Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers A preliminary version of this paper was published in the proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011).

Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers thanks: A preliminary version of this paper was published in the proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011).

Saeed Alaei Department of Computer Science, University of Maryland, College Park, MD 20742, email: saeed@cs.umd.edu. Part of this work was done when the author was visiting Microsoft Research, New England. This work was partially supported by the NSF grant CCF-0728839
Date: 2012-06-08 03:38:29 -0400 (Fri, 08 Jun 2012) 111Id: bca.tex 201 2012-06-08 07:38:29Z saeed
Abstract

For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to single buyer sub-problems. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) buyers’ types must be distributed independently (not necessarily identically), (ii) objective function must be linearly separable over the buyers, and (iii) except for the supply constraints, there should be no other inter-buyer constraints. Our framework is general in the sense that it makes no explicit assumption about buyers’ valuations, type distributions, and single buyer constraints (e.g., budget, incentive compatibility, etc).

We present two generic multi buyer mechanisms which use single buyer mechanisms as black boxes; if an -approximate single buyer mechanism can be constructed for each buyer, and if no buyer requires more than of all units of each item, then our generic multi buyer mechanisms are -approximation of the optimal multi buyer mechanism, where is a constant which is at least . Observe that is at least (for ) and approaches as . As a byproduct of our construction, we present a generalization of prophet inequalities. Furthermore, as applications of our framework, we present multi buyer mechanisms with improved approximation factor for several settings from the literature.

1 Introduction

The main challenge of stochastic optimization arises from the fact that all instances in the support of the distribution are relevant for the objective and this support is exponentially big in the size of problem. This paper aims to address this challenge by providing a general decomposition technique for assignment problems on independently distributed inputs where the objective is linearly separable over the inputs. The main challenge faced by such a decomposition approach is that the feasibility constraint of an assignment problem introduces correlation in the outcome of the optimal solution. In mechanism design problems, such constraints are typically the supply constraints. For example, when buyers are independent, a revenue maximizing seller with unlimited supply can decompose the problem over the buyers and optimize for each buyer independently. However, in the presence of supply constraints, a direct decomposition is not possible. Our decomposition technique can be roughly described as the following: (i) Construct a mechanism that satisfies the supply constraints only in expectation (ex-ante); the optimization problem for constructing such a mechanism can be fully decomposed over the set of buyers. (ii) Convert the mechanism from the previous step to another mechanism that satisfies the supply constraint at every instance.

We restrict our discussion to Bayesian combinatorial auctions. We are interested in mechanisms that allocate a set of heterogenous items with limited supply to a set of buyers in order to maximize the expected value of a certain objective function which is linearly separable over the buyers (e.g., welfare, revenue, etc). The buyers’ types are assumed to be distributed independently according to publicly known priors. We defer the formal statement of our assumptions to §2.

The following are the main challenges in designing mechanisms for multiple buyers.

  1. The decisions made by the mechanism for different buyers should be coordinated because of supply constraints.

  2. The decisions made by the mechanism for each buyer should be optimal (or approximately optimal).

Making coordinated optimal decisions for multiple buyers is challenging as it requires optimizing over the joint type space of all buyers, the size of which grows exponentially in the number of buyers. The second challenge is usually due to incentive compatibility (IC) constraints, specially in multi-dimensional settings where these constraints cannot be encoded compactly. In this paper, we mostly address the first challenge by providing a framework for approximately decomposing the mechanism design problem for multiple buyers to sub-problems dealing with each buyer individually.

Our framework can be summarized as follows. We start by relaxing the supply constraints, i.e., we consider the mechanisms for which only the ex-ante expected number of allocated units of each item is no more than the supply of that item. Note that “ex-ante” means that the expectation taken over all possible inputs (i.e., all possible types of the buyers). We show that the optimal mechanism for the relaxed problem can be constructed by independently running single buyer mechanisms, where each single buyer mechanism is subject to an ex-ante probabilistic supply constraint. In particular, we show that if one can construct an -approximate mechanism for each single buyer problem, then running these mechanisms simultaneously and independently yields an -approximate mechanism for the relaxed multiple buyer problem. We then present two methods for converting the mechanism for the relaxed problem to a mechanism for the original problem while losing a small constant factor in the approximation. We present two generic multi buyer mechanisms that use the single buyer mechanisms from the previous step as blackboxes 222Note that the single buyer mechanisms can be different for different buyers, e.g., to accommodate different classes of buyers.. In the first mechanism, we serve buyers sequentially by running, for each buyer, the corresponding single buyer mechanism from the previous step. However, we sometimes randomly preclude some of the items from the early buyers in order to ensure that late buyers get the same chance of being offered with those items; we ensure that the ex-ante expected probability of preclusion is equalized over all buyers, regardless of the order in which they are served (i.e., we simultaneously minimize the preclusion probability for all buyers). In the second mechanism, we run all of the single buyer mechanisms simultaneously and then modify the outcomes by deallocating some units of the over-allocated items at random while adjusting the payments respectively; we ensure that the ex-ante probability of deallocation is equalized among all units of each item and therefore simultaneously minimized for all buyers.

