Simple posted pricing mechanisms for selling a divisible item
aa r X i v : . [ c s . G T ] J u l Simple posted pricing mechanismsfor selling a divisible item ∗ Ioannis Caragiannis † Apostolis Kerentzis ‡ Abstract
We study the problem of selling a divisible item to agents who have concave valuationfunctions for fractions of the item. This is a fundamental problem with apparent applicationsto pricing communication bandwidth or cloud computing services. We focus on simplesequential posted pricing mechanisms that use linear pricing, i.e., a fixed price for the wholeitem and proportional prices for fractions of it. We present results of the following form thatcan be thought of as analogs of the well-known prophet inequality of Samuel-Cahn (1984).For ρ ≈ ρ -fractionof the item, this results in a ρ -approximation of the optimal social welfare. The value of ρ can be improved to approximately 42% if sequential posted pricing considers the agents inrandom order. We also show that the best linear pricing yields an expected revenue that is atmost O ( κ ) times smaller than the optimal one, where κ is a bound on the curvature of thevaluation functions. The proof extends and exploits the approach of Alaei et al. (2019) andbounds the revenue gap by the objective value of a mathematical program. The dependenceof the revenue gap on κ is unavoidable as a lower bound of Ω(ln κ ) indicates. Selling a single item to potential buyers is a popular problem in microeconomics, with amazingrelated discoveries during the last sixty years. In the most standard model, there are n potentialbuyers (the agents), with private values for the item. The celebrated Vickrey auction [33] isoptimal with respect to social welfare, in the sense that the agent who values the item themost gets it. Revenue maximization is possible when agents draw their private values fromindependent probability distributions. Statistical information about these distributions is knownto the seller, who can use it and run the ideal auction to sell the item (and maximize her expectedrevenue). Such revenue-maximizing auctions were first presented (among other results) in theseminal paper of Myerson [26].Even though welfare- and revenue-maximizing auctions are success stories (see [14] for adetailed coverage), the corresponding auctions are relatively complex and usually far from theones used in practice. A different format of intermediate complexity is known as sequentialposted pricing [8]. According to it, the seller approaches the buyers in some order and proposesa price to each of them. When approached, a buyer can either refuse to buy if the proposedprice exceeds her value for the item (and the process continues with the next buyer), or buy theitem at the proposed price (in this case, the process stops). In general, sequential posted pricingcan become notoriously complex (pricing in the airline industry is an annoying example from ∗ This work was partially supported by the COST Action 16228 “European Network for Game Theory”. † Computer Technology Institute “Diophantus” & Department of Computer Engineering and Informatics,University of Patras, 26504 Rion, Greece. Email: [email protected] . ‡ Department of Computer Engineering and Informatics, University of Patras, 26504 Rion, Greece. Email: [email protected] . anonymous price (computedusing statistical information about the valuations) is proposed to all agents.A well-known result by Samuel-Cahn [29], known as the prophet inequality , assumes agentsdrawing their values from independent probability distributions and states a simple but remark-able result with numerous applications: if there is a price in which the probability that someagent will buy the item at a sequential posted pricing process is 1 /
2, then the same processyields a 50% of the optimal social welfare. Constant approximations to optimal revenue (or aconstant revenue gap ) by sequential posted pricing with an anonymous price is possible under aregularity assumption for the valuations. Such statements (e.g., in [16]) usually read as follows:for any set of buyers who draw their valuations from independent and regular probability distri-butions, there is a price depending only on these distributions that yields an expected revenuethat 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 itembut gets value from fractions of it as well. In particular, we assume that each agent has avaluation function that indicates the value of the agent for fractions of the item between 0and 1. 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 sellercan define different prices for different fractions of the item and could further discriminateamong agents.We are interested in the design of simple sequential posted pricing mechanisms that specif-ically use linear pricing . In particular, we restrict pricing to a fixed price for the whole itemand proportional prices for fractions. Do simple prophet inequality statements, like the oneof Samuel-Cahn [29] 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 theprophet inequality of Samuel-Cahn. If there is a linear pricing that results in selling an expectedfraction of approximately 32% of the item using sequential posted pricing, this yields a 32%-approximation to the social welfare. This result holds for any sequential posted pricing ordering.For random orderings, the guarantee improves to 42%. Regarding revenue, we show that the bestlinear pricing recovers a fraction of optimal revenue that depends polynomially on a curvaturemeasure of the valuations functions. In particular, this revenue gap is a small