We propose a simple two-disease epidemic model where one disease exhibits only a drug-sensitive strain, while the other exhibits both drug-sensitive and drug-resistant strains. Treatment for the first disease may select for resistance in the other. We model antibiotic use as a mathematical game through the study of individual incentives and community welfare. The basic reproduction number is derived and the existence and local stability of the model equilibria are analyzed. When the force of infection of each disease is unaffected by the presence of the other, we find that there is a conflict of interest between individual and community, known as a tragedy of the commons, under targeted treatment toward persons infected by the single strain disease, but there is no conflict under mass treatment. However, we numerically show that individual and social incentive to use antibiotics may show disaccord under mass treatment if the restriction on the transmission ability of the dually infected people is removed, or drug resistant infection is worse than drug sensitive infection, or the uninfected state has a comparative disutility over the infected states.