Diffusion of Behavior and Equilibrium Properties in Network Games

Situations in which agents’ choices depend on choices of those in close proximity, be it social or geographic, are ubiquitous. Selecting a new computer platform, signing a political petition, or even catching the flu are examples in which social interactions have a significant role. While some behaviors or states propagate and explode within the population (e.g., Windows OS, the HIV virus) others do not (e.g., certain computer viruses). Our goal in this paper is twofold. First, we provide a general dynamic model in which agents’ choices depend on the underlying social network of connections. Second, we show the usefulness of the model in determining when a given behavior expands within a population or disappears as a function of the environment’s fundamentals. We study a framework in which agents face a choice between two actions, 0 and 1 (e.g., whether to pursue a certain level of education, switch to Linux OS, etc.). Agents are linked through a social network, and an agent’s payoffs from each action depend on the number of neighbors she has and her neighbors’ choices. The diffusion process is defined so that at each period, each agent best responds to the actions taken by her neighbors in the previous period, assuming that her neighbors follow the population distribution of actions (a mean-field approximation). Steady states correspond to equilibria of the static game. Under some simple conditions, equilibria take one of two forms. Some are stable, so that a slight perturbation to any such equilibrium would lead the diffusion process to converge back to that equilibrium point. Other equilibria are unstable, so that a slight change in the distribution of actions leads to a new distribution of actions and eventually to a stable steady state. We call such equilibria tipping points. We analyze how the environment’s fundamentals (cost distribution, payoffs, and network structure) affect the set of equilibria, and characterize the adoption patterns within the network. The paper relates to recent work on network games and network diffusion, including work by Stephen Morris (2000); Pastor-Satorras and Vespignani (2000); Mark E. J. Newman (2002); Dunia López-Pintado (2004); Jackson and Brian W. Rogers (2007); Jackson and Yariv (2005); and Andrea Galeotti et al. (2005, henceforth GGJVY). Its contribution is in characterizing diffusion of strategic behavior and analyzing the stability properties of equilibria, and employing methods that allow us to make comparisons across general network structures and settings. Given that social networks differ substantially and systematically in structure across settings (e.g., ethnic groups, professions, etc.), understanding the implications of social structure on diffusion is an important undertaking for a diverse set of applications