constant whenthe valuation functions are close to linear. Such a dependency is shown to be unavoidable, as alogarithmic lower bound shows.When comparing the welfare of sequential posted pricing to the optimal one, we follow astandard approach in the price of anarchy literature (e.g., see [28] for a survey that focuses onauctions). The contribution of every agent to the welfare is split into two parts: the utility ofthe agent and her payment. As agents are utility maximizers, the important information thatallows to compare to the optimal social welfare is that alternative decisions similar to the onesin a welfare-maximizing allocation are not profitable for the agent. The idea that has madethe particular bounds possible is to consider many different alternative decisions (also calleddeviations in the price of anarchy literature), each with a different weight. This idea has beenused in the analysis of auctions in the past (e.g., in [3, 4, 10, 31, 32]). To the best of ourknowledge, 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 valuations2or it. Consequently, Myerson’s characterization of the revenue-maximizing allocation does notcarry over. To tackle this issue, we resort to the approach of Alaei et al. [2], who use the notionof an ex-ante relaxation previously considered in [1, 7]. Instead of comparing the revenue of thebest linear pricing to the optimal revenue, we compare to the best possible revenue of an ex-anterelaxation of the original problem, in which an expected fraction of at most 1 is sold to agents.We are able to formulate the question of the gap between the two revenues as a mathematicalprogram that has similarities to the one of [2] and allows us to use the result of [2] as a blackbox. Our result shows that the revenue gap is at most O ( κ ), where κ is the maximum ratioof the slope of a valuation function at point 0 over its value at point 1. Our lower bound ofΩ(ln κ ) is shown on a single valuation function, indicating that high curvature of valuations isnot compatible with high revenue. As mentioned above, our work on social welfare is related to the literature on prophet inequal-ities in optimal stopping theory. The first prophet inequalities were obtained by Krengel andSucheston [21, 22] while the result of Samuel-Cahn [29] mentioned above is the most relatedto ours in the sense that it uses a simple threshold strategy. In the TCS literature, prophetinequalities were first studied by Hajiaghayi et al. [13] and have since been proved very usefulin social welfare maximization in quite complex domains (e.g., see [1, 11, 12, 20]) as well as inrevenue maximization [7, 8]. The survey by Lucier [24] provides an excellent overview of theseresults. A result of Duetting et al. [11] implies a close to 50% approximation of optimal socialwelfare in our setting. However, it does not correspond to linear pricing. We remark that a50% approximation is best possible by adapting a folklore lower bound on prophet inequalities(e.g., see [24]).Sequential posted pricing with an anonymous price has received much attention recently.Alaei et al. [2] prove that this mechanism achieves a constant approximation of 2 .
72 of theoptimal revenue. They use a general strategy that we discuss in more detail later in Section 4.The tight bound of 2 .
62 follows by two recent papers by Jin et al. [16, 17]. We remark that allresults for anonymous pricing carry over to our setting if the valuation functions are restrictedto be linear. Indeed, the behavior of an agent with a linear valuation function against a linearpricing with a price of p per unit is either to buy the whole item if her value for the whole itemis higher than p or refuse to buy otherwise. We mostly focus on non-linear valuation functionswhere divisibility differentiates our problem a lot.In the economics literature, divisible items have received attention, with the focus beingmostly on whether bundling can be beneficial or not [27] and on how to structure the sale asmany auctions of shares [34]. Perfect divisibility is considered in [23, 30]. In another more relateddirection, the operations research community has considered resource allocation mechanisms todivide an item based on signals received by the agents, that are further used to impose paymentsto the agents. Among them, the proportional mechanism, first defined by Kelly [19] and analyzedby Johari and Tsitsiklis [18] is the most popular one. Even though its social welfare has beenanalyzed extensively in stochastic settings that are very similar to ours [18, 5, 9] (see also [6]and the references therein), no revenue guarantees are known. The rest of the paper is structured as follows. We begin with preliminary definitions andnotation in Section 2. Our results for the social welfare appear in Section 3 and for the revenuein Section 4. We conclude with a short discussion on open problems in Section 5.3
Preliminaries
