An indivisible object may be sold to one of n agents who know their valuations of the object. The seller would like to use a revenue-maximizing mechanism but her knowledge of the valuations' distribution is scarce: she knows only the means (which may be different) and an upper bound for valuations. Valuations may be correlated. Using a constructive approach based on duality, we prove that a mechanism that maximizes the worst-case expected revenue among all deterministic dominant-strategy incentive compatible, ex post individually rational mechanisms takes the following form: (1) the bidders submit bids; (2) for each bidder, a bidder-specific linear function of the bid is calculated (a "linear score"); (3) the object is awarded to the agent with the highest score, provided it's nonnegative; (4) the winning bidder pays the minimal amount he would need to bid to still win in the auction. The set of optimal mechanisms includes other mechanisms but all those have to be close to the optimal linear score auction in a certain sense. When means are high, all optimal mechanisms share the linearity property. Second-price auction without a reserve is an optimal mechanism when the number of symmetric bidders is sufficiently high.
A seller chooses a reserve price in a second-price auction to maximize worst-case expected revenue when she knows only the mean of value distribution and an upper bound on either values themselves or variance. Values are private and iid. We prove that it is always optimal to set the reserve price to seller's own valuation. However, the maxmin reserve price may not be unique. If the number of bidders is sufficiently high, all prices below the seller's valuation, including zero, are also optimal. A second-price auction with the reserve equal to seller's value (or zero) is an asymptotically optimal mechanism (among all mechanisms) as the number of bidders grows without bound.
A principal is uncertain about a technology mapping an agent's effort to the distribution of output. The agent is risk neutral and there is a participation constraint but no limited liability constraint. Transfers can be costly. An example of this setting is the case where the principal is a society trying to properly incentivize a firm to carry out innovation. We first show that when the principal employs minimax-regret criterion in the face of the technological uncertainty, an optimal contract is affine. We then characterize the full set of optimal contracts. A contract is optimal if and only if it lies within certain affine, increasing bounds that collapse to a point when output reaches its maximum value.