# Structural Holes in Social Networks with Exogenous Cliques

## Abstract

**:**

## 1. Introduction

## 2. The Model

**Definition**

**1.**

#### Equilibrium Concept

**Definition**

**2.**

- (i)
- for any $i\in N$ and every ${s}_{i}\in {S}_{i}$, ${\Pi}_{i}({s}^{PN})\ge {\Pi}_{i}({s}_{i},{s}_{-i}^{PN})$
- (ii)
- for any pair of players $i,j\in N$ and every strategy pair $({s}_{i},{s}_{j})$ in which ${s}_{il}={s}_{il}^{PN},\forall l\ne j\mathrm{and}{s}_{jk}={s}_{jk}^{PN},\forall k\ne i$,$${\Pi}_{i}({s}_{i},{s}_{j},{s}_{-i-j}^{PN})>{\Pi}_{i}({s}_{i}^{PN},{s}_{j}^{PN},{s}_{-i-j}^{PN})\Rightarrow {\Pi}_{j}({s}_{i},{s}_{j},{s}_{-i-j}^{PN})<{\Pi}_{j}({s}_{i}^{PN},{s}_{j}^{PN},{s}_{-i-j}^{PN}).$$

**Definition**

**3.**

- (i)
- for any $i\in N$ and every ${s}_{i}\in {S}_{i}$, ${\Pi}_{i}({s}^{B})\ge {\Pi}_{i}({s}_{i},{s}_{-i}^{B})$
- (ii)
- for any pair of players $i,j\in N$ and every strategy pair $({s}_{i},{s}_{j})$,$${\Pi}_{i}({s}_{i},{s}_{j},{s}_{-i-j}^{B})>{\Pi}_{i}({s}_{i}^{B},{s}_{j}^{B},{s}_{-i-j}^{B})\Rightarrow {\Pi}_{j}({s}_{i},{s}_{j},{s}_{-i-j}^{B})<{\Pi}_{j}({s}_{i}^{B},{s}_{j}^{B},{s}_{-i-j}^{B}).$$

## 3. Results

**Lemma**

**1.**

- (i)
- non-essential agents do not have external links and
- (ii)
- $$c>\frac{1}{2}+\frac{{m}_{j}-1}{3}+\sum _{l\in {C}_{i}(g)}\frac{1}{e(i,l;g)+3}+({m}_{j}-1)\sum _{l\in {C}_{i}(g)}\frac{1}{e(i,l;g)+4},$$

**Lemma**

**2.**

**Corollary**

**1.**

**Proposition**

**1.**

**Proposition**

**2.**

**Corollary**

**2.**

**Example**

**1.**

**Proposition**

**3.**

**Example**

**2.**

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs

**Lemma**

**A1.**

**Lemma**

**A2.**

**Proof**

**of**

**Lemma**

**1.**

- (a)
- Some non-essential department in ${C}_{i}(g)$ (say ${M}_{i}$) is extreme.Part (i) of Lemma 1 implies that ${M}_{i}$ can only have one external link (say $ik$) in a PNE network. This link should be profitable for both i and k. Therefore, creating a new critical link $jk$ bridging ${C}_{i}(g)$ and ${C}_{j}(g)$ would be (weakly) profitable for both j and k whenever ${m}_{j}\ge {m}_{i}$. Thus, ${M}_{j}$ should be smaller than the extreme department ${M}_{i}$ in any PNE network.
- (b)
- All non-essential departments in ${C}_{i}(g)$ are non-extreme.Let ${M}_{i}$ be a non-essential department in ${C}_{i}(g)$. Notice that Part (i) of Lemma 1 implies that only one agent (say i) in ${M}_{i}$ can have external links. Since ${M}_{i}$ is non-extreme, player i has two or more external links, i.e., ${\eta}_{i}(g)\ge 2$. Player i’s payoff can be written as:$$\begin{array}{ccc}\hfill {\Pi}_{i}& =& \sum _{r=0}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{r+2}+({m}_{i}-1)\left(\frac{{\eta}_{i}^{0}(g)-({m}_{i}-1)}{3}+\sum _{r=1}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{r+3}\right)-{\eta}_{i}(g)c\hfill \end{array}$$$$\frac{1}{{\eta}_{i}(g)}\left[\sum _{r=0}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{r+2}+({m}_{i}-1)\left(\frac{{\eta}_{i}^{0}(g)-({m}_{i}-1)}{3}+\sum _{r=1}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{r+3}\right)\right]\ge c$$$$\begin{array}{ccc}\hfill \Delta {\Pi}_{j}& >\hfill & \frac{1}{2}+\sum _{r=0}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{r+3}+({m}_{j}-1)\left(\frac{1}{3}+\sum _{r=0}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{r+4}\right)-c\hfill \\ & >\hfill & \sum _{r=0}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{2r+4}+({m}_{j}-1)\left(\sum _{r=0}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{2r+6}\right)-c\hfill \\ & \ge \hfill & \frac{1}{{\eta}_{i}(g)}\left[\sum _{r=0}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{r+2}+({m}_{j}-1)\sum _{r=0}^{{R}_{i}(g)}\frac{{\eta}_{i}^{r}(g)}{r+3}\right]-c\hfill \end{array}$$