We also introduce a toy problem, the magician’s problem, in §4, along with a near optimal solution for it, which is used as the main ingredient of our multi buyer mechanisms. As a byproduct, we present improved generalized prophet inequalities for maximizing the sum of multiple choices.

As applications of our framework, in §6, we present mechanisms with improved approximation factor for several settings from the literature. For each setting we present a single buyer mechanism that satisfies the requirements of our framework, and can be plugged in one of our generic multi buyer mechanisms.

1.1 Related Work

In single dimensional settings, the related works form the CS literature are mostly focused on approximating the VCG mechanism for welfare maximization and/or approximating the Myerson’s mechanism Myerson (1981) for revenue maximization (e.g., Bulow and Roberts (1989); Babaioff et al. (2006); Blumrosen and Holenstein (2008); Hartline and Roughgarden (2009); Dhangwatnotai et al. (2010); Chakraborty et al. (2010); Yan (2011)). Most of them consider mechanisms that have simple implementation and are computationally efficient. For welfare maximization in single dimensional settings, Hartline and Lucier (2010) gives a blackbox reduction from mechanism design to algorithmic design.

In multidimensional setting, for welfare maximization, Hartline et al. (2011) presents a blackbox reduction from mechanism design to algorithm design which subsumes the earlier work of Hartline and Lucier (2010). For revenue maximization, Chawla et al. (2010) presents several sequential posted pricing mechanisms for various settings with different types of matroid feasibility constraints. These mechanisms have simple implementation and approximate the revenue of the optimal mechanism. For unit-demand buyers whose valuations’ for the items are distributed according to product distributions, Chawla et al. (2010) present a sequential posted pricing mechanism that obtains in expectation at least -fraction of the revenue of the optimal posted pricing mechanism. In §6.2, we present an improved sequential posted pricing mechanism for this setting with an approximation factor of in which is the number of units available of each item, and is a constant which is at least . For combinatorial auctions with additive/correlated valuations with budget and demand constraints, Bhattacharya et al. (2010) presents all-pay -approximate BIC mechanisms for revenue maximization and a similar mechanism for welfare maximization. In subsection 6.4, we present an improved mechanism for this setting with an approximation factor of . Note that is at least and approaches as . Bhattacharya et al. (2010) also presents sequential posted pricing mechanisms for the same setting, obtaining approximation factors. For a similar setting, in §6.3, we present an improved sequential posted pricing mechanism with an approximation factor of . Finally, Chawla et al. (2011) also considers various settings with hard budget constraints.

Prophet inequalities have been extensively studied in the past (e.g. Hill and Kertz (1992)). Prior to this work, the best known bound for the generalization to sum of choices was by Hajiaghayi et al. (2007). We improve this to . Note that the current bound is tight for , and is useful even for small values of .

2 Preliminaries

The framework of this paper is presented for combinatorial auctions, but it can be readily applied to Bayesian mechanism design in other contexts. We begin by defining the model and some notation.

Model

We consider the problem of selling indivisible heterogenous items to buyers where there are units of each item . All the relevant private information of each buyer is represented by her type where is the type space of buyer . Let be the space of all type profiles. The buyers’ type profile is distributed according to a publicly known prior . We use and to denote the random variables333Note that these random variables are often correlated. Furthermore, for a deterministic mechanism, these variables take deterministic values as a function of . respectively for the allocation of item to buyer and the payment of buyer , for type profile . For a mechanism , the random variables for allocations and payments are denoted respectively by and . We are interested in computing a mechanism that (approximately) maximizes444All of the results of this paper can be applied to minimization problems by simply maximizing the negation of the objective function. the expected value of a given objective function where , , and respectively represent the types, the allocations, and the payments of all buyers. We are only interested in mechanisms which are within a given space of feasible mechanisms. Formally, we aim to compute a mechanism that (approximately) maximizes .

Assumptions

We make the following assumptions.

  1. Independence. The buyers’ types must be distributed independently, i.e., where is the distribution of types for buyer . Note that for a buyer who has multidimensional types, itself does not need to be a product distribution.

  2. Linear Separability of Objective. The objective function must be linearly separable over the buyers, i.e., where , , and respectively represent the type, the allocations, and the payment of buyer .

  3. Single–Unit Demands. No buyer should ever need more than one unit of each item, i.e., for all . This assumption is not necessary and is only to simplify the exposition; it can be removed as explained in §8.

