Again, let’s pretend to be the auctioneer. You know that each bidder is bidding her expected rival’s value conditional on being the winning bidder. Now the bidder with value V1:n is in fact the winning bidder (although, she does not know that while bidding). Her bid is what you receive as the auctioneer. You expect her value to be V1:n. You also realize that based on her belief she will estimate her highest rival bidder’s value which is E[V2:n]. The expectation calculation is based on the winning bidder’s probability belief on other bidders’ values. If you share the same belief as the bidders on the value distributions of the different bides then from your point of view the expected bid that the winning bidder submits is E[V2:n], as well. A good way to see this is to consider the following example.
Uniform distribution.
Suppose that there are 4 bidders. Let’s assume that the auctioneer and he bidders all believe that the values are realizations of i.i.d. standard uniform random variables. In that situation if the winning bidder has value v the price in the auction will be 3v/4. The winning bidder is expected to have a value 4/5. So, the price in the auction is expected to be 3(4/5)/4 = 3/5 = E[V2:4].