Mean-field models in network game theory [electronic resource]
Keywords: Management Science and Engineering.

In many contexts, the interactions between a large number of agents are governed by a network structure. Examples include online social networks, computer networks or societies in which each agent interacts only with a given set of individuals. In many situations those agents must make strategic decisions such as whether to form a certain opinion, whether to adopt a new product or technology, whether to get vaccinated or whether to invest in computer security solutions. While these so-called network games have many useful applications, they can pose important challenges in terms of tractability and place an unrealistic cognitive burden upon agents. In this thesis, we study such large network games and propose a mean-field equilibrium concept (MFE) as an approximation that allows for both tractability and realism in terms of the cognitive burden placed on agents. We study different applications. In Chapter 2, we study a network game of technology adoption. When a product or technology is first introduced, there is uncertainty about its value or quality. This quality can be learned by trying the product, at a risk. It can also be learned by letting others try it and free-riding on the information that they generate. We propose a class of dynamic games to study the adoption of technologies of uncertain value, when agents are connected by a network. This class of games allows for referral incentives, whereby an agent can earn rewards by encouraging his neighbors to adopt. We derive a mean-field equilibrium (MFE) and show that a pricing policy that involves referral incentives leads to a double-threshold strategy by which both low and high-degree agents may choose to experiment with the technology of uncertain value whereas the middle-degree agents free-ride on the information revealed by that experimentation. We characterize how different dynamic pricing mechanisms affect the pattern of early/late adoption and information diffusion. Pricing mechanisms that allow a monopolist to guarantee early adoption by agents of high or low degrees are proposed. We illustrate how referral incentives can be preferable on certain networks while inter-temporal price discrimination (i.e. price discounts) may be preferable on others. In Chapter 3, we study cascading failures in networks and the incentives that agents have to invest in costly protection against failure. A finite set of agents are connected through a network and can fail either intrinsically or as a result of the failure of a subset of their neighbors. Particular applications of this game include vaccination, investment in computer security solutions, airport security as well as investments in cash buffers in an interbank system. We derive a Bayes-Nash equilibrium in which agents form a belief about the joint probability of failure of their neighbors. We characterize the equilibrium based on an agent's effective probability of failure and derive conditions under which equilibrium strategies are monotone in degree. We show that this monotonicity is reversed, depending on whether the investment in protection insulates an agent against the failure of his neighbors or just against his own intrinsic failure. The former case defines a game of strategic substitutes in which some agents free-ride on the investment in protection of others, while the latter case defines a game of strategic complements in which agents pool their investments in protection. When failure risk is increasing in degree, protection against the failure of neighbors induces more investment by higher-degree agents whereas protection against intrinsic failure induces more investment by lower-degree agents. Welfare implications are discussed. Finally in Chapter 4, given the difficulty of expressing the beliefs in the previous chapter, we introduce a bounded-rationality solution concept as an approximation: a mean-field equilibrium (MFE). Agents simply consider a mean-field approximation of the cascading process when making their decision of whether to invest in protection. We study the game with binary actions --- where an agent can make a costly investment in protection against failure --- and show that the equilibria are characterized by thresholds in degree, above or below which agents choose to invest. When the costly investment protects them against the failure of their neighbors, the equilibrium is unique, whereas when it protects them only against their own intrinsic failure, there can be multiple equilibria. The mean-field model conveniently allows for comparative statics in terms of the degree distribution. It is shown that a more interconnected system can either induce more or less investment in protection, depending on whether the failure risk is decreasing or increasing in degree.