How Good Are Simple Mechanisms for Selling
Multiple Goods?∗
Sergiu Hart† and Noam Nisan‡
June 25, 2014
Abstract
Maximizing the revenue from selling two goods (or items) is a notoriously difficult problem, in stark contrast to the single-good case. We
show that simple “one-dimensional” mechanisms, such as selling the
two goods separately, guarantee at least 73% of the optimal revenue
when the valuations of the two goods are independent and identically
distributed, and at least 50% when they are independent. However, in
the general case where the valuations may be correlated, simple mechanisms cannot guarantee any positive fraction of the optimal revenue.
We also introduce a “measure of complexity” for mechanisms—the
menu size—and show that it is naturally related to the fraction of the
optimal revenue that can be guaranteed.
∗
This paper is based on results from two working papers: Hart and Nisan (2012)
and Hart and Nisan (2013).
Research partially supported by a European Research Council Advanced Investigator grant (Hart) and by an Israel Science Foundation grant (Nisan). We thank Motty Perry and Phil Reny for introducing us
to the subject and for many useful discussions.
A presentation is available at
http://www.ma.huji.ac.il/hart/abs/2good-p.html
†
The Hebrew University of Jerusalem (Center for the Study of Rationality, Department
of Economics, and Institute of Mathematics). E-mail : [email protected] Web site:
http://www.ma.huji.ac.il/hart
‡
The Hebrew University of Jerusalem (Center for the Study of Rationality, and
School of Computer Science and Engineering) and Microsoft Research.
E-mail :
[email protected] Web site: http://www.cs.huji.ac.il/~noam
1
Contents
1 Introduction
1.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Preliminaries
2.1 The Model . . . . . . . . . . . . . . . . . . . . . .
2.2 Guaranteed Fraction of Optimal Revenue (GFOR)
2.3 No Positive Transfer (NPT) and Subdomain . . .
2.4 The Equal Revenue Distribution . . . . . . . . . .
3
8
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10
10
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17
18
20
21
22
23
4 Correlated Goods
4.1 Proof for Correlated Goods . . . . . . . . . . . . . . . . . . .
4.2 Simple Mechanisms and Menu Size . . . . . . . . . . . . . . .
4.3 Menu Size and Revenue . . . . . . . . . . . . . . . . . . . . .
24
25
28
31
5 Summary and Open Problems
34
References
35
A Appendix
A.1 Proof for I.I.D. Goods . . . . . . . . . . .
A.2 Distributions Where Bundling Is Optimal .
A.3 ER Goods . . . . . . . . . . . . . . . . . .
A.4 Multiple Buyers . . . . . . . . . . . . . . .
38
38
43
45
47
3 Independent Goods
3.1 Proof for Independent Goods
3.2 Upper Bound on GFOR . . .
3.3 Multiple Buyers . . . . . . . .
3.4 Bundling . . . . . . . . . . . .
3.5 More Than Two Goods . . . .
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1
Introduction
Suppose that a seller has one good (or item) to sell to a single buyer whose
willingness to pay (or “value”) for the good is x. While x is known to the
buyer, it is unknown to the seller, who only knows its distribution (given by
a cumulative distribution function F ). If the seller offers to sell the good for
a price p then the probability that the buyer will buy is 1 − F (p), and the
seller’s revenue will be p · (1 − F (p)). The seller will choose a price p∗ that
maximizes this expression.
This problem is the classical monopolist-pricing problem. Looking at it
from an auction point of view, one may ask whether there are mechanisms
for selling the good that yield a higher revenue. Such mechanisms could be
indirect, could offer different prices for different probabilities of getting the
good, and perhaps there are others. Yet, the characterization of optimal
auctions of Myerson (1981) concludes that the take-it-or-leave-it offer at the
above price p∗ yields the optimal revenue among all mechanisms. Even more,
Myerson’s result also applies when there are multiple buyers, in which case
p∗ would be the reserve price in a second-price auction.
Now suppose that the seller has two (different) goods that he wants to
sell to a single buyer. Furthermore, consider the simplest case where the
buyer’s values for the two goods are independently and identically distributed
according to the distribution F (“i.i.d.-F ” for short), and where, furthermore,
his valuation is additive: if the value of the first good is y and of the second
is z, then the value of the bundle consisting of both goods is1 y + z. It would
seem that since the two goods are completely independent of each other,
then the best one should be able to do is to sell each of them separately in
the optimal way, and thus extract exactly twice the revenue one would make
from a single good. Yet this turns out to be false.
Example E1: Consider the one-good distribution F taking values 1 and 2,
each with probability 1/2. Let us first look at selling a single good optimally:
1
Our buyer’s demand is not limited to one item (as is the case in some of the existing
literature; see Section 1.1).
3
the seller can either choose to price it at 1, selling always2 and getting a
revenue of 1, or choose to price the good at 2, selling it with probability 1/2,
again obtaining an expected revenue of 1, and so the optimal revenue from
a single good is 1. Now consider the following mechanism for selling both
goods: bundle them together, and sell the bundle for price 3. The probability
that the sum of the buyer’s values for the two goods is at least 3 is 3/4, and
so the revenue is 3 · 3/4 = 2.25—larger than the revenue of 2 that is obtained
by selling them separately.
However, that is not always so: bundling may sometimes be worse than
selling the goods separately.
