To delay or not: Temporal vaccination games on networks

Interventions such as vaccinations or installing anti-virus software are common strategies for controlling the spread of epidemics and malware on complex networks. Typically, nodes decide whether to implement such an intervention independently, depending on the costs they incur. A node can be protected by herd immunity, if enough other nodes implement such an intervention, making the problem of determining strategic decisions for vaccination a natural game-theoretical problem. There has been a lot of work on vaccination and network security game models, but all these models assume the vaccination decisions are made at the start of the game. However, in practice, a lot of individuals defer their vaccination decision, and the reasons for this behavior are not well understood, especially in network models. In this paper, we study a novel repeated game formulation, which considers vaccination decisions over time. We characterize Nash equilibria and the social optimum in such games, and find that a significant fraction of vaccinations might be deferred, in general. This depends crucially on the network structure, and the information and the vaccination delay. We show that finding Nash equilibria and the social optimum are NP-hard in general, and we develop an approximation algorithm for the social optimum whose approximation guarantee depends on the delay.