Bayesian Combinatorial Auctions: Expanding Single Buyer
Mechanisms to Many Buyers
Saeed Alaei ∗
Dept. of Computer Science, University of Maryland
College Park, MD 20742. †
Date: 2011-09-22 18:39:42 -0400 (Thu, 22 Sep 2011)
‡
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) the buyers’ types must be distributed independently (not necessarily identically), (ii) the
objective function must be linearly separable over the set of buyers, and (iii) there should be
no constraints involving multiple buyers except for the supply constraints. Our framework is
general in the sense that it makes no explicit assumption about any of the following: (i) the
buyers’ valuations (e.g., submodular, additive, etc), (ii) the distribution of types for each buyer,
and (iii) the other constraints involving individual buyers (e.g., budget constraints, etc).
We present two generic n-buyer mechanisms that use 1-buyer mechanisms as black boxes.
Assuming that an α-approximate 1-buyer mechanism can be constructed for each buyer1 and
assuming that no buyer ever needs more than k1 of all copies of each item for some integer
k ≥ 1, then our generic n-buyer mechanisms are γk · α-approximation of the optimal n-buyer
1
mechanism, in which γk is a constant which is at least 1 − √k+3
. Observe that γk is at least
1
2 (for k = 1) and approaches 1 as k → ∞. As a byproduct of our construction, we present
a generalization of prophet inequalities. Furthermore, as applications of our framework, we
improve several results 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 addresses this challenge by giving 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,
∗
email: [email protected]
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
‡
Id: bca.tex 144 2011-09-22 22:39:42Z saeed
1
Note that the 1-buyer mechanisms do not need to be the same for different buyers, i.e., different 1-buyer mechanisms can be used to accommodate different classes of buyers.
†
1
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 looking for mechanisms
for allocating 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 that 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 section 2.
The following are the main challenges in designing mechanisms for multiple buyers.
(I) The decisions made by the mechanism for different buyers should be coordinated because of
supply constraints.
(II) The decisions made by the mechanism for each buyer has to 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 copies
of each item is no more than the supply of that item. Note that “ex-ante” means this expectation
taken over all possible inputs (i.e., all possible types of all buyers). We show that the optimal
mechanism for the relaxed problem can be constructed by independently running n 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 α-approximate mechanism for each
single buyer problem, then running these mechanisms simultaneously and independently yields an
α-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. To do this we present two generic mechanisms
that use the single buyer mechanisms from the previous step as blackboxes. 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 being
offered to some buyers in order to ensure that buyers that are served later also get a chance of being
offered with those items. We do this in such a way that would ensure that the ex-ante expected
probability of preclusion is equalized over all buyers, and therefore simultaneously minimized for
all buyers. In the second mechanism, we run all of the single buyer mechanisms simultaneously
and then modify the outcomes by deallocating some copies of the over-allocated items at random
while adjusting the payments respectively. We do this in such a way that would ensure the ex-ante
expected probability of deallocation for each item is equalized among all copies of that item and
therefore simultaneously minimized for all buyers.
2
We also introduce a toy problem, the magician’s problem, along with a near optimal solution
for it. The solution of this problem is used as the main ingredient for converting mechanisms for
the relaxed problem to mechanisms for the original problem. It also yields improved generalized
prophet inequalities through a direct reduction.
As applications of our framework, in section 5, we construct improved mechanisms for several
settings from the literature. For each setting we only construct a single buyer mechanism that
satisfies the requirements of our framework, and then our generic construction can be applied to
construct a mechanism for multiple buyers, using the single buyer mechanism as the building block.
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 [Mye81] for revenue maximization (e.g., [BR89, BLP06, BH08, HR09, DRY10,
CEDG+ 10, Yan11]). Most of them consider mechanisms that have simple implementation and are
computationally efficient. For welfare maximization in single dimensional settings, [HL10] gives a
blackbox reduction from mechanism design to algorithmic design.
In multidimensional setting, for welfare maximization, [HKM11] presents a blackbox reduction from mechanism design to algorithm design which subsumes the earlier work of [HL10]. For
revenue maximization, [CHMS10] 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, [CHMS10]
1
present a sequential posted pricing mechanism that obtains in expectation at least 6.75
-fraction
of the revenue of the optimal posted pricing mechanism. In subsection 5.4, we present an improved sequential posted pricing mechanism for this setting with an approximation factor of 12 γk
in which k is the number of copies available of each item, and γk is a constant which is at least
1
. For combinatorial auctions with additive/correlated valuations with hard budget and
1 − √k+3
capacity constraints, [BGGM10] presents all-pay BIC mechanisms for revenue maximization and
also for welfare maximization. They obtain an approximation factor of 14 in each case. In subsection 5.1, we present an improved mechanism for this setting with an approximation factor of γk .
Note that γk is at least 21 and as k → ∞, γk approaches 1. [BGGM10] also presents sequential
posted pricing mechanisms for the same setting, obtaining O(1) approximation factors. For the
same setting2 , in subsection 5.3, we present an improved sequential posted pricing mechanism with
an approximation factor of (1 − 1e )γk . [CMM11] also considers various settings with hard budget
constraints.
Prophet inequalities have been extensively studied in the past (e.g. [HK92]).
Prior to this work,
√
ln
k
the best known bound for the generalization to sum of k choices was 1 − O( √k ) by [HKS07]. We
1
improve this to 1 − √k+3
. Note that the current bound is not only asymptotically better than the
previous bound, but is also tight for k = 1, whereas the previous bound would be useful only for
large k.
2
Model & Overview of Approach
Model: We consider mechanisms for selling m indivisible heterogenous items to n buyers where
there are kj copies of each item j ∈ [m]. All the relevant private information of each buyer i ∈ [n]
2
We actually consider a slightly less general setting by not allowing capacity constraints.
3
is represented by her type ti ∈ Ti where Ti is the type space of buyer i. Let T = T1 × · · · × Tn
be the space of profiles of types. The profile of types t ∈ T is distributed according to a publicly
known prior D. We are looking for the optimal mechanism from a given space of mechanisms M.
For a mechanism M, we use XijM (t) and PiM (t) to denote the random variables3 respectively for
the allocation of item j to buyer i and the payment of buyer i, when the reported profile of types is
t. We are looking for mechanisms that maximize the expected value of a given objective function
W (t, x, p) where t, x, and p respectively represent the types, the allocations, and the payments
of all buyers. Formally, we are looking for a mechanism M ∈ M that (approximately) maximizes
Et∼D [W (t, X M (t), P M (t))].
Assumptions: We make the following assumptions.
(A1) The buyers’ types must be distributed independently, i.e., D = D1 × · · · × Dn where Di is the
distribution of types for buyer i. Note that for a buyer i who has multidimensional types, Di
itself does not need to be a product distribution.
P
(A2) The objective function must be linearly separable over the buyers, i.e., W (t, x, p) = i Wi (ti , xi , pi )
where ti , xi , and pi respectively represent the type, the allocation, and the payment of buyer
i.
(A3) No buyer must ever need more than one copy of each item, i.e., XijM (t) ∈ {0, 1} for all t. This
assumption is not necessary and can be lifted as explained in section 7.
(A4) M must be restricted to (Bayesian) incentive compatible mechanisms. By direct revelation
principle this assumption is without loss of generality4 ,
(A5) M must be a convex space. In other words, every convex combination of every two mechanisms
from M must itself be a mechanism in M. A convex combination of two mechanisms M, M0 ∈
M is another mechanism M00 which simply runs M with probability α and runs M0 with
probability 1 − α, for some α ∈ [0, 1]. In particular, if M is restricted to deterministic
mechanisms, then it is not convex. 5 .
(A6) The set of constraints that specify M must be decomposable to supply constraints and single
buyer constraints. Note that incentive compatibility constraints, budget constraints, etc., are
single buyer constraints. We define this assumption formally as follows. For any mechanism
M, let [[M]]i be the single buyer mechanism perceived by buyer i, as if the other buyers are
part of the mechanism. Let Mi = {[[M]]i |M ∈ M} be the space of single buyer mechanisms
perceived by buyer i resulting from mechanisms in M. We require that for any arbitrary
mechanism M the following holds: if M satisfies the supply constraints and also [[M]]i ∈ Mi
(for all i ∈ [n]), then M ∈ M.
We shall clarify the last assumption by giving an example. Suppose M is the space of all truthful
buyer specific item pricing mechanisms, then M satisfies the last assumption. On the other hand,
if M is the space of mechanisms that offer the same set of prices to every buyer, then it does not
satisfy the decomposability assumption.
3
Note that these random variables are often correlated. Furthermore, if M is a deterministic mechanism then for
any given t these variables take deterministic values as a function of t.
4
It is WLOG, given that we are only interested in mechanisms that have Bayes-Nash equilibria.
5
As an example of a randomized space of mechanisms without this property, consider the space of mechanisms
where the expected payment of every type must be either less than $2 or more than $4
4
Formally, the problem we are looking at is to find a mechanism M that is a solution to the
following program:
X
maximize:
Et∼D [Wi (ti , XiM (t), PiM (t))]
(P )
i
subject to:
X
∀t ∈ T, ∀j ∈ [m] :
XijM (t) ≤ kj
(S)
i
∀i ∈ [n] :
[[M]]i ∈ Mi
(M )
Throughout the rest of this paper, we adopt the P
conventionP
of omittingP
the range P
of the sums
whenever the range is clear from the context (e.g., i means i∈[n] , and j means j∈[m] ).
Summary of Approach: We now present an overview of our framework for constructing approximately optimal mechanisms for the above program. We start by relaxing the supply constraints
to hold only in expectation. We show that the optimal mechanism for the relaxed problem can
be constructed by combining n independent single buyer mechanisms. We then present two approaches for converting the mechanism constructed in the previous step to a mechanism for the
original problem. Each step is explained in more detail next.
The problem is initially relaxed by requiring the supply constraints to hold only in expectation.
In other words, the constraints (S) are replaced with the following constraints:
X
∀j ∈ [m] :
Et∼D [
XijM (t)] ≤ kj
(S 0 )
i
We show that an optimal mechanism for the relaxed problem can be constructed by combining n
independent single buyer mechanisms. First, we define special classes of single buyer mechanisms.
Definition 1 (Primary Mechanism/Primary Benchmark). A primary mechanism for buyer i is a
single buyer mechanism Mi that allows specifying an upper bound on the ex-ante expected probability
of allocating each item. For every q̄i ∈ [0, 1]m , Mi (q̄i ) is a single buyer mechanism from Mi for
which the ex-ante expected probability6 of allocating a copy of item j to buyer i is at most q̄ij . The
optimal primary mechanism is the one that has the highest expected objective value.
A primary benchmark for buyer i is a function Ri : [0, 1]m → R+ such that Ri (q̄i ) is an upper
bound on the expected objective value of the optimal primary mechanism for buyer i subject to q̄i .
If Mi and Ri are the optimal primary mechanism and the optimal primary benchmark for buyer i,
then Ri (q̄i ) is exactly equal to the expected objective value of Mi (q̄i ).
We show that an optimal mechanism for the relaxed problem can be constructed from n independent optimal primary mechanisms. Let M∗ be any optimal mechanism for the relaxed problem
∗ =E
M∗
∗
and let qij
t∼D [Xij (t)] be the expected probability that M allocates a copy of item j to buyer
i. Let Mi denote the optimal primary mechanism for each buyer i. In section 4, we prove that
the mechanism that runs Mi (qi∗ ), independently for each buyer i, has the same expected objective
value as M∗ . Therefore, if we can construct the optimal primary mechanism for each buyer, then
we can construct an optimal mechanism for the relaxed problem by simply using Mi (qi∗ ) indepen∗ . We will show that q ∗ is
dently for each buyer i, assuming that we know how to compute the qij
ij
the optimal assignment for the following program in which Ri is the optimal primary benchmark
6
Note that the expectation is taken over all possible types of buyer i
5
for buyer i:
X
maximize:
Ri (q̄i )
(CPR )
i
X
∀j ∈ [m] :
q̄ij ≤ kj
i
∀i ∈ [n], ∀j ∈ [m] : q̄ij ∈ [0, 1]
In particular, in section 4, we prove that the optimal primary benchmarks Ri (·) are always concave,
and therefore the above program is a convex program. Consequently, this program can be efficiently
∗ . Note that usually each function R is itself the optimal objective value of
solved to compute qij
i
a linear or convex program, and does not have a closed form; in that case, all the corresponding
linear/convex programs can be merged into one. So far, we have explained how the problem
of constructing an optimal mechanism for the relaxed problem can be reduced to the problem
of constructing the optimal primary mechanisms/primary benchmarks. Next, we explain how to
convert it to a mechanism for the original (non-relaxed) problem.