  4. Incentive Compatibility. must be restricted to (Bayesian) incentive compatible mechanisms. By direct revelation principle this assumption is without loss of generality555It is WLOG, given that we are only interested in mechanisms that have Bayes-Nash equilibria.,

  5. Convexity. must be a convex space. In other words, every convex combination of every two mechanisms from must itself be a mechanism in . A convex combination of two mechanisms is another mechanism which simply runs with probability and runs with probability , for some . In particular, if is restricted to deterministic mechanisms, it is not convex; however if also includes mechanisms that randomize over deterministic mechanisms, then it is convex 666For an example of a randomized non-convex space of mechanisms, consider the space of mechanisms where the expected payment of every type must be either less than or more than ..

  6. Decomposability. The set of constraints specifying must be decomposable to supply constraints (i.e., , for each item ) and single buyer constraints(e.g., incentive compatibility, budget, etc). We define this assumption formally as follows. For any mechanism , let be the single buyer mechanism perceived by buyer , by simulating777The single buyer mechanism induced on buyer can be obtained by simulating all buyers other than by drawing a random from and running on buyer and the simulated buyers with types ; note that this is a single-buyer mechanism because the simulated buyers are just part of the mechanism. the other buyers according to their respective distributions . Define to be the space of all feasible single buyer mechanisms for buyer . The decomposability assumption requires that for any arbitrary mechanism the following holds: if satisfies the supply constraints and also (for all buyers ), then it must be that .

We shall clarify the last assumption by giving an example. Suppose is the space of all buyer specific item pricing mechanisms, then satisfies the last assumption. On the other hand, if is the space of mechanisms that offer the same set of prices to every buyer, it does not satisfy the decomposability assumption, because there is an implicit inter-buyer constraint that the same prices should be offered to different buyers.

Throughout the rest of this paper, we often omit the range of the sums whenever the range is clear from the context (e.g., means , and means ).

Multi buyer problem

Formally, the multi buyer problem is to find a mechanism which is a solution to the following program.

maximize ()
subject to

Observe that, in the absence of the first set of constraints, we could optimize the mechanism for each buyer independently. This observation is the key to our multi to single buyer decomposition, which allows us to approximately decompose/reduce the multi buyer problem to single buyer problems. A mechanism is an -approximation of the optimal mechanism if it is a feasible mechanism for the above program and obtains at least -fraction of the optimal objective value of the program.

Ex ante allocation rule

For a multi-dimensional mechanism , the ex ante allocation rule is a vector in which is the expected probability of allocating a unit of item to buyer , where the expectation is taken over all possible type profiles. Note that for any feasible mechanism , by linearity of expectation, the ex ante allocation rule satisfies , for every item .

Single buyer problem

The single buyer problem, for buyer , is to compute an optimal single buyer mechanism and its expected objective value, subject to a given upper bound on the ex ante allocation rule; in other words, the single buyer mechanism may not allocate a unit of item to buyer with an expected probability of more than , where the expectation is taken over . Formally, the single buyer problem is to compute the optimal value of the following program along with a corresponding solution (i.e., the optimal ), for a given .

maximize ()
subject to

We typically denote an optimal single buyer mechanism for buyer , subject to a given , by , and denote its expected objective value (i.e., the optimal value of the above program as a function of ) by . Later, we prove that , which we refer to as the optimal benchmark for buyer , is a concave function of . In the case of approximation, we say that a single buyer mechanism together with a concave benchmark provide an -approximation of the optimal single buyer mechanism/optimal benchmark, if the expected objective value of is at least and if is an upper bound on the optimal benchmark, for every .

To make the exposition more concrete, consider the following single buyer problem as an example. Suppose there is only one type of item (i.e., ) and the objective is to maximize the expected revenue888The optimal multi buyer mechanism for this setting is given by Myerson (1981); yet we consider this setting to keep the example simple and intuitive.. Suppose buyer ’s valuation is drawn from a regular distribution with CDF, . The optimal single buyer mechanism for , subject to , is a deterministic mechanism which offers the item at some fixed price, while ensuring that the probability of sale (i.e., the probability of buyer ’s valuation being above the offered price) is no more than . In particular, the optimal benchmark is the optimal value of the following convex program as a function of .

maximize
subject to

Furthermore, the optimal single buyer mechanism offers the item at the price where is the optimal assignment for the above convex program. Note that, for a regular distribution, is concave in , so the above program is a convex program.

3 Decomposition via Ex ante Allocation Rule

In this section we present general methods for approximately decomposing/reducing the multi buyer problem to single buyer problems. Recall that a single buyer problem is to compute the optimal single buyer mechanism and its expected objective value (i.e., the optimal benchmark), subject to an upper bound on the ex ante allocation rule. We present two methods for constructing an approximately optimal multi buyer mechanism, using and as black box. Furthermore, we show that if we can only compute an -approximation of the optimal single buyer mechanism/optimal benchmark for each buyer , then the factor simply carries over to the approximation factor of the final multi buyer mechanism.

