Mechanism Design for a Risk Averse Seller

We design efficient algorithms to compute dominant strategy truthful mechanisms for a risk-averse seller, in Bayesian single-parameter and multi-parameter settings. We model risk aversion by a concave utility function, where the the seller maximizes its expected utility. Further, we argue that dominant strategy incentive compatible mechanisms should not be affected by the risk aversion of buyers. Much earlier work on Bayesian mechanism design has focused on maximizing expected revenue, and no succinct characterization of expected utility maximizing mechanism is known even for multi-unit auctions.
Our first contribution is a poly-time algorithm for multi-unit auctions, to compute a sequential posted pricing mechanism (SPM) that for any ε 0, yields a (1 – 1/e – ε )-approximation to the expected utility of an optimal mechanism. In comparison, the best known approximation factor using SPM is (1-1/e) even for the expected revenue objective (linear utility). Our result uses a correlation gap bound and then applies concentration bounds to show the existence of an order-independent SPM for large inventory, and an approximation scheme for optimal adaptive SPM for small inventory. Our result extends to single parameter settings with partition matroid feasibility constraints.
For the multi-parameter setting with unit demand buyers, we design a poly-time algorithm to compute a (1 – 1/e – ε)/6.75-approximation.
In comparison, the best known guarantee of an efficient mechanism to maximize expected revenue in unit-demand setting is 1/6.75. Our mechanism is an order-independent SPM, and is based on a clever application of a correlation gap bound, along with splitting and merging of random variables to redistribute probability mass across buyers. This allows us to reduce the problem to a simpler covering problem while losing a factor of (1-1/e), for which we develop an approximation scheme. We believe that our techniques will be useful in handling risk aversion in other stochastic optimization problems.