Z}}_{V}$.

**Figure 2.**The core ${\mathcal{R}}^{*}\left(\{1,4,5\}\right)$ of the subgame $\mathsf{\Omega}(\{1,4,5\},{\widehat{f}}_{{R}^{*}})$ of the 5-user system in Figure 1.

**Figure 3.**For the core ${\mathcal{R}}^{*}\left(\{1,4,5\}\right)$ of the subgame $\mathsf{\Omega}(\{1,4,5\},{\widehat{f}}_{{R}^{*}})$, the extreme point set is $\mathrm{EX}\left(\{1,4,5\}\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{(\frac{3}{2},4,0),(\frac{3}{2},\frac{3}{2},\frac{5}{2}),(1,\frac{9}{2},0),(1,2,\frac{5}{2})\}$, the mean value of which is the Shapley value ${\widehat{\mathbf{r}}}_{\{1,4,5\}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}(\frac{5}{4},3,\frac{5}{4})$. We apply the random permutation method twice as in Example 5. We randomly generate 3 permutations of 1, 4 and 5 each time and get the two approximations of ${\widehat{\mathbf{r}}}_{\{1,4,5\}}$. In this figure, the path to $(1,\frac{9}{2},0)$ shows an example of how the Edmond algorithm (Algorithm 3 in [8]) finds the vertex $(1,\frac{9}{2},0)$ corresponding to the permutation $(4,5,1)$.

**Figure 4.**The error measured by the ${\ell}_{1}$-norm ${\parallel {\mathbf{r}}_{V}^{\left(n\right)}-{\mathbf{r}}_{V}^{*}\parallel}_{1}$ of the estimation sequence $\left\{{\mathbf{r}}_{V}^{\left(n\right)}\right\}$ generated by the SDA algorithm in Example 7 to determine the fractional egalitarian solution in ${\mathcal{R}}^{*}\left(V\right)$, the minimizer of $min\left\{{\sum}_{i\in V}{r}_{i}^{2}:{\mathbf{r}}_{V}\in {\mathcal{R}}^{*}\left(V\right)\cap {\mathbb{Q}}_{{\left|\mathcal{P}\right|}^{*}-1}^{\left|V\right|}\right\}$. The error linearly decreases to zero with gradient $-1$; i.e., the ${\ell}_{1}$-norm ${\parallel {\mathbf{r}}_{V}^{\left(n\right)}-{\mathbf{r}}_{V}^{*}\parallel}_{1}$ is reduced by $\frac{2}{|{\mathcal{P}}^{*}|-1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1$ in each iteration.

**Figure 5.**By applying the SDA algorithm to the subgame $\mathsf{\Omega}(\{1,4,5\},{\widehat{f}}_{{R}^{*}})$ of the 5-user system in Example 1 with the initial point ${\mathbf{r}}_{\{1,4,5\}}^{\left(0\right)}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}(1,\frac{9}{2},0)$, we get the estimation sequence $\left\{{\mathbf{r}}_{\{1,4,5\}}^{\left(n\right)}\right\}$ resulting an update path toward the fractional egalitarian solution ${\mathbf{r}}_{\{1,4,5\}}^{*}$, the minimizer of $min\left\{{\sum}_{i\in \{1,4,5\}}{r}_{i}^{2}:{\mathbf{r}}_{\{1,4,5\}}\in {\mathcal{R}}^{*}\left(\{1,4,5\}\right)\cap {\mathbb{Q}}_{|{\mathcal{P}}^{*}|-1}^{3}\right\}$.

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**MDPI and ACS Style**

Ding, N.; Sadeghi, P.; Smith, D.; Rakotoarivelo, T.
Attaining Fairness in Communication for Omniscience. *Entropy* **2022**, *24*, 109.
https://doi.org/10.3390/e24010109

**AMA Style**

Ding N, Sadeghi P, Smith D, Rakotoarivelo T.
Attaining Fairness in Communication for Omniscience. *Entropy*. 2022; 24(1):109.
https://doi.org/10.3390/e24010109

**Chicago/Turabian Style**

Ding, Ni, Parastoo Sadeghi, David Smith, and Thierry Rakotoarivelo.
2022. "Attaining Fairness in Communication for Omniscience" *Entropy* 24, no. 1: 109.
https://doi.org/10.3390/e24010109