We denote by n the number of agents and use the integers in set [ n ] = { , , ..., n } to identifythem. A valuation function for agent i is a monotone non-decreasing concave function v i :[0 , → R ≥ . We assume that each agent i draws her valuation function v i , independentlyfrom the other agents, from a publicly known probability distribution F i . We use F to denotethe product distribution F × F × ... × F n . We denote by v = h v , v , ..., v n i a vector of thevaluation functions of the agents (or valuation profile) and write v ∼ F to denote that sucha vector is drawn at random according to the joint probability distribution F . We use thestandard notation v − i to refer to the subvector of v that consists of the valuation functions ofall agents besides agent i .We consider sequential posted pricing mechanisms that use linear pricing with a price p perunit and an ordering π of the agents (i.e., a permutation of the elements in [ n ]). The ordering π defines the order in which the agents act. We use the notation j ≻ π i to indicate that agent i acts prior to agent j according to π . The notation j (cid:23) π i allows the possibility that j = i .When it is agent i ’s turn to act, she can buy any fraction of the item that has not been givento agents that acted before her at a price of p per unit of the item purchased. We assume thatagents are utility maximizers. Agent i has a utility of v i ( z ) − p · z when buying a fraction of z at price p per unit. Since the valuation function v i is concave, the utility derivative v ′ i ( z ) − p is a monotone non-increasing function. Hence, if a fraction of α of the item is available whenit is agent i ’s turn to act, she will either buy a fraction of z that nullifies the derivative of herutility, i.e., v ′ i ( z ) = p , or a fraction of 0 if v ′ i (0) < p , or the whole remaining item if v ′ i ( α ) > p .For a valuation profile v , we denote by y i ( v , p, π ) the fraction of the item that agent i buysat price p per unit when her turn comes according to the ordering π . We also denote by y ∗ i ( v i , p )the fraction of the item agent i would get at price p per unit if she were the only agent competingfor the item. By the discussion above, we have y i ( v , p, π ) = y ∗ i ( v i , p ) , if P j (cid:23) π i y ∗ i ( v i , p ) ≤ − P j ≻ π i y ∗ i ( v i , p ) , if P j ≻ π i y ∗ i ( v i , p ) ≤ < P j (cid:23) π i y ∗ i ( v i , p )0 , if P j ≻ π i y ∗ i ( v i , p ) > X j ∈ [ n ] y j ( v , p, π ) = min , X j ∈ [ n ] y ∗ j ( v j , p ) . (1)We denote by u i ( v , p, π ) = v i ( y i ( v , p, π )) − p · y i ( v , p, π ) the utility of agent i . We alsouse SW( v , p, π ) = P i ∈ [ n ] v i ( y i ( v , p, π )) for the social welfare achieved by the execution of themechanism. We also denote by x i ( v ) the fraction of the item that agent i gets in an socialwelfare-maximizing assignment and by SW ∗ ( v ) = P i ∈ [ n ] v i ( x i ( v )) the optimal social welfare.Clearly, the optimal social welfare depends only on the valuation profile and not on the price p or the ordering π . The sub-optimality of the social welfare achieved by the mechanism isactually due to these latter two characteristics.We denote by REV( v , p, π ) = p · P i ∈ [ n ] y i ( v , p, π ) the revenue of the mechanism. We do notdefine the optimal revenue here; this definition will be given implicitly in Section 4 togetherwith a refinement of the setting as described above and additional notions. We first present our results for social welfare. We distinguish between mechanisms that useadversarial (worst-case) and random ordering. In the analysis of both mechanisms, we use4he following lemma which bounds the expected utility of an agent by her contribution to theoptimal social welfare. The proof follows by comparing the decision of the agent with severalalternative ones that are defined in terms of her decision in a hypothetical welfare-maximizingallocation of the item.
Lemma 1.
For every agent i , price p per unit, ordering π , and β > , it holds: E v ∼ F [ u i ( v , p, π )] ≥ β (cid:18) E v ∼ F [ v i ( x i ( v ))] − p E v ∼ F [ x i ( v )] (cid:19) · − e − /β − Z − e − /β Pr v ∼ F X j ≻ π i y ∗ j ( v j , p ) ≥ t d t . Proof.
Observe that the rightmost parenthesis in the expression above is non-negative. If theleftmost parenthesis is negative, the lemma clearly follows. In the following, we assume that E v ∼ F [ v i ( x i ( v ))] ≥ p · E v ∼ F [ x i ( v )].Let v be a vector of valuation functions and t ∈ [0 , P j ≻ π i y ∗ j ( v j , p ) ≤ t ,a fraction of at least 1 − t of the item is unallocated when it is agent i ’s turn to act. Since agent i prefers a fraction of y i ( v , p, π ) to a fraction of (1 − t ) E v ′− i ∼ F − i [ x i ( v i , v ′− i )], we have u i ( v , p, π ) ≥ v i (1 − t ) E v ′− i ∼ F − i [ x i ( v i , v ′− i )] ! − (1 − t ) · p E v ′− i ∼ F − i [ x i ( v i , v ′− i )] ≥ (1 − t ) E v ′− i ∼ F − i [ v i (cid:0) x i ( v i , v ′− i ) (cid:1) ] − (1 − t ) · p E v ′− i ∼ F − i [ x i ( v i , v ′− i )] . (2)The second inequality follows by Jensen inequality due to the concavity of function v i .Clearly, E v ∼ F [ u i ( v , p, π )] ≥ E v ∼ F h u i ( v , p, π ) nP j ≻ π i y ∗ j ( v j , p ) ≤ t oi for every t ∈ [0 , β R − e − /β t − t = 1. Using these observations and inequality (2), wehave E v ∼ F [ u i ( v , p, π )] ≥ β Z − e − /β E v ∼ F u i ( v , p, π ) X j ≻ π i y ∗ j ( v j , p ) ≤ t d t − t ≥ β Z − e − /β E v ∼ F E v ′− i ∼ F − i [ v i (cid:0) x i ( v i , v ′− i ) (cid:1) ] − p E v ′− i ∼ F − i [ x i ( v i , v ′− i )] ! X j ≻ π i y ∗ j ( v j , p ) ≤ t d t = β Z − e − /β E v i ∼ F i " E v ′− i ∼ F − i [ v i (cid:0) x i ( v i , v ′− i ) (cid:1) ] − p E v ′− i ∼ F − i [ x i ( v i , v ′− i )] · Pr v ∼ F X j ≻ π i y ∗ j ( v j , p ) ≤ t d t ≥ β (cid:18) E v ∼ F [ v i ( x i ( v ))] − p E v ∼ F [ x i ( v )] (cid:19) · − e − /β − Z − e − /β Pr v ∼ F X j ≻ π i y ∗ j ( v j , p ) ≥ t d t . The equality follows since the quantity E v ′− i ∼ F − i [ v i (cid:0) x i ( v i , v ′− i ) (cid:1) ] − p E v ′− i ∼ F − i [ x i ( v i , v ′− i )]does not depend on the condition P j ≻ π i y ∗ j ( v j , p ) ≤ t . The last inequality follows since E v ∼ F [ v i ( x i ( v ))] ≥ p E v ∼ F [ x i ( v )]. 5 .1 Using an adversarial ordering Our first main result for the social welfare is the following.
