We study undominated mechanisms with transfers for regulating a monopolist who privately observes the marginal cost of production. A mechanism is undominated if no other mechanism gives the regulator a strictly higher payoff at some marginal cost of the monopolist without lowering the regulator’s payoff at other costs. We show that an undominated mechanism has a quantity floor: whenever the monopolist is allowed to operate, it produces above a threshold quantity. Moreover, the regulator’s operation decision is stochastic only if the monopolist produces at the quantity floor. We provide a near-complete characterization of the set of undominated mechanisms and use it to (a) derive a max-min optimal regulatory mechanism, (b) provide a foundation for deterministic mechanisms, and (c) show that the efficient mechanism is dominated.