We now present two approaches for converting a mechanism for the relaxed problem to a
mechanism for the original problem, while losing only a small constant fraction of the expected
objective value. Let M be the mechanism constructed in the previous step, which uses Mi (q̄i )
independently for each buyer i. Since M satisfies the supply constraints only in expectation, it
most likely violates those constraints in some instances. We propose two separate approaches
for dealing with this issue, each one yielding a generic mechanism for the original (non-relaxed)
problem. The following is a high level description of the two generic mechanisms.
1. Pre-Rounding: This mechanism serves buyers sequentially in an arbitrary order. For each
buyer i, it runs Mi (q̄i0 ) in which q̄i0 is the same as q̄i except that some of its entries are set
to 0 as explained next. The outcome of Mi (q̄i0 ) is taken as the final outcome for buyer i.
0 to 0 effectively precludes M (q̄ 0 ) from allocating a copy of item j to buyer i. The
Setting q̄ij
i i
0 to 0 for any item j that is sold out prior to
supply constraints are enforced by setting q̄ij
serving buyer i. Moreover, for each item, the mechanism tries to minimize simultaneously for
all buyers the expected probability of preclusion by equalizing this expected probability for
all buyers. Effectively, the mechanism sometimes precludes some items from being offered to
earlier buyers in order to make sure that later buyers get the same chance of being offered
with those items. Note that, for any given pair of buyer and item, we only care about the
expected probability of preclusion where the expectation is taken over the types of other
buyers. In particular, an item might be precluded from the current buyer with probability
1 if certain scenarios of outcomes have been realized for buyers served prior to the current
buyer. We show that if there are at least k copies of each item then the expected probability
1
of preclusion of each item for each buyer is no more than √k+3
.
2. Post-Rounding: This mechanism runs Mi (q̄i ) for each buyer i independently and then
modifies the outcomes by deallocating some of the items at random to ensure that the supply
constraints are met at every instance. This is done in such a way that would minimize the
expected probability of deallocation observed by each buyer by equalizing this probability
over all copies of each item. The payments are also scaled down accordingly by the same
probability. Note that, for any given pair of buyer and item, we only care about the expected
probability of deallocation, where the expectation is taken over the types of other buyers. In
particular, a buyer who faces a small expected probability of deallocation could still face a
deallocation probability of 1 for some items, when certain profiles of types are reported by
6
other buyers. We show that if there are at least k copies of each item, then the expected
1
probability of deallocation is no more than √k+3
for each copy.
In section 4 we explain the above mechanisms in more detail and present some technical as1
sumptions that are sufficient to ensure that they retain at least a 1 − √k+3
fraction of the expected
objective value of M.
Throughout the above discussion, we have assumed that we can construct the optimal primary
mechanisms and the optimal primary benchmarks. However, it is likely that we can only construct
an approximation of them. Suppose for each buyer i, we only have an α-approximate primary
mechanism and a corresponding concave primary benchmark Ri (i.e., the expected objective value
of Mi (q̄i ) is at least α · Ri (q̄i ) for every q̄i ∈ [0, 1]m ). Then we can still used Mi and Ri in the
above construction, but the final approximation factor will be multiplied by α.
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 1 (Market Expansion). Suppose for each buyer i ∈ [n], we have an α-approximate
primary mechanism Mi and a corresponding concave primary benchmark Ri . Then, with some
further assumptions (explained later), a mechanism M ∈ M can be constructed by using the primary
mechanisms as building blocks, such that the expected objective value of M is at least γk · α-fraction
of the expected objective value of the optimal mechanism from M, where k = minj kj and γk is a
1
constant which is at least 1 − √k+3
.
In order to explain our construction in more detail, we shall first describe the magician’s problem
and its solution, which is used in equalizing the expected probabilities of preclusion/deallocation
over all buyers.
3
The Magician’s Problem
In this section, we present an abstract online stochastic toy problem and a near-optimal solution for
it. The solution to this problem is the main ingredient for combining single buyer mechanisms to
construct mechanisms for multiple buyers. It is also used to prove a generalized prophet inequality.
Definition 2 (The Magician’s Problem). A magician is presented with a series of boxes one by
one. There is a prize hidden in one of the boxes. He has k magic wands that can be used to open the
boxes. On each box is written a probability. If a wand is used on a box, it opens, but with at most
the written probability the wand breaks. Let qi denote this probability for the ith box. 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
P no prior information about them (not even the number of the boxes). However, it is
given that i qi ≤ k, and that the villain 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 magic wands before coming 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 expected 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 γ, he has to devise a strategy
that guarantees an ex-ante expected probability of at least γ for opening each box. Notice that
the amount of the prize (or whether it can be split among multiple boxes) does not affect the
problem. It is easy to show the following strategy ensures an ex-ante expected probability of at
7
least 41 for opening each box: for each box randomize and use a wand with probability 21 . But
can we do better? Next, we present an algorithm that takes a parameter γ and tries to ensure a
minimum ex-ante expected probability of γ for opening each box. In Theorem 2, we show that
1
for any γ ≤ 1 − √k+3
this algorithm indeed guarantees that the ex-ante expected probability of
opening each box is at least γ.
Algorithm 1 (γ-Conservative Magician). The magician constructs a strategy table yij using the
dynamic programs given below. yij specifies the probability with which the magician should choose
to use a wand on the ith box if j wands have been broken prior to seeing the ith box. So if yij = 0
or yij = 1, then the magician makes a deterministic decision, otherwise he should randomize and
open the ith box with probability yij . We use Yi as the indicator random variable which is 1 iff the
magician chooses to use a wand to open the ith box. The strategy table can be computed using the
following dynamic programs (note that γ is a parameter that is given):


i ≥ 1, 0 ≤ j < θi
1
j
θi −1
θi
θi −1
yi = (γ − φi )/(φi − φi )
(DP.y)
i ≥ 1, j = θi


0
otherwise.
θi = min{j|φji ≥ γ}


1
j
j
j−1
j
φi = yi−1
qi−1 φi−1
+ (1 − yi−1
qi−1 )φji−1


0
(DP.θ)
i = 1, j ≥ 0
i ≥ 2, j ≥ 0
otherwise.
(DP.φ)
Note that computing yij only requires the knowledge of q1 , · · · , qi−1 , so computing yij and making a
decision about the ith box can be done even before seeing the ith box itself.
Interpretation of The γ-Conservative Magician(Alg. 1) The main idea of the algorithm is
the following. After seeing the first i − 1 boxes and prior to the arrival of the ith box, the magician
computes a threshold θi as follows. θi is the smallest integer such that the ex-ante expected
probability of having broken no more than θi wands, on the first i − 1 boxes, is at least γ. In other
words, if Si is the random variable that represents the number of magic wands broken prior to
seeing the ith box, then θi is chosen to be the smallest integer such that P r[Si ≤ θi ] ≥ γ. Observe
that if the magician always opens the ith box when the number of wands broken so far is no more
than θi , and otherwise discards the box, then he can guarantee an ex-ante probability of at least
γ for opening the ith box. Furthermore, if P r[Si ≤ θi ] > γ, i.e. if the inequality is strict, then
in the event of having broken exactly θi wands prior to the ith box, the magician randomizes and
opens the ith box with a probability strictly less than 1, which is just enough to ensure that the
total ex-ante expected probability of opening the ith box is at least γ. It can be verified that φji ,
as defined by the dynamic program, is a lower bound on P r[Si ≤ j]. In fact, if q1 , · · · , qi−1 are the
exact probabilities of breaking a wand for each of the first i − 1 boxes, then P r[Si ≤ j] = φji . In
order to prove that the above strategy ensures that each box is opened with an ex-ante expected
probability of at least γ, we need to show that yij = 0 for all j ≥ k and all i. In other words, we
need to show that the strategy table of the magician does not instruct him to open a box if he has
broken all of his k wands. In Theorem 2 we present a sufficient condition on γ that ensures yij = 0
for all j ≥ k and all i.
1
Theorem 2 (γ-Conservative Magician). For any γ ≤ 1 − √k+3
, a γ-conservative magician guarantees that each box is opened with an ex-ante expected probability at least γ. Furthermore, if qi are
8
the exact probabilities of breaking a wand, then the γ-conservative magician opens each box with an
ex-ante expected probability exactly γ 7
Proof. See section 6.
Definition 3 (γk ). We define γk to be the largest probability such that for any instance of the
magician’s problem with k 0 wands, where k 0 ≥ k, a γk -conservative magician with k 0 wands can
guarantee that each box is opened with an ex-ante expected probability at least γk . By Theorem 2,
1
1
because for any k 0 ≥ k obviously 1 − √k+3
≤ 1 − √k10 +3 .
we know that γk must be at least 1 − √k+3
Observe that γk is a non-decreasing function in k which is at least 12 (when k = 1) and approaches
1
on γk is not far from the
1 as k → ∞. The next theorem shows that the lower bound of 1 − √k+3
optimal.
Theorem 3 (Optimal Magician). For any > 0, it is not possible to guarantee an ex-ante expected
k
probability of 1 − ekk k! + or better for opening each box(i.e., no magician can guarantee it). Note
that 1 −
kk
ek k!
≈1−
√1 .
2πk
Proof. See section A.
Next, we prove a generalization of prophet inequalities by a direct reduction to the magician’s
problem.
Definition 4 (Sum of k Choices). A sequence of n non-negative random variables V1 , · · · , Vn are
presented to a gambler one by one in an arbitrary order. The gambler knows n and the distribution
of each random variable in advance, but not the order in which they are presented. Upon being
presented with the random variable Vi , the gambler observes the actual draw of Vi , and has to
decide whether to keep it or to discard it. This decision cannot be changed later. The gambler
must select k of the random draws from the sequence. His objective is to maximize the sum of the
selected draws. The prophet knows all the actual draws in advance, so he chooses the k highest
draws. We assume that the order in which the random variables are presented to the gambler is
fixed in advance and does not change during the process.
√
[HKS07] proved that there is a strategy for the gambler that guarantees at least 1 − O( √lnkk )
fraction of the payoff of the prophet, in expectation, by using a non-decreasing sequence of k
stopping rules (thresholds) 8 . Next, we construct a gambler that obtains at least γk fraction of the
prophet’s payoff, in expectation, by using a γk -conservative magician as a black box. Note that
1
γk ≥ 1 − √k+3
. This gambler uses only a single threshold. However, he may skip some of the
random variables at random.
7
In particular the fact that the probability of breaking a wand for the ith box is exactly qi conditioned on any
sequence of prior events implies that for each box the event of breaking a wand has to be independent of the sequence
of past events and independent of other boxes.
8
A gambler with stopping rules τ1 , · · · , τk works as follows. Upon seeing Vi , he selects it iff Vi ≥ τj+1 where j is
the number of random draws selected so far.
9
Theorem 4 (Prophet Inequalities for Sum of k Choices). The following strategy ensures that the
gambler obtains at least γk fraction of payoff of the prophet in expectation. 9
P
• Find the threshold τ such that i P r[Vi > τ ] = k. This can be done by doing a binary search
on τ .
• Use a γk -conservative magician with k magic wands. Upon seeing each Vi , create a box and
write qi = P r[Vi > τ ] on the box and present it to the magician. If the magician chooses to
open the box and also Vi > τ , then select Vi and break the magician’s wand, otherwise skip Vi .
