We consider multi-attribute Bayesian optimization, where each design in an optimization problem’s feasible space is associated with a vector of attributes that can be evaluated via a time-consuming computer code, and each vector of attributes is assigned a utility according to a decision-maker’s utility function. A standard Bayesian optimization approach could be applied if the utility function were known to us: we would place a Bayesian prior distribution over the composition of the objective function, which returns a design’s vector of attributes, and the utility function, which maps those attributes onto a utility. In contrast, we assume the utility function cannot be evaluated and is known implicitly only to the decision-maker. We propose a Bayesian optimization algorithm that chooses the designs to evaluate, such that the expected utility of the design chosen by the decision-maker, according to our algorithm’s estimate of the objective function, is large. In contrast with existing approaches for multi-attribute optimization that focus on estimating a Pareto frontier, our approach can take advantage of prior information about the decision-maker’s utility, obtained from past experiences with the decision-maker or from a utility elicitation process.