Theorem 2.
Let β be such that e /β = 2 + 1 /β , i.e., β ≈ . , and ρ = e − /β ≈ . .Let p be such that the mechanism with linear posted pricing that uses price p per unit andprocesses the agents in any order satisfies E v ∼ F [ P i ∈ [ n ] y i ( v , p, π )] = ρ . Then, this mechanismyields a ρ -approximation of the optimal social welfare.Proof. To prove the theorem, we will bound the expected utility of agent i as follows: E v ∼ F [ u i ( v , p, π )] ≥ ρ E v ∼ F [ v i ( x i ( v ))] − ρ · p E v ∼ F [ x i ( v )] . (3)Then, we will have E v ∼ F [SW( v , p, π )] = E v ∼ F X i ∈ [ n ] v i ( y i ( v , p, π )) = E v ∼ F X i ∈ [ n ] ( v i ( y i ( v , p, π )) − p · y i ( v , p, π )) + p · E v ∼ F X i ∈ [ n ] y i ( v , p, π ) = X i ∈ [ n ] E v ∼ F [ u i ( v , p, π )] + ρ · p ≥ ρ X i ∈ [ n ] E v ∼ F [ v i ( x i ( v ))] − ρ · p X i ∈ [ n ] E v ∼ F [ x i ( v )] + ρ · p = ρ E v ∼ F [SW ∗ ( v )] . The last equality follows by linearity of expectation and since P i ∈ [ n ] x i ( v ) = 1 for every valua-tion vector v .To show inequality (3), we assume that E v ∼ F [ v i ( x i ( v ))] − p E v ∼ F [ x i ( v )] is non-negative (ob-serve that (3) trivially holds otherwise) and use Lemma 1. Using the property R α Pr[ X ≥ t ] d t = E [min { α, X } ] for every non-negative random variable X and equation (1), we have Z − e − /β Pr v ∼ F X j ≻ π i y ∗ j ( v j , p ) ≥ t d t ≤ Z − e − /β Pr v ∼ F X j ∈ [ n ] y ∗ j ( v j , p ) ≥ t d t = E v ∼ F min − e − /β , X j ∈ [ n ] y ∗ j ( v j , p ) ≤ E v ∼ F X j ∈ [ n ] y j ( v , p, π ) = e − /β . (4)By our assumption that E v ∼ F [ v i ( x i ( v ))] − p E v ∼ F [ x i ( v )] is non-negative, Lemma 1 and inequality(4) yield E v ∼ F [ u i ( v , p, π )] ≥ β (1 − e − /β ) (cid:18) E v ∼ F [ v i ( x i ( v ))] − p E v ∼ F [ x i ( v )] (cid:19) = ρ E v ∼ F [ v i ( x i ( v ))] − ρ · p E v ∼ F [ x i ( v )] , as desired. The last equality follows by the definition of β and ρ .6 .2 Using a random ordering Now, we assume that the mechanism selects the ordering π uniformly at random among allpermutations of [ n ]. We denote by Π this probability distribution. In addition to Lemma 1,the proof of the main result of this section (Theorem 4) uses the following technical lemma. Lemma 3.
For every agent i , price p per unit, and α > , it holds: E v ∼ F π ∼ Π min { α, X j ≻ π i y ∗ j ( v j , p ) } ≤ max { α, / } E v ∼ F π ∼ Π X j ∈ [ n ] y j ( v , p, π ) . Proof.