Proof. First, we compute an upper bound on the expected payoff of the prophet. Let qi be the
probability that the prophet chooses Vi (i.e. the probability that Vi is among the k highest draws).
Now let ui (qi ) denote the maximum possible contribution of the random variable Vi to the expected
payoff of the prophet if Vi is selected with probability qi . Note that ui (qi ) is equal to the expected
value of Vi conditioned on being above the 1 − qRi quantile. Let Fi (·) and fi (·) denote the CDF
∞
and PDF of Vi . ui (qi ) can be defined as ui (qi ) = F −1 (1−qi ) vfi (v)dv. By changing the integration
i R
q
variable and applying the chain rule we get ui (qi ) = 0 i Fi−1 (1 − q)dq. Observe that dqdi ui (qi ) =
P
Fi−1 (1 − qi ) is a non-increasing function, so ui (qi ) is a concave function. Furthermore, i qi ≤ k
because the prophet cannot choose more than k random draws. So the optimal objective value of
the following convex program is an upper bound on the payoff of the prophet.
X
maximize:
ui (qi )
(U )
qi ≤ k
(τ )
i
X
i
∀i ∈ [n] :
qi ≥ 0
(µi )
P
P
P
Let L(q, τ, µ) = − i ui (qi ) + τ ( i qi − k) − i µi qi be the Lagrangian for the above convex
program. By KKT stationarity condition, at the optimal assignment, it must be ∂q∂ i L(q, τ, µ) =
0. On the other hand, ∂q∂ i L(q, τ, µ) = −Fi−1 (1 − qi ) + τ − µi . Assuming that qi > 0, then
by complementary slackness µi = 0, which then implies that qi = 1 − Fi (τ ), so qi = P r[Vi >
τP
]. Furthermore, it is easy to show that the first constraint must be tight, which implies that
i P r[Vi > τ ] = k. Observe that the contribution of each Vi to the objective value of the convex
program is exactly E[Vi |Vi > τ ]P r[Vi > τ ]. Now, by using a γk -conservative magician we can ensure
that each box is opened with probability at least γk which implies the contribution of each Vi to the
expected payoff of the gambler is E[Vi |Vi > τ ]P r[Vi > τ ]γk which proves that the expected payoff
of the gambler is at least γk fraction of optimal objective value of the convex program, which was
itself and upper bound on the expected payoff of the prophet.
4
The Two Generic Mechanisms
In this section, we present the details of the approach that was outlined in section 2. The model and
assumptions were explained in that section. We start by proving that an (approximately) optimal
mechanism for the relaxed problem can always be constructed from n (approximately) optimal
primary mechanisms.
9
To simplify the exposition we assume that the distributions do not have point masses. Our theorem holds with
slight modifications if we allow point masses.
10
Theorem 5. Suppose for each buyer i, we have an α-approximate primary mechanism Mi and a
matching concave primary benchmark Ri , as defined in Def. 1 (i.e., for every q̄i ∈ [0, 1]m , Mi (q̄i )
obtains α · Ri (q̄i ) in expectation, and Ri (q̄i ) is an upper bound on the expected objective value of the
optimal primary mechanism subject to q̄i ). Consider the mechanism M which simply uses Mi (q̄i )
independently for each buyer i, where q̄i is the optimal assignment of the following convex program.
Then, M is a feasible mechanism for the relaxed problem, and its expected objective value is at least
an α-fraction of the optimal objective value of the convex program. Furthermore, the optimal value
of this program is an upper bound on the expected objective value of the optimal mechanism in M.
X
maximize:
Ri (q̄i )
(CPR )
i
X
∀j ∈ [m] :
q̄ij ≤ kj
i
∀i ∈ [n], ∀j ∈ [m] :
q̄ij ∈ [0, 1]
Proof. Let M∗ be any optimal mechanism for the relaxed problem. For each buyer i, we construct a
single buyer mechanism M∗i as follows. M∗i creates n − 1 dummy buyers whose types are randomly
drawn from D−i . It then runs M∗ on buyers i and the n − 1 other dummy buyers. Note that buyer
i cannot tell the different between M∗i and the original M∗ , because buyers types are distributed
independently. Observe that the expected contribution of buyer i to the objective value of M∗ is the
same as her expected contribution to the objective value of M∗i . So the mechanism that runs each
one of M∗1 , · · · , M∗n independently has the same expected objective value and the same expected
∗ = E
M∗
probabilities of allocation as M∗ . Now let qij
t∼D [Xij (t)] be the expected probability that
∗ is a feasible assignment for the convex
M∗ allocates a copy of item j to buyer i. Observe that qij
program. Furthermore, the expected objective value of M∗ P
is equal to the sum of the expected
∗
∗
objective values of M1 , · · · , Mn which is upper bounded by i Ri (qi∗ ). So the optimal objective
∗
value of the convex program may only be higher than the expected objective
P value of M . Now
observe that the expected objective value of the mechanism M is at least i α · Ri (q̄i ) where q̄i is
the optimal assignment for the convex program. So the expected objective value of M is at least
α-fraction of the expected objective value of M∗ .
Note that in Theorem 5, Ri are concave functions by definition. However, we shall show that
the optimal primary benchmarks are also concave.
Theorem 6. The optimal primary benchmarks are always concave.
Proof. We prove this for an arbitrary buyer i. Let Mi and Ri denote the optimal primary
mechanism and the optimal primary benchmark for buyer i. To show that Ri is concave, it is
enough to show that for any q, q 0 ∈ [0, 1]m and any α ∈ [0, 1], the following inequality holds:
Ri (αq + (1 − α)q 0 ) ≥ αRi (q) + (1 − α)Ri (q 0 ). Consider the single buyer mechanism M00 that works
as follows: M00 uses Mi (q) with probability α and uses Mi (q 0 ) with probability 1 − α. Note that
Mi is a convex space (this follows from A5 and A6), therefore M00 ∈ Mi . Observe that by linearly
of expectation, the expected probabilities of allocation for M00 is no more than αq + (1 − α)q 0 and
the expected objective value of M00 is αRi (q) + (1 − α)Ri (q 0 ). So the expected objective value of the
optimal primary mechanism, subject to the upper bound of αq + (1 − α)q 0 on the expected probabilities of allocation, may only be higher. That implies Ri (αq + (1 − α)q 0 ) ≥ αRi (q) + (1 − α)Ri (q 0 )
which proves our claim.
Next, we present a formal description of the two generic mechanisms that were outlined in
section 2. Throughout the rest of this section, we assume that for each buyer i, we have an α11
approximate primary mechanism Mi and a corresponding concave primary benchmark Ri . First,
we present the pre-rounding mechanism.
Mechanism 1 (γ-Pre-Rounding).
(I) Solve the convex program of (CPR ) and let q̄ij denote an optimal assignment for it.
(II) For each item j ∈ 1 · · · m: create an instance of γ-conservative magician (see Alg. 1) with
kj magic wands (this will be referred to as the j th magician). We will use these magicians
through the rest of the mechanism. Note that γ is a parameter that is given.
(III) For each buyer i ∈ 1 · · · n:
(a) For each j ∈ 1 · · · m: write q̄ij on a box and present it to the j th magician. Let Yij
denote the indicator random variable which is 1 iff the magician opens the box. Set
0 ← q̄ Y .
q̄ij
ij ij
(b) Run the mechanism Mi (q̄i0 ) on buyer i and use its outcome as the final outcome for
buyer i. Furthermore, let Xi1 , . . . , Xim denote the indicator random variables for the
allocation of Mi (q̄i0 ).
(c) For each j ∈ 1 · · · m: if Xij = 1, then break the wand of the j th magician.
In order for Mech. 1 to retain at least a γ-fraction of the the expected objective value of each
Mi (q̄i ), we have to make further technical assumptions. We show that it is enough to assume that
each Ri has a budget-balanced and cross monotonic cost sharing scheme, and γ ≤ γk . Next we
define this formally.
Definition 5 (Budget Balanced Cross Monotonic Cost Sharing Scheme). For any vector q ∈ [0, 1]m
and any subset S ⊂ [m], let q[S] denote a vector of length m in which the j th component is equal
to qj if j ∈ S and is 0 otherwise. A function R(q) has a budget balanced cross monotonic cost
sharing scheme iff there exists a cost share function ξ : [m] × [0, 1]m → R+ with the following two
properties:
P
(i) ξ must be budget balanced which means for all q ∈ [0, 1]m and S ⊂ [m],
j∈S ξ(j, q[S]) =
R(q[S]).
(ii) ξ must be cross monotonic which means for all q ∈ [0, 1]m , j ∈ [m] and S, T ⊂ [m], ξ(j, q[S]) ≥
ξ(j, q[S ∪ T ]).
Theorem 7 (γ-Pre-Rounding). Suppose for each buyer i we have an α-approximate primary mechanism Mi and a corresponding concave primary benchmark Ri that has a budget balanced cross
monotonic cost sharing scheme. Then, for any γ ∈ [0, γk ], the γ-pre-rounding mechanism (Mech. 1)
is a γ · α-approximation of the optimal mechanism in M. Furthermore, the resulting mechanism is
always a dominant strategy incentive compatible (DSIC) mechanism in M.
Proof. See section A.
The above requirement on Ri can be interpreted as follows. Roughly it means that the contribution of each item j to the expected objective value of the primary mechanism for buyer i should not
decrease when items other than j are being precluded from buyer i. In particular, this assumption
holds if Ri (q̄i [S]) is a submodular function of S. The following are two examples of environments
where this assumption holds: (a) for welfare maximization, assuming that the buyers’ valuations
are submodular), and (b) for revenue maximization, when the buyers’ valuations are submodular
and M is restricted to mechanisms that can be interpreted as buyer specific item pricing.
12
Remark 1. The γ-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 M includes all feasible BIC
mechanisms, the gap between the optimal DSIC mechanism and the optimal BIC mechanism is at
most 1/γk . This gap is at most 2 (for k = 1) and vanishes as k → ∞. That is because Mech. 1 is
always DSIC, yet it approximates the optimal mechanism in M.
Next, we present the post-rounding mechanism.
Mechanism 2 (γ-Post-Rounding).
(I) Solve the convex program of (CPR ) and let q̄ij denote an optimal assignment for it.
(II) For each buyer i ∈ 1 · · · n: run the corresponding primary mechanism Mi (q̄i ) on buyer i and
let Xi1 , . . . , Xim and Pi denote the random variables for the allocation and the payment of
Mi . Furthermore, let q̂ij be the actual marginal probability of allocating item j to buyer i by
Mi (q̄i ). Note that q̂ij ≤ q̄ij .
(III) For each item type j ∈ 1 · · · m:
(a) Create a new instance of the γ-conservative magician (see Alg. 1) with kj magic wands
(this will be referred to as the j th magician).
(b) For each i ∈ 1 · · · n: create a box corresponding to Xij and write q̂ij on the box and
present it to the j th magician. Let Yij denote the indicator random variable which is 1
iff the magician chooses to open the box. Set Xij0 ← Xij Yij . If Xij0 = 1 then break the
magician’s wand.
(IV) For each buyer i ∈ 1 · · · n: charge buyer i a payment of Pi0 ← γPi and for each j ∈ 1 · · · m,
allocate a copy of item j to buyer i iff Xij0 = 1.
In order for Mech. 2 to be applicable, we need to make further technical assumptions. The
following assumptions are sufficient to ensure that Mech. 2 is truthful and retains at least a γfraction of the expected objective value of each Mi (q̄i ), given that γ ∈ [0, γk ].
(A0 1) The actual expected probabilities of allocation for each Mi (q̄i ) must be available (i.e., efficiently computable). Note that q̄i is only an upper bound on these probabilities.
(A0 2) The objective functions must be of the form Wi (ti , xi , pi ) = Wi (ti , xi , 0) + ci pi in which
ci ∈ R is an arbitrary fixed constant. Also, each Wi must have a cross monotonic and budget
balanced cost sharing scheme in xi .
(A0 3) The resulting mechanism must be in M. In particular, that implies M may not be restricted
to any from of incentive compatibility stronger than Bayesian incentive compatibility (BIC),
because Mech. 2 is only BIC.
