Is 2nd-price with a discount auction truth-telling? I know that 2nd-price auction is truth-telling, but 3rd-price auction is not. What If I run the regular 2nd-price auction, in the end, the winner is charged at the 2nd bidding price with a discount, say 10% off. Is truth-telling is a dominant strategy for this type of auction? For example, with bids being `10, 5, 4`, and a fixed discount `10%`, the winner will be charged `5*0.9 = 4.5` (This is an extension to this question: Third and average price auction)

An equilibrium will be one where everyone bids 11.11% more than their valuation. That is, a bidder with private valuation $p$ reports $p/0.9$. Everyone is bidding more than private valuation, but the information revealed is same as in a regular 2nd-price auction.