I am having trouble understanding a second price auction with a reserve price, i.e. a second price auction where each player’s valuation is uniformly distributed on $[0, 1]$, and the two valuations are independent random variables. Here we have a fixed reserve price $r$ > 0 that is common knowledge with the buyers: if the two bids are below the threshold $r$, then there’s no winner.
This thread about second price auctions explained fairly well the logic in finding the expected revenue for the auctioneer, however, I don't understand how we can find the expected payment for the winner.
To me it seems like we have 3 possible cases for the winner:
- Both players bid under $r$ and there is no winner, therefore, expected payment is zero.
- One player bids below $r$ and the other exceeds $r$, making the expected payment $r$.
- Both players bid above the reserve, making the game reduce to a regular second price auction, so the expected payment by the winner is the loser's bid.
Does my logic make sense? How then do I find the winner's payment on expectation? I having trouble setting these problems up so any advice on that is welcome as well.