When a famous painting becomes available for sale, it is often known which museum or collector will be the likely winner. Yet, the auctioneer actively woos representatives of other museums that have no chance of winning to attend anyway.Suppose a piece of art has recently become available for sale and will be auctioned off to the highest bidder, with the winner paying an amount equal to the second highest bid. Assume that most collectors know that Kate places a value of $100,000 on the art piece and that she values this art piece more than any other collector. Suppose that if no one else shows up, Kate simply bids $100,0002=$50,000$100,0002=$50,000and wins the piece of art.

The expected price paid by Kate, with no other bidders present, is

.

Suppose the owner of the artwork manages to recruit another bidder, Manuel, to the auction. Manuel is known to value the art piece at $80,000.

The expected price paid by Kate, given the presence of the second bidder Manuel, is

.