Example E2: Consider the one-good distribution F taking values 0 and 1,
each with probability 1/2. Selling the two goods separately yields a revenue
of 1/2 from each good (set the price at 1), and so 1 in total, whereas the
revenue from selling the bundle is only 3/4 (the optimal price for the bundle
is 1).
In other cases neither selling separately nor bundling is optimal.
Example E3: Consider the one-good distribution F taking the values 0, 1,
and 2, each with probability 1/3. The unique optimal mechanism for two
such i.i.d. goods turns out to be to offer to the buyer the choice between
any single good at price 2 and the bundle of both goods at a “discount”
price of 3. This mechanism gets a revenue of 13/9 = 1.44..., which is larger
than the revenue of 4/3 = 1.33... obtained from either selling the two goods
separately, or from selling them as a single bundle.
A similar situation obtains for the uniform distribution on [0, 1], for which
neither bundling nor selling separately is optimal (Manelli and Vincent 2006).
In still other cases the optimal mechanism is not even deterministic and
2
Since we want to maximize revenue we can always assume without loss of generality
that ties are broken in a way that maximizes revenue. This “seller-favorable” property
can always be achieved by appropriate small perturbations; see Hart and Reny (2012).
4
must offer lotteries for the goods. This happens for instance in the following
example, taken from Hart and Reny (2012):3
Example E4: Consider the distribution taking the values 1, 2, and 4, with
probabilities 1/6, 1/2, and 1/3, respectively. It turns out that the unique
optimal mechanism for two such i.i.d. goods offers the buyer a choice of one
of the following options: buying a lottery ticket that gives the first good with
probability 1/2 for a price of 1, buying a similar lottery ticket for good 2,
buying the bundle of both goods (surely) for a price of 4, and buying nothing;
any deterministic mechanism has a strictly lower revenue.4
So, it is not clear what optimal mechanisms for selling two goods look
like, and indeed characterizations of optimal auctions even for this simple case
are not known (see Section 1.1). The two-dimensional problem is extremely
difficult, and the mechanisms that amount to solving only one-dimensional
problems—such as separate selling and bundling—do not maximize the revenue in general.
This leads to the following question: how good are simple mechanisms
for selling two goods? In particular, how much of the optimal revenue can
be obtained by the simple mechanism of selling the goods separately and
independently, where (by Myerson’s 1981 result) the seller posts a price for
each good and the buyer chooses which goods to buy?
Thus, we consider the fraction of the optimal revenue that can be guaranteed by separate selling. More generally, given a class of mechanisms
N —such as separate selling or bundled selling—and a class of environments
X—such as two independent and identically distributed goods or two correlated goods—we define the Guaranteed Fraction of Optimal Revenue
(GFOR) as that maximal fraction α between 0 and 1 such that for every
environment in X there is a mechanism in N that yields a revenue of at least
3
Examples where randomization increases the revenue appear in the literature, starting
with Thanassoulis (2004) in the somewhat different setup of unit demand, and Manelli
and Vincent (2006, 2007, and 2012 for a correction).
4
Computing the optimal revenue when there are finitely many possible valuations
amounts to solving a linear programming problem; restricting to deterministic mechanisms adds integer constraints (and may be solved, for instance, by a simple enumeration
of all possibilities).
5
the fraction α of the optimal revenue; see Section 2.2 for a formal definition
and the connection to notions often used in the computer science literature.
Our main result is:
Theorem 1 In the case of one buyer and two goods:
(i) If the valuations of the two goods are independent and identically distributed, then selling each good separately at the optimal price guarantees at least 73% of the optimal revenue:
GFOR(separate) ≥
e
≈ 0.731.
e+1
(ii) If the valuations of the two goods are independent, then selling each good
separately at its optimal price guarantees at least 50% of the optimal
revenue:
1
GFOR(separate) ≥ .
2
(iii) If the valuations of the two goods are correlated, then selling each good
separately at its optimal price cannot guarantee any positive fraction
of the optimal revenue:
GFOR(separate) = 0.
(“Optimal price” refers to the Myerson 1981 optimal price in the one-good
case.)
When the goods are independent, our results (i) and (ii) are “positive”:
separate selling yields at least 73%, or 50%, of the optimal revenue. This
result turns out to be very robust. First, it applies to any distribution of values (we make no assumptions on the distributions, such as monotone hazard
rate).5 Second, it holds for any number of buyers. Third, it extends (with
5
One may argue that there is no need for results that are uniform with respect to the
valuations’ distributions, on the grounds that the seller knows that distribution. However,
as we have shown above, knowing the distribution does not help finding the optimal
mechanism (even for simple distributions), whereas separate mechanisms are always easy
6
different fractions) to more than two goods. And fourth, similar bounds
are obtained for bundling, the other class of simple one-dimensional mechanisms.6
When the goods are correlated, our result (iii) is “negative”: separate
selling cannot guarantee any positive fraction of the optimal revenue. Indeed, there exist two-good valuations for which selling separately yields an
arbitrarily small fraction of the optimal revenue. This result is very robust as
well: it holds also for bundling, and even for the larger class of deterministic
mechanisms. It further attests to the difficulty of the problem of maximizing
revenue for two (or more) goods.
The proof of the aforementioned negative result suggests that the difference between “complex” mechanisms that yield the optimal revenue and
“simple” ones whose revenues are significantly smaller is not due just to the
fact that the former use randomizations and the latter do not. Rather, it
turns out that the true limiting factor is the number of options that the
buyer is choosing from, which we call the menu size of the mechanism.
