#
Additively Separable Hedonic Games with Social Context^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Our Results

#### 1.2. Related Work

#### 1.3. Paper Organization

## 2. Model

## 3. Nash Stable Outcomes

**Theorem**

**1.**

**Proof.**

## 4. Price of Anarchy

**Proposition**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

- ${v}_{i,j}={v}_{j,i}=\frac{1}{1+\alpha}$ for every $(i,j)\in E$;
- ${v}_{i,j}={v}_{j,i}=1$ for all pairs $(i,j)\notin E$ such that $i\in A$ and $j\in B$;
- ${v}_{i,j}=0$ for all remaining pairs.

## 5. Price of Stability

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 6. Open Problems

- the problem in which valuations can be different from zero only between players $i,j$ for which edge $(i,j)$ belongs to the social graph (and not for all pairs of players);
- the setting in which the valuations are not symmetric;
- the problem in which an edge-weighted social graph is considered, with the weights denoting how important is a player for another one;
- different combinations of the classical utilities of the friends, that in this first work have been combined in an additive way;
- the problem defined with player-specific parameters ${\alpha}_{1},\dots ,{\alpha}_{n}$, in which every player has a different degree of altruism.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A social network G, a coalition structure $\mathcal{C}$ and the non-null valuations ${v}_{i,j}$.

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**MDPI and ACS Style**

Monaco, G.; Moscardelli, L.; Velaj, Y.
Additively Separable Hedonic Games with Social Context. *Games* **2021**, *12*, 71.
https://doi.org/10.3390/g12030071

**AMA Style**

Monaco G, Moscardelli L, Velaj Y.
Additively Separable Hedonic Games with Social Context. *Games*. 2021; 12(3):71.
https://doi.org/10.3390/g12030071

**Chicago/Turabian Style**

Monaco, Gianpiero, Luca Moscardelli, and Yllka Velaj.
2021. "Additively Separable Hedonic Games with Social Context" *Games* 12, no. 3: 71.
https://doi.org/10.3390/g12030071