9410747 Stacchetti Mechanism design problems arise naturally in numerous economic situations, including the regulation of a monopolist, auctions, government procurement, nonlinear pricing, and the provision of public goods. There is a vast literature in each one of these major fields of application by now, but for the most part, the analysis has been reduced to the case in which uncertainty is modelled by a single parameter, and the number of goods is restricted to two (one of the two goods is typically money). The technical difficulties presented by higher dimensional cases have impeded progress, and delayed our understanding of economic applications. The project develops new techniques in the theory of multidimensional mechanism design, and applies these techniques to several important economic problems. More specifically, the techniques for multidimensional mechanism design developed by this project are based on the work of Rochet and Armstrong. For problems with a certain linear structure (including auctions with externalities and pricing in multiproduct monopolies) conjugate duality leads to an envelope condition and a full characterization of incentive compatible mechanisms. The divergence theorem takes the place of integration by parts, which is used in the one-dimensional case to simplify the objective function of the principal. In a symmetric auction with externalities, these techniques allow one to identify the only buyer type whose participation constraint binds at the optimal solution. Myerson's notion of "regularity", which is so crucial in one-dimensional problems, has a more complicated but useful analog in higher dimensions. The symmetric auction model with externalities of Jehiel, Moldovanu and Stacchetti is extended by relaxing the assumption of symmetry and considering the case where buyers experience externalities even when the seller keeps the object. Dynamic games are investigated where there is a strategic motive for buying t he objective, namely, to resell it later. The generalized regularity analysis alluded to above would extend Armstrong's results on pricing and regulation to multiproduct monopolies to a much wider range of economic environments.