Indeed, we will show that the revenue may strictly increase with menu size,
and that, in general, mechanisms with arbitrarily large menu size are needed
to obtain the optimal revenue. However, deterministic mechanisms (which
include separate and bundled selling) have menu size at most7 3. The upshot
of this analysis is that the menu size is a good measure of the complexity of
mechanisms and auctions.
to compute (as they use only optimal prices for one-dimensional distributions). It is thus
important to know how far from optimum we are, particularly when we do not know what
that optimum is or how to find it.
6
When there are many independent goods, the Law of Large Numbers implies that
the value of the bundle of all goods together is close to its expectation, and so selling the
bundle allows the seller to extract most of the surplus (see Section 3.4). The difficult case
is thus when there is more than one but not too many goods—the prime example being
that of two goods, which is already most complex.
7
The 3 options that the buyer faces in a deterministic mechanisms are: buy only good
1 for a price p1 , or buy only good 2 for a price p2 , or buy both goods at a price p12
(in addition to the “outside” zero option of getting nothing and paying nothing, which
we assume is always available, and so is not counted in the menu size). Separate selling
corresponds to p12 = p1 + p2 , and bundled selling to dropping altogether the two options
of buying only one good (make p1 and p2 infinite).
7
The structure of the paper is as follows. In Section 2 we present our model,
concepts, and notations, followed by a number of preliminary results. Section
3 is devoted to the “positive” results for independent goods of Theorem 1 (i)
and (ii) together with various extensions (to multiple buyers, to more than
two goods, and to bundling auctions). Section 4 deals with the “negative”
result for correlated goods of Theorem 1 (iii), following which the menu-size
complexity measure is introduced and its effect on revenues is studied. We
conclude in Section 5 with a summary and several open problems. Some of
the proofs are relegated to the appendices.
The reader may find additional related results in the two working papers
Hart and Nisan (2012, 2013), from which most of this material is taken; the
first working paper (2012) deals with independent goods, and the second
(2013) with correlated goods and menu size.
1.1
Literature
We briefly describe some of the previous work on these issues. McAfee and
McMillan (1988) identify cases where the optimal mechanism is deterministic. However, Thanassoulis (2004) and Manelli and Vincent (2006) found
a technical error in the paper and exhibit counterexamples. These last two
papers contain good surveys of the work within economic theory, with more
recent analysis by Fang and Norman (2006), Jehiel, ter Vehn, and Moldovanu
(2007), Pycia (2006), Lev (2011), Pavlov (2011), and Hart and Reny (2012).
In the past few years algorithmic work on these types of topics was carried
out. One line of work (e.g., Briest, Chawla, Kleinberg, and Weinberg 2010;
Cai, Daskalakis, and Weinberg 2011) shows that for discrete distributions
the optimal auction can be found by linear programming in rather general
settings. This is certainly true in our simple setting where the direct representation of the auction constraints provides a polynomial size linear program.
Thus we emphasize that the difficulty in our case is not computational, but
is rather that of characterizing and understanding the results of the explicit
computations: this is certainly so for continuous distributions, but also for
8
discrete ones.8,9 Another line of work in computer science (Chawla, Hartline, and Kleinberg 2007; Chawla, Hartline, Malec, and Sivan 2010; Chawla,
Malec, and Sivan 2010; Daskalakis and Weinberg 2011) attempts to approximate the optimal revenue by simple mechanisms. This was done for various
settings, especially unit-demand settings and some generalizations. One conclusion from this line of work is that for many subclasses of distributions (such
as those with monotone hazard rate) various simple mechanisms can extract
a constant fraction of the expected value of the goods.10 This is true in
our simple setting, where for such distributions selling the goods separately
provides a constant fraction of the expected value and thus of the optimal
revenue.
In unit-demand settings, Briest, Chawla, Kleinberg, and Weinberg (2010)
show that the gap between the revenue of randomized and deterministic
mechanisms may be unbounded for k ≥ 3 goods, and leave the case of two
goods open, while showing that for a restricted class of randomized auctions,
the gap is only constant. Since the revenues in their model and in ours are
within constant factors (that depend on the number of goods; cf. Appendix
1 in Hart and Nisan 2013), a GFOR of zero in one model yields the same in
the other. Therefore our construction in Section 4—which is in fact inspired
by the Briest, Chawla, Kleinberg, and Weinberg (2010) construction—settles
their open problem for two goods.
Since the circulation of early versions of our 2012 and 2013 working pa8
If we limit ourselves to deterministic auctions (and discrete distributions), finding
the optimal one is computationally easy in the case of one buyer (just enumerate), in
contrast to the general case of multiple buyers with correlated values (whose computational
complexity has been established by Papadimitriou and Pierrakos 2011).
9
This suggests that a notion of “conceptual complexity” may be appropriate here. The
usual computational complexity may not capture all the difficulty of a problem, since,
even after computing the precise solution, one may not understand its structure, what it
means and represents, and how it varies with the given parameters (e.g., Hart and Reny
2012, where it is shown that the optimal revenue may not be monotonic with respect to
the valuations).
10
In our setting this is true even more generally, for instance, whenever the ratio between
the median and the expectation is bounded (which happens in particular when the tail of
the distribution is “thinner” than x−α for α > 1). Indeed, posting a price equal to the
median yields a revenue of one-half the median, hence at least a constant fraction of the
expectation.