(A0 4) The valuations of each buyer must be in the form of a weighted rank function of some
matroid.
13
Note that the above assumptions are not strictly necessary, but sufficient. Observe that A0 2
obviously holds for revenue maximization (i.e., Wi (ti , xi , pi ) = pi ), and also for welfare maximization
when buyers have quasilinear utilities and submodular valuations (i.e., Wi (ti , xi , pi ) = vi (xi ) where
vi (xi ) is the valuation of buyer i for allocation xi 10 ). Next, we formally define A0 4.
Definition 6 (Valuations as Matroid Weighted Rank Functions). Valuations of a buyer for bundles
of items can be represented as a weighted rank function of a matroid if there is a matroid with the
ground set being the set of items, and such that for any bundle S of items:
• If S is an independent set of the matroid, then the valuation of the buyer for S is just the
sum of her valuations for each item in S.
• If S is not an independent set, then the valuation of the buyer for S is equal to her valuation
for the highest valued independent subset of S.
In particular, additive valuations with capacities, unit demand valuations, etc. can be represented
as matroid weighted rank functions 11 .
Theorem 8 (γ-Post-Rounding). Suppose for each buyer i we have an α-approximate primary
mechanism Mi and a corresponding concave primary benchmark Ri . Suppose the assumptions A0 1
through A0 4 hold. Then, for any γ ∈ [0, γk ], the γ-post-rounding mechanism (Mech. 2) is a γ · αapproximation of the optimal mechanism in M. Furthermore, the resulting mechanism is always a
Bayesian incentive compatible (BIC) mechanism in M.
Proof. See section A.
5
Primary Mechanisms
In this section, we present several approximation primary mechanisms. Note that a primary mechanism (see Def. 1) only considers a single buyer. Each one of the primary mechanisms presented in
this section satisfies the requirements of one of the generic mechanisms of section 4. So they can
be readily converted to mechanisms for many independent buyers using either Mech. 1 or Mech. 2.
Except for subsection 5.1, we restrict the space of mechanisms to asymmetric item pricing (AIP)
mechanisms as defined below.
Definition 7 (Asymmetric Item Pricing (AIP)). A mechanism is asymmetric item pricing (AIP)
if from the perspective of each buyer each item has a price which does not depend on the report of
that buyer. The prices observed by different buyers could be different. In the presence of budget
constraints, a buyer may pay a fraction of the price of an item and then receive the item with a
probability proportional to the paid fraction of the price12 . A mechanism is AIP if its outcome can
be interpreted in such a way. In other words, an AIP 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 optimal consumption bundle according to her
observed prices.
10
Note that the payment terms cancel out because the utility of the seller is counted toward the social welfare of
the mechanism
11
Note that budget constraints are not part of the valuations.
12
It is easy to see that a rational buyer, with submodular valuations and with a budget constraint, always pays
the full price for any item they purchase, except for the last item purchased, for which they must have run out of
budget. In other words, such a buyer never fractionally purchases more than one item.
14
Note that a primary AIP mechanism is just a single buyer item pricing mechanism, so we will
instead use the term primary IP mechanism. In particular, a primary IP mechanism simply chooses
a price distribution irrespective of the type of the buyer and then draws the prices of items from
that distribution and offers the items at those prices to the buyer.
All the primary IP mechanisms presented throughout the rest of this section satisfy the requirements of γ-pre-rounding, so they can be used in Mech. 1 to yield sequential posted pricing
mechanisms that approximate the optimal AIP mechanism.
Table 1 lists several settings considered in this section along with the results obtained.
Primary Mech/Expanded by
5.1 Mech. 3 / Mech. 2
5.2 Mech. 4/ Mech. 1
5.3 Mech. 5 / Mech. 1
5.4 Mech. 6/ Mech. 1
Setting
additive correlated valuations with polynomial number of types, budget, capacity, revenue or welfare
unit demand, single item(multi unit),
budget, revenue
additive valuations, product distribution,
budget, revenue
unit demand, product distribution (regular), revenue
Resulting Mech
general(BIC)
Approximation Factor
γk of optimal BIC
Sequential IP
γk of optimal AIP
Sequential IP
γk (1 − 1e ) of optimal AIP
Sequential IP
1
γ
2 k
of optimal AIP
Table 1: Summary of mechanisms obtained using the framework of this paper.
5.1
Correlated Valuations with Capacity and Hard Budget Constraints
We present the optimal revenue maximizing primary mechanism for a buyer with correlated and
additive valuations with a capacity and a hard budget constraint. Since a primary mechanism only
interacts with a single buyer, we shall drop the subscript i. Let T denote the type space of the
buyer. For each t ∈ T , let f (t) denote the probability that the buyer’s type is t. We assume that
T is a discrete type space and f (t) is explicitly given for each t as part of the input. Let vtj denote
the valuation of the buyer for item j when her type is t. Furthermore, suppose that the buyer has a
total budget of B and is interested in at most C items. We assume that the only private information
of the buyer is her type and everything else is publicly known. For this environment, we present an
optimal revenue maximizing primary mechanism. This primary mechanism can be used in the γpost-rounding mechanism (Mech. 2) to yield a γk -approximate BIC mechanism for multiple buyers.
Remember that γk is at least 12 , furthermore as k → ∞, γk approaches 1 which means the resulting
mechanism approaches the optimal mechanism. The previous best approximation mechanism for
this environment was a 14 -approximate BIC mechanism by [BGGM10]. Note that the construction
is only polynomial in |T |, which means it may not be polynomial in the input size if f (·) is encoded
in the input in a compact form.
Consider the following linear program in which xtj and pt are the variables respectively corresponding to the probability of allocating a copy of item j and the payment when the buyer’s
reported type is t. The optimal objective value of this LP is an upper bound on the revenue of
the optimal primary mechanism subject to an upper bound of q̄j on the probability of allocating a
copy of item j.
15
X
maximize:
f (t)pt
(LPrev )
t
X
∀j ∈ [m] :
f (t)xtj ≤ q̄j
t
X
∀t ∈ T :
xtj ≤ C
j
X
∀t, t0 ∈ T :
vtj xtj − pt ≥
j
X
vtj xt0 j − pt0
j
∀t ∈ T, ∀j ∈ [m] :
xtj ∈ [0, 1]
∀t ∈ T :
pt ∈ [0, B]
We construct the optimal primary mechanism as follows.
Mechanism 3.
• Define the optimal primary benchmark R(q̄) to be the optimal objective value of (LPrev ) as a
function of q̄ = (q̄1 , · · · , q̄m ).
• Given q̄1 , · · · , q̄m , solve the linear program of (LPrev ) to compute xtj and pt .
• If the buyer reports her type as t, then charge her a payment of pt and allocate each item j
with probability xtj as explained next. Use the dependent randomized rounding algorithm of
[GKS02] to round each xtj to either
P 0 or 1 such that if Xj is the result of rounding the xtj
then E[Xj ] = xtj and such that j Xj ≤ C. Then, for each j allocate a copy of item j to the
buyer iff Xj = 1.
Theorem 9. The primary mechanism Mech. 3 is a truthful optimal revenue maximizing primary
mechanism that satisfies all the requirements of the γ-post-rounding (Mech. 2).
Proof. The proof of truthfulness and optimality of Mech. 3 trivially follows from the (LPrev ). So, we
only focus on proving that this mechanism satisfies the requirements of Theorem 8. First, observe
that the benchmark function, R(q̄), is concave (this follows by applying Lemma. 1). Second, observe
that the valuations of the buyer can be represented as a weighted rank function of a uniform matroid
of rank C. Third, notice that given q̄1 , · · · , q̄m , the
P exact marginal probabilities of allocation, i.e.
q̂1 , · · · , q̂m , can be computed as follows: q̂j =
t f (t)xtj . So the mechanism Mech. 3 and it
benchmark satisfy the requirements of Theorem 8 for γ-post-rounding.
Lemma 1. Consider any convex program of the following form, in which u(·) is a concave function,
gj (·) are convex functions, and X is a convex set. Let R(q̄) denote the optimal objective value of
this program as a function of q̄ = (q̄1 , · · · , q̄m ). Then R(q̄) is concave.
maximize:
u(x)
∀j :
gj (x) ≤ q̄j
x∈X
Proof. See section A.
16
(CPu )
P
Remark 2. Observe that by replacing the objective function of (LPrev ) with t,j f (t)vtj xtj we get
a truthful optimal welfare maximizing primary mechanism, which can also be used in the γ-postrounding mechanism (Mech. 2) to yield a γk -approximate welfare maximizing BIC mechanism for
multiple buyers.
5.2
Single-Item with Hard Budget Constraints
We present an optimal revenue maximizing primary IP mechanism for a unit-demand buyer with
a publicly known budget B where there is only one type of item (i.e., m = 1). The only private
information of the buyer is her valuation for the item, which is drawn from a publicly known
distribution with CDF F (·). To avoid complicating the proofs, we assume that F (·) is continuous
and strictly increasing in its domain13 . Next, we present the optimal revenue maximizing primary
IP mechanism in this environment. Note that this mechanism is optimal among the primary IP
mechanisms. We start by defining the modified CDF function F B (·) as follows:
(
F (v)
v≤B
B
F (v) =
(F B )
B
1 − (1 − F (v)) v v ≥ B
Intuitively, 1 − F B (p) is the probability of allocating the item to the buyer if we offer the item
at price p. Note that the buyer only buys if her valuation is more than p which happens with
probability 1 − F (p) .If p > B then she will pay her whole budget and only get the item with
probability Bp . Observe that if we want to allocate the item with probability q we can offer a price
of F B (−1) (1 − q) in which case it can be easily verified that we get a revenue of q · F B (−1) (1 − q)
b
in expectation. Now define the function H(q) = q · F B (−1) (1 − q) and let H(q)
denote its concave
closure (i.e., the smallest concave function that is an upper bound on H(q) for every q). We address
b
the problem of efficiently computing H(q)
later in Lemma. 2. Next, we show that the revenue of
the optimal primary IP mechanism is no more than the optimal objective value of the following
convex program, when the probability of allocating the item is restricted to be no more than q̄.
maximize:
b
H(q)
(CPrev−1 )
q ≤ q̄
q≥0
Theorem 10. The revenue of the optimal primary IP mechanism, when restricted to allocate
the item with probability at most q̄, is equal to the optimal objective value of the convex program
(CPrev−1 ). Furthermore, assuming that q ∗ is the optimal assignment for the convex program, if
b ∗ ) = H(q ∗ ) then the optimal primary IP mechanism uses a single price p = F B (−1) (1 − q ∗ )
H(q
otherwise, it randomizes between two prices but the probability of allocation is still q ∗ .
Proof. First we prove that the expected revenue of the optimal primary IP mechanism which we
b ∗ ). Then we construct a distribution over prices that obtains
denote by R∗ is upper bounded by H(q
this revenue. Note that any primary IP mechanism can be specified as a distribution over prices.
Let P be the optimal price distribution. So R∗ = Ep∼P [p(1 − F B (p))]. Note that every price p
corresponds to an allocation probability q = 1 − F B (p). So any probability distribution over p
can be specified as a probability distribution over q. Let Q denote the probability distribution
for q that corresponds to price distribution P, then we can write R∗ = Eq∼Q [q · F B (−1) (1 − q)] =
b
Eq∼Q [H(q)] ≤ Eq∼Q [H(q)]
also notice that by Jensen’s inequality this is less than or equal to
13
The proofs can be modified to work without this assumption.
17
b q∼Q [q]). Note that Eq∼Q [q] is exactly the probability of allocating the item if we use the
H(E
price distribution P so it must be no more than q̄ which implies that q = Eq∼Q [q] is a feasible
b q∼Q [q]) ≤ H(q
b ∗ ) because q ∗ was the optimal assignment
assignment for (CPrev−1 ) and therefore H(E
for (CPrev−1 ). That completes the first part of the proof. Next, we construct the optimal price
b ∗ ) = H(q ∗ ) then the optimal price distribution is just a single price p =
distribution. If H(q
F B (−1) (1 − q ∗ ). Otherwise, by definition of concave closure, there are two points q − and q + and
b ∗ ) = θH(q − ) + (1 − θ)H(q + ). In this case the
θ ∈ (0, 1) such that q ∗ = θq − + (1 − θ)q + and H(q
optimal price distribution offers price p− with probability θ and offers price p+ with probability
1 − θ.
