Strategic immunization and group structure

We consider the spread of a harmful state through a population divided into two groups. Interaction patterns capture the full spectrum of assortativity possibilities. We show that a central planner who aims for eradication optimally either divides equally the resources across groups, or concentrates entirely on one group, depending on whether there is positive or negative assortativity, respectively. We study a game in which agents can, at a cost, immunize. Negative assortative interactions generate highly asymmetric equilibrium outcomes between ex-ante identical groups. When groups have an underlying difference, even a small amount of inter-group contacts generates large asymmetries. We study the diffusion of a harmful state through a population. Immunity is available, but is costly. The state is meant to capture various kinds of choices or risky behaviors such as, for example, tobacco use, in which case immunity is interpreted as a commitment to avoid the temptation of smoking. The state can also capture the presence of an electronic virus on a computer network; in this case immunity represents the purchase of anti-virus software, or other costly measures taken to avoid the virus. But perhaps the most conventional interpretation is that the state represents human infection of various communicable diseases that spread through social contacts; in this case immunity captures a decision to vaccinate oneself.