  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.
