Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / In this work, letâs apply auction theory to solve a parsimonious procurement model in a situation where there is a contracting firm by the Government. Itâs a natural extension of the bidimensional auction model proposed by Yeon-Koo Che (1993) to model actions of the Department of Defense (DoD) in United States. The extension will be made based in a secret reservation price model proposed by Laffont et at. (1994) to unidimensional case. At end, we will make a short numerical analysis of the model proposed in this work. / Neste trabalho, vamos aplicar a teoria dos leilÃes para resolver um modelo parcimonioso de uma licitaÃÃo pÃblica em uma situaÃÃo em que hà a contrataÃÃo de uma firma pelo Governo. Trata-se de uma extensÃo natural do modelo de leilÃo bidimensional proposto por Yeon-Koo Che (1993) para modelar leilÃes do Departamento de Defesa dos Estados Unidos. A extensÃo serà feita baseada em um modelo de preÃo reserva secreto proposto por Laffont et al. (1994) para o caso unidimensional. Por fim, faremos uma breve anÃlise numÃrica do modelo desenvolvido neste trabalho.

In this work we present a general duality-theory framework for revenue maximization in additive Bayesian auctions involving multiple items and many bidders whose values for the goods follow arbitrary continuous joint distributions over some multi-dimensional real interval. Although the single-item case has been resolved in a very elegant way by the seminal work of Myerson [1981], optimal solutions involving more items still remain elusive. The framework extends linear programming duality and complementarity to constraints with partial derivatives. The dual system reveals the natural geometric nature of the problem and highlights its connection with the theory of bipartite graph matchings. We demonstrate the power of the framework by applying it to various special monopoly settings where a seller of multiple heterogeneous goods faces a buyer with independent item values drawn from various distributions of interest, to design both exact and approximately optimal selling mechanisms. Previous optimal solutions were only known for up to two and three goods, and a very limited range of distributional priors. The duality framework is used not only for proving optimality, but perhaps more importantly, for deriving the optimal mechanisms themselves. Some of our main results include: the proposal of a simple deterministic mechanism, which we call Straight-Jacket Auction (SJA) and is defined in a greedy, recursive way through natural geometric constraints, for many uniformly distributed goods, where exact optimality is proven for up to six items and general optimality is conjectured; a scheme of sufficient conditions for exact optimality for two-good settings and general independent distributions; a technique for upper-bounding the optimal revenue for arbitrarily many goods, with an application to uniform and exponential priors; and the proof that offering deterministically all items in a single full bundle is the optimal way of selling multiple exponentially i.i.d. items.