The optimal revenue maximizing primary IP mechanism can be constructed as follows.
Mechanism 4.
• Define the benchmark R(q̄) to be the optimal objective value of (CPrev−1 ) as a function of q̄.
• Given q̄, solve the convex program of (CPrev−1 ) to compute the optimal assignment for q.
b
• Using the optimal q compute the optimal price as follows: if H(q)
= H(q) then offer the single
(−1)
B
price p = F
(1 − q) otherwise randomize between two prices p− and p+ with probabilities
θ and 1 − θ as explained in the proof of Theorem 10.
Theorem 11. Mech. 4 is the optimal revenue maximizing primary IP mechanism. Furthermore,
this mechanism satisfies the requirements of γ-pre-rounding.
Proof. The proof of the optimality of the mechanism follows from Theorem 10. Furthermore,
the benchmark function, R(q̄), is concave (this follows by applying Lemma. 1) and it is trivially
submodular (because there is only one item), therefore it meets the requirements of DSIC expansion.
b
Now we get back to the problem of efficiently computing H(·).
b
b 1+ (·) can be constructed using
Lemma 2. A (1 + )-approximation of H(·)
which we denote by H
log L
a piece-wise linear function with ` = log(1+) pieces and in time O(` log `) in which L is the ratio of
the maximum valuation to minimum non-zero valuation. Note that we need at least log2 L bits just
to represent such valuations so this construction is polynomial in the input size for any constant .
Proof. WLOG, assume that all possible non-zero valuations of the buyer are in the range of [1, L].
log L
Let ` = b log(1+)
c. For r = 0 · · · `, consider the prices pr = (1+)`−r and compute the corresponding
b 1+ (·) by constructing the convex hall of the points:
qr = 1 − F B (pr ). Construct H
(0, 0), (q1 , p1 q1 ), (q2 , p2 q2 ), · · · , (q` , p` q` ), (1, 0). This can be done in time O(` log `). Note that
p = F B (−1) (1 − q) is a decreasing function of q so at every point q ∈ [qr , qr+1 ], the corresponding
b
price is F B (−1) (q) ∈ [pr+1 , pr ] but pr = (1 + )pr+1 therefore at every point q: H1+ (q) ≤ H(q)
≤
(1 + )H1+ (q) which completes the proof.
b 1+ (·) in Mech. 4, we need to use (1 + )H
b 1+ (·) in the objective funcRemark 3. In order to use H
b
tion of the (CPrev−1 ) instead of H(·) for computing the benchmark. Furthermore, the mechanism
will be a (1 − )-approximation of the optimal primary IP mechanism. Also notice that finding p−
b 1+ (·) is trivial.
and p+ by using the H
18
5.3
Additive Independent Valuations with Hard Budget Constraints
We present a (1 − 1e )-approximate revenue maximizing primary IP mechanism for a buyer with
publicly known budget B who has independent and additive valuations for m items (i.e., her
valuation for a bundle of items is the sum of her valuations for individual items in the bundle). We
assume the buyer’s valuation for each item j is distributed independently according to a publicly
known distribution with CDF Fj (·). To avoid complicating the proofs, we assume that each Fj (·)
is continuous and strictly increasing in its domain14 . This primary mechanism can be used in the
γ-pre-rounding mechanism (Mech. 1) to yield a (1 − 1e )γk -approximate sequential posted pricing
mechanism for multiple buyers. The previous best approximation mechanism for this environment
was an O(1)-approximate15 sequential posted pricing mechanism by [BGGM10]. We should note
that the mechanism in [BGGM10] is slightly more general as it allows the buyers to have capacity
constraints as well.
As in subsection 5.2, we start by defining the modified CDF function FjB (·) for each item j as
follows.
(
Fj (v)
v≤B
FjB (v) =
(FjB )
B
1 − (1 − Fj (v)) v v ≥ B
b j (·) be its concave closure
Furthermore, for each item j, let Hj (q) = q · FjB −1 (1 − q) and let H
as in subsection 5.2. Similarly, for each j, define Rj (q̄j ) to be the optimal objective value of the
following convex program as a function of q̄j .
maximize:
b j (q)
H
(CPrev−j )
q ≤ q̄j
q≥0
The next theorem presents an upper bound on the revenue of the optimal primary IP mechanism.
P
Theorem 12. The revenue of the optimal primary IP mechanism is upper bounded by min( j Rj (q̄j ), B),
when the probability of allocating each item j is restricted to be no more than q̄j .
Proof. For any j if we were only to sell the item j, by Theorem 10, the maximum revenue we could
get using an IP primary mechanism would be Rj (q̄j ). Now observe that if we compute the optimal
price distribution for each item separately, we might only get less revenue because the budget is
shared among all items and the buyer might not be able to buy some of the items that she would
otherwise buy if there were no other items. That means the actually probability of allocating
each item j would then be less than q̄j . So the optimal joint price distribution might sell at lower
prices, but the extra revenue may only come from lower types which were originally excluded by
the optimal primary mechanism of each individual item. So the overall revenue from each item j
cannot be more than R(q̄P
j ). Finally, observe that the expected revenue of the mechanism cannot
be more that B, so min( j Rj (q̄j ), B) is an upper bound on the revenue of the optimal primary
IP mechanism subject to q̄.
14
The proofs can be modified to work without this assumption.
15 1
96
19
We construct a (1 − 1e )-approximate revenue maximizing primary IP mechanism as follows.
Mechanism 5.
• Define the benchmark R(q̄) = min(
P
j
Rj (q̄j ), B).
• Given q̄, solve the convex program of (CPrev−j ) for each item j. Let qj denote the optimal
assignment for the convex program of item j.
b j (qj ) = Hj (qj )
• Using the optimal q1 , · · · , qn compute the prices as follows: for each item j if H
then offer the single price pj = FjB (−1) (1 − qj ) otherwise randomize between two prices p−
j
and p+
with
probabilities
θ
and
1
−
θ
,
as
explained
in
the
proof
of
Theorem
10.
Note
that
j
j
j
the randomization is done for each item independently.
Theorem 13. Mech. 5 obtains at least 1 − 1e fraction of the revenue of the optimal primary IP
mechanism. Furthermore, this mechanism satisfies the requirements of γ-pre-rounding.
Proof. First, we show that Mech. 5 obtains at least 1 − 1e fraction of its benchmark R(q̄) which by
Theorem 12 is an upper bound on the revenue of the optimal primary IP mechanism. Consider
an imaginary replica of the original buyer who has exactly the same valuations as the original
buyer but has a separate budget B for each item. We call this imaginary buyer the “jumbo
replica”. Furthermore, suppose that any payment received from the jumbo replica beyond B is
lost (i.e., if the jumbo replica pays Z the mechanism receives only min(Z, B)). Observe that for
any assignment of prices, the payment received from the original buyer and the payment received
from the jumbo replica are exactly the same because if the original buyer has’t hit his budget
limit then both the original buyer and the jumbo replica will buy the same items and pay the
exact same amount. Otherwise, if the original buyer hits his budget limit, then the mechanism
receives exactly B from both the original buyer and the jumbo replica. So we only need to show
that the revenue received from the jumbo replica by using the price distribution of Mech. 5 is at
least 1 − 1e of R(q̄). Observe that from the view point of the jumbo replica there is no connection
between different items so he makes a decision for each item independently. Let Zj be the random
variable that denotes the amount paid by the jumbo replica for item j using the price distribution
of Mech. 5. By Theorem 10,
P we know that E[Zj ] = Rj (q̄j ) and the total revenue received by
the mechanism is Z = min( j Zj , B). Notice that Z1 , · · · , Zm are independentPrandom variables
in the range of [0, B]. By applying Lemma. 3, we can conclude that E[min( j Zj , B)] ≥ (1 −
P
1
1
e ) min( j E[Zj ], B) = (1 − e )R(q̄) which proves our claim. Next, we show that Mech. 5 satisfies
the requirements of γ-pre-rounding.
Observe that all Rj (·) are concave so R(q̄) is also concave.
P
Furthermore, R(q̄[S]) = min( j∈S Rj (q̄j ), B) is submodular in S, so it has a cross monotonic
budget balanced cost share scheme (see Def. 5). Therefore, Mech. 5 meets the requirements of
γ-pre-rounding.
Lemma 3. Let B be an arbitrary positive number and let Z1 , · · · , Zm be independent random
variables such that Zj ∈ [0, B], for all j. Then the following inequality holds.
X
E[min(
Zj , B)] ≥ (1 −
j
1
P
)B
( j E[Zj ])/B
e
Proof. See section A.
20
X
1
≥ (1 − ) min(
E[Zj ], B)
e
j
5.4
Unit Demand Independent Valuations
We present a 21 -approximate revenue maximizing primary IP mechanism for a unit demand buyer
with independent valuations for m items. We assume that for each item j, the buyer’s valuation
is distributed independently according to a publicly known distribution with CDF Fj (·). To avoid
complicating the proofs, we assume that each Fj (·) is continuous and strictly increasing in its domain. Furthermore, we require the distributions to be regular. This primary mechanism can be
used in the γ-pre-rounding mechanism (Mech. 1) to yield a 12 γk -approximate sequential posted pricing mechanism for many buyers. The previous best approximation mechanism for this environment
1
was a 6.75
-approximate sequential posted pricing mechanism by [CHMS10]16 .
We start by defining Hj (q) = q · Fj−1 (1 − q) for each item j. Because Fj (·) is the CDF of a
regular distribution the function Hj (·) is concave as stated in the following lemma.
Lemma 4. If F (·) is the CDF of a regular distribution then the function H(q) = q · F −1 (1 − q) is
concave.
∂
H(q) is non-increasing in q.
Proof. To show that H(q) is concave it is enough to show that ∂q
q
∂
−1
But ∂q H(q) = F (1 − q) − f (F −1 (1−q)) in which f (·) is the derivative of F (·). If we substitute
q = 1 − F (p), then it is enough to show that the resulting function is non-decreasing in p because
(p)
∂
H(q) = p − 1−F
q is itself non-increasing in p. However, by this substitution we get ∂q
f (p) which is
non-decreasing in p by definition of regularity.
Intuitively, Hj (qj ) is the maximum revenue that can be obtained by a mechanism that allocates
item j with probability qj . Next, we show that the revenue of the optimal primary IP mechanism
when restricted to allocate the items with probabilities at most q̄1 , · · · , q̄m is no more than the
optimal objective value of the following convex program:
X
Hj (qj )
maximize:
(CPrev−u )
j
∀j :
X
qj ≤ q̄j
(λj )
qj ≤ 1
(τ )
qj ≥ 0
(µj )
j
∀j :
Theorem 14. The revenue of the optimal primary IP mechanism, when restricted to allocate the
items with probabilities at most q̄1 , · · · , q̄m , is upper bounded by the optimal objective value of the
convex program (CPrev−u ).
∗ be the probabilities of allocating the items to the buyer by the optimal
Proof. Let q1∗ , · · · , qm
primary IP mechanism. For each item j, the expected revenue that can be obtained if item j is
b j (q ∗ ) (the proof of this claim is essentially
allocated with probability qj∗ is upper bounded by H
j
the same as the proof of Theorem 10).
Therefore,
the
expected
revenue of the optimal primary
P b ∗
IP mechanism cannot be more that j H
(q
).
Furthermore,
observe
that q ∗ is a feasible solution
j j
for the convex program
P ∗ (CPrev−u ) because the optimal primary ∗mechanism never allocates more
than one item so j qj ≤ 1 and also for each j it must be that qj ≤ q̄j . Therefore, the revenue of
the optimal primary IP mechanism cannot be more that the optimal objective value of the convex
program.
16
The corresponding mechanism in [CHMS10] does not work for non-regular distributions either.
