Mechanism Design and Risk Aversion

We develop efficient algorithms to construct utility maximizing mechanisms in the presenceof risk averse players (buyers and sellers) in Bayesian single parameter and multi-parametersettings. We model risk aversion by a concave utility function, and players play strategically tomaximize their expected utility. Bayesian mechanism design has usually focused on maximizingexpected revenue in a risk neutral environment, i.e. where all the buyers and the seller havelinear utility, and no succinct characterization of expected utility maximizing mechanisms isknown even for single-parameter multi-unit auctions.We first consider the problem of designing optimal DSIC (dominant strategy incentive com-patible) mechanism for a risk averse seller in the case of multi-unit auctions, and we give apoly-time computable deterministic sequential posted pricing mechanism (SPM) that for anyǫ > 0, yields a (1 − 1/e − ǫ)-approximation to the expected utility of the seller in an optimalDSIC mechanism. Our result is based on a novel application of a correlation gap bound, alongwith splitting and merging of random variables to redistribute probability mass across buyers.This allows us to reduce our problem to that of checking feasibility of a small number of distinctconfigurations, each of which corresponds to a covering LP. A feasible solution to the LP givesus the distribution on prices for each buyer to use in a randomized SPM. We get a deterministicSPM by sampling from this randomized SPM. Our techniques extend to the multi-parametersetting with unit demand buyers.We next consider the setting when buyers as well as the seller are risk averse, and theobjective is to maximize the seller’s expected utility. We design a truthful-in-expectation mech-anism whose utility is a( (1− 1e)2×max( 1− 1e, 1− 1√2πk)) -approximation to the optimal BICmechanism under two mild assumptions: (a) ex post individual rationality and (b) no positivetransfers. Our mechanism consists of multiple rounds. It considers each buyer in a round withsmall probability, and when a buyer is considered, it allocates an item to the buyer accordingto payment functions that are computed using stochastic techniques developed for DSIC mech-anisms. Lastly, we consider the problem of revenue maximization for a risk neutral seller inpresence of risk averse buyers, and give a poly-time algorithm to design an optimal mechanismfor the seller.We believe that the techniques developed in this work will be useful in handling otherstochastic optimization problems with a concave objective function.