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$($eBay Auction with Buy$-$It$-$Now Option$)$ Consider a simple auction where a seller runs

an eBay auction with two buyers, each with private valuation $1,$ $2$ independently

drawn from the uniform distribution on $[0,1].$

$($a$)$ In a standard ascending$-$price auction, the seller increases the price from $0$ to $1,$

with bidders choosing whether to stay or leave. The auction ends when only one

bidder remains, who wins and pays the final price.

Question: A bidder$$s strategy s : $[0,1][0,1]$ describes the price s$($$)$ at which

the bidder drops out if no one else has dropped out, given her valuation $.$

Find the symmetric Bayes$-$Nash equilibrium. Also, find the expected utility of a

bidder in equilibrium as a function of her valuation $.$