Nash's generalization of Gittins’ classic index result to so-called generalized bandit problems (GBPs) in which returns are dependent on the states of all arms (not only the one which is pulled) has proved important for applications. The index theory for special cases of this model in which all indices are positive is straightforward. However, this is not a natural restriction in practice. An earlier proposal for the general case did not yield satisfactory index-based suboptimality bounds for policies — a central feature of classical Gittins index theory. We develop such bounds via a notion of duality for GBPs which is of independent interest. The index which emerges naturally from this analysis is the reciprocal of the one proposed by Nash.