  Motivated by practical constraints in online advertising, we investigate
single-parameter auction design for bidders with constraints on their Return On
Investment (ROI) -- a targeted minimum ratio between the obtained value and the
payment. We focus on ex post ROI constraints, which require the ROI condition
to be satisfied for every realized value profile. With ROI-constrained bidders,
we first provide a full characterization of the allocation and payment rules of
dominant-strategy incentive compatible (DSIC) auctions. In particular, we show
that given any monotone allocation rule, the corresponding DSIC payment should
be the Myerson payment with a rebate for each bidder to meet their ROI
constraints. Furthermore, we also determine the optimal auction structure when
the item is sold to a single bidder under a mild regularity condition. This
structure entails a randomized allocation scheme and a first-price payment
rule, which differs from the deterministic Myerson auction and previous works
on ex ante ROI constraints.