Multi buyer benchmark

We start by showing that the optimal value of the following convex program gives an upper bound on the expected objective value of the optimal multi buyer mechanism.

maximize ()
subject to

We first show that the above program is indeed a convex program.

Theorem 1.

The optimal benchmarks are always concave.

Proof.

We prove this for an arbitrary buyer . Let and denote the optimal single buyer mechanism and the optimal benchmark for buyer . To show that is concave, it is enough to show that for any and any , the following inequality holds.

Consider the single buyer mechanism that works as follows: runs with probability and runs with probability . Note that is a convex space (this follows from 5 and 6), therefore . Observe that by linearly of expectation, the ex ante allocation rule of is no more than and the expected objective value of is exactly . So the expected objective value of the optimal single buyer mechanism, subject to , may only be higher. That implies which proves the claim. ∎

Theorem 2.

The optimal value of the convex program () is an upper bound on the expected objective value of the optimal multi buyer mechanism.

Proof.

Let be an optimal multi buyer mechanism. Let denote the ex ante allocation rule corresponding to , i.e., . Observe that is a feasible assignment for the convex program and yields an objective value of which is upper bounded by the optimal value of the convex program. So to prove the theorem it is enough to show that the contribution of each buyer to the expected objective value of is upper bounded by . Consider , i.e, the single buyer mechanism induced by on buyer . can be obtained by simply running on buyer and simulating the other buyers with random types ; Observe that is a feasible single buyer mechanism subject to and obtains the same expected objective value as from buyer , so the expected objective value of the optimal single buyer mechanism subject to could only be higher. ∎

Constructing multi buyer mechanisms

Theorem 2 suggests that by computing an optimal assignment of for the convex program () and running the single buyer mechanism for each buyer , one might obtain a reasonable multi buyer mechanism; however such a multi buyer mechanism would only satisfy the supply constraints in expectation; in other words, there is a good chance that some items are over allocated with a non-zero probability. We present two generic multi buyer mechanisms for combining the single buyer mechanisms and resolving the conflicts in the allocations in such a way that would ensure the supply constraints are met at every instance and not just in expectation. In both approaches we first solve the convex program () to compute the optimal . The high level idea of each mechanism is explained below.

  1. Pre-Rounding. This mechanism serves the buyers sequentially (arbitrary order); for each buyer , it selects a subset of available items and runs the single buyer mechanism , where denotes the vector resulting from by zeroing the entries corresponding to items not in . In particular, this mechanism sometimes precludes some of the available items from early buyers to make them available to late buyers. We show that if there are at least units of each item, then includes item with probability at least , for each buyer and each item .

  2. Post-Rounding. This mechanism runs for all buyers simultaneously and independently. It then modifies the outcomes by deallocating the over allocated items at random in such a way that the probability of deallocation observed by all buyers are equal, and therefore minimized over all buyers. The payments are adjusted respectively. We show that if there are at least units of each item, every allocation is preserved with probability from the perspective of the corresponding buyer.

We will explain the above mechanisms in more detail in §5 and present some technical assumptions that are sufficient to ensure that they retain at least fraction of the expected objective value of each .

Approximately optimal single buyer mechanisms

Throughout the above discussion, we assumed that we can compute the optimal single buyer mechanisms and the corresponding optimal benchmarks. However, it is likely that we can only compute an approximation of them. Suppose for each buyer , and , instead of being optimal, only provide an -approximation of the optimal single buyer mechanism/optimal benchmark, and suppose is concave; then we can still use and in the above construction, but the final approximation factor will be multiplied by .

Main result

The following informal theorem summarizes the main result of this paper. The formal statement of this result can be found in Theorem 7 and Theorem 8.

Theorem 3 (Market Expansion).

If for each buyer , an -approximate single buyer mechanism and a corresponding concave benchmark can be constructed in polynomial time, then, with some further assumptions (explained later), a multi buyer mechanism can be constructed in polynomial time by using as building blocks, such that is -approximation of the the optimal multi buyer mechanism in , where and is a constant which is at least .

In order to explain the construction of the multi buyer mechanism, we shall first describe the magician’s problem and its solution, which is used in both pre-rounding and post-rounding for equalizing the expected probabilities of preclusion/deallocation over all buyers.

4 The Magician’s Problem

In this section, we present an abstract online stochastic toy problem and a near-optimal solution for it which provides the main ingredient for combining single buyer mechanisms to form multi buyer mechanisms; it is also used to prove a generalized prophet inequality.

Definition 1 (The Magician’s Problem).

