Social and Economic Network Formation: A Dynamic Model

Abstract

We study the dynamics of a game-theoretic network formation model that yields large-scale small-world networks. So far, mostly stochastic frameworks have been utilized to explain the emergence of these networks. On the other hand, it is natural to seek for game-theoretic network formation models in which links are formed due to strategic behaviors of individuals, rather than based on probabilities. Inspired by Even-Dar and Kearns’ model [8], we consider a more realistic framework in which the cost of establishing each link is dynamically determined during the course of the game. Moreover, players are allowed to put transfer payments on the formation and maintenance of links. Also, they must pay a maintenance cost to sustain their direct links during the game. We show that there is a small diameter of at most 4 in the general set of equilibrium networks in our model. We achieved an economic mechanism and its dynamic process for individuals which firstly; unlike the earlier model, the outcomes of players’ interactions or the equilibrium networks are guaranteed to exist. Furthermore, these networks coincide with the outcome of pairwise Nash equilibrium in network formation. Secondly; it generates large-scale networks that have a rational and strategic microfoundation and demonstrate the main characterization of small degree of separation in real-life social networks. Furthermore, we provide a network formation simulation that generates small-world networks.

Appendix: Omitted Proofs

Proof of Lemma 2

The proof is inspired by the arguments in [10]. First, we show the related inequality in Definition 5 holds for the case when the subset $l_i(G)$ consists of two distinct links $ij$ and $ik$, which is indicated in below inequality.

$$\begin{aligned} mu_i(G, ij+ik) \ge mu_i(G, ij) + mu_i(G, ik) \end{aligned}$$

(5)

If we consider any player such as u in network $G$, the distance between $i$ and u ($d_G(i,u)$) contributes to the distance expenses in $i$’s utility. It is important to note that removing any link such as $ij$ or $ik$ from the network $G$ cannot decrease this distance, however if the removed link belongs to the shortest path between $i$ and u in $G$, then the distance would be increased. This argument can be extended to removing two links such as $ij$ and $ik$ from $G$.

$$\begin{aligned} d_G(i,u) \le d_{G-ij}(i,u) \le d_{G-ij-ik}(i,u) \end{aligned}$$

(6)

$$\begin{aligned} d_G(i,u) \le d_{G-ik}(i,u) \le d_{G-ij-ik}(i,u) \end{aligned}$$

(7)

In computing the marginal utilities of networks $G-ik, G-ij$, and $G-ij-ik$, we should note that the link-prices of removed links cannot be refunded for player $i$.

$$\begin{aligned} mu_i(G, ij) = - \sum _{u \ne i} ( d_{G}(i,u) - d_{G-ij}(i,u) ) - c_{ij} - t_{ij}^i \end{aligned}$$

(8)

$$\begin{aligned} mu_i(G, ik) = - \sum _{u \ne i} ( d_{G}(i,u) - d_{G-ik}(i,u) ) - c_{ik} - t_{ik}^i \end{aligned}$$

(9)

$$\begin{aligned} mu_i(G, ij+ik) = - \sum _{u \ne i} ( d_{G}(i,u) - d_{G-ij-ik}(i,u) ) - c_{ij} - c_{ik} - t_{ij}^i - t_{ik}^i \end{aligned}$$

(10)

According to Inequalities (6) and (7), we can simply imply the Inequality (5). Moreover, we can easily extend this argument for any subset of links $l_i(G)$.       $\square$

Proof of Lemma 4

The set $S_i^d$ consists of players in the neighborhood of $i$ within a distance at most $d$. Furthermore, for each of these players such as $k$ in the set $S_i^d$, according to Lemma 3, we consider the set $T_{G(\mathbf t )}(i,k)$. All players outside of this set should use one of players such as $k$ in their shortest path to $i$. As a result, we can cover all players outside the set $S_i^d$ by allocating a set $T_{G(\mathbf t )}(i,k)$ to $i$ for all players in set $S_i^d$. By doing so, an upper bound of $|T_{G(\mathbf t )}(i,k)||S_i^d| + |S_i^d|$ for the number players in network ($n$) is achieved.

In order to obtain an upper bound for the set $T_{G(\mathbf t )}(i,k)$ in wide range of different possible choices for $i$ and $k$, we define $c$ to be the maximum maintenance cost for all possible links in network. According to Remark 1, this is an upper bound for the all possible direct transfer payments in network as well, hence, $|T_{G(\mathbf t )}(i,k)|\le \dfrac{d^\alpha + 2c}{d-1}$. By substituting the upper bounds of $T_{G(\mathbf t )}(i,k)$ and $S_i^d$ in $|T_{G(\mathbf t )}(i,k)||S_i^d|+|S_i^d|\ge n$, the desired inequality can be achieved.       $\square$

Proof of Lemma 5

Let $G$ be an arbitrary instance from the set of equilibrium networks in our model, which are the set of pairwise stable networks with transfer ($G\in PS^t(u)$), given the utility function $u(.)$ in (1). Also, let $\mathbf t$ be the profile of strategies for players that forms $G$. Further, assume that the largest distance between any two players (or diameter) in network $G$ exists between two players $i$ and $j$. We denote $\Delta$ to be the size this distance. Note that the pair of $i$ and $j$ is not necessarily unique.

Based on the stable state, we can imply that creation $ij$ is not beneficial for neither $i$ nor $j$. If $j$ wants to establish a link to $i$, the left side of Inequality (12) is a lower bound for the $j$’s benefit that comes from the reduced distances to players in $S_i^2$. This set includes $i$ itself and two subsets of players that are in distance 1 (type 1) and 2 (type 2) from $i$. First, let $k$ represents players in $S_i^2$ such that their distances to $j$ can be reduced by adding $ij$, as a fraction with respect to all players in $|S_i^2|$. Moreover, let $h_1$ represents player $i$ itself as a fraction with respect to all players in $|S_i^2|$. By establishing $ij$, $j$’s distance to $i$ reduced by $\Delta -1$.

Furthermore, let $h_2$ and $h_3$ represent the fractions of the number of type 1 players and type 2 players, respectively, in $S_i^2$. Their reduced distances for $j$ is computed according to the initial distances of these two types of players in $S_i^2$ from $j$. Among the type 1 players, there are two subsets of players that $g_1$ and $g_2$ are their fractions with distance of $\Delta -1$ and $\Delta$ from $j$, respectively. Furthermore, in type 2 players, there are three subsets of players in terms of their distance from $j$ with fractions of $f_1, f_2, f_3$ that are in distance of $\Delta -2, \Delta -1, \Delta$ from $j$, respectively.

$$\begin{aligned} k|S_i^2|\bigg (h_1(\Delta -1)&+ h_2\Big (g_1 (\Delta -3) + g_2(\Delta -2)\Big ) + h_3\Big (f_1 (\Delta -5) + f_2(\Delta -4) + f_3 (\Delta -3)\Big )\bigg )\nonumber \\&\le \Delta ^ \alpha + c_{ji} + t_{ij}^{j} \le \Delta ^ \alpha + 2c \end{aligned}$$

(11)

$$\begin{aligned} \implies |S_i^2| \le \Delta ^ \alpha + 2c \Big / k\bigg ( \Delta - \Big ( h_1+h_2(g_1 + 2)+h_3 (2f_1 + f_2 + 3) \Big ) \bigg ) \end{aligned}$$

(12)

where

$$\begin{aligned} 0<f_1+f_2+f_3=g_1+g_2=h_1+h_2+h_3=1, \text { and } 0\le k, f_i, g_i, h_i\le 1. \end{aligned}$$

(13)

      $\square$