# Mechanism Design for a Risk Averse Seller

## Abstract

We develop efficient algorithms to construct approximately utility maximizing mechanisms for a risk averse seller in the presence of potentially risk-averse buyers in Bayesian single parameter and multi-parameter settings. We model risk aversion by concave utility function. Bayesian mechanism design has usually focused on revenue maximization in a *risk-neutral* environment, and while some work has regarded buyers’ risk aversion, very little of past work addresses the seller’s risk aversion.

We first consider the problem of designing a DSIC mechanism for a risk-averse seller in the case of multi-unit auctions. We give a poly-time computable pricing mechanism that is a (1 − 1/*e* − *ε*)-approximation to an optimal DSIC mechanism, for any *ε* > 0. Our result is based on a novel application of correlation gap bound, that involves *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 distinct configurations, each of which corresponds to a covering LP.

DSIC mechanisms are robust against buyers’ risk aversion, but may yield arbitrarily worse utility than the optimal BIC mechanisms, when buyers’ utility functions are assumed to be known. For a risk averse seller, we design a truthful-in-expectation mechanism whose utility is a small constant factor approximation to the utilty of the optimal BIC mechanism under two mild assumptions: (a) ex post individual rationality and (b) no positive transfers. Our mechanism simulates several rounds of sequential offers, that are computed using stochastic techniques developed for our DSIC mechanism. We believe that our techniques will be useful for other stochastic optimization problems with concave objective functions.

## Keywords

Utility Function Risk Aversion Expected Utility Revenue Maximization Split Operation## Preview

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