**Proof**

**of**

**Lemma**

**2.**

**Proof**

**of**

**Proposition**

**1.**

- (a)
- Agent i is contained in an essential department ${M}_{i}$.Let ${M}_{j}$ and ${M}_{k}$ be two departments, such that the essential player i is part of all paths between the members of ${M}_{j}$ and ${M}_{k}$. Since there does not exist any additional essential agent, individual i must have two or more links to at least one of the ${M}_{i}$-groups containing ${M}_{j}$ or ${M}_{k}$. Take two agents from ${M}_{j}$ and ${M}_{k}$ linked to the essential department ${M}_{i}$. Notice that at least one of them is linked to i. Then, consider the deviation consisting of deleting their links to ${M}_{i}$ and forming a link between them.8 Without increasing their costs, they will circumvent the essential player i to access the other ${M}_{i}$-group. Therefore, they will strictly increase their access payoff. Therefore, they have incentives to deviate, contradicting the initial statement.
- (b)
- Player i is contained in a non-essential department ${M}_{i}$.There are two types of non-essential departments. Next, I develop them in turn:
- (i)
- ${M}_{i}$ is an extreme department. Let i be an essential agent in ${M}_{i}$. Since i is the unique essential agent in ${M}_{i}$, she/he must have at least two links, say $ik,ij\in g$, where $k,j\in {M}_{j}$. Consider the deviation in which j severs the link $ij$ and forms a new link $jl$ where $l\in {M}_{i}$ and $l\ne i$. The marginal payoff for agent j will be positive, given that without increasing her/his linking costs, the essential agent i will be avoided in order to reach the members of ${M}_{i}$. On the other hand, by forming such a link, agent l would eliminate the essential agent i to reach the rest of the component at the cost of one additional link (c). For a sufficiently large component, agent l would also have incentives to deviate. Consequently, an extreme department cannot include essential agents in a BE network.
- (ii)
- ${M}_{i}$ is a non-extreme and non-essential department. Notice that $i\in {M}_{i}$ can be essential if and only if she/he is the unique agent in ${M}_{i}$ with external links. Let $ij\in g$ where $j\notin {M}_{i}$. Consider that j severs the link $ij$ and simultaneously forms a link $jk$ where $k\in {M}_{i}$ and $k\ne i$. With this deviation, j will have two different paths to communicate with any member of ${M}_{i}$ without any additional cost. Then, the marginal payoff of agent j will be positive since she/he avoids one essential agent. On the other hand, agent k will circumvent the essential agent i to access the rest of the component with an additional cost of c. Thus, for a sufficiently large component, k will also deviate. Consequently, this kind of department cannot contain essential agents either.

**Proof**

**of**

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**3.**

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1. | For example, a strategy profile in which no player announces a link (resulting in the empty network) is always a Nash equilibrium. |

2. | |

3. | Condition (ii) of Lemma 1 specifies the upper-bound of the size of isolated departments as a function of c. |

4. | For example, the empty network is a PNE if $c>\frac{1}{2}+\frac{1}{3}({m}_{j}-1)+\frac{1}{3}({m}_{i}-1)+\frac{1}{4}({m}_{j}-1)({m}_{i}-1)$ for any pair of isolated departments ${M}_{i}$ and ${M}_{j}$. |

5. | In the spirit of Goyal and Vega-Redondo (2007) (see their Footnote 10), this requirement can be stated more precisely by saying that there is a function $F(c)$ such that if the size of the multi-department component exceeds $F(c)$, this result is obtained. This is the interpretation of most of the results in this paper. The particular form of the lower-bound F would depend on the result under consideration, and its specification is beyond the scope of that paper. |

6. | This contrasts with [9], where the unique equilibrium network is efficient. |

7. | An ${M}_{j}$-group with t agents is sufficiently small if $c>\tilde{c}(t)$. |

8. | Notice that such a deviation is not allowed under the PNE concept. |

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**MDPI and ACS Style**

Rubí-Barceló, A. Structural Holes in Social Networks with Exogenous Cliques. *Games* **2017**, *8*, 32.
https://doi.org/10.3390/g8030032

**AMA Style**

Rubí-Barceló A. Structural Holes in Social Networks with Exogenous Cliques. *Games*. 2017; 8(3):32.
https://doi.org/10.3390/g8030032

**Chicago/Turabian Style**

Rubí-Barceló, Antoni. 2017. "Structural Holes in Social Networks with Exogenous Cliques" *Games* 8, no. 3: 32.
https://doi.org/10.3390/g8030032