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

LARROSA, JUAN MANUEL; TOHMÉ, FERNANDO

Game Theory: Strategies, Equilibria, and Theorems

Nova Science Publishers

Lugar: New York; Año: 2009; p. 319 - 336

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 configuration of thesocial structure. Political upheaveals, sudden fads and the fast adaptationof innovations are just examples of this. We intend to provide some clueson how this might happen, but instead of assuming that it is an unintendedconsequence of random actions we consider here a game theoreticframework in which rational agents make decisions aimed to maximizetheir payoffs.Starting with a framework very much like Bala & Goyal’s (2000) weconsider a finite society in which agents are endowed with some amountof a private but reproducible good (information) that upon contact canbe copied or transmitted from one agent to another. While there is acost of establishing a connection, there are also gains in accessing newinformation. The difference between these two yields the payoff of a connection.Rational agents will behave strategically and the Nash equilibriawill provide the network architecture.As it is well known from Erd¨os and Renyi (1959) seminal treatmentof random graphs, new connections may lead to phase transitions in thedensity of the graph. That is, jumps in the number of clusters frommany to a single major one. While for social networks the frameworkof random graph is not quite cogent, similar results may arise varyingthe nature of the probability distribution on potential connections (Newmanet al., 2002). In this paper we will show how the same is true in our non-probabilistic, game-theoretic framework. By slight changes inthe information carried by individual agents (representing the influence ofnon-social sources) the equilibrium networks may vary suddenly. We willsee that in the end, if each agent has an information endowment largerthan the cost of establishing connections, a minimally connected networkbecomes the unique outcome. That means that a highly organized structurearises when everyone is “valuable”. On the other hand, if the value istoo low for every agent (i.e. there is no gain in connecting to others) theonly efficient outcome is the empty network. In the middle, we will show,there exist some critical agents to which most of the others will want toestablish contact and yield components in a disconnected network