A magician is presented with a series of boxes one by one, in an online fashion. There is a prize hidden in one of the boxes. The magician has magic wands that can be used to open the boxes. If a wand is used on box , it opens, but with a probability of at most , which written on the box, the wand breaks. The magician wishes to maximize the probability of obtaining the prize, but unfortunately the sequence of boxes, the written probabilities, and the box in which the prize is hidden are arranged by a villain, and the magician has no prior information about them (not even the number of the boxes). However, it is guaranteed that , and that the villain has to prepare the sequence of boxes in advance (i.e., cannot make any changes once the process has started).

The magician could fail to open a box either because (a) he might choose to skip the box, or (b) he might run out of wands before getting to the box. Note that once the magician fixes his strategy, the best strategy for the villain is to put the prize in the box that has the lowest ex ante probability of being opened, based on the magician’s strategy. Therefore, in order for the magician to obtain the prize with a probability of at least , he has to devise a strategy that guarantees an ex ante probability of at least for opening each box. Notice that allowing the prize to be split among multiple boxes does not affect the problem. It is easy to show the following strategy ensures an ex ante probability of at least for opening each box: for each box randomize and use a wand with probability . But can we do better? We present an algorithm parameterized by a probability which guarantees a minimum ex-ante probability of for opening each box while trying to minimize the number of wands broken. In Theorem 4, we show that for this algorithm never requires more than wands.

Definition 2 (-Conservative Magician).

The magician adaptively computes a sequence of thresholds and makes a decision about each box as follows: let denote the number of wands broken prior to seeing the box; the magician makes a decision about box by comparing against ; if , it opens the box; if , it does not open the box; and if , it randomizes and opens the box with some probability (to be defined). The magician chooses the smallest threshold for which where the probability is computed ex ante (i.e., not conditioned on past broken wands). Note that is a parameter that is given. Let denote the ex ante CDF of random variable , and let be the indicator random variable which is iff the magician opens the box . Formally, the probability with which the magician should open box condition on is computed as follows.

()
()

Observe that is in fact computed before seeing box itself.

Define ; the CDF of can be computed from the CDF of and as follows (assume is the exact probability of breaking a wand for box ).

()

Furthermore, if each is just an upper bound on the probability of breaking a wand on box , then the above definition of stochastically dominates the actual CDF of , and the magician opens each box with a probability of at least .

In order to prove that a -conservative magician does not fail for a given choice of , we must show that the thresholds are no more than . The following theorem states a condition on that is sufficient to guarantee that for all .

Theorem 4 (-Conservative Magician).

For any , a -conservative magician with wands opens each box with an ex ante probability of at least . Furthermore, if is the exact probability (not just an upper bound) of breaking a wand on box for each , then each box is opened with an ex ante probability exactly 999In particular the fact that the probability of the event of breaking a wand for the box is exactly , conditioned on any sequence of prior events, implies that these events are independent for different boxes.

Proof.

See §7. ∎

Definition 3 ().

We define to be the largest probability such that for any and any instance of the magician’s problem with wands, the thresholds computed by a -conservative magician are less than . In other words, is the optimal choice of which works for all instances with wands. By Theorem 4, we know that must be101010Because for any obviously . at least .

Observe that is a non-decreasing function in which is at least (when ) and approaches as . The next theorem shows that the lower bound of on cannot be far from the optimal.

Theorem 5 (Hardness of Magician’s Problem).

For any , it is not possible to guarantee an ex ante probability of for opening each box (i.e., no magician can guarantee it). Note that by Stirling’s approximation.

Proof.

See §A. ∎

Prophet Inequalities

We prove a generalization of prophet inequalities by a direct reduction to the magician’s problem.

Definition 4 (-Choice Sum).

A sequence of non-negative random numbers are drawn from arbitrary distributions one by one in an arbitrary order. A gambler observes the process and may select of the random numbers, with the goal of maximizing the sum of the selected ones; a random number may only be selected at the time it is drawn, and it cannot be unselected later. The gambler knows all the distributions in advance, and observes from which distribution the current number is drawn, but not the order in which the future numbers are drawn. On the other hand, a prophet knows all the actual draws in advance, so he chooses the highest draws. We assume that the order in which the random numbers are drawn is fixed in advance (i.e., may not change based on the decisions of the gambler).

Hajiaghayi et al. (2007) proved that there is a strategy for the gambler that guarantees in expectation at least fraction of the payoff of the prophet, using a non-decreasing sequence of stopping rules (thresholds) 111111A gambler with stopping rules works as follows. Upon seeing , he selects it iff where is the number of random draws selected so far.. Next, we construct a gambler that obtains in expectation at least fraction of the prophet’s payoff, using a -conservative magician as a black box. Note that . This gambler uses only a single threshold. However, he may skip some of the random variables at random.

