A deferred-acceptance auction is an auction in which the allocation is chosen by repeatedly rejecting the least attractive bids. It is a truthful mechanism with strategic properties that make it particularly suitable to complex auctions such as the radio spectrum reallocation auction.
The government wants to maximize the social welfare. In this case, there are two feasible allocations: either give all rights to Alice, or give the North to Bob and the South to Carl. Since the valuations are private information of the agents, the government needs to use a truthful mechanism in order to induce the agents to reveal their true valuations. We compare two types of truthful mechanisms.
Vickrey–Clarke–Groves solution
The Vickrey–Clarke–Groves algorithm finds the socially-optimal allocation, which is to give both areas to Alice. Alice should pay a price determined by the externalities it imposes on the other agents. In this case, Alice pays $2M, since without her, the welfare of Bob and Carl would have been $2M. Bob and Carl receive nothing and pay nothing. A similar outcome can be implemented by an immediate acceptance auction. This auction iteratively accepts the highest-valued agent that can still be feasibly selected, and charges them the threshold payments. In this case, Alice is selected first, so Bob and Carl can no longer be selected. Alice pays her threshold value which is $1M.
Deferred-acceptance auction solution
2. The deferred-acceptance auction iteratively rejects the lowest-valued agent that can be rejected while keeping an optimal set of active agents. So, Carl is rejected first, then Bob. Alice remains and she is accepted. She pays the threshold value which is $1M. Both auction types are truthful - no single agent could gain by reporting a different value. However, they differ when agents can form coalitions. Suppose that Bob and Carl together increase their bid to $4M. Now, the VCG auction will accept Bob and Carl, and charge each of them a price of 0 ! In contrast, the DAA will reject Alice, then accept Bob and Carl, and charge each of them his threshold price, which is $3M - so they do not gain anything from their misreport.
