Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures

We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable con dence intervals on the optimal value of such stochastic programs using the Robust Stochastic Approximation and the Stochastic Mirror Descent (SMD) algorithms. When the objective functions are uniformly convex, we also propose a multistep extension of the Stochastic Mirror Descent algorithm and obtain con dence intervals on both the optimal values and optimal solutions. Numerical simulations show that our con dence intervals are much less conservative and are quicker to compute than previously obtained con dence intervals for SMD and that the multistep Stochastic Mirror Descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart. Our con dence intervals are also more reliable than asymptotic con dence intervals when the sample size is not much larger than the problem size.

Optimal auctions with multidimensional types and the desirability of exclusion

Within the context of a single-unit, independent private values auction model, we show that if bidder types are multidimensional, then under the optimal auction exclusion of some bidder types will occur. A second contribution of the paper is methodological in nature. In particular, we identify conditions under which an auction model with multidimensional types can be reduced to a model with one dimensional types without loss of generality. Reduction results of this type have achieved the status of folklore in the mechanism design literature. Here, we provide a proof of the reduction result for auctions.

Optimal auction with a general distribution: virtual valuation without densities

We characterize the optimal auction in an independent private values framework for a completely general distribution of valuations. We do this introducing a new concept: the generalized virtual valuation. To show the wider applicability of this concept we present two examples showing how to extend the classical models of Mussa and Rosen and Baron and Myerson for arbitrary distributions

Private or public? a taxonomy of optimal ownership and management regimes

We develop a theory of public versus private ownership based on value diversion by managers. Government is assumed to face stronger institutional constraints than has been assumed in previous literature. The model which emerges from these assumptions is fexible and has wide application. We provide amapping between the qualitative characteristics of an asset, its main use - including public goods characteristics, and spillovers toother assets values - and the optimal ownership and management regime. The model is applied to single and multiple related assets. We address questions such as; when is it optimal to have one of a pair ofr elated assets public and the other private; when is joint management desirable; and when should a public asset be managed by the owner of a related private asset? We show that while private ownership can be judged optimal in some cases solely on the basis of qualitative information, the optimality of any other ownership and management regimes relies on quantitative analysis. Our results reveal the situations in which policy makers will have difficulty in determining the opimal regime.