Solution
The Second-Price Auction
Show the problem again
In a sealed-bid auction, the highest bidder wins but pays the second-highest bid. What is the optimal bidding strategy, regardless of what the other bidders do?
Worked solution
Bid exactly your true value: it is a dominant strategy. Your bid affects only whether you win, never the price you pay. Bidding below your value can only lose you auctions you'd have profited from; bidding above can only win you auctions at a price exceeding your value. So truthful bidding weakly dominates everything else: the key insight behind Vickrey auctions and much of mechanism design.
Source: The Vickrey auction, from William Vickrey (1961). Statement written for AxiomIQ.