Then, Bob will not participate in the second auction at all, so Alice will win the second item and pay 0, and her net utility will be 5 − 0 = 5.
In effect, a bidder may wish to “raise a rival’s costs” in one market in order to gain advantage in another.
Such considerations seem to have played a significant role in the auctions for radio spectrum licenses conducted by the Federal Communications Commission.
Assessment of rival bidders’ budget constraints was a primary component of the pre-bidding preparation of GTE’s bidding team.
A natural question to ask for such a game is when there exists a subgame perfect equilibrium in pure strategies (SPEPS).
The proof is by backward induction:[1]: 872–874 Notes: Once we know that a subgame perfect equilibrium exists, the next natural question is how efficient it is – does it obtain the maximum social welfare?
Numerical results show that, when there are many bidders with concave valuation functions, the efficiency loss decreases as the number of users increases.
[1] All bidders regard all items as pure substitute goods, so their valuations are unit demand.
This implies linear inefficiency for many natural settings, including: Case 6: unit-demand bidders with incomplete information.
When all bidders have unit demand valuations, the PoA of a Bayesian Nash equilibrium in a SAFP is at most 3.
An important practical question for sellers selling several items is how to design an auction that maximizes their revenue.
For example, a reserve price can raise the seller's revenue even though it is set at such a low level that it is never binding in equilibrium.
Sequential-auctions and simultaneous-auctions are both special case of a more general setting, in which the same bidders participate in several different mechanisms.
The class of smooth mechanisms – mechanisms that generate approximately market clearing prices – result in high-quality outcome both in equilibrium and in learning outcomes in the full information setting, as well as in Bayesian equilibrium with uncertainty about participants.
They are approximately efficient under the no-overbidding assumption, and the weak smoothness property is also maintained by composition.