9
pers, there has been a flurry of work regarding optimal mechanisms for multiple goods for various cases: Daskalakis, Deckelbaum, and Tzamos (2013),
Wang and Tang (2013), Giannakopoulos and Koutsoupias (2014), and Giannakopoulos (2014). The general problem was studied from a computational perspective in Daskalakis, Deckelbaum, and Tzamos (2014), where it
is shown to be computationally intractable (formally, #P -hard). Several
developments have occurred regarding GFOR for multiple goods. Li and
Yao (2013) improve our lower bound on GFOR(separate) for k goods
(see Section 3.5) from c/ log2 k to the tight c/ log k. For the case of k independent and identically distributed goods, Li and Yao (2013) proved that
GFOR(bundled) is bounded from below by a constant that is independent
of the number of goods k. Babaioff, Immorlica, Lucier, and Weinberg (2014)
show, for k independent (but not necessarily identically distributed goods),
that GFOR({separate, bundled}) is bounded from below by a constant
that is independent of k (that is, there is c > 0 such that for every number
k and any k independent goods, either separate selling or bundling yields at
least the fraction c of the optimal revenue).11 The menu-size complexity of
mechanisms and auctions, as well as a complexity measure that uses more
general ways of describing auctions, was further studied in Dughmi, Han,
and Nisan (2014).
2
Preliminaries
In this section we present the model formally and define the concepts that
we use, together with a number of preliminary results.
2.1
The Model
One seller (or “monopolist”) is selling a number k ≥ 1 of “goods” (or “items,”
“objects,” etc.) to one buyer.
The goods have no value nor cost to the seller. Let x1 , x2 , ..., xk ≥ 0
11
Formally, max{SRev(X), BRev(X)} ≥ c·Rev(X) for every k ≥ 2 and every X =
(X1 , ..., Xk ) where X1 , ..., Xk are independent (see Section 2.1 for notations).
10
be the buyer’s values for the goods. The value for getting a set of goods is
P
additive: getting the subset L ⊂ {1, 2, ..., k} of goods is worth ℓ∈L xℓ to
the buyer (and so, in particular, the buyer’s demand is not restricted to one
good only). The values are given by a random variable X = (X1 , X2 , ..., Xk )
that takes values in Rk+, ; we will refer to X as a k-goods valuation. The
realization x = (x1 , x2 , ..., xk ) ∈ Rk+ of X is known to the buyer, but not
to the seller, who only knows its distribution F (which may be viewed as
the seller’s belief). The buyer and the seller are assumed to be risk-neutral
and to have quasi-linear utilities (i.e., the utility is additive with respect to
monetary transfers; e.g., getting good 1 with probability 1/2 and paying 8
with probability 1/4 is worth (1/2) · x1 − (1/4) · 8 to the buyer and (1/4) · 8
to the seller).
The objective is to maximize the seller’s (expected) revenue.
As has been well established by the so-called “Revelation Principle”
(starting with Myerson 1981; see for instance the book of Krishna 2010),
we can restrict ourselves to “direct mechanisms” and “truthful equilibria.”
A (direct)12 mechanism µ consists of a pair of functions (q, s), where q =
(q1 , q2 , ..., qk ) : Rk+ → [0, 1]k and s : Rk+ → R, which prescribe the allocation
of goods and the payment, respectively. Specifically, if the buyer reports
a valuation vector x ∈ Rk+ , then qℓ (x) ∈ [0, 1] is the probability that the
buyer receives good13 ℓ (for ℓ = 1, 2, ..., k), and s(x) is the payment that
the seller receives from the buyer. When the buyer reports his valuation x
P
truthfully, his payoff is14 b(x) = kℓ=1 qℓ (x)xℓ − s(x) = q(x) · x − s(x), and
the seller’s payoff is15 s(x). The mechanism µ = (q, s) satisfies individual rationality (IR) if b(x) ≥ 0 for every x ∈ Rk+ , and incentive compatibility (IC)
if b(x) ≥ q(x̃) · x − s(x̃) for every alternative report x̃ ∈ Rk+ of the buyer when
12
“Mechanism” will always mean “direct mechanism.”
When the goods are infinitely divisible and the valuations are linear in quantities (i.e.,
the value of a quantity λ of good ℓ is λxℓ ), we may interpret qℓ also as the quantity of
good ℓ that the buyer gets.
14
When y = (y
Pin)i=1,...,n and z = (zi )i=1,...,n are n-dimensional vectors, y ·z denotes their
scalar product i=1 yi zi .
15
In the literature the payment to the seller is called transfer, cost, price, revenue, and
so on, and denoted by t, c, p, ...; this plethora of names and notations applies to the buyer
as well. We hope that using the mnemonic s for the seller’s final payoff and b for the
buyer’s final payoff will avoid confusion.
13
11
his valuation is x, for every x ∈ Rk+ . Let M denote the class of all IC and IR
mechanisms µ = (q, s). The expected revenue from a buyer with valuation X
using a mechanism µ = (q, s) ∈ M is16 R(µ; X) := E [s(X)] , and the optimal
revenue from X is, by the Revelation Principle, Rev(X) := supµ∈M R(µ; X),
the highest revenue that can be obtained by any IC and IR mechanism17 µ.
