We consider a buyer–seller interaction where the seller has one object to allocate and the buyer has private values and private budgets. We study the performance, in terms of expected revenue, of approximation mechanisms in this setting. Approximation mechanisms are obtained from the set of incentive compatible, individually rational, and budget feasible mechanisms, denoted M, in two different ways: (i) by imposing additional constraints, thus restricting M to a subclass N; or (ii) by relaxing some of the original constraints, thus expanding M to a superclass N′. We measure
the performance of a restricted class N ⊆ M using the guaranteed fraction of optimal revenue ratio introduced by Hart and Nisan (2017). To measure the performance of a relaxation class M ⊆ N′ we introduce a related ratio, which we call the maximal value of relaxation. We focus on a restricted class of simple mechanisms, consisting of those whose associated menus have a small number of entries. We also pay attention to a restricted class of mechanisms that involves stronger incentive constraints and that has received attention in the literature for its computational tractability. Finally, we consider a relaxed class of mechanisms where the seller can costlessly prevent the buyer from over-reporting the budget. In a few cases, we obtain good approximation results. In others, the revenue gap between
optimal mechanisms and optimal approximation mechanisms is large. Overall, our research points to the advantages and limitations of using approximation mechanisms in settings with private values and private budgets.