We study the design of procurement contracts in environments where the buyer faces uncertainty over the product's demand and the seller's cost. The buyer has a conjecture (model) but does not fully trust it. She first identifies all worst-case optimal contracts, which deliver the largest payoff guarantee over a set of plausible demand and cost functions. She then selects the contract that maximizes the expected payoff (under her model) over such a restricted set. We term contracts surviving this two-step procedure as robustly optimal contracts. In usual screening problems, just like the one considered here, there are many worst-case optimal contracts, and our approach offers a selection criterion.

A set of jobs need to be served by a single server which can serve only one job at a time. Jobs have processing times and incur waiting costs (linear in their waiting time). The jobs share their costs through compensation using monetary transfers. We characterize the Shapley value rule for this model using fairness axioms. Our axioms include a bound on the cost share of jobs in a group, efficiency, and some independence properties on the the cost share of a job.

de Vries et al. [3] show that if valuation function of each buyer satisfies gross substitutes condition, then we can design primal-dual auctions that implement the Vickrey-Clarke-Groves (VCG) outcome. But they require prices in these auctions to be non-linear and non-anonymous. In this note, we show that such a requirement is necessary. This means, if valuation function of each buyer satisfies gross substitutes, then there does not exist a primal-dual auction which will always converge to the VCG outcome if prices in the auction are (i) linear and non-anonymous, OR (ii) non-linear and anonymous.