Bayesian-optimal pricing (BO 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, in which the price is determined in advance without collecting actual buyers' bids.
Single item and single buyer.
In the simplest setting, the seller has a single item to sell (with zero cost), and there is a single potential buyer. The highest price that the buyer is willing to pay for the item is called the "valuation" of the buyer. The seller would like to set the price exactly at the buyer's valuation. Unfortunately, the seller does not know the buyer's valuation. In the Bayesian model, it is assumed that the buyer's valuation is a random variable drawn from a known probability distribution.
Suppose the cumulative distribution function of the buyer is formula_1, defined as the probability that the seller's valuation is less than formula_2. Then, if the price is set to formula_3, the expected value of the seller's revenue is:
formula_4
because the probability that the buyer will want to buy the item is formula_5, and if this happens, the seller's revenue will be formula_3.
The seller would like to find the price that maximizes formula_7. The first-order condition, that the optimal price formula_8 should satisfy, is:
formula_9
where formula_10 the probability density function.
For example, if the probability distribution of the buyer's valuation is uniform in formula_11, then formula_12 and formula_13 (in formula_11). The first-order condition is formula_15 which implies formula_16. This is the optimal price only if it is in the range formula_11 (i.e., when formula_18).
Otherwise (when formula_19), the optimal price is formula_20.
This optimal price has an alternative interpretation: it is the solution to the equation:
formula_21
where formula_22 is the virtual valuation of the agent. So in this case, BO pricing is equivalent to the Bayesian-optimal mechanism, which is an auction with reserve-price formula_8.
Single item and many buyers.
In this setting, the seller has a single item to sell (with zero cost), and there are multiple potential buyers whose valuations are a random vector drawn from some known probability distribution. Here, different pricing methods come to mind:
In the multiple-buyer setting, BO pricing is no longer equivalent to BO auction: in pricing, the seller has to determine the price/s in advance, while in auction, the seller can determine the price based on the agents' bids. The competition between the buyers may enable the auctioneer to raise the price. Hence, in theory, the seller can obtain a higher revenue in an auction.
Example. There are two buyers whose valuations are distributed uniformly in the range formula_24. 
In practice, however, an auction is more complicated for the buyers since it requires them to declare their valuation in advance. The complexity of the auction process might deter buyers and ultimately lead to loss of revenue. Therefore, it is interesting to compare the optimal pricing revenue to the optimal auction revenue, to see how much revenue the seller loses by using the simpler mechanism.
Buyers with independent and identical valuations.
Blumrosen and Holenstein study the special case in which the buyers' valuations are random variables drawn independently from the same probability distribution. They show that, when the distribution of the buyers' valuations has "bounded support", BO-pricing and BO-auction 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. For example, when the distribution is uniform in [0,1] and there are formula_25 potential buyers:
In contrast, when the distribution of the buyers' valuations has "unbounded support", the BO-pricing and the BO-auction might not converge to the same revenue. E.g., when the cdf is formula_29:
Buyers with independent and different valuations.
Chawla and Hartline and Malec and Sivan study the setting in which the buyers' valuations are random variables drawn independently from different probability distributions. Moreover, there are constraints on the set of agents that can be served together (for example: there is a limited number of units). They consider two kinds of discriminatory pricing schemes:
Their general scheme for calculating the prices is:
Prob[the valuation of agent formula_33 is at least formula_39] = formula_42 Prob[the BO mechanism serves agent formula_33].
If formula_44 then the marginal-probability that an agent is served by the SPM is equal to the marginal-probability that it is served by the BO auction.
The approximation factors obtainable by an OPM depend on the structure of the constraints:
Moreover, they show two lower bounds:
The approximation factors obtainable by an SPM are naturally better:
The lower bound (proved by ) is approximately 1.25.
Yan explains the success of the sequential-pricing approach based on the concept of correlation gap, in the following way. The revenue of a mechanism is related to a set function formula_48. E.g, in a k-unit auction, the function is formula_49
Both "Winners" and "Demand" are random-sets, determined by the agents' valuations. Moreover, by carefully setting the price, it is possible to ensure that each agent formula_33 has the same probability formula_34 to be in "Winners" and to be in "Demand". However, in "Winners", there is high correlation between different agents (if one agent wins, there is more probability that other agents lose), while in "Demand", the agents are independent. Therefore, the correlation gap is an upper bound on the loss of performance when using BO SPM instead of BO auction. This gives the following approximation factors:
Different items and one unit-demand buyer.
In this setting, the seller has several different items for sale (e.g. cars of different models). There is one potential buyer, that is interested in a single item (e.g. a single car). The buyer has a different valuation for each item-type (i.e., he has a valuation-vector). Given the posted prices, the buyer buys the item that gives him the highest net utility (valuation minus price).
The buyer's valuation-vector is a random-vector from a multi-dimensional probability distribution. The seller wants to compute the price-vector (a price per item) that gives him the highest expected revenue.
Chawla and Hartline and Kleinberg study the case in which the buyer's valuations to the different items are independent random variables. They show that:
They also consider the computational task of calculating the optimal price. The main challenge is to calculate formula_60, the inverse of the virtual valuation function.
Different items and many unit-demand buyers.
In this setting, there are different types of items. Each buyer has different valuations for different items, and each buyer wants at most one item. Moreover, there are pre-specified constraints on the set of buyer-item pairs that can be allocated together (for example: each item can be allocated to at most one buyer; each buyer can get at most one item; etc).
Chawla and Hartline and Malec and Sivan study two kinds of discriminatory pricing schemes:
A sequential-pricing mechanism is, in general, not a truthful mechanism, since an agent may decide to decline a good offer in hopes of getting a better offer later. It is truthful only when, for every buyer, the buyer-item pairs for that buyer are ordered in decreasing order of net-utility. Then, it is always best for the buyer to accept the first offer (if its net utility is positive). A special case of that situation is the "single-parameter setting": for every buyer, there is only a single buyer-item pair (e.g, there is a single item for sale).
To every multi-parameter setting corresponds a single-parameter setting in which each buyer-item pair is considered an independent agent. In the single-parameter setting, there is more competition (since the agents that come from the same buyer compete with each other). Therefore, the BO revenue in the single-parameter setting is an upper bound on the BO revenue in the multi-parameter setting. Therefore, if an OPM is an "r"-approximation to the optimal mechanism for a single-parameter setting, then it is also an "r"-approximation to the corresponding multi-parameter setting. See above for approximation factors of OPMs in various settings.
See Chapter 7 "Multi-dimensional Approximation" in for more details.
Many unit-demand buyers and sellers.
Recently, the SPM scheme has been extended to a double auction setting, where there are both buyers and sellers. The extended mechanism is called 2SPM. It is parametrized by an order on the buyers, an order on the sellers, and a matrix of prices - a price for each buyer-seller pair. The prices are offered to in order to buyers and sellers who may either accept or reject the offer. The approximation ratio is between 3 and 16, depending on the setting.
References.
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