A Game-Theoretic Analysis of the Tipping Point Phenomenon: Informational Phase Transitions in Social Networks

LARROSA, JUAN M.C.; TOHMÉ, FERNANDO

Game Theory: Strategies, Equilibria, and Theorems.

Nova Science Publishers

Lugar: New York; Año: 2008; p. 350 - 366

A well-known phenomenon (popularized by Malcolm Gladwell's book) in real world social networks is the existence of `tipping points'. That is, thresholds that once surpassed lead to a whole new con¯guration of the social structure. Political upheaveals, sudden fads and the fast adaptation of innovations are just examples of this. We intend to provide some clues on how this might happen, but instead of assuming that it is an unintended consequence of random actions we consider here a game theoretic framework in which rational agents make decisions aimed to maximize their payo®s. Starting with a framework very much like Bala & Goyal's (2000) we consider a ¯nite society in which agents are endowed with some amount of a private but reproducible good (information) that upon contact can be copied or transmitted from one agent to another. While there is a cost of establishing a connection, there are also gains in accessing new information. The di®erence between these two yields the payo® of a connection. Rational agents will behave strategically and the Nash equilibria will provide the network architecture. As it is well known from Erdos and Renyi (1959) seminal treatment of random graphs, new connections may lead to phase transitions in the density of the graph. That is, jumps in the number of clusters from many to a single major one. While for social networks the framework of random graph is not quite cogent, similar results may arise varying the nature of the probability distribution on potential connections (Newman et al., 2002). In this paper we will show how the same is true in our non-probabilistic, game-theoretic framework. By slight changes in the information carried by individual agents (representing the in°uence of non-social sources) the equilibrium networks may vary suddenly. We will see that in the end, if each agent has an information endowment larger than the cost of establishing connections, a minimally connected network becomes the unique outcome. That means that a highly organized structure arises when everyone is \valuable". On the other hand, if the value is too low for every agent (i.e. there is no gain in connecting to others) the only e±cient outcome is the empty network. In the middle, we will show, there exist some critical agents to which most of the others will want to establish contact and yield components in a disconnected network.information) that upon contact can be copied or transmitted from one agent to another. While there is a cost of establishing a connection, there are also gains in accessing new information. The di®erence between these two yields the payo® of a connection. Rational agents will behave strategically and the Nash equilibria will provide the network architecture. As it is well known from Erdos and Renyi (1959) seminal treatment of random graphs, new connections may lead to phase transitions in the density of the graph. That is, jumps in the number of clusters from many to a single major one. While for social networks the framework of random graph is not quite cogent, similar results may arise varying the nature of the probability distribution on potential connections (Newman et al., 2002). In this paper we will show how the same is true in our non-probabilistic, game-theoretic framework. By slight changes in the information carried by individual agents (representing the in°uence of non-social sources) the equilibrium networks may vary suddenly. We will see that in the end, if each agent has an information endowment larger than the cost of establishing connections, a minimally connected network becomes the unique outcome. That means that a highly organized structure arises when everyone is \valuable". On the other hand, if the value is too low for every agent (i.e. there is no gain in connecting to others) the only e±cient outcome is the empty network. In the middle, we will show, there exist some critical agents to which most of the others will want to establish contact and yield components in a disconnected network.