Theorem 6 (Prophet Inequalities – -Choice Sum).

The following strategy ensures that the gambler obtains at least fraction of payoff of the prophet in expectation. 121212To simplify the exposition we assume that the distributions do not have point masses. The result holds with slight modifications if we allow point masses.

  • Find a threshold such that (e.g., by doing a binary search on ).

  • Use a -conservative magician with wands. Upon seeing each , create a box and write on it and present it to the magician. If the magician chooses to open the box and also , then select and break the magician’s wand, otherwise skip .

Proof.

First, we compute an upper bound on the expected payoff of the prophet. Let be the ex ante probability (i.e., before any random number is drawn) that the prophet chooses (i.e. the probability that is among the highest draws). Let denote the maximum possible contribution of the random variable to the expected payoff of the prophet if is selected with an ex ante probability . Note that is equal to the expected value of conditioned on being above the quantile, multiplied by the probability of being above that quantile. Assuming and denote the CDF and PDF of , we can write . By changing the integration variable and applying the chain rule we get . Observe that is a non-increasing function, so is a concave function. Furthermore, because the prophet cannot choose more than random draws. So the optimal value of the following convex program is an upper bound on the payoff of the prophet.

maximize
subject to ()
()

Define the Lagrangian for the above convex program as

By KKT stationarity condition, at the optimal assignment, it must be . On the other hand, . Assuming that , by complementary slackness , which then implies that , so . Furthermore, it is easy to show that the first constraint must be tight, which implies that . Observe that the contribution of each to the objective value of the convex program is exactly . By using a -conservative magician we can ensure that each box is opened with probability at least which implies the contribution of each to the expected payoff of the gambler is which proves that the expected payoff of the gambler is at least fraction of optimal objective value of the convex program, which was itself and upper bound on the expected payoff of the prophet. ∎

5 Generic Multi Buyer Mechanisms

In this section, we present a formal description of the two generic multi buyer mechanisms outlined toward the end of §3. Throughout the rest of this section we assume that for each buyer we can compute a single buyer mechanism and a corresponding concave benchmark , which together provide -approximation of the optimal single buyer mechanism/optimal benchmark for buyer . We show that the resulting multi buyer mechanism will be -approximation of the the optimal multi buyer mechanism in , where and is the optimal magician parameter which is at least (Definition 3) .

5.1 Pre-Rounding

This mechanism serves the buyers sequentially (arbitrary order); for each buyer , it selects a subset of available items and runs the single buyer mechanism , where is an optimal assignment for the benchmark convex program (), and denotes the vector resulting from by zeroing the entries corresponding to items not in . In particular, this mechanism sometimes precludes some items from early buyers to make them available to late buyers. For each item, the mechanism tries to minimize the probability of preclusion for each buyer by equalizing it for all buyers. Note that, for any given pair of buyer and item, we only care about the probability of preclusion in expectation, where the expectation is taken over the types of other buyers and the random choices of the mechanism. The mechanism is explained in detail in Definition 5.

Definition 5 (-Pre-Rounding).
  1. Solve the convex program () and let be an optimal assignment.

  2. For each item , create an instance of -conservative magician (Definition 2) with wands (this will be referred to as the magician). We will use these magicians through the rest of the mechanism. Note that is a parameter that is given.

  3. For each buyer :

    1. For each item , write on a box and present it to the magician. Let be the set of items where the corresponding magicians opened the box.

    2. Run on buyer and use its outcome as the final outcome for buyer .

    3. For each item , if a unit of item was allocated to buyer in the previous step, break the wand of the magician.

Note that since is a feasible assignment for convex program (), it must satisfy , so by setting and by Theorem 4 and Definition 3 we can argue that each includes each item with probability at least where is at least .

In order for the above mechanism to retain at least a -fraction of the the expected objective value of each , further technical assumptions are needed in addition to . We show that it is enough to assume each has a budget-balanced and cross monotonic cost sharing scheme.

Definition 6 (Budget Balanced Cross Monotonic Cost Sharing Scheme).

A function has a budget balanced cross monotonic cost sharing scheme iff there exists a cost share function with the following two properties:

  1. must be budget balanced which means for all and , .

  2. must be cross monotonic which means for all , and , .

Intuitively, a cost share function associates a fraction of the expected objective value returned by the benchmark function to each item; and ensures that the fraction associated with each item does not decrease when other items are excluded. In particular, the above assumption holds if is a submodular function of (e.g., for welfare maximization, assuming that buyers’ valuations are submodular131313We conjecture that it holds in general for revenue maximization, when buyers’ valuations are submodular and is restricted to mechanisms which use buyer specific item pricing.). Note that it is enough to show that such a cost sharing function exists; however it is never used in the mechanism and its computation is not required.

