Bayesian-optimal pricing

Bayesian-optimal pricing is a kind of algorithmic pricing in which a seller determines the sell-prices based on probabilistic assumptions on the valuations of the buyers. It is a simple kind of a Bayesian-optimal mechanism.

Pricing versus auctions

An auction mechanism is similar to a pricing algorithm in that its goal is to determine prices maximizing the seller's revenue. However, an auction is more complicated and has several steps: the potential buyers bid, the winner/s is/are selected, and only then the prices are determined.

The advantage of an auction is that, theoretically, it enables the seller to get higher revenue by properly selecting the set of winners based on their bids. Moreover, the competition between the buyers may enable the auctioneer to raise the prices. For example,^[1]^{: chapter 7, page 124} consider a seller who has two different items and wants to sell one of them. There are two buyers: each buyer wants a single item. The seller wants a revenue of at least 4. He considers two options:

posted pricing: the seller sets the price of both items to 4. The first buyer that comes, buys the item that he wants, pays 4, and goes home. The seller's revenue is exactly 4.
auction: the seller does a Vickrey auction with a reserve-price of 4. Suppose the buyers' valuations are 5 and 7; then the buyer with valuation 7 will win the item that he wants, and pay 5. The seller's revenue is now 5. The auction allowed the seller to utilize the competition between the buyers in order to increase his revenue.

The disadvantage of an auction is that it is more complicated for the buyers, which may deter buyers and ultimately lead to loss of revenue.^[2]^[3] Moreover, often the revenue using algorithmic posted-pricing is a good approximation to the revenue of the Bayesian optimal mechanisms. This was studied in several settings.

A single-item auction, where the agents' values are i.i.d. random variables from a known probability distribution with bounded support. The optimal posted-pricing mechanism and the Bayesian-optimal mechanism converge to the same revenue. The convergence rate is asymptotically the same when discriminatory prices are allowed, and slower by a logarithmic factor when symmetric prices must be used. E.g, when the distribution is uniform in [0,1], the revenue of the Bayesian-optimal mechanism is $1-2/n$ ; of discriminatory prices - $1-4/n$ ; and of symmetric prices - $1-\log(n)/n$ .^[4]
A single-item auction, where the agents' values are i.i.d. random variables from a known probability distribution with unbounded support. Here, the optimal pricing and the Bayesian-optimal mechanism might not converge to the same revenue. E.g, when the cdf is $F(x)=1-1/x^{2}$ , the expected revenue on the Bayesian-optimal auction is $.88{\sqrt {n}}$ , of discriminatory prices - $.7{\sqrt {n}}$ , and of symmetric prices - $.64{\sqrt {n}}$ .
A multi-item unit-demand auction: there are different types of items, each agent has different valuations for different items, and each agent wants to buy at most one item (depending on the valuation and price). A sequential-posted-pricing mechanism attains a constant fraction of the revenue of the Bayesian-optimal mechanism. The approximation factor ranges from 1/1.5 to 1/8, depending on the feasibility constraints (constraints on the set of buyers that can be served together).^[5]^[6]

^ Jason D. Hartline (2012). Approximation in Economic Design (PDF).
^ Ausubel, Lawrence M.; Milgrom, Paul (2005). "The Lovely but Lonely Vickrey Auction". Combinatorial Auctions. p. 17. doi:10.7551/mitpress/9780262033428.003.0002. ISBN 9780262033428.
^ Catherine Holahan (June 3, 2008). "Auctions on eBay: A Dying Breed". Retrieved 1 July 2016.
^ Blumrosen, Liad; Holenstein, Thomas (2008). "Posted prices vs. Negotiations". Proceedings of the 9th ACM conference on Electronic commerce - EC '08. p. 49. doi:10.1145/1386790.1386801. ISBN 9781605581699.
^ Chawla, Shuchi; Hartline, Jason D.; Malec, David L.; Sivan, Balasubramanian (2010). "Multi-parameter mechanism design and sequential posted pricing". Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10. p. 311. doi:10.1145/1806689.1806733. ISBN 9781450300506.
^ Yan, Qiqi (2011). "Mechanism Design via Correlation Gap". Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms. p. 710. doi:10.1137/1.9781611973082.56. ISBN 978-0-89871-993-2.

[hartline12-1] Jason D. Hartline (2012). Approximation in Economic Design (PDF).

[am06-2] Ausubel, Lawrence M.; Milgrom, Paul (2005). "The Lovely but Lonely Vickrey Auction". Combinatorial Auctions. p. 17. doi:10.7551/mitpress/9780262033428.003.0002. ISBN 9780262033428.

[h08-3] Catherine Holahan (June 3, 2008). "Auctions on eBay: A Dying Breed". Retrieved 1 July 2016.

[bh08-4] Blumrosen, Liad; Holenstein, Thomas (2008). "Posted prices vs. Negotiations". Proceedings of the 9th ACM conference on Electronic commerce - EC '08. p. 49. doi:10.1145/1386790.1386801. ISBN 9781605581699.

[chms10-5] Chawla, Shuchi; Hartline, Jason D.; Malec, David L.; Sivan, Balasubramanian (2010). "Multi-parameter mechanism design and sequential posted pricing". Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10. p. 311. doi:10.1145/1806689.1806733. ISBN 9781450300506.

[y10-6] Yan, Qiqi (2011). "Mechanism Design via Correlation Gap". Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms. p. 710. doi:10.1137/1.9781611973082.56. ISBN 978-0-89871-993-2.

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Pricing versus auctions

See also