  We study efficiency loss in Bayesian revenue optimal auctions. We quantify
this as the worst case ratio of loss in the realized social welfare to the
social welfare that can be realized by an efficient auction. Our focus is on
auctions with single-parameter buyers and where buyers' valuation sets are
finite. For binary valued single-parameter buyers with independent (not
necessarily identically distributed) private valuations, we show that the worst
case efficiency loss ratio (ELR) is no worse than it is with only one buyer;
moreover, it is at most 1/2. Moving beyond the case of binary valuations but
restricting to single item auctions, where buyers' private valuations are
independent and identically distributed, we obtain bounds on the worst case ELR
as a function of number of buyers, cardinality of buyers' valuation set, and
ratio of maximum to minimum possible values that buyers can have for the item.
