I agree with Samuelson's 'solution'!

"Under Mankiw’s proposal, the airline would have to offer passengers enough money so that four of them would decide to take the money or some combination of a free ticket and cash. Let’s assume that two passengers require $1,000 plus a free ticket, a third accepts a $1,200 buyout and the last sets a price of $1,500.

"I would tweak Mankiw’s proposal by requiring that all the bumped passengers receive the highest payment. In this case, it’s $1,500. (On this issue, it’s not clear where Mankiw stands.) So the airline’s total cost would be $6,000, plus the cost of vouchers for new tickets. I would also put an upper limit for each passenger, calculated as some multiple of a full ticket price, but it would be high."