Bayesian-optimal mechanism
Bayesian optimal mechanism design (BOMD) is a challenge in mechanism design and auction theory.
A typical application is a seller who wants to sell an item in an auction. There are many different types of auctions, and the seller wants to design the auction rules in a way that will maximize his profits. Naturally, the profits depend on the amount that each buyer is willing to pay for the item. The seller does not know these amounts, but he assumes that they are drawn from a certain known probability distribution. The phrase "Bayesian optimal mechanism design" has the following meaning:[1]: 335–338
- Bayesian means that we know the probability-distribution from which the agents' valuations are drawn (in contrast to prior-free or worst-case mechanisms, which do not assume any prior probability distribution).
- Optimal means that we want to maximize the expected revenue of the auctioneer, where the expectation is over the randomness in the agents' valuations.
- Mechanism design means that we want to design rules that define a truthful mechanism, in which each agents has an incentive to act according to his true value.
Example
There is one item for sale. There are two potential buyers. The valuation of each buyer is drawn i.i.d. from the uniform distribution on [0,1].
The Vickrey auction is a truthful mechanism and its expected profit in this case is 1/3 (the first-price sealed-bid auction is a non-truthful mechanism and its expected profit is the same).
This auction is not optimal. It is possible to get a better profit by setting a reservation price. The Vickrey auction with a reservation price of 1/2 achieves an expected profit of 5/12, which in this case is optimal.
References
- ^ Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.