The goal of contact tracing is to reduce the likelihood of transmission, particularly to individuals who are at greatest risk for developing complications of infection, as well as identifying individuals who are in need of medical treatment of other interventions. In this paper, we develop a simple mathematical model of contact investigations among a small group of individuals and apply game theory to explore conflicts of interest that may arise in the context of perceived costs of disclosure. Using analytic Kolmogorov equations, we determine whether or not it is possible for individual incentives to drive noncooperation, even though cooperation would yield a better group outcome. We found that if all individuals have a cost of disclosure, then the optimal individual decision is to simply not disclose each other. With further analysis of (1) completely offsetting the costs of disclosure and (2) partially offsetting the costs of disclosure, we found that all individuals disclose all contacts, resulting in a smaller basic reproductive number and an alignment of individual and group optimality. More data are needed to understand decision making during outbreak investigations and what the real and perceived costs are.