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.
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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
because the probability that the buyer will want to buy the item is
The seller would like to find the price that maximizes
where
For example, if the probability distribution of the buyer's valuation is uniform in
This optimal price has an alternative interpretation: it is the solution to the equation:
where
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
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
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
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:
If
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
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.