  We develop efficient algorithms to construct utility maximizing mechanisms in
the presence of risk averse players (buyers and sellers) in Bayesian settings.
We model risk aversion by a concave utility function, and players play
strategically to maximize their expected utility. Bayesian mechanism design has
usually focused on maximizing expected revenue in a {\em risk neutral}
environment, and no succinct characterization of expected utility maximizing
mechanisms is known even for single-parameter multi-unit auctions.
  We first consider the problem of designing optimal DSIC mechanism for a risk
averse seller in the case of multi-unit auctions, and we give a poly-time
computable SPM that is $(1-1/e-\eps)$-approximation to the expected utility of
the seller in an optimal DSIC mechanism. Our result is based on a novel
application of a correlation gap bound, along with {\em splitting} and {\em
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.
A feasible solution to the LP gives us the distribution on prices for each
buyer to use in a randomized SPM.
  We next consider the setting when buyers as well as the seller are risk
averse, and the objective is to maximize the seller's expected utility. We
design a truthful-in-expectation mechanism whose utility is a $(1-1/e
-\eps)^3$-approximation to the optimal BIC mechanism under two mild
assumptions. Our mechanism consists of multiple rounds that processes each
buyer in a round with small probability. Lastly, we consider the problem of
revenue maximization for a risk neutral seller in presence of risk averse
buyers, and give a poly-time algorithm to design an optimal mechanism for the
seller.