21
We construct a 12 -approximate revenue maximizing primary IP mechanism as follows.
Mechanism 6.
• Define the benchmark R(q̄) to be the optimal objective value of (CPrev−u ) as a function of q̄.
• Given q̄, solve the convex program of (CPrev−u ) and let q denote the optimal assignment of
the convex program.
• For each item j: assign the price pj = Fj−1 (1 − qj ). WLOG, assume that items are labeled
such that p1 ≤ · · · ≤ pm .
• For each item j, define rj = max(qj pj + (1 − qj )rj+1 , rj+1 ) and let rm+1 = 0. Let T be the
subset of items defined as follows: T = {j|pj ≥ rj }. Only offer the items in T at prices
computed in the previous step (i.e., set the price of other items to infinity).
Theorem 15. Mech. 6 obtains at least 21 fraction of the revenue of the optimal primary IP mechanism. Furthermore, this mechanism satisfies the requirements of γ-pre-rounding.
Proof. First, we show that Mech. 5 obtains at least 12 fraction of its benchmark R(q̄) which by
Theorem 14 is an upper P
bound on the revenue of the optimal primary IP mechanism. Observe
that in Mech. 6, R(q̄) = j qj pj and notice that qj is exactly the probability that the valuation
of the buyer for item j is at least pj . Now consider a replica of the buyer which has the exact
same valuations as the original buyer but always buys the item that has the lowest price among the
items that are priced below her valuations. We call this imaginary replica the “malevolent replica”.
Notice that for any assignment of prices, the revenue obtained from the malevolent replica is a
lower bound on the revenue obtained fromP
the original buyer. So it is enough to show that the
mechanism obtains a revenue of at least 21 j qj pj from the malevolent replica. Observe that rj ,
as defined in Mech. 6, is exactly the expected revenue obtained from the malevolent replica when
offered the items in T ∩ {j, · · · , m}. Basically, the item j is offered to the buyer (i.e., j ∈ T ) if
pj ≥ rj which also implies that rj = qj pj + (1 − qj )rj+1 , otherwise item j is not offered at all and
rj = rj+1 . Note that if we only offer the items in T ∩ {j, · · · , m} and if j ∈ T then the malevolent
replica buys item j with probability qj and generates a revenue of pj , otherwise with probability
1 − qj an expected revenue of rj+1 is obtained from items in T ∩ {j + 1, · · · , m}. Now observe that
the expected revenue obtained fromPthe malevolent replica by Mech. 6 is exactly r1 . By applying
Lemma. 5 we conclude that r1 ≥ 21 j pj qj which completes the proof of the first claim.
Next, we show that this mechanism satisfies the requirements of γ-pre-rounding. Observe that
by Lemma. 1, the optimal objective value of (CPrev−u ) is a concave function of q̄1 , · · · , q̄m so the
benchmark function R(q̄) is concave. It only remains to show that R(q̄) has a budget balanced
cross monotonic cost sharing scheme. Define the cost share function ξ(j, q̄) = Hj (qj (q̄)) in which
qj (q̄) is the optimal assignment of variable qj as a function of q̄ in the convex program (CPrev−u ).
We shall show that ξ is budget balanced and cross monotonic (see Def. 5).
m
• To show that
P ξ is budget balanced, we must show that for any q̄ ∈ [0, 1] and S ⊂ [m],
R(q̄[S]) =
j∈S ξ(j, q̄[S]). Note that
P R(q̄[S]) is defined as the optimal objective value of
(CPrev−u ) which means R(q̄[S]) = j Hj (qj (q̄[S])). However, for any j 6∈ S, it must be that
qj (q̄[S]) = 0 which implies that Hj (qj (q̄[S])) = 0. So the claim follows.
• To show that ξ is cross monotonic, we must show that for any q̄ ∈ [0, 1]m and S, T ⊂ [m],
ξ(j, q̄[S]) ≥ ξ(j, q̄[S ∪ T ]). Let L(q, λ, τ, µ), as defined below, be the Lagrangian of (CPrev−u ).
X
X
X
X
L(q, λ, τ, µ) = −
Hj (qj ) +
λj · (qj − q̄j ) + τ · (
qj − 1) −
µj qj
j
j
j
22
j
By KKT stationarity conditions, at the optimal assignment the following holds:
∂
∂
L(q, λ, τ, µ) = −
Hj (qj ) + λj + τ − µj = 0
∂qj
∂qj
Note that (a) all dual variables must be non-negative, (b) by complementarity slackness λj
may be non-zero only if qj = q̄j , and (c) complementarity slackness implies that µj may
be non-zero only if qj = 0. Therefore, if τ is given, then the optimal assignment for qj
is uniquely17 determined by the above equation and the aforementioned complementarity
slackness conditions. Let qj (τ ) denote the optimal assignment of qj as a function of τ . Given
the concavity of Hj (·), it is easy to verify that qj (τ ) is a non-increasing function of τ .
We prove that ξ is cross monotonic by contradiction. Suppose ξ is not cross monotonic, i.e.
ξ(j, q̄[S ∪ T ]) > ξ(j, q̄[S]) for some j = j ∗ . Let τ (q̄) denote the optimal assignment of τ as a
function of q̄. Remember that we defined ξ(j, q̄) = Hj (qj (q̄)) = Hj (qj (τ (q̄))) and Hj (qj (τ ))
is a non-increasing function of τ . So it must be that qj ∗ (τ (q̄[S ∪ T ])) > qj ∗ (τ (q̄[S])), and
consequently P
τ (q̄[S ∪ T ]) < τ (q̄[S]). But then τ (q̄[S]) > 0 and complementary slackness
implies that j qj (τ (q̄[S])) = 1. Observe that because qj (τ ) is a non-increasing function of
τ , it must be P
that for all j, qj (τ (q̄[S ∪ T ])) ≥ qj (τ (q̄[S])) and the inequity is strict for j = j ∗
which means j qj (τ (q̄[S ∪ T ])) > 1 which is a contradiction.
Lemma
P 5. Let p1 , · · · , pm and q1 , · · · , qm be two sequences of non-negative real numbers and suppose j qj ≤ 1. For each j ∈ [m] define rj = max(qj pj + (1 − qj )rj+1 , rj+1 ) and let rm+1 = 0.
P
Then r1 ≥ 21 j qj pj .
Proof. See section A.
6
Analysis of The γ-Conservative Magician
In this section, we present the proof of Theorem 2. We prove the theorem in two parts. First we
show that, assuming the γ-conservative magician has infinitely many magic wands, the dynamic
programming based strategy of Alg. 1 indeed guarantees that each box is opened with an ex-ante
probability at least γ. The second part, which is the challenging part, is to show that for any
1
γ ≤ 1 − √k+3
the strategy table of the γ-conservative magician never instructs him to open a box if
1
he has broken all of his k magic wands. In other words, we show that if γ ≤ 1 − √k+3
then yij = 0
for every j ≥
√ k and every i. We should emphasis that it is not possible to get a bound of better
than 1 − O( √lnkk ) by using ordinary concentration bounds or martingale inequalities. Furthermore,
in Theorem 3 we show that it is not possible to guarantee that each box is opened with probability
1
1
of 1 − √2πk
+ or better, so the bound of 1 − √k+3
is not far from optimal.
Part 1: We show that if the γ-conservative magician has infinitely many magic wands, then the
strategy table of Alg. 1 indeed ensures that each box is opened with probability at least γ. Below,
17
To avoid complicating the proof, we assume that Hj (·) are strictly concave. The proof can be modified to work
without this assumption.
23
we repeat the dynamic program for computing the strategy table.


i ≥ 1, 0 ≤ j < θi
1
j
θi −1
θi
θi −1
yi = (γ − φi )/(φi − φi )
i ≥ 1, j = θi


0
otherwise.
θi = min{j|φji ≥ γ}


1
j
j
j−1
j
φi = yi−1
qi−1 φi−1
+ (1 − yi−1
qi−1 )φji−1


0
Part 1(a):
For i > 1:
(DP.y)
(DP.θ)
i = 1, j ≥ 0
i ≥ 2, j ≥ 0
otherwise.
(DP.φ)
First we prove that P r[Si ≤ j] ≥ φji by induction on i. The base case i = 1 is trivial.
j
P r[Si ≤ j] ≥ P r[Si−1 ≤ j − 1] + P r[Si−1 = j](1 − yi−1
qi−1 )
j
j
= P r[Si−1 ≤ j − 1]yi−1
qi−1 + P r[Si−1 ≤ j](1 − yi−1
qi−1 )
j
j
j
≥ φj−1
i−1 yi−1 qi−1 + φi−1 (1 − yi−1 qi−1 )
=
φji
by induction hypothesis
by (DP.φ)
(1.a)
Observe that all of the above inequalities are met with equality if each qi is the exact probability
of breaking a magic wands for box i instead of being just an upper bound on it.
Next, we show that P r[Yi = 1] ≥ γ:
X
P r[Yi = 1] =
P r[Yi = 1|Si = j]P r[Si = j]
Part 1(b):
j
=
θi
X
yij P r[Si = j]
j=0
because yij = 1 for j < θi
= P r[Si < θi ] + yiθi P r[Si = θi ]
= (1 − yiθi )P r[Si < θi ] + yiθi P r[Si ≤ θi ]
≥ (1 − yiθi )φiθi −1 + yiθi φθi i
by (1.a)
= φiθi −1 + yiθi (φθi i − φiθi −1 )
by substituting yiθi from (DP.y)
=γ
Observe that all of the above inequalities are met with equality if each qi is the exact probability
of breaking a magic wand for box i instead of being just an upper bound on it.
1
, the γ-conservative magician never tries to open
Part 2: Next, we show that when γ ≤ 1 − √k+3
a box after breaking k magic wands. In other words, we show that in the strategy table computed
by the dynamic program (DP.y), yij = 0 for every j ≥ k and every i.
Instead of proving the claim directly, we proceed as follows: (a) we present a closely related
stochastic process on an infinite tape with one unit of infinitely divisible sand and a barrier, (b) we
establish the connection between this process and the strategy table of the magician, and (c) we
24
prove a theorem about this stochastic process and use it to prove second part of the theorem. The
process is as follows.
Process 1 (Sand/Barrier).
• A parameter
γ ∈ (0, 1) and a sequence of probabilities q1 , · · · , qn are given as the input, such
P
that i qi ≤ k.
• There are (a) a tape of infinite length, (b) one unit of infinitely divisible sand, and (c) a
barrier. Initially, the barrier is at position 1 on the tape and all the sand is at position 0 (on
the left of the barrier). During the process, the sand and the barrier are gradually moved to
the right, distributing the sand over the tape, but never crossing the barrier. The barrier is
moved one position to the right whenever the amount of sand at the position of the barrier
increases beyond 1 − γ. The process runs in n rounds. For each i, let λi denote the position
of the barrier at the beginning of the round i, and for each j let sji denote the amount of sand
at position j at the beginning of round i. So, λ1 = 1, s01 = 1 and sj1 = 0 for all j > 0. During
each round i we do the following:
– We select a fraction of the sand from each position to the left of the barrier such that the
total amount of selected sand is exactly γ. Note that we can always do this because the
amount of sand at the position of the barrier is no more than 1 − γ, so the rest of the
sand must be to the left of the barrier. Let yij ∈ [0, 1] denote the fraction of the sand from
position j that gets selected. We start selecting all the sand greedily from P
left to right
j
until the total amount selected is γ. In other words, we choose yi such that j yij sji = γ
and such that for some index θi we have yij = 1 for all j < θi , and yij = 0 for all j > θi .
– We then move qi fraction of the selected sand as follows. For each j we move qi fraction
of the sand that was selected from position j to position j + 1, we do this simultaneously
for all positions (i.e. the amount of sand that is moved to j + 1 is exactly yij sji qi ).
– If the total amount of sand at the position of the barrier is more than 1 − γ, then we
move the barrier one position to the right (i.e., to position λi + 1).