For every permutation π of [ n ], denote by π ′ its reverse. Then, using the fact that eachpermutation is selected equiprobably by Π and equation (1), we have E v ∼ F π ∼ Π min { α, X j ≻ π i y ∗ j ( v j , p ) } = E v ∼ F π ∼ Π
12 min { α, X j ≻ π i y ∗ j ( v j , p ) } + 12 min { α, X j ≻ π ′ i y ∗ j ( v j , p ) } ≤ E v ∼ F π ∼ Π min { α, X j ∈ [ n ] \{ i } y ∗ j ( v j , p ) } ≤ max { α, / } E v ∼ F π ∼ Π min { , X j ∈ [ n ] y ∗ j ( v j , p ) } = max { α, / } E v ∼ F π ∼ Π X j ∈ [ n ] y j ( v , p, π ) , as desired. Theorem 4.
Let ρ = (1 + 2 ln 2) − ≈ . and p be such that the mechanism with linearposted pricing that uses price p per unit and processes the agents in a random order satis-fies E v ∼ F ,π ∼ Π [ P i ∈ [ n ] y i ( v , p, π )] = ρ . Then, this mechanism yields a ρ -approximation of theoptimal social welfare.Proof. The proof is similar to that of Theorem 2. We will bound the expected utility of agent i as follows: E v ∼ F π ∼ Π [ u i ( v , p, π )] ≥ ρ E v ∼ F [ v i ( x i ( v ))] − ρ · p E v ∼ F [ x i ( v )] . (5)Then, by expressing the contribution of each agent to the social welfare as the sum of her utilityand her payment, we will have E v ∼ F π ∼ Π [SW( v , p, π )] = E v ∼ F π ∼ Π X i ∈ [ n ] v i ( y i ( v , p, π )) = E v ∼ F π ∼ Π X i ∈ [ n ] ( v i ( y i ( v , p, π )) − p · y i ( v , p, π )) + p · E v ∼ F π ∼ Π X i ∈ [ n ] y i ( v , p, π ) = X i ∈ [ n ] E v ∼ F π ∼ Π [ u i ( v , p, π )] + ρ · p ρ X i ∈ [ n ] E v ∼ F [ v i ( x i ( v ))] − ρ · p X i ∈ [ n ] E v ∼ F [ x i ( v )] + ρ · p = ρ E v ∼ F [SW ∗ ( v )] . The last equality follows by linearity of expectation and since P i ∈ [ n ] x i ( v ) = 1 for every valua-tion vector v .To show inequality (5), we assume that E v ∼ F [ v i ( x i ( v ))] − p E v ∼ F [ x i ( v )] is non-negative(observe that (5) trivially holds otherwise) and use Lemma 1 with β = . Using the property R α Pr[ X ≥ t ] d t = E [min { α, X } ] for every non-negative random variable X , we have Z / Pr v ∼ F π ∼ Π X j ≻ π i y ∗ j ( v j , p ) ≥ t d t = E v ∼ F π ∼ Π min / , X j ≻ π i y ∗ j ( v j , p ) ≤ · E v ∼ F π ∼ Π X j ∈ [ n ] y j ( v , p, π ) = ρ . (6)By our assumption that E v ∼ F [ v i ( x i ( v ))] − p E v ∼ F [ x i ( v )] is non-negative, Lemma 1 with β = and inequality (6) yield E v ∼ F π ∼ Π [ u i ( v , p, π )] ≥ β (cid:16) − e − /β − ρ (cid:17) (cid:18) E v ∼ F [ v i ( x i ( v ))] − p E v ∼ F [ x i ( v )] (cid:19) = ρ E v ∼ F [ v i ( x i ( v ))] − ρ · p E v ∼ F [ x i ( v )] , as desired. The last equality follows by the definition of β and ρ . For convenience, we assume that the valuation functions that are drawn by the agents aredifferentiable in [0 ,
1] and have bounded curvature. In particular, we use the ratio v ′ (0) /v (1)as a measure of a curvature of the valuation function v and consider agents that draw randomvaluation functions with curvature at most κ ≥
1. Then, the approximations of optimal revenuewe present are expressed as functions of κ .As results in anonymous pricing mechanisms imply (see, e.g., [2]), reasonable revenue ap-proximations are not possible if the valuation functions do not satisfy a regularity condition. Inthe classical setting of selling an indivisible item, each agent i has a scaller valuation v i drawnfrom a probability distribution wth regular commulative density function H i . Regularity meansthat the revenue-quantile curve q · H − i (1 − q ) is concave in q . In our setting, valuations arenot scalars but functions, so the regularity assumption needs to be extended. One modelingassumption that has been followed in [30] is to define valuation functions as v i ( z ) = t i · h i ( z )with a scalar part t i and a function part h i ( z ). The function part is known to the seller andcan be used by the mechanism. The scalar part is drawn from a probability distribution, inwhich regularity can be imposed. This is essentially a single-parameter environment , where itis a simple exercise to apply Myerson’s approach [26] of maximizing revenue by maximizingvirtual welfare. This means that revenue-optimal mechanisms have a well-known structure inthis setting.The setting we consider is multi-parameter and more general than the one we just described.For every agent i ∈ [ n ] and every x ∈ [0 , v ′ i ( x ) of theconcave valuation function is drawn from a regular probability distribution with cummulative8ensity function F i,x . In our case, regularity means that the revenue-quantile curve q · F − i,x (1 − q )is concave in q for every agent and every x ∈ [0 , Theorem 5.