Theorem 7 (-Pre-Rounding).

Suppose for each buyer , is an -approximate incentive compatible single buyer mechanism, and is the corresponding concave benchmark. Also suppose has a budget balanced cross monotonic cost sharing scheme. Then, for any , the -pre-rounding mechanism (Definition 5) is dominant strategy incentive compatible (DSIC) mechanism which is in and is a -approximation of the optimal mechanism in .

Proof.

See §A. ∎

Remark 1.

The -pre-rounding mechanism assumes no control and no prior information about the order in which buyers are visited. The order specified in the mechanism is arbitrary and could be replaced by any other ordering which may be unknown in advance. In particular, this mechanism can be adopted to online settings in which buyers are served in an unknown order.

Corollary 1.

In any setting where Theorem 7 is applicable and when includes all feasible BIC mechanisms, the gap between the optimal DSIC mechanism and the optimal BIC mechanism is at most . This gap is at most (for ) and vanishes as . That is because Definition 5 is always DSIC, yet it approximates the optimal mechanism in .

5.2 Post-Rounding

This mechanism runs simultaneously and independently for all buyers to compute a tentative allocation/payment for each buyer; it then deallocates some of the items at random to ensure that the supply constraints are met at every instance; it ensures that the probability of deallocation perceived by each buyer (i.e., in expectation over the types of other buyers and random choices of the mechanism) is equalized and therefore simultaneously minimized for all buyers. The payments are also adjusted respectively. The mechanism is explained in detail in Definition 7.

Definition 7 (-Post-Rounding).
  1. Solve the convex program () and let denote an optimal assignment.

  2. Run simultaneously and independently for all buyers , and let and denote respectively the allocation (subset of items) and payment computed by for buyer .

  3. For each item , create an instance of -conservative magician (Definition 2) with wands (this will be referred to as the magician). We will use these magicians through the rest of the mechanism. Note that is a parameter that is given.

  4. For each buyer :

    1. For each item , write on a box and present it to the magician, where is the exact probability141414Note that is only an upper bound on the probability of allocation, so of allocating a unit of item to buyer ; let be the set of items where the corresponding magicians opened the box.

    2. Let and . The final allocation and payment of buyer is given by and respectively.

    3. For each item , break the wand of the magician.

Note that ; so by setting and by Theorem 4 and Definition 3 we can argue that each includes each item with probability at least where is at least . Consequently, any item that is in will also be in with probability exactly .

In order for -post-rounding to retain at least a -fraction of the the expected objective value of each and preserve incentive compatibility, further technical assumptions are needed in addition to ; next, we present a set of assumptions which is sufficient for this purpose151515I.e., one might come up with other sets of assumptions that are also sufficient..

  1. The exact ex ante allocation rule for each (i.e., ) must be available (i.e., efficiently computable). Note that is only an upper bound on the ex ante allocation rule.

  2. The objective functions must be of the form in which is an arbitrary fixed constant. Also, each must have cost sharing scheme in which is cross monotonic and budget balanced.

  3. The resulting mechanism must be in . In particular, that implies may not be restricted to any from of incentive compatibility stronger than Bayesian incentive compatibility (BIC), because the -post-rounding is only BIC.

  4. The valuations of each buyer must be in the form of a weighted rank function of some matroid.

Observe that 2 obviously holds for revenue maximization (because ), and also for welfare maximization with quasilinear utilities and submodular valuations (because where is the valuation of buyer for allocation 161616Note that the payment terms cancel out because the utility of the seller is counted toward the social welfare of the mechanism). Next, we formally define 4.

Definition 8 (Matroid Weighted Rank Valuation).

A valuation function is a matroid weighted rank valuation iff there exists a matroid , and a weight function such that is equal to the weight of a maximum weight independent subset of , i.e,

Matroid weighted rank valuations include additive valuations with demand constraints, unit demand valuations, etc.

Theorem 8 (-Post-Rounding).

Suppose for each buyer , is an -approximate incentive compatible single buyer mechanism, and is a corresponding concave benchmark. Also suppose the assumptions 1 through 4 hold. Then, for any , the -post-rounding mechanism (Definition 7) is a Bayesian incentive compatible (BIC) mechanism which is in and is a -approximation of the optimal mechanism in .

Proof.

See §A. ∎

6 Single Buyer Mechanisms

In this section, we present approximately optimal single buyer mechanisms for several common settings. Each one of the single buyer mechanisms presented in this section satisfies the requirements of one of the generic multi buyer mechanisms of §5, so they can be readily converted to a multi buyer mechanisms. Except for §6.4, we restrict the space of mechanisms to item pricing mechanisms with budget randomization as defined next.

Definition 9 (Item Pricing with Budget Randomization (IPBR)).

