The 4G auction in the UK used a highly complex mechanism called the combinatorial clock auction. Here is a simple example of how an combinatorial auction works (using the the Vickrey-Clarke-Groves rule).
Little Jimmy has two chickens to sell. His friends are little Dave, little Fred and big Phil. Little Dave wants one chicken and is prepared to pay £5 for it. Little Fred wants one chicken and is prepared to pay £3 for it. Big Phil wants both chickens for a total of £7, but it not prepared to buy just one on its own.
How do we decide who gets the chickens? How do we create an auction where Dave, Fred and Phil have an incentive to bid their maximum valuation?
Little Jimmy has two chickens to sell. His friends are little Dave, little Fred and big Phil. Little Dave wants one chicken and is prepared to pay £5 for it. Little Fred wants one chicken and is prepared to pay £3 for it. Big Phil wants both chickens for a total of £7, but it not prepared to buy just one on its own.
How do we decide who gets the chickens? How do we create an auction where Dave, Fred and Phil have an incentive to bid their maximum valuation?
Answer: Maximise the value of the bids and make the winners pay the opportunity cost of winning. (Price = total bids of all winning bids if winner didn't bid - total bids of all other winning bids)...
We maximise the sum of the bids by having little Dave and little Fred winning one chicken each. But how much do they pay?
Dave: Fred is the other winner with utility £3. If Dave was removed Phil would win both with £7. (7-3=4)
Fred: Dave is the other winner with utility £5. If Fred was removed Phil would win both with £7. (7-5=2)
Therefore...
Little Dave wins one chicken and pays £4
Little Fred wins one chicken and pays £2
Big Phil wins nothing. Big Phil is sad. Big Phil is slightly less big now.
Little Fred wins one chicken and pays £2
Big Phil wins nothing. Big Phil is sad. Big Phil is slightly less big now.
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