The revenue can never exceed the expected valuation of all goods together:
P
P
Rev(X) ≤ E [ ℓ Xℓ ] (since s(x) ≤ q(x) · x ≤ ℓ xℓ by IR and x ≥ 0).
When there is only one good, i.e., when k = 1, Myerson’s (1981) result is
that
Rev(X) = sup p · P [X ≥ p] = sup p · (1 − F (p)),
(1)
p≥0
p≥0
where F is the cumulative distribution function of X. Optimal mechanisms
correspond to the seller “posting” a price p and the buyer buying the good
for the price p whenever his value is at least p; in other words, the seller
makes the buyer a “take-it-or-leave-it” offer to buy the good at price p.
The menu of a mechanism µ = (q, s) is its range M ≡ M (µ) := {(q(x), s(x)) :
x ∈ Rk+ } ⊂ [0, 1]k × R, where each “entry” (g, t) on the menu is an outcome
consisting of an allocation of goods g ∈ [0, 1]k and a payment t ∈ R. When µ
is IC and IR, this may be interpreted as the seller “posting” a menu M and
then the buyer, given his valuation, “choosing” one entry18 from this menu.
Conversely, a closed set M0 ⊂ [0, 1]k × R of outcomes generates an IC and
IR mechanism µ = (q, s) by taking (q(x), s(x)) ∈ arg max(g,t)∈M0 (g · x − t) for
every x ∈ Rk+ ; its menu M is clearly a subset of the given set M0 . Therefore
there is no loss of generality in assuming that the mechanism µ is defined
and satisfies IC and IR on the whole space Rk+ , rather than just on some
domain D ⊂ Rk+ , such as the set of possible valuations of X (take M0 to be
the closure of {(q(x), s(x)) : x ∈ D}; see Hart and Reny 2012).
Besides the maximal revenue, we are also interested in what can be ob16
In Hart and Reny (2012) it is shown that for IC and IR mechanisms one may assume
without loss of generality that s is measurable; since s is bounded from below by s(0)
(which follows from IC at 0), the expected revenue E [s(X)] is well defined (but may be
infinite).
17
Since clearly only the distribution F of X matters, we will sometime write R(µ; F )
and Rev(F ) instead of R(µ; X) and Rev(X), respectively.
18
Entry, not entrée—a mechanism is after all not a restaurant.
12
tained from certain classes of mechanisms. Thus, given a class N ⊂ M of
IC and IR mechanisms, let N -Rev(X) := supν∈N R(ν; X) be the maximal
revenue that can be extracted from a buyer with valuation X when restricted
to mechanisms ν in the class N . Some classes of mechanisms:
• Separate: Each good ℓ is sold separately. The maximal revenue from
separate mechanisms is denoted SRev, and so
SRev(X) := Rev(X1 ) + Rev(X2 ) + ... + Rev(Xk ).
• Bundled: All goods are sold together in one “bundle.” The maximal
revenue from bundled mechanisms is denoted BRev, and so
BRev(X) := Rev(X1 + X2 + ... + Xk ).
• Deterministic: Each good ℓ is either fully allocated or not at all,
i.e., qℓ (x) ∈ {0, 1} (rather than qℓ (x) ∈ [0, 1]) for every x ∈ Rk+ and
ℓ = 1, ..., k. The maximal revenue from deterministic mechanisms is
denoted DRev.
For additional classes of mechanisms, related to menu size, see Section 4.2
below. The separate and the bundled revenues are obtained by solving onedimensional problems (where one uses (1)), whereas the deterministic revenue
is a multi-dimensional problem. Examples where these classes yield revenues
that are smaller than the maximal revenue are well known (see also Examples
E1–E4 in the Introduction, and the references in Section 1.1).
We conclude with a useful characterization of incentive compatibility that
is well known (starting with Rochet 1985).
Proposition 2 Let µ = (q, s) be a mechanism for k goods with buyer payoff
function b (i.e., q : Rk+ → [0, 1]k , s : Rk+ → R, b : Rk+ → R, and b(x) =
q(x) · x − s(x) for every x ∈ Rk+ ). Then µ = (q, s) satisfies IC if and only if
b is a convex function and for all x the vector q(x) is a subgradient of b at x
(i.e., b(x̃) − b(x) ≥ q(x) · (x̃ − x) for all x̃).
13
Proof. µ is IC if and only if b(x) = q(x) · x − s(x) = maxx̃∈Rk+ (q(x̃) · x − s(x̃))
for every x, which implies that b is a convex function of x (as the maximum
of a collection of affine functions in x). Moreover, for every x and x̃ we have
b(x̃) − b(x) − q(x) · (x̃ − x) = b(x̃) − (q(x) · x̃ − s(x)), and so the subgradient
inequalities are precisely the IC inequalities.
Thus s(x) = q(x) · x − b(x) = ∇b(x) · x − b(x), where ∇b(x) stands for a
(sub)gradient of b at x, and so the revenue can be expressed in terms of the
buyer payoff function19 b.
2.2
Guaranteed Fraction of Optimal Revenue (GFOR)
Let X be a class of valuations (such as two independent goods or k i.i.d.
goods; formally, it is a class of random variables X with values in some Rk ),
and let N be a class of IC and IR mechanisms (such as separate selling or
deterministic mechanisms; formally, N is a subset of the class M of all IC and
IR mechanisms). The Guaranteed Fraction of Optimal Revenue (GFOR) for
the class of valuations X and the class of mechanisms N is defined as the
maximal fraction α such that, for any valuation X in X, mechanisms in the
class N yield at least the fraction α of the maximal revenue. Formally,
N -Rev(X)
supν∈N R(ν; X)
= inf
.