P
Let φji = jr=0 sri be the total amount of sand in positions 0, · · · , j at the beginning of round
i. Observe that by the above process we have:
j
j
j−1
φji = φj−1
i−1 + (1 − yi−1 qi−1 )(φi−1 − φi−1 )
j
j
j
= yi−1
qi−1 φj−1
i−1 + (1 − yi−1 qi−1 )φi−1
Notice that the state of the process defined above can be computed using exactly the same
dynamic program that was used to compute the strategy table of the γ-conservative magician.
Furthermore, in order to show that yij = 0 for every j ≥ k and every i, it is enough to show that
the barrier is never moved past position k of the tape. Next, we present a theorem which is the
main step of the proof and is also interesting on its own.
Theorem 16 (Sand/Barrier). Throughout the process defined in Proc. 1, the average distance of
1
the sand from the barrier is strictly less than 1−γ
. In particular, at the beginning of round i this
λi
1−γ
distance is strictly less
P than 1−γ , and this is true for any sequence of probabilities q1 , · · · , qn ,
regardless of how big i qi is.
25
We use the above theorem to derive a sufficient condition on γ that ensures the barrier is never
moved past position k. At the beginning of round i, let di denote the average distance of the sand
from the origin and d0i denote the average distance of the sand from the barrier. Observe that
λi = di + d0i . Furthermore, notice that di = γqi−1 + di−1 , in other words, the average distance of
the sand from the origin is increased exactly by γqi−1 during round i − 1 (because the amount of
selected sand is exactly γ and qi−1 fraction of the selected sand is moved one position to the right).
This can be further combined with Theorem 16 to get the following inequality:
λi = di + d0i <
i−1
X
qr γ +
r=1
1 − γ λi
1 − γ λi
≤ kγ +
1−γ
1−γ
In order to show that the barrier is never moved past position k, it is enough to show that the
above inequality cannot hold for λi = k + 1. Equivalently, it is enough to prove that the following
inequality always holds:
k + 1 ≥ kγ +
1 − γ k+1
1−γ
(Λ)
1
Instead of the above inequality, we can consider the stronger inequality k + 1 ≥ kγ + 1−γ
which
√1
is quadratic in γ and can be solved to get a bound of γ ≤ 1 −
. This bound is in fact
1/2+
a weaker constraint than γ ≤ 1 −
√1
k+3
k+1/4
when k ≥ 7. It can also be verified that for k < 7 and
1
1
the inequality (Λ) holds. So we have proved that for γ ≤ 1 − √k+3
, the barrier is
γ ≤ 1 − √k+3
never moved past position k, which implies that the γ-conservative magician never tries to open a
box after breaking k magic wands. That completes the proof of Theorem 2.
Next, we present the proof of the sand theorem.
Proof of Theorem 16. First, we show that throughout the process of Proc. 1, the following invariant
holds:
∀i, ∀j ∈ [1, λi − 1] :
φj−1
< γφji
i
(φineq )
We prove this by induction on i. The base case of the induction is trivial. For i ≥ 2:
j
j
j−1
φji = φj−1
i−1 + (1 − yi−1 qi−1 )(φi−1 − φi−1 )
j−1
j−1
≥ φi−1
+ (1 − yi−1
qi−1 )(φji−1 − φj−1
i−1 )
because yij is non-increasing in j
j−1
j−1
j
= yi−1
qi−1 φj−1
i−1 + (1 − yi−1 qi−1 )φi−1
1
1
j−1
j−1
> yi−1
qi−1 φj−2
+ (1 − yi−1
qi−1 ) φj−1
γ i−1
γ i−1
1
= φij−1
γ
by induction hypothesis
by (DP.φ)
We also need to consider the case in which the barrier is moved. Suppose the barrier is moved at
the end of round i, i.e. λi+1 = λi + 1, then we must show that the invariant also holds for position
j = λi . Notice that the barrier is moved only if the amount of sand at the position of the barrier is
more than 1 − γ. So the total sand to the left of the barrier, just before it was moved, must have
26
i −1
i
been strictly less than γ which means φλi+1
< γ, furthermore φλi+1
= 1 because the barrier just
λi −1
λi
moved to position λi + 1. So φi+1 < γφi+1 .
Now, we prove the main claim of the theorem. Let d0i denote the average distance of the sand
from the barrier at the beginning of round i, then:
d0i
=
λX
i −1
φji
sand at position j is counted exactly λi − j times
j=0
<
λX
i −1
γ λi −1−j φλi i −1
by (φineq )
j=0
≤
λX
i −1
because φλi i −1 ≤ 1
γ λi −1−j
j=0
=
7
1 − γ λi
1−γ
When Buyers Need More Than One Copy of Each Item
In this section, we show that the more general model, in which each buyer may need more than
one copy of each item, but no more than k1 of all copies of an item, can be reduced to the simpler
model in which there are at least k copies of every item and no one demands more than 1 copy of
each item.
Definition 8 (Multi-Demand Market Transformation). Let kj denote the number of copies of item
k
j. Define cj = b kj c and divide the copies of item j almost equally into cj bins (i.e., each bin will
contain either cj or cj + 1 copies). Create a new item type for each bin (i.e., items from the same
bin has the same type, but items from different bins are treated as different types of item).
Theorem 17. Let M be the space of feasible mechanisms, in the original (multi-demand) market,
which do not allocate more than k1 of all copies of each item to any single buyer. Similarly, let
M(1) be the space of feasible mechanisms, in the transformed market, which do not allocate more
than one copy of each item to any single buyer. Then, any mechanism in M can be interpreted as a
mechanism in M(1) and vice-versa with the same actual allocations/payments. Therefore, in order
to find the optimal mechanism in the original market, it is enough to find the optimal mechanism
in the transformed market.
Proof. First, we show that any mechanism in M ∈ M(1) can be interpreted as a mechanism in M.
That is trivially true because M allocates to each buyer at most one copy from each bin, which is
at most cj copies of each item j of the original market, which is no more than k1 of all copies of
item j.
Next, we show that any mechanism any M ∈ M can be interpreted as a mechanism in M(1) .
We proceed as follows. For every j, we create a list Lj of all the bins of item j. Lj is initially sorted
in decreasing order of the size of the bins. Let XijM be the number of copies of item j allocated to
buyer i by M. We specify the allocations in the transformed market as follows. For each buyer i
we repeat the following XijM times: Allocate one copy from the bin that is in the front of the list
Lj and then move the bin back to the end of the list. It is easy to see that no two copies from the
same bin are allocated to the same buyer.
27
Note that by Theorem 17, any mechanism in the original market is equivalent to a mechanism in
the transformed market with the exact same allocations/payments from the perspective of buyers.
So WLOG, we can work with the transformed market and only consider the mechanisms in this
market. However, note that to use our generic mechanisms in the transformed market, the underlying primary mechanisms should be capable of handling correlated valuations, because copies of
the same item, even when labeled with different types, are perfect substitutes from the view point
of a buyer. Among the primary mechanisms presented in section 5, only Mech. 3 (subsection 5.1)
can handle correlated valuations.
8
Conclusion
In this paper, for Bayesian combinatorial auctions, we presented an approximate reduction from
n-buyer mechanisms to 1-buyer mechanisms, which leads to the following conclusions.
• The Important Parameter: As the ratio of the maximum demand to supply (i.e., k1 ) decreases,
less coordination is required on decisions made by the mechanism for different buyers. In other
words, as k1 → 0, the optimal mechanism treats each buyer almost independently of other
buyers. Observe that all of the approximation factors in this paper only depend on k (i.e., γk )
and not on n. This suggests that for characterizing the properties of such markets, the right
parameter to consider is perhaps the ratio of the maximum demand to supply. In particular,
notice that the number of buyers is irrelevant.
• Computational Hardness: In mechanism design, for a variety of settings, the difficulty of
making coordinated optimal decisions for buyers can be avoided by losing a small constant
1
factor of the objective (i.e., √k+3
), therefore the main difficulty of constructing constant
factor approximation mechanisms in multi-dimensional settings stems from the difficulty of
designing single buyer mechanisms, which ultimately stems from the IC constraints in the
single buyer problem.
9
Acknowledgment
I thank Jason Hartline for many helpful suggestions and comments. I also thank Azarakhsh
Malekian for a fruitful discussion that lead to the proof of the sand theorem (Theorem 16).
References
[BGGM10] Sayan Bhattacharya, Gagan Goel, Sreenivas Gollapudi, and Kamesh Munagala. Budget constrained auctions with heterogeneous items. In STOC, pages 379–388, 2010.
[BH08]
Liad Blumrosen and Thomas Holenstein. Posted prices vs. negotiations: an asymptotic
analysis. In ACM Conference on Electronic Commerce, page 49, 2008.
[BLP06]
Moshe Babaioff, Ron Lavi, and Elan Pavlov. Single-value combinatorial auctions and
implementation in undominated strategies. In SODA, pages 1054–1063, 2006.
[BR89]
Jeremy Bulow and John Roberts. The simple economics of optimal auctions. Journal
of Political Economy, 97(5):1060–90, October 1989.
28
[CEDG+ 10] Tanmoy Chakraborty, Eyal Even-Dar, Sudipto Guha, Yishay Mansour, and
S. Muthukrishnan. Approximation schemes for sequential posted pricing in multiunit auctions. In WINE, 2010.
[CHMS10]
Shuchi Chawla, Jason D. Hartline, David L. Malec, and Balasubramanian Sivan.
Multi-parameter mechanism design and sequential posted pricing. In STOC, pages
311–320, 2010.
[CMM11]
Shuchi Chawla, David Malec, and Azarakhsh Malekian. Bayesian mechanism design
for budget-constrained agents. In ACM Conference on Electronic Commerce, 2011.
[DRY10]
Peerapong Dhangwatnotai, Tim Roughgarden, and Qiqi Yan. Revenue maximization
with a single sample. In ACM Conference on Electronic Commerce, pages 129–138,
2010.
[GKS02]
Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy 0002, and Aravind Srinivasan.
Dependent rounding in bipartite graphs. In FOCS, pages 323–332, 2002.
[HK92]
Theodore P. Hill and Robert P. Kertz. A survey of prophet inequalities in optimal
stopping theory. In Contemporary Mathematics, 1992.
[HKM11]
Jason D. Hartline, Robert Kleinberg, and Azarakhsh Malekian. Bayesian incentive
compatibility and matchings. In SODA, 2011.
[HKS07]
Mohammad Taghi Hajiaghayi, Robert D. Kleinberg, and Tuomas Sandholm. Automated online mechanism design and prophet inequalities. In AAAI, pages 58–65,
2007.
[HL10]
Jason D. Hartline and Brendan Lucier. Bayesian algorithmic mechanism design. In
STOC, pages 301–310, 2010.
[HR09]
Jason D. Hartline and Tim Roughgarden. Simple versus optimal mechanisms. In ACM
Conference on Electronic Commerce, pages 225–234, 2009.
[Mye81]
Roger B. Myerson. Optimal auction design. 1981.
[Yan11]
Qiqi Yan. Mechanism design via correlation gap. In SODA, 2011.
A
Missing Proofs
Proof of Theorem 3. Suppose we create n boxes and in each box, independently, we put $1 with
probability nk . If the magician opens a box containing a $1 then he gets the $1 but we break his
wand (i.e., qi = nk ). Observe that the expected total prize is k dollars but because we put a dollar
in each box independently, there are some instances in which there are more than k non-empty
boxes but the magician cannot win more than k dollars at any instance. Let Xi be the
Pindicator
random variable which is 1 iff there is a dollar in box i. The expected
total
prize
is
E[
i Xi ] = k
P
but the expected prize that the magician can win is at most E[min( i Xi , k)]. It can be verified
P
k
that E[min( i Xi , k)] ≈ (1 − ekk k! ) · k asymptotically as n → ∞. In fact, for any positive , there is
P
k
a large enough n such that E[min( i Xi , k)] < (1 − ekk k! + ) · k. On the other hand, observe that if
a magician can guarantee that every box is opened with probability at least γ = 1 −
29
kk
ek k!
+ , then
the magician will win at least
can make such a guarantee.
