Algorithmic pricing

Algorithmic pricing is the practice of automatically setting the practice of items for sale, in order to maximize the seller's profits.

Dynamic pricing algorithms usually rely on one or more of the following data.

Statistical information on potential buyers, similarly to Bayesian-optimal mechanisms and random-sampling mechanisms.
Prices of competitors. E.g., a seller of an item may automatically detect the lowest price currently offered for that item, and suggest a price within $1 of that price.^[1]^[2]
Personal information of the currently active buyer, such as his demographics and his interest in the product. If the seller detects that you are about to buy, your price goes up.^[3]
Business information of the seller, such as the expected date in which he is going to receive new stocks, or his target selling velocity in units per day.^[4]

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,^[5]^{: 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:

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.^[6]^[7] Moreover, in several settings, it is possible to bound the loss of revenue when using algorithmic-pricing instead of Bayesian optimal mechanisms:

In a single-item auction, where the agents' values are i.i.d. random variables from a known probability distribution with bounded support, the optimal pricing 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 - $1-\log(n)/n$ .^[8]
In a single-item auction, where the agents' values are i.i.d. random variables from a known probability distribution with unbounded support, 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 - $.64{\sqrt {n}}$ .