Bayesian optimal knapsack procurement

•We study a binary knapsack problem where the weight is chosen by strategic players.•There has recently been interest in the OR and computer science literature.•We derive a Bayesian optimal mechanism.•Individual take-it-or-leave-it offers are optimal. Ex post competition is not needed.

A budget-constrained buyer wants to purchase items from a shortlisted set. Items are differentiated by observable quality and sellers have private reserve prices for their items. The buyer’s problem is to select a subset of maximal quality. Money does not enter the buyer’s objective function, but only his constraints. Sellers quote prices strategically, inducing a knapsack game. We report the Bayesian optimal mechanism for the buyer’s problem. We find that simultaneous take-it-or-leave-it offers are interim optimal.