P
i γE[Xi ]
k
= (1 − ekk k! + ) · k in expectation. Therefore no magician
Proof of Theorem 7. First, we show that the expected objective value of Mech. 1 is at least γ · αfraction of the expected objective value of the optimal mechanism in M. Note that by Theorem 5,
the expected objective
P value of the optimal mechanism in M is upper bounded by the optimal value
of (CPR ) which is i Ri (q̄i ). Therefore, it is enough to show that the γ-pre-rounding mechanism
obtains an expected objective value of at least γαRi (q̄i ) from each buyer i. Let Si = {j|Yij = 1},
where Yij is the indicator random variable corresponding to the decision of the j th magician on
the ith box presented to that magician, as defined in Mech. 1. Note that Si is the subset of items
available to Mi (q̄i0 ), i.e., q̄i0 = q̄i [Si ]. Given that γ ∈ [0, γk ], we show that each item j ∈ [m] is
included in Si with probability at least γ. We then show that Mi (q̄i [Si ]) obtains an objective value
of at least γαRi (q̄i ) in expectation. Observe that for each item j, a sequence of n boxes with magic
wand breakage probability upper bounds q̄1j , · · · q̄nj are presented to the j th magician. By applying
Theorem 2, given that γ ∈ [0, γk ], each magician guarantees that each box is opened with probability
at least γ; which means P r[Yij = 1] ≥ γ, or equivalently P r[j ∈ Si ] ≥ γ. Observe that Mi (q̄i [Si ])
obtains αE[Ri (q̄i [Si ])] in expectation. So it is enough to show that E[Ri (q̄i [Si ])] ≥ γRi (q̄i ). Let ξi
be a budget balanced cross monotonic cost share function for Ri (·). Then:
E[Ri (q̄i [Si ])] = E
X
ξi (j, q̄i [Si ])
because ξi is budget balanced
ξi (j, q̄i [{1, · · · , m}])
because ξi is cross monotonic
j∈Si
≥E
X
j∈Si
=
X
P r[j ∈ Si ] · ξi (j, q̄i )
j∈[m]
≥
X
γξi (j, q̄i )
j∈[m]
= γRi (q̄i )
because ξi is budget balanced
Next, we show that Mech. 1 is in M and is dominant strategy incentive compatible (DSIC). The
fact that this mechanism is in M follows from assumption A6 and the fact that for each item j, the
corresponding magician breaks no more than kj wands, which means no more than kj copies are
allocated at any instance. Note that the outcome of Mi (q̄i0 ) is the final outcome for buyer i. Also
note that Mi (q̄i0 ) is a mechanism from Mi , for any q̄i0 , and therefore it must be incentive compatible,
because Mi only includes incentive compatible single buyer mechanisms for buyer i (this follows
from assumption A4, and the definition of Mi ). Also note that the only way the reports of other
buyers could affect the outcome of buyer i is by affecting q̄i0 , and yet Mi (q̄i0 ) is truthful for any
choice of q̄i0 . Therefore Mech. 1 is DSIC. Observe that this mechanism also preserves all of the
ex-post properties of each Mi (e.g., if Mi is ex-post individually rational, then so is the final
mechanism).
Proof of Theorem 8. First, we show that the expected objective value of Mech. 2 at least γ · αfraction of the expected objective value of the optimal mechanism in M. Note that by Theorem 5,
the expected objective
P value of the optimal mechanism in M is upper bounded by the optimal value
of (CPR ) which is i Ri (q̄i ). Therefore, it is enough to show that the γ-post-rounding mechanism
obtains an expected objective value of at least γαRi (q̄i ) from each buyer i. Let Si = {j|Yij = 1},
where Yij is the indicator random variable corresponding to the decision of the j th magician on
30
the ith box presented to that magician, as defined in Mech. 2. Note that Si is the subset of items
that would not be deallocated from buyer i, i.e., Xi0 = Xi [Si ], where Xi and Xi0 are as defined in
Mech. 2. In particular, Si does not depend on the report of buyer i, and is independent of Xi and
Pi . Given that γ ∈ [0, γk ], we show that each item j ∈ [m] is included in Si with probability exactly
γ. We then show that the mechanism obtains an expected objective value of at least γαRi (q̄i )
from each buyer i. Observe that for each item j, a sequence of n boxes with magic wand breakage
probabilities q̂1j , · · · q̂nj are presented to the j th magician. By applying Theorem 2, given that
γ ∈ [0, γk ], each magician guarantees that each box is opened with probability exactly γ; which
means P r[Yij = 1] = γ, or equivalently P r[j ∈ Si ] = γ. Observe that Mi (q̄i ) would obtain an
expected objective value of E[Wi (ti , Xi , Pi )] which would be at least αRi (q̄i ). Therefore, to show
that the final mechanism obtains an expected objective value of γαRi (q̄i ), it is enough to show that
E[Wi (ti , Xi0 , γPi )] ≥ γE[Wi (ti , Xi , Pi )]. Let ξi be a budget balanced cross monotonic cost share
function for Wi as required by A0 2. Then:
E[Wi (ti , Xi0 , γPi )] = E[Wi (ti , Xi [Si ], 0) + γPi ]
X
= E[
ξi (j, ti , Xi [Si ])] + γE[Pi ]
By A0 2
because ξi is budget balanced
j∈Si
≥ E[
X
ξi (j, ti , Xi [{1, · · · , m}])] + γE[Pi ]
because ξi is cross monotonic
j∈Si
=
X
P r[j ∈ Si ] · E[ξi (j, ti , Xi )] + γE[Pi ]
j∈[m]
= γE[
X
ξi (j, ti , Xi ) + Pi ]
j∈[m]
= γE[Wi (ti , Xi , Pi )]
because ξi is budget balanced
Next, we show that Mech. 2 is Bayesian incentive compatible (BIC) and does not over allocate
any item. Consider any arbitrary buyer i. As stated in the first part of the proof, the final allocation
for buyer i is Xi0 = Xi [Si ], where Si = {j|Yij = 1}. We showed that Si is a random subset of items
that includes each item j with a marginal probability of exactly γ. Furthermore, by A0 4, valuations
of buyer i can be interpreted as a weighted rank function of a matroid. So WLOG we may assume
that Xi specifies an independent set of this matroid (otherwise we could modify Xi by deallocating
some of the items to make it an independent set without decreasing the valuation of the buyer).
Observe that when Xi specifies an independent set, then the valuation of the buyer for the items
specified by Xi are additive. Consequently, the expected valuation of the buyer i for Xi0 = Xi [Si ] is
exactly γ times her expected valuation for Xi . Observe that both the expected valuation and the
expected payment of buyer i are multiplied by γ for any outcome of Mi (q̄i ). Therefore, the resulting
mechanism is incentive compatible because Mi (q̄i ) is in Mi and is therefore incentive compatible.
However, the final mechanism is only Bayesian incentive compatible because Si depends on the
reports of buyers other that i, and P r[j ∈ Si ] = γ only in expectation over other buyers’ reports.
Also note that the mechanism does not over allocate any item, because for each copy of item j
being allocated one of the kj wands of the j th magician breaks.
Proof of Lemma. 1. The proof is very similar to the proof of Theorem 6. Let R(q̄) denote the
optimal value of the convex program as a function of q̄. To show that R(q̄) is concave, it is
enough to show that for any q̄ and q̄ 0 and any α ∈ [0, 1] if we define q̄ 00 = αq̄ + (1 − α)q̄ 0 , then
31
R(q̄ 00 ) ≥ αR(q̄) + (1 − α)R(q̄ 0 ). Let x and x0 be the optimal assignments for the program subject to
q̄ and subject to q̄ 0 respectively. It is easy to verify that x00 = αx + (1 − α)x0 is a feasible assignment
for the convex program subject to q̄ 00 . Therefore, R(q̄ 00 ) must be at least u(x00 ). On the other hand
u(·) is a concave function, so u(x00 ) ≥ αu(x) + (1 − α)u(x0 ) = αR(q̄) + (1 − α)R(q̄ 0 ). That proves
the claim.
P
Proof of Lemma. 3. Let µ = j E[Zj ]. Define the random variables Wj = max(Wj−1 − Zj , 0) and
P
P
W0 = B. Observe that for each j, Wj = max(B − jr=1 Zr , 0) so min( jr=1 Zr , B) + Wj = B.
P
Therefore E[min( jr=1 Zr , B)] + E[Wj ] = B and to prove the theorem it is enough to show that
1
E[Wm ] ≤ eµ/B
· B. To show this we will prove the following inequality:
E[Zj ]
)Wj−1
B
Wj ≤ (1 −
(Wj )
Assuming that (Wj ) is true, we can conclude the following which proves the claim.
Wm ≤ B ·
m
Y
(1 −
j=1
≤B·
1
eµ/B
The last inequality follows from the fact that
E[Zj ]
B
E[Zj ]
)
B
P
j
E[Zj ]
B
=
µ
B,
therefore the right hand side takes
µ
mB
its maximum when for all j :
=
and m → ∞.
To prove the second inequality in the statement of the lemma we can use the fact that (1−xa ) ≥
1
1
(1 − x)a for any a ≤ 1 and conclude that (1 − eµ/B
)B ≥ (1 − emin(µ,B)/B
)B ≥ (1 − 1e ) min(µ,B)
B=
B
1
(1 − e ) min(µ, B).
At the end, we prove the inequality (Wj ) as follows.
E[Wj ] = E[max(Wj−1 − Zj , 0)]
Wj−1
≤ E[max(Wj−1 − Zj
, 0)]
B
Wj−1
= E[Wj−1 − Zj
]
B
1
= E[Wj−1 ] − E[Zj Wj−1 ]
B
1
≤ E[Wj−1 ] − E[Zj ]E[Wj−1 ]
B
E[Zj ]
= (1 −
)E[Wj−1 ]
B
Wj−1
≤1
B
Zj
because
≤1
B
because
because Zj and Wj−1 are independent.
That completes the proof.
Proof of Lemma. 5. To prove the claim, it is enough to show that P rq1j pj ≥ 12 . WLOG, we may
j
P
assume that j pj qj = 1 since we can scale p1 , · · · , pm by a constant c = P 1qj pj and this will also
j
scale r1 , · · · , rm by the same constant c so the ratio P rq1j pj will not be affected. Now consider the
j
following LP and observe that qj , pj , and rj , as defined in the statement of the lemma, form a
feasible assignment for this LP. If we show that the optimal objective value of the LP is bounded
32
below by
therefore
1
2 , then it means any feasible assignment
P r1
≥ 12 . In the following LP, pj and rj
j qj pj
yields an objective value of at least 21 , and
are variables and everything else is constant.
minimize:
r1
∀j ∈ {1 · · · m} :
rj ≥ qj pj + (1 − qj )rj+1
(αj )
rj ≥ rj+1
(βj )
∀j ∈ {1 · · · m} :
X
qj p j ≥ 1
(γ)
j
pj ≥ 0,
rj ≥ 0
To prove that the optimal objective value of the above LP is bounded below by 21 , we construct
a feasible solution for its dual that obtains an objective value of 12 . The following is the dual LP.
maximize:
γ
∀j ∈ {1 · · · m} :
γ ≤ αj
∀j ∈ {2 · · · m} :
(pj )
α1 + β1 ≤ 1
(r1 )
αj + βj ≤ (1 − qj−1 )αj−1 + βj−1
(rj )
0 ≤ (1 − qm )αm + βm
αj ≥ 0,
βj ≥ 0,
(rm+1 )
γ≥0
Suppose for all j we set αj = γ and set βj = βj−1 − qj−1 γ except for j = 1 we set β1 = 1 − γ.
P
From this assignment, we get βj = 1 − γ − γ j−1
k=1 qk . Observe that we get a feasible solution
as long as all βj resulting
from
this
assignment
are
P non-negative. Furthermore, it is 1easy to see
Pm
that βj ≥ 1 − γ − γ k=1 qk ≥ 1 − 2γ because j qj ≤ 1. Therefore, by setting γ = 2 , all βj are
non-negative and we always get a feasible solution for the dual LP with an objective value of 12
which completes the proof.
33