Bayesian-optimal mechanism

A Bayesian-optimal mechanism (BOM) is a mechanism in which the designer does not know the valuations of the agents for whom the mechanism is designed, but the designer knows that they are random variables and knows the probability distribution of these variables.

A typical application is a seller who wants to sell some items to potential buyers. The seller wants to price the items in a way that will maximize their profit. The optimal prices depend on the amount that each buyer is willing to pay for each item. The seller does not know these amounts, but 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 mechanism design, 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 means that we want to design rules that define a truthful mechanism, in which each agent has an incentive to report their true value.

^ Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.

[agt-1] Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.

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