The private entrepreneurs are not forced to limit themselves to the standard auction rules, and in case of the procurement auctions one can observe many hybrid or quasi-auction rules spontaneously introduced. The paper analyzes two of them, that are based on the common assumption: the second-best bidder is given an additional right to improve her initial bid, and therefore win the auction. The focus of the paper is on the analysis of price and allocative efficiency of such modifications, to determine whether they can serve as an alternative to the standard auction rules. Theoretical analyses, followed by the laboratory experiments, provide no proof for that conclusion though, as none of the rules under study beats the classical first-price sealed-bid auction.