  One method to offer some bidders a discount in a first-price auction is to
augment their bids when selecting a winner but only charge them their original
bids should they win. Another method is to use their original bids to select a
winner, then charge them a discounted price that is lower than their bid should
they win. We show that the two methods have equivalent auction outcomes, for
equal additive discounts and for multiplicative ones with appropriate
adjustments to discount amounts. As a result, they have corresponding
equilibria when equilibria exist. We also show that with the same level of
multiplicative adjustments, bidders with discounts should prefer an augmented
bid to a discounted price. Then we estimate optimal bid functions for valuation
distributions based on data from online advertising auctions, and show how
different discount levels affect auction outcomes for those bid functions.
