In case of the private procurement auctions the entrepreneurs are not forced to limit themselves to the standard auction rules, and in practice one can observe many hybrid or quasi-auction mechanisms spontaneously introduced. The paper analyzes two of them, which start as a first-price sealed-bid auction, followed by a run-off in a form of an English auction, and which differ by the transparency of rules concerning the initiation of the second stage. The focus of the paper is on the analysis of the price and allocative efficiency of these mechanisms, in order to determine whether they can serve as an alternative to the standard auction rules. Theoretical analyses are followed by the laboratory experiments, which demonstrate that the mechanisms under study are characterized by both high price and allocative efficiency, and therefore could be considered an interesting substitute of the standard auction rules.