The revenue gap when selling a divisible item to n agents with valuations ofmaximum curvature κ ≥ using linear pricing is at most O ( κ ) . To bound the maximum revenue gap, we adapt the approach of [2] for selling an indivisible itemusing sequential posted pricing and an anonymous price.Alaei et al. [2] use the mathematical program below to bound the gap between the optimalexpected 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-upwork by Jin et al. [16]), 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 itemto at most one agent in expectation . Their mathematical program uses as variables (1) theregular cummulative density function H i of the valuation of agent i for the item and (2) theprobability r i that the item is given to agent i in the revenue-maximizing allocation for theex-ante relaxation. maximize X i ∈ [ n ] r i H − i (1 − r i ) (7)subject to X i ∈ [ n ] r i ≤ p − Y i ∈ [ n ] H i ( p ) ≤ R, ∀ p > Rr i ∈ [0 , , ∀ i ∈ [ n ] H i is a regular cdf , ∀ i ∈ [ n ]Viewed together with the first constraint, the objective of mathematical program (7) is tomaximize the revenue of the ex-ante relaxation. The second set of constraints implements therestriction that no anonymous pricing p yields a revenue higher than R . The ratio between theobjective value of (7) and R is an upper bound to the revenue gap. The main result of [2] is asfollows. Theorem 6 (Alaei et al. [2]) . For every R ≥ , the objective value of the mathematical program(7) is at most e · R . Our proof of the revenue gap when selling a divisible item with linear pricing will use theresult of [2] as a black box. We remark that the original result in [2] uses specifically R = 1.The extension we consider here is without loss of generality and is used for simplicity of ourexposition. Let us now return to our setting. We also use an ex-ante relaxation to upper-bound the optimalrevenue and to relate it to the maximum revenue that can be achieved with linear pricing. Let9s consider pricing functions p i : [0 , → R + for each agent i ∈ [ n ] that maximize the revenue ofthe ex-ante relaxation. For every agent i ∈ [ n ] and every x ∈ [0 , q i ( x ) be the probabilitythat a fraction of at least x is bought by agent i with optimal pricing. For i ∈ [ n ] and x ∈ [0 , i will get a fraction of at least x if her utility derivative is non-negative atpoint x , i.e., v ′ i ( x ) ≥ p ′ i ( x ). Hence, q i ( x ) = Pr[ v ′ i ( x ) ≥ p ′ i ( x )] = 1 − F i,x ( p ′ i ( x )) and, equivalently, p ′ i ( x ) = F − i,x (1 − q i ( x )). Using the quantities q i ( x ) and p ′ i ( x ), we can express the expectedpayment by agent i as R q i ( x ) p ′ i ( x ) d x = R q i ( x ) F − i,x (1 − q i ( x )) d x . Intuitively, q i ( x ) denotesthe probability that agent i buys her x -th point of the item, p ′ i ( x ) denotes the payment increasedue to this point, and the quantity q i ( x ) F − i,x (1 − q i ( x )) represents its contribution to the revenue.Hence, denoting by E [REV ∗ ( v )] the optimal expected revenue for our original problem, we get E [REV ∗ ( v )] ≤ X i ∈ [ n ] Z q i ( x ) F − i,x (1 − q i ( x )) d x. (8)The constraint X i ∈ [ n ] Z q i ( x ) d x ≤ 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 anylinear pricing. We will need the following lemma. Lemma 7.
