Herd immunity is a "tragedy of the commons" type of game. Case closed. Unless you can prove that a rational actor would chose in his own self-interest to cooperate in which case you might get considered for a Nobel prize in economics.
Whether or not we can achieve herd immunity is a tragedy of the commons type game, though it's hardly "case closed" - I know several researchers working in that particular area. The existence of herd immunity is a relatively easy calculus problem.