Abstract: Equilibrium bidding strategies in first-price combinatorial auctions are not well understood. Ex-post split-award auctions are a widespread form of combinatorial procurement auctions in which the demand for some quantity to be procured is split into two or more shares. Similar to the seminal article by Anton and Yao (1992) who analyze markets with two bidders, we focus on diseconomies of scale, but with an arbitrary number of bidders. These markets are practically relevant and strategically challenging because bidders must coordinate to achieve an efficient outcome. Dutch combinatorial auction formats have received little attention so far. We show that, unlike in single-object auctions, first-price sealed-bid and the Dutch combinatorial auctions are not strategically equivalent. While the former exhibits a coordination problem for bidders, the latter has a unique and efficient equilibrium. Motivated by industry practice, we also analyze a combination of a Dutch and a first-price sealed-bid format that has attractive properties, and the efficiency and purchasing costs of these auctions are compared in a welfare analysis. The price information revealed during the Dutch auction formats avoids equilibrium selection problems and this also helps bidders coordinate in the lab. Overall, the theoretical predictions explain important aspects of the empirical bidder behavior remarkably well, and first-price auctions with more than two bidders achieve high levels of efficiency also in the lab.