On Bidding with Securities: Risk Aversion and Positive Dependence
  DeMarzo et al. (2005) consider auctions in which bids are selected from a completely ordered family of securities whose values are tied to the resource being auctioned. The paper defines a notion of relative steepness of families of securities and shows that a steeper family provides greater expected revenue to the seller. Two assumptions are: the buyers are risk-neutral; the random variables through which values and signals of the buyers are realized are affiliated. We show that this revenue ranking holds for the second price auction in the case of risk-aversion. However, it does not hold if affiliation is relaxed to a less restrictive form of positive dependence, namely first order stochastic dominance (FOSD). We define the relative strong steepness of families of securities and show that it provides a necessary and sufficient condition for comparing two families in the FOSD case. All results extend to the English auction. 
Vineet Abhishek, Bruce Hajek and Steven R. Williams
cs.GT