We identify two possible equilibrium configurations for a non-renewable resource duopoly in a discrete-time framework. For the purpose of illustration, we suppose initial endowments of firms that allow for a maximum of two extraction periods. In the first possible equilibrium, the duopoly exists for two periods, while in the second possible equilibrium, the duopoly lasts only for one period and the firm with the higher initial endowment becomes a monopolist in the second and last period. As neither equilibrium configuration dominates the other for both firms at the same time, it is unclear whether firms acting simultaneously can coordinate on one particular configuration.