X∈X Rev(X)
X∈X supµ∈M R(µ; X)
GFOR ≡ GFOR(N ; X) := inf
Thus GFOR ≥ α if and only if for every valuation X in X there is a
mechanism ν in N such that its revenue is R(ν; X) ≥ α·Rev(X) (we are
ignoring here the trivial issues of “max” vs. “sup”), and GFOR ≤ α if there
exists a valuation X in X such that for every mechanism ν in N its revenue
is R(ν; X) ≤ α·Rev(X).
Remarks. (a) Ratios. Why are we considering ratios and fractions? The
19
The function b, being convex, is differentiable almost everywhere, and so ∇b(x) is the
gradient (∂b(x)/∂xℓ )ℓ=1,...,k for almost every x. As pointed out in Hart and Reny (2012),
when maximizing revenue one may use “seller-favorable” mechanisms and replace the term
∇b(x) · x with b′ (x; x), the directional derivative of b at x in the direction x, which is well
defined for every x.
14
reason is that the revenue is covariant with rescalings, but not with translations. Indeed, Rev(λX) = λ·Rev(X) for any λ > 0, but Rev(X + c) is in
general different from Rev(X) + c for constant c > 0 (this happens already
in the one-good case; cf. (1)).
(b) Competitive ratio. The computer science literature uses the concepts
of a “competitive ratio” and “approximation ratio,” which are just the reciprocal 1/GFOR of GFOR. While the two notions are clearly equivalent,
using the optimal revenue as benchmark and measuring everything relative
to this base—as GFOR does—seems to come more naturally.
2.3
No Positive Transfer (NPT) and Subdomain
A mechanism µ = (q, s) satisfies “No Positive Transfer ”20 (NPT) if s(x) ≥ 0
for every x ∈ Rk+ . Proposition 3 below collects a number of useful results
related to NPT. In particular, NPT can always be assumed without loss of
generality when maximizing revenue.21 Moreover, the revenue from any IC
and IR mechanism on a subdomain of valuations cannot exceed the overall
maximal revenue—even when the mechanism does not satisfy NPT (this is
used in our proofs, e.g., in Section 3.1, where we use ŝ that may take negative
values).
Proposition 3 Let µ = (q, s) be an IC and IR mechanism, and let X be a
k-good valuation with values in Rk+ , where k ≥ 1. Then:
(i) µ satisfies NPT if and only if s(0) = 0.
(ii) There is a mechanism µ̂ = (q, ŝ) with the same q and ŝ(x) ≥ s(x) for
all x ∈ Rk+ , such that µ̂ satisfies IC, IR, and NPT.
(iii) Rev(X) = supµ R(µ; X) where the supremum is taken over all IC, IR,
and NPT mechanisms µ.
20
“Transfer” refers to −s(x), i.e., a transfer from the seller to the buyer.
This is not true in more general setups; for instance, when there are multiple buyers
that are correlated, Bayesian–Nash implementation may require using positive transfers,
i.e., s(x) < 0 (cf. Crémer and McLean 1988; see Appendix A.4).
21
15
(iv) Let A ⊂ Rk+ be a set of values of X; then22
E [s(X) 1X∈A ] ≤ Rev(X 1X∈A ) ≤ Rev(X).
(v) Rev(X) = limN →∞ Rev(X 1kXk≤N ).
Proof. (i) IC at 0 yields s(x) ≥ s(0) for all x, and IR at 0 yields s(0) ≤ 0; the
minimal payment is thus s(0), which cannot be positive. Therefore s(x) ≥ 0
for all x if and only if s(0) = 0.
(ii) Put ŝ(x) := s(x) − s(0) ≥ s(x) for all x (recall that s(0) ≤ 0 by IR at
0). Then µ̂ satisfies IC since the payment differences have not changed (i.e.,
ŝ(x) − ŝ(x̃) = s(x) − s(x̃) for all x, x̃), it satisfies IR since q(x) · x − s(x) ≥
q(0) · x − s(0) ≥ −s(0) (by IC), and it satisfies NPT since ŝ(0) = 0.
(iii) Follows from (ii) since µ̂ yields at least as much revenue as µ (because
ŝ(x) ≥ s(x) for all x).
(iv) For the first inequality: use (ii) to get E [s(X) 1X∈A ] ≤ E [ŝ(X) 1X∈A ] =
E [ŝ(X 1X∈A )] ≤Rev(X 1X∈A ) (the equality since ŝ(0) = 0 by (i)). For the
second inequality: E [s(X 1X∈A )] = E [s(X) 1X∈A ] ≤ E [s(X)] for any µ that
satisfies NPT; apply (iii).
£
¤
(v) When µ is NPT (and so s(0) = 0) we have E s(X 1kXk≤N ) =
¤
£
E s(X) 1kXk≤N , which converges to E [s(X)] as N → ∞; apply (iv) and
(iii).