Let X , X , ..., X k be random variables with X i ∈ [0 , for i ∈ [ k ] and let X = P i ∈ [ k ] X i . Then, E [min { , X } ] ≥ − Q i ∈ [ k ] (1 − E [ X i ]) .Proof. We claim that E [min { , X } ] is minimized when X , X , ..., X k are Bernoulli randomvariables. Then, it will be E [min { , X } ] = 1 − Y i ∈ [ k ] Pr[ X i = 0] = 1 − Y i ∈ [ k ] (1 − E [ X i ]) , and the lemma will follow.To prove this claim, we show that by replacing the random variable X k with the Bernoullirandom variable Y k with Pr[ Y k = 1] = E [ X k ] and Pr[ Y k = 0] = 1 − E [ X k ] the expectation E [min { , X } ] can only become smaller. The claim will follow by repeating this argument andreplacing each random variable X i with a Bernoulli one that has the same expectation.Formally, let Y = X + ... + X k − + Y k = X ′ + Y k ; we will show that E [min { , X } ] ≥ E [min { , Y } ]. Denoting by G the cdf of the random variable Y k , we have that, conditioned on X ′ = w , the expected contribution of X k to min { , X } is E [min { , X } − X ′ | X ′ = w ] = Z − w (1 − G ( z )) d z ≥ (1 − w ) Z (1 − G ( z )) d z = (1 − w ) E [ X k ] = (1 − w ) Pr[ Y k = 1] = E [min { , Y } − X ′ | X ′ = w ] . The inequality follows since 1 − G ( z ) is non-increasing in z and, subsequently, R t (1 − G ( z )) d z is concave in t and has no point below the line t R t (1 − G ( z )) d z for t ∈ [0 , X ′ by f , we have E [min { , X } ] = Z f ( w )( w + E [min { , X } − X ′ | X ′ = w ]) d w Z f ( w )( w + E [min { , Y } − X ′ | X ′ = w ]) d w = E [min { , Y } ] , as desired.By applying Lemma 7 with k = n , X i = y ∗ i ( v i , p ) for i ∈ [ n ] and using equation (1), we havethat the expected revenue at a price p per unit using an ordering π of the agents is E [REV( v , p, π )] = p · E v ∼ F X i ∈ [ n ] y i ( v , p, π ) = p · E v ∼ F min , X i ∈ [ n ] y ∗ i ( v i , p ) ≥ p · − Y i ∈ [ n ] (cid:18) − E v i ∼ F i [ y ∗ i ( v i , p )] (cid:19) . (10)To determine the revenue gap, we can require that any linear pricing with price per unit p has revenue at most 1 and ask: “how large can the revenue of the optimal ex-ante relaxationcan be?” Bounding the RHS of (10) is sufficient for bounding the revenue of linear pricingby 1. Then, the maximum value the optimal revenue can get is upper-bounded by the valuethe RHS of (8) can get under the constraint (9). So, we will bound the revenue gap using thefollowing mathematical program that has as variables the cdf’s F i,x ( t ) for i ∈ [ n ], x ∈ [0 ,
1] and t ∈ [0 , + ∞ ) and the probabilities q i ( x ) for i ∈ [ n ] and x ∈ [0 , X i ∈ [ n ] Z q i ( x ) F − i,x (1 − q i ( x )) d x (11)subject to X i ∈ [ n ] Z q i ( x ) d x ≤ p · − Y i ∈ [ n ] (cid:18) − E v i ∼ F i [ y ∗ i ( v i , p )] (cid:19) ≤ , ∀ p > q i ( x ) ∈ [0 , , ∀ i ∈ [ n ] , x ∈ [0 , F i,x is a regular cdf , ∀ i ∈ [ n ] , x ∈ [0 , We now prove an upper bound on the objective value of mathematical program (11) by relatingit to mathematical program (7), and exploiting the revenue gap of [2] for anonymous itempricing (Theorem 6).Our main tool is the following transformation. We use the notation ( q , F ) as abbreviationfor the probabilities q i ( x ) for every agent i ∈ [ n ] and x ∈ [0 ,
1] and the cummulative densityfunctions F i,x ( t ) for every agent i ∈ [ n ], x ∈ [0 , t ≥
0. Given the pair ( q , F ), we define r i = R q i ( x ) d x and H i ( t ) = F i, (2 κt ) for every agent i ∈ [ n ] and t ≥
0, and use the pair ( r , H )as their abbreviation.Bounding the objective value of mathematical program (11) has two steps; the first one isimplemented by the next lemma. 11 emma 8. Given a feasible solution ( q , F ) of the mathematical program (11), the solution ( r , H ) is a feasible solution of the mathematical program (7) with R = 2 κ − .Proof. Clearly, the solution ( r , H ) satisfies the first constraint of the mathematical program(7) since ( q , F ) satisfies the first constraint of (11). In the following, we show that the secondconstraint is satisfied as well. We will need two technical lemmas. Lemma 9.
Let k be an integer and ≤ z , z , ..., z k ≤ . Then, for every t ∈ (0 , , it holdsthat − Y i ∈ [ k ] (1 − t · z i ) ≥ t − Y i ∈ [ k ] (1 − z i ) Proof.
It suffices to show that the LHS is concave as a function of t ; then it is at least as highas the line that connects points (0 ,
0) and (1 , − Q i ∈ [ k ] (1 − z i )), i.e., the RHS of the aboveinequality. Indeed, the first derivative of the LHS is equal to Y i ∈ [ k ] (1 − t · z i ) · X i ∈ [ k ] z i − t · z i and its second derivative is equal to − Y i ∈ [ k ] (1 − t · z i ) X i ∈ [ k ] z i − t · z i − X i ∈ [ k ] (cid:18) z i − t · z i (cid:19) < , as desired. Lemma 10.
