The article deals with auctions in relation to the placement of government orders. The game-theoretic models of the reverse auction of the first price and the reverse Vickrey auction are described. The analysis of the considered models is carried out on the basis of certain principles of optimality of the mathematical theory of games, namely, the concepts of non-dominated and dominant strategies. Practical recommendations are given for using the proposed competitive mechanisms to solve the problem of overestimation of future costs by auction participants.