2.4
The Equal Revenue Distribution
A useful valuation for one good is the random variable X that takes values
X ≥ 1 and satisfies P [X ≥ x] = 1/x for every x ≥ 1. It is precisely that
valuation with revenue normalized at 1 (i.e., Rev(X) = 1) for which any
posted price p ≥ 1 is optimal (recall Myerson’s 1981 result (1)). The distribution of X is called the Equal Revenue (ER) distribution (it is the Pareto
distribution with index 1 and scale 1).23 Its cumulative distribution function
22
We write 1W for the indicator of the event W : it takes the value 1 when W occurs
and the value 0 otherwise.
23
The Pareto distribution with index α > 0 and scale c > 0 is given by F (x) = 1−(c/x)α
for x ≥ c (for index α = 1 it is similar to the one-sided Cauchy distribution). Interestingly,
16
and density function are FER (x) = 1 − 1/x and fER (x) = 1/x2 for all24 x ≥ 1.
While the revenue of ER is finite, its expectation is infinite: Rev(ER) = 1
R∞
and E [ER] = 1 x · x−2 dx = ∞.
Myerson’s (1981) result (1) for one good may be restated as follows:
Rev(Y ) ≤ 1 if and only if Y ≤SD1 ER, where ≤SD1 stands for first-order
stochastic dominance (indeed, Rev(Y ) ≤ 1 if and only if 1 − FY (p) ≤ 1/p
for all p ≥ 0, which is the same as FY (p) ≤ FER (p) for all p). That is, the revenue from a one-good valuation Y is at most 1 if and only if one can increase
the values of Y and obtain a new valuation X (i.e., X ≥ Y everywhere) that
is ER-distributed.
For some results on ER distributions and revenues, see Section 3.2 and
Appendices A.2 and A.3.
3
Independent Goods
This section is devoted to the case of independent goods. Our main results
on the fraction of the optimal revenue that can be guaranteed by selling
separately two independent goods are equivalently stated as follows.
Restatement of Theorem 1 (i) and (ii). Let X = (X1 , X2 ) be a two-good
valuation, with X1 and X2 one-dimensional nonnegative random variables.
(i) If X1 and X2 are independent and identically distributed then
Rev(X1 ) + Rev(X2 ) ≥
e
Rev(X1 , X2 ).
e+1
(ii) If X1 and X2 are independent then
Rev(X1 ) + Rev(X2 ) ≥
1
Rev(X1 , X2 ).
2
X is ER if and only if 1/X is Uniform on the interval (0, 1].
24
The “virtual valuation” x − (1 − F (x))/f (x) (used when there are multiple buyers) is
everywhere zero for the ER distribution (and that happens only for ER and its rescalings).
17
The proof in the independent case (ii) is provided in Section 3.1; the
proof for the independent and identically distributed case (i), which goes
along similar lines but is significantly more intricate, is relegated to Appendix
A.1. These inequalities provide the lower bounds on GFOR(separate) of
Theorem 1 (i) and (ii); an upper bound (of approximately 78%) is provided
in Section 3.2.
The result turns out to be robust, as it extends and generalizes to any
number of buyers (Section 3.3), and to more than two goods (Section 3.5).
We also study the other class of simple mechanisms that amount to onedimensional optimization, namely, bundled selling, and obtain bounds on its
GFOR; in addition, we identify a useful class of distributions where bundling
is in fact optimal (Section 3.4).
3.1
Proof for Independent Goods
In this section we prove the result of Theorem 1 (ii) that separate selling
guarantees at least 1/2 of the maximal revenue for two independent goods.
Proof of Theorem 1(ii). Let X = (Y, Z) where Y, Z ≥ 0 are independent
real random variables (we write Y for X1 and Z for X2 ), and let µ = (q, s) be
an IC and IR mechanism for two goods. We want to show that the expected
revenue R(µ; X) satisfies R(µ; X) ≤ 2Rev(Y ) + 2Rev(Z). This will be done
by splitting R(µ; X) = E [s(Y, Z)] into two parts, according to which one of
Y and Z is higher, i.e., E [s(Y, Z)] = E [s(Y, Z) 1Y ≥Z ] + E [s(Y, Z) 1Z>Y ], and
showing that
E [s(Y, Z) 1Y ≥Z ] ≤ 2Rev(Y ),
E [s(Y, Z) 1Z>Y ] ≤ 2Rev(Z).
and
(2)
(3)
To prove (2), for every fixed value z ≥ 0 of the second good define a
mechanism (q̂, ŝ) ≡ (q̂ z , ŝz ) for the first good by replacing the allocation of the
second good with an equivalent decrease in payment; that is, the allocation of
the first good is unchanged, i.e., q̂(y) := q1 (y, z), and the payment is ŝ(y) :=
s(y, z) − q2 (y, z) · z, for every y ≥ 0. The one-good mechanism µ̂ = (q̂, ŝ) is
18
IC and IR for y, since µ = (q, s) was IC and IR for (y, z) (for IC, only the
constraints (ỹ, z) vs. (y, z) matter; and for IR, b̂(y) = b(y, z), where b̂ is the
buyer’s payoff function in µ̂). We have s(y, z) = ŝ(y) + q2 (y, z) · z ≤ ŝ(y) + z
(since z ≥ 0 and q2 ≤ 1), and so, since Y is independent of Z,
E [ s(Y, Z) 1Y ≥Z | Z = z] = E [ s(Y, z) 1Y ≥z ] ≤ E [ŝ(Y ) 1Y ≥z ] + E [z 1Y ≥z ] .
