In optimal stopping problems, decision makers are assumed to search randomly to learn the utility of alternatives; in contrast, in one-shot multi-attribute utility optimization, decision makers are assumed to have perfect knowledge of utilities. We point out that these two contexts represent the boundaries of a continuum, of which the middle remains uncharted: How should people search intelligently when they possess imperfect information about the alternatives? We assume that decision makers first estimate the utility of each available alternative and then search the alternatives in order of their estimated utility until expected benefits are outweighed by search costs. We considered three well-known models for estimating utility: (i) a linear multi-attribute model, (ii) equal weighting of attributes, and (iii) a single-attribute heuristic. We used 12 real-world decision problems, ranging from consumer choice to industrial experimentation, to measure the performance of the three models. The full model (i) performed best on average but its simplifications (ii and iii) also had regions of superior performance. We explain the results by analyzing the impact of the models’ utility order and estimation error.