For every agent i ∈ [ n ] and any price p , it holds that E v i ∼ F i [ y ∗ i ( v i , p )] ≥ − H i ( p )2 κ − .Proof. Assume that v ′ i (0) ≥ κp ; hence, v i (1) ≥ p . By the definition of y ∗ i ( v i , p ), we have that v ′ i ( z ) ≤ p for z ≥ y ∗ i ( v i , p ). Hence, v i ( y ∗ ( v i , p )) ≥ v i (1) − v ′ i ( y ∗ i ( v i , p )) · (1 − y ∗ i ( v i , p )) ≥ v i (1) − p · (1 − y ∗ i ( v i , p )) (12)Furthermore, by our assumption on the curvature of the valuation functions, v ′ i ( z ) ≤ κ · v i (1)for z ≤ y ∗ i ( v i , p ). Hence, v i ( y ∗ ( v i , p )) ≤ κ · v i (1) · y ∗ ( v i , p ) . (13)Now, (12) and (13) yield y ∗ ( v i , p ) ≥ v i (1) − pκ · v i (1) − p ≥ κ − . Thus, E v i ∼ F i [ y ∗ i ( v i , p )] ≥ E v i ∼ F i [ y ∗ i ( v i , p ) { v ′ i (0) ≥ κp } ] ≥ − F i, (2 κp )2 κ − − H i ( p )2 κ − . Now, using the second constraint of the mathematical program (11) for p > κ −
1, Lemma 9and Lemma 10, we have 1 ≥ p − Y i ∈ [ n ] (cid:18) − E v i ∼ F i [ y ∗ i ( v i , p )] (cid:19) p − Y i ∈ [ n ] (cid:18) − − H i ( p )2 κ − (cid:19) ≥ p κ − − Y i ∈ [ n ] H i ( p ) , i.e., equivalently, p − Y i ∈ [ n ] H i ( p ) ≤ R, ∀ p > R, with R = 2 κ −
1, as the second contstraint of the mathematical program (7) requires.The second step of our proof is to relate the objective values of the two mathematicalprograms (11) and (7).
Lemma 11.
The objective value of the mathematical program (11) at solution ( q , F ) is at most κ times the objective value of the mathematical program (7) with R = 2 κ − at solution ( r , H ) .Proof. Notice that, by the definition of H , we have F − i, ( t ) = 2 κH − i ( t ). Hence, Z q i ( x ) F − i,x (1 − q i ( x )) d x ≤ Z q i ( x ) F − i, (1 − q i ( x )) d x ≤ Z q i ( x ) d x · F − i, (cid:18) − Z q i ( x ) d x (cid:19) = 2 κr i H − i (1 − r i ) . The first inequality follows by the concavity of valuations which implies F − i,x ( t ) is non-increasingin x . The second inequality follows by applying Jensen inequality; recall that, by the regularityof v ′ i (0), the function q · F − i, (1 − q ) is concave in q .Theorem 5 now follows by Theorem 6 and Lemmas 8 and 11. Indeed, these three statementsimply a revenue gap of at most 2 κ (2 κ − e . We now show that a dependence of the revenue gap on κ is unavoidable. Theorem 12.
For every κ > , there exists a concave valuation function with curvature κ sothat any linear pricing recovers at most a /ρ -fraction of the optimal revenue, with ρ ≥ κ .Proof. Let ρ be such that ρ − ln ρ = 1 + ln κ and define the valuation function v ( x ) = ( κz, ≤ x ≤ κρ ρ ln x, κρ ≤ x ≤ v is concave with curvature κ . In particular, it is linear in [0 , κρ ] and has strictydecreasing derivative in [ κρ , κ is κρ . Also, the whole item is sold at price p < /ρ since v ′ (1) = 1 /ρ . Atprice p ∈ [1 /ρ, κ ), the fraction x of the item that is sold is such that v ′ ( x ) = p , i.e., equal to pρ . In every case, the revenue is at most 1 /ρ . In contrast, a non-linear pricing p ( x ) = v ( x ) canrecover the whole value of 1 for the item as revenue.13 Open problems
Motivated by recent work on anonymous pricing for the standard setting of selling a single(indivisible) item, we have explored the power of linear posted pricing when selling a perfectlydivisible item. Our work leaves several interesting open problems. Regarding social welfare,can the parameters ρ and ρ in Theorems 2 and 4 be improved to 50%? Duetting et al. [11](see also the discussion in [24]) show that a 50% approximation to optimal social welfare ispossible using linear pricing when valuations are of the form v i ( z ) = v i · min { z/s i , } but theirtechniques do not seem to yield the same bound with linear pricing for more general concavevaluation functions.Our results on the revenue gap leave even more space for improvements. Clearly, it would beinteresting to determine the tight bound on the revenue gap for linear pricing, closing the gapbetween O ( κ ) and Ω(ln κ ). More importantly, the optimal revenue for the ex-ante relaxationof the multi-parameter setting in Section 4.2 could be used to bound the revenue gap of othermechanisms for allocating a divisible item, such as the proportional mechanism [18, 19], whichhas been extensively studied in terms of social welfare [5, 6, 9]. This could lead to results onthe Bayes-Nash price of anarchy for revenue which, with a few exceptions such as [3, 15, 25],are rather sporadic in the literature. References [1] S. Alaei. Bayesian combinatorial auctions: Expanding single buyer mechanisms to manybuyers.
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