The first term is the revenue from a subdomain of values of y, and so is
at most the maximal revenue Rev(Y ) by Proposition 3 (iv).25 As for the
second term, we have
E [z 1Y ≥z ] = z · P [Y ≥ z] ≤ Rev(Y ),
(4)
since posting a price of z, and the buyer buying when y ≥ z, constitutes an
IC and IR mechanism for26 y. Thus
E [ s(Y, Z) 1Y ≥Z | Z = z] ≤ 2Rev(Y )
for every value z of Z; taking expectation over z yields (2).
The proof above applies also when replacing the inequality Y ≥ Z with
Y > Z throughout (the only place where it matters is in (4), which clearly
holds also in this case), and so27 E [s(Y, Z) 1Y >Z ] ≤ 2Rev(Y ). Interchanging
Y and Z yields (3), and the proof is complete.
Remark. Inequality (4) implies that Rev(X1 )+Rev(X2 ) ≥ E [min{X1 , X2 }]
(take expectation of (4) over the values z of Z, interchange Y and Z, and add
the two resulting inequalities). Thus, while in the single-good case one cannot guarantee any positive fraction of the expected value as revenue (take for
instance the ER distribution, for which E [ER] = ∞ but Rev(ER) = 1), in
the case of two independent goods one can at least guarantee the expectation
25
µ̂ need not satisfy NPT, as ŝ may take negative values (to get NPT, replace ŝ(y) with
ŝ(y) − ŝ(0); cf. the proof of Proposition 3 (iv)).
26
We are not using here Myerson’s (1981) characterization (1) of optimal one-good
mechanisms as posting-price mechanisms, but only that such mechanisms are IC and IR.
27
Alternatively, we could assume that s ≥ 0 (by NPT; cf. Section 2.3), and then
E [s(Y, Z) 1Y >Z ] ≤ E [s(Y, Z) 1Y ≥Z ] .
19
of the minimum of the values of the two goods. A mechanism that yields the
revenue E [min{X1 , X2 }] posts random prices p1 and p2 for the two goods,
where p1 and p2 are independent random variables, p1 is distributed like X2 ,
and p2 is distributed like X1 (there is no typo here!); this is a randomized
separate mechanism.28
3.2
Upper Bound on GFOR
Our main result gives lower bounds on GFOR (73% or 50%). But how high
can GFOR(separate) actually be? The best estimate we have is that it
cannot exceed approximately 78%: there exist two-good valuations where
selling separately yields only that fraction of the optimal revenue.
Proposition 4 In the case of two goods,
GFOR(separate) ≤
1
≈ 0.782...,
w+1
where29 w ≡ W (1/e) is the solution of the equation wew = 1/e.
Proof. Let X = (X1 , X2 ) with X1 and X2 i.i.d.-ER (see Section 2.4).
Selling separately yields SRev(X) = Rev(X1 )+Rev(X2 ) = 2, and selling
the bundle yields BRev(X) = Rev(X1 + X2 ) = 2(w + 1); this computation is relegated to Appendix A.3. Therefore Rev(X) ≥ 2(w + 1) and
SRev(X)/Rev(X) ≤ 1/(w + 1).
Remarks. (a) We will see in Section 3.4 below (Theorem 6) that bundling
is in fact optimal for two i.i.d. goods with distributions in a certain class—
that includes the ER distribution. Therefore in the proof above we have
Rev(X) = BRev(X) = 2(w + 1) and SRev(X)/Rev(X) = 1/(w + 1).
28
The inequality Rev(X1 ) + Rev(X2 ) ≥ E [min{X1 , X2 }] is tight, as it can be shown
that it becomes an equality when X1 , X2 are i.i.d.-ER. It does not hold when X1 and X2
are not independent (for an extreme case take the fully correlated case with X1 = X2 being
ER); the correct inequality here is E [Rev(X1 |X2 )] + E [Rev(X2 |X1 )] ≥ E [min{X1 , X2 }]
(where E [Rev(X1 |X2 )] is the expectation over the values x2 of X2 of the revenue from a
random variable with distribution
given by
£ X1 |X2 = x2 ).
¤ All this generalizes to any k ≥ 2
P
goods, where we obtain ℓ Rev(Xℓ ) ≥ E (m − 1)X (m) for every m = 2, ..., k, with X (m)
denoting the m-th order statistic of X1 , ..., Xk (so X (1) = maxℓ Xℓ and X (k) = minℓ Xℓ ).
29
W is the Lambert function: W (x) is the solution of the equation wew = x.
20
(b) The upper bound of 0.78 is useful only when the goods are independent, as Theorem 1 (iii) shows that it decreases all the way down to 0 when
the goods are allowed to be correlated.
3.3
Multiple Buyers
Up to now we dealt with a single buyer, but our result for two independent goods turns out to hold also when there are multiple buyers. Unlike the
simple decision-theoretic problem facing a single buyer, we now have a multiperson game among the buyers. Two main notions of equilibrium are considered: “dominant strategy” equilibrium and “Bayesian–Nash” equilibrium
(corresponding to “ex-post” and “interim” implementations, respectively);
see Appendix A.4 for details. Our result holds for both concepts.
Theorem 5 In the case of n independent buyers and two goods: if the valuations of the two goods are independent, then selling each good separately
using its optimal one-good mechanism guarantees at least 50% of the optimal
revenue:
1
GFOR(separate) ≥ ;
2
this holds when the optimal revenue is taken