An item pricing mechanism is a possibly randomized mechanism that offers a menu of prices to each buyer and allows each buyer to choose their favorite bundle. The payment of a buyer is equal to the total price of the items in her purchased bundle. Note that the prices offered to different buyers do not need to be identical and buyers can be served sequentially. In the presence of budget constraints, a buyer is allowed to pay a fraction of the price of an item and receive the item with a probability equal to the paid fraction171717A utility maximizing buyer, with submodular valuations and budget constraint, always pays the full price for any item she purchases, except potentially for the last item purchased, for which she must have run out of budget.. A mechanism is considered an item pricing mechanism if its outcome can be interpreted as such181818I.e., an item pricing mechanism may collect all the reports and compute the final outcome along with buyer specific prices, such that the outcome of each buyer would be the same as if each buyer purchased their favorite bundle according to her observed prices, and the prices observed by each buyer should be independent of her report..

Item pricing mechanisms are simple and practical as opposed to optimal BIC mechanisms which often involve lotteries. Also budget randomization allows us to get around the hardness of the knapsack problem faced by the budgeted buyers; in particular, assuming that budgets are large compared to prices, budget randomization can be safely ignored since the optimal integral solution of the knapsack problem approaches its optimal fractional solution.

Table 1 lists several settings for which we obtain a multi buyer mechanism with an improved approximation factor compared to previous best known approximations. For each setting, we present a single buyer mechanism that satisfies the requirements of one of the generic multi buyer mechanisms of §5. The corresponding single buyer mechanisms are presented in detail throughout the rest of this section. Note that the final approximation factor for each multi buyer mechanism is equal to the approximation factor of the corresponding single buyer mechanism multiplied by ; recall that which approaches as .

Setting Approx Space of Mechanisms Ref
single item(multi unit), unit demand, budget constraint, revenue maximization item pricing with budget randomization §6.1
multi item(heterogenous), unit demand, product distribution, revenue maximization deterministic §6.2
multi item(heterogenous), additive valuations, product distribution, budget constraint, revenue maximization item pricing with budget randomization §6.3
multi item(heterogenous), additive valuations, correlated distribution with polynomial number of types, budget constraint, matroid constrains, revenue or welfare maximization randomized (BIC) §6.4
Table 1: Summary of mechanisms obtained using the framework of this paper.

For each single buyer mechanism presented in this paper, the single buyer benchmark function is defined as the optimal value of some convex program of the following general form, in which is some concave function, are some convex functions, and is some convex polytope (in the rest of this section we only consider a single buyer, so we will omit the subscript ).

maximize ()
subject to
Lemma 1.

is concave, i.e., the optimal value of a convex program of the form () is always concave in .

Proof.

See Appendix A. ∎

Note that we can substitute each in the multi buyer benchmark convex program () with the corresponding single buyer benchmark convex program to obtain a combined convex program which can be solved efficiently. If each is captured by a linear program, the combined multi buyer program will also be a linear program.

6.1 Single Item, Unit Demand, Budget Constraint

In this section, we consider a unit-demand buyer with a publicly known budget and one type of item (i.e., ). The only private information of the buyer is her valuation for the item, which is drawn from a publicly known distribution with CDF . To avoid complicating the proofs, we assume that is continuous and strictly increasing in its domain191919The proofs can be modified to work without this assumption.. We present a single buyer mechanism which is optimal among item pricing mechanisms with budget randomization (IPBR). We start by defining the modified CDF function as follows.

()

Intuitively, is the probability of allocating the item to the buyer if we offer the item at price . Note that the buyer only buys if her valuation is more than which happens with probability ; if , she will pay her whole budget and only get the item with probability , otherwise she pays the full price and receives the item with probability 1. Observe that if we want to allocate the item with probability we can offer a price of which yields a revenue of in expectation. Define and let denote its concave closure (i.e., the smallest concave function that is an upper bound on for every ). We will address the problem of efficiently computing later in Lemma 2. Next, we show that the optimal value of the following convex program is equal to the expected revenue of the optimal single buyer IPBR mechanism subject to ; therefore we will define the single buyer benchmark function to be equal to the optimal value of this program as a function of .

maximize ()
subject to
Theorem 9.

The revenue of the optimal single buyer IPBR mechanism, subject to an upper bound of on the ex ante allocation rule, is equal to the optimal value of the convex program (). Furthermore, assuming that is the optimal assignment for the convex program, if , then the optimal mechanism uses a single price otherwise, it randomized between two prices with probabilities and for some and .

Proof.

First, we prove that the expected revenue of the optimal single buyer IPBR mechanism, subject to , is upper bounded by . We then construct a price distribution that obtains this revenue. Note that any single buyer IPBR mechanism can be specified as a distribution over prices. Let be the optimal price distribution. So the optimal revenue is . Note that every price