Selling a single item to potential buyers is a popular problem in microeconomics, with amazing related discoveries during the last sixty years. In the most standard model, there are potential buyers (the agents), with private values for the item. The celebrated Vickrey auction 
is optimal with respect to social welfare, in the sense that the agent who values the item the most gets it. Revenue maximization is possible when agents draw their private values from independent probability distributions. Statistical information about these distributions is known to the seller, who can use it and run the ideal auction to sell the item (and maximize her expected revenue). Such revenue-maximizing auctions were first presented (among other results) in the seminal paper of Myerson.
Even though welfare- and revenue-maximizing auctions are success stories (see  for a detailed coverage), the corresponding auctions are relatively complex and usually far from the ones used in practice. A different format of intermediate complexity is known as sequential posted pricing . According to it, the seller approaches the buyers in some order and proposes a price to each of them. When approached, a buyer can either refuse to buy if the proposed price exceeds her value for the item (and the process continues with the next buyer), or buy the item at the proposed price (in this case, the process stops). In general, sequential posted pricing can become notoriously complex (pricing in the airline industry is an annoying example from practice) as the best possible price for an agent can depend on statistical information about the agents’ valuations and on the decisions of agents that were approched previously. However, sequential posted pricing usually works very well when the same anonymous price (computed using statistical information about the valuations) is proposed to all agents.
A well-known result by Samuel-Cahn , known as the prophet inequality, assumes agents drawing their values from independent probability distributions and states a simple but remarkable result with numerous applications: if there is a price in which the probability that some agent will buy the item at a sequential posted pricing process is , then the same process yields a of the optimal social welfare. Constant approximations to optimal revenue (or a constant revenue gap) by sequential posted pricing with an anonymous price is possible under a regularity assumption for the valuations. Such statements (e.g., in ) usually read as follows: for any set of buyers who draw their valuations from independent and regular probability distributions, there is a price depending only on these distributions that yields an expected revenue that is a constant fraction of the expected revenue returned by Myerson’s auction.
In this work, we deviate from the above setting and assume that the item is perfectly divisible. So, different agents can get fractions of the item while some fraction of the item can stay unsold. Agent behavior is more refined now. Each agent is still interested in obtaining the whole item but gets value from fractions of it as well. In particular, we assume that each agent has a valuation function that indicates the value of the agent for fractions of the item between and . Concavity is a typical assumption here, corresponding to non-increasing marginal value. Pricing of a divisible item can become very complicated in this setting. For example, the seller can define different prices for different fractions of the item and could further discriminate among agents.
We are interested in the design of simple sequential posted pricing mechanisms that specifically use linear pricing. In particular, we restrict pricing to a fixed price for the whole item and proportional prices for fractions. Do simple prophet inequality statements, like the one of Samuel-Cahn  mentioned above, hold in this setting? Do such mechanisms have nearly-optimal revenue? These are the two questions that we study.
For the first question, we show the following results, which are similar in spirit to the prophet inequality of Samuel-Cahn. If there is a linear pricing that results in selling an expected fraction of approximately of the item using sequential posted pricing, this yields a -approximation to the social welfare. This result holds for any sequential posted pricing ordering. For random orderings, the guarantee improves to . Regarding revenue, we show that the best linear pricing recovers a fraction of optimal revenue that depends polynomially on a curvature measure of the valuations functions. In particular, this revenue gap is a small constant when the valuation functions are close to linear. Such a dependency is shown to be unavoidable, as a logarithmic lower bound shows.
When comparing the welfare of sequential posted pricing to the optimal one, we follow a standard approach in the price of anarchy literature (e.g., see  for a survey that focuses on auctions). The contribution of every agent to the welfare is split into two parts: the utility of the agent and her payment. As agents are utility maximizers, the important information that allows to compare to the optimal social welfare is that alternative decisions similar to the ones in a welfare-maximizing allocation are not profitable for the agent. The idea that has made the particular bounds possible is to consider many different alternative decisions (also called deviations in the price of anarchy literature), each with a different weight. This idea has been used in the analysis of auctions in the past (e.g., in [4, 3, 10, 32, 31]). To the best of our knowledge, it is used to prove prophet-inequality-type results for the first time.
Bounding the revenue is even more challenging. First, our setting deviates from the single-parameter environment of an indivisible single item and agents with single-valued valuations for it. Consequently, Myerson’s characterization of the revenue-maximizing allocation does not carry over. To tackle this issue, we resort to the approach of Alaei et al. , who use the notion of an ex-ante relaxation previously considered in [2, 7]. Instead of comparing the revenue of the best linear pricing to the optimal revenue, we compare to the best possible revenue of an ex-ante relaxation of the original problem, in which an expected fraction of at most is sold to agents. We are able to formulate the question of the gap between the two revenues as a mathematical program that has similarities to the one of  and allows us to use the result of  as a black box. Our result shows that the revenue gap is at most , where is the maximum ratio of the slope of a valuation function at point over its value at point . Our lower bound of is shown on a single valuation function, indicating that high curvature of valuations is not compatible with high revenue.
1.1 Related work
As mentioned above, our work on social welfare is related to the literature on prophet inequalities in optimal stopping theory. The first prophet inequalities were obtained by Krengel and Sucheston [21, 22] while the result of Samuel-Cahn  mentioned above is the most related to ours in the sense that it uses a simple threshold strategy. In the TCS literature, prophet inequalities were first studied by Hajiaghayi et al.  and have since been proved very useful in social welfare maximization in quite complex domains (e.g., see [2, 11, 12, 20]) as well as in revenue maximization [7, 8]. The survey by Lucier  provides an excellent overview of these results. A result of Duetting et al.  implies a close to approximation of optimal social welfare in our setting. However, it does not correspond to linear pricing. We remark that a approximation is best possible by adapting a folklore lower bound on prophet inequalities (e.g., see ).
Sequential posted pricing with an anonymous price has received much attention recently. Alaei et al.  prove that this mechanism achieves a constant approximation of of the optimal revenue. They use a general strategy that we discuss in more detail later in Section 4. The tight bound of follows by two recent papers by Jin et al. [16, 17]. We remark that all results for anonymous pricing carry over to our setting if the valuation functions are restricted to be linear. Indeed, the behavior of an agent with a linear valuation function against a linear pricing with a price of per unit is either to buy the whole item if her value for the whole item is higher than or refuse to buy otherwise. We mostly focus on non-linear valuation functions where divisibility differentiates our problem a lot.
In the economics literature, divisible items have received attention, with the focus being mostly on whether bundling can be beneficial or not  and on how to structure the sale as many auctions of shares . Perfect divisibility is considered in [23, 30]. In another more related direction, the operations research community has considered resource allocation mechanisms to divide an item based on signals received by the agents, that are further used to impose payments to the agents. Among them, the proportional mechanism, first defined by Kelly  and analyzed by Johari and Tsitsiklis  is the most popular one. Even though its social welfare has been analyzed extensively in stochastic settings that are very similar to ours [18, 5, 9] (see also  and the references therein), no revenue guarantees are known.
We denote by the number of agents and use the integers in set to identify them. A valuation function for agent is a monotone non-decreasing concave function . We assume that each agent draws her valuation function , independently from the other agents, from a publicly known probability distribution . We use to denote the product distribution . We denote by
a vector of the valuation functions of the agents (or valuation profile) and writeto denote that such a vector is drawn at random according to the joint probability distribution . We use the standard notation to refer to the subvector of that consists of the valuation functions of all agents besides agent .
We consider sequential posted pricing mechanisms that use linear pricing with a price per unit and an ordering of the agents (i.e., a permutation of the elements in ). The ordering defines the order in which the agents act. We use the notation to indicate that agent acts prior to agent according to . The notation allows the possibility that . When it is agent ’s turn to act, she can buy any fraction of the item that has not been given to agents that acted before her at a price of per unit of the item purchased. We assume that agents are utility maximizers. Agent has a utility of when buying a fraction of at price per unit. Since the valuation function is concave, the utility derivative is a monotone non-increasing function. Hence, if a fraction of of the item is available when it is agent ’s turn to act, she will either buy a fraction of that nullifies the derivative of her utility, i.e., , or a fraction of if , or the whole remaining item if .
For a valuation profile , we denote by the fraction of the item that agent buys at price per unit when her turn comes according to the ordering . We also denote by the fraction of the item agent would get at price per unit if she were the only agent competing for the item. By the discussion above, we have
We denote by the utility of agent . We also use for the social welfare achieved by the execution of the mechanism. We also denote by the fraction of the item that agent gets in an social welfare-maximizing assignment and by the optimal social welfare. Clearly, the optimal social welfare depends only on the valuation profile and not on the price or the ordering . The sub-optimality of the social welfare achieved by the mechanism is actually due to these latter two characteristics.
We denote by the revenue of the mechanism. We do not define the optimal revenue here; this definition will be given implicitly in Section 4 together with a refinement of the setting as described above and additional notions.
3 Approximating the optimal social welfare
We first present our results for social welfare. We distinguish between mechanisms that use adversarial (worst-case) and random ordering. In the analysis of both mechanisms, we use the following lemma which bounds the expected utility of an agent by her contribution to the optimal social welfare. The proof follows by comparing the decision of the agent with several alternative ones that are defined in terms of her decision in a hypothetical welfare-maximizing allocation of the item.
For every agent , price per unit, ordering , and , it holds:
Observe that the rightmost parenthesis in the expression above is non-negative. If the leftmost parenthesis is negative, the lemma clearly follows. In the following, we assume that .
Let be a vector of valuation functions and . Recall that when , a fraction of at least of the item is unallocated when it is agent ’s turn to act. Since agent prefers a fraction of to a fraction of , we have
The second inequality follows by Jensen inequality due to the concavity of function .
Clearly, for every . Furthermore, observe that . Using these observations and inequality (2), we have
The equality follows since the quantity does not depend on the condition . The last inequality follows since . ∎
3.1 Using an adversarial ordering
Our first main result for the social welfare is the following.
Let be such that , i.e., , and . Let be such that the mechanism with linear posted pricing that uses price per unit and processes the agents in any order satisfies . Then, this mechanism yields a -approximation of the optimal social welfare.
To prove the theorem, we will bound the expected utility of agent as follows:
Then, we will have
The last equality follows by linearity of expectation and since for every valuation vector .
for every non-negative random variableand equation (1), we have
as desired. The last equality follows by the definition of and . ∎
3.2 Using a random ordering
Now, we assume that the mechanism selects the ordering uniformly at random among all permutations of . We denote by this probability distribution. In addition to Lemma 1, the proof of the main result of this section (Theorem 4) uses the following technical lemma.
For every agent , price per unit, and , it holds:
For every permutation of , denote by its reverse. Then, using the fact that each permutation is selected equiprobably by and equation (1), we have
as desired. ∎
Let and be such that the mechanism with linear posted pricing that uses price per unit and processes the agents in a random order satisfies . Then, this mechanism yields a -approximation of the optimal social welfare.
The proof is similar to that of Theorem 2. We will bound the expected utility of agent as follows:
Then, by expressing the contribution of each agent to the social welfare as the sum of her utility and her payment, we will have
The last equality follows by linearity of expectation and since for every valuation vector .
as desired. The last equality follows by the definition of and . ∎
4 Approximating the optimal revenue
For convenience, we assume that the valuation functions that are drawn by the agents are differentiable in and have bounded curvature. In particular, we use the ratio as a measure of a curvature of the valuation function and consider agents that draw random valuation functions with curvature at most . Then, the approximations of optimal revenue we present are expressed as functions of .
As results in anonymous pricing mechanisms imply (see, e.g., ), reasonable revenue approximations are not possible if the valuation functions do not satisfy a regularity condition. In the classical setting of selling an indivisible item, each agent has a scaller valuation drawn from a probability distribution wth regular commulative density function . Regularity means that the revenue-quantile curve
revenue-quantile curveis concave in . In our setting, valuations are not scalars but functions, so the regularity assumption needs to be extended. One modeling assumption that has been followed in  is to define valuation functions as with a scalar part and a function part . The function part is known to the seller and can be used by the mechanism. The scalar part is drawn from a probability distribution, in which regularity can be imposed. This is essentially a single-parameter environment, where it is a simple exercise to apply Myerson’s approach  of maximizing revenue by maximizing virtual welfare. This means that revenue-optimal mechanisms have a well-known structure in this setting.
The setting we consider is multi-parameter and more general than the one we just described. For every agent and every , we assume that the derivative value of the concave valuation function is drawn from a regular probability distribution with cummulative density function . In our case, regularity means that the revenue-quantile curve is concave in for every agent and every . Unfortunately, Myerson’s machinery cannot be applied in our setting, so we do not have any clean form of the revenue-maximizing mechanism. This makes the proof of our main result for the revenue gap that follows more challenging.
The revenue gap when selling a divisible item to agents with valuations of maximum curvature using linear pricing is at most .
4.1 Interlude: a quick overview of the approach of Alaei et al. 
To bound the maximum revenue gap, we adapt the approach of  for selling an indivisible item using sequential posted pricing and an anonymous price.
Alaei et al.  use the mathematical program below to bound the gap between the optimal expected revenue and the best expected revenue that can be achieved with an anonymous price. Actually, instead of comparing directly to the optimal revenue (in fact, this is done in follow-up work by Jin et al. ), they compare to the best possible revenue of the ex-ante relaxation, i.e., the maximum expected revenue that can be achieved by a mechanism that sells the item to at most one agent in expectation. Their mathematical program uses as variables (1) the regular cummulative density function of the valuation of agent for the item and (2) the probability that the item is given to agent in the revenue-maximizing allocation for the ex-ante relaxation.
Viewed together with the first constraint, the objective of mathematical program (7) is to maximize the revenue of the ex-ante relaxation. The second set of constraints implements the restriction that no anonymous pricing yields a revenue higher than . The ratio between the objective value of (7) and is an upper bound to the revenue gap. The main result of  is as follows.
Theorem 6 (Alaei et al. ).
For every , the objective value of the mathematical program (7) is at most .
4.2 A mathematical program for the revenue gap
Let us now return to our setting. We also use an ex-ante relaxation to upper-bound the optimal revenue and to relate it to the maximum revenue that can be achieved with linear pricing. Let us consider pricing functions for each agent that maximize the revenue of the ex-ante relaxation. For every agent and every , let be the probability that a fraction of at least is bought by agent with optimal pricing. For and , observe that agent will get a fraction of at least if her utility derivative is non-negative at point , i.e., . Hence, and, equivalently, . Using the quantities and , we can express the expected payment by agent as . Intuitively, denotes the probability that agent buys her -th point of the item, denotes the payment increase due to this point, and the quantity represents its contribution to the revenue. Hence, denoting by the optimal expected revenue for our original problem, we get
requires that, on average, no more than the whole item is available for purchase by the agents, and thus guarantees that the RHS of equation (8) is indeed the revenue of the ex-ante relaxation.
Our next step is to include additional constraints for bounding the revenue obtained by any linear pricing. We will need the following lemma.
Let , , …, be random variables with for and let . Then, .
We claim that is minimized when , , …, are Bernoulli random variables. Then, it will be
and the lemma will follow.
To prove this claim, we show that by replacing the random variable with the Bernoulli random variable with and the expectation can only become smaller. The claim will follow by repeating this argument and replacing each random variable with a Bernoulli one that has the same expectation.
Formally, let ; we will show that . Denoting by the cdf of the random variable , we have that, conditioned on , the expected contribution of to is
The inequality follows since is non-increasing in and, subsequently, is concave in and has no point below the line for . Denoting the pdf of the random variable by , we have
as desired. ∎
To determine the revenue gap, we can require that any linear pricing with price per unit has revenue at most and ask: “how large can the revenue of the optimal ex-ante relaxation can be?” Bounding the RHS of (10) is sufficient for bounding the revenue of linear pricing by . Then, the maximum value the optimal revenue can get is upper-bounded by the value the RHS of (8) can get under the constraint (9). So, we will bound the revenue gap using the following mathematical program that has as variables the cdf’s for , and and the probabilities for and :
4.3 Bounding the objective value of the mathematical program
We now prove an upper bound on the objective value of mathematical program (11) by relating it to mathematical program (7), and exploiting the revenue gap of  for anonymous item pricing (Theorem 6).
Our main tool is the following transformation. We use the notation as abbreviation for the probabilities for every agent and and the cummulative density functions for every agent , , and . Given the pair , we define and for every agent and , and use the pair as their abbreviation.
Bounding the objective value of mathematical program (11) has two steps; the first one is implemented by the next lemma.
Clearly, the solution satisfies the first constraint of the mathematical program (7) since satisfies the first constraint of (11). In the following, we show that the second constraint is satisfied as well. We will need two technical lemmas.
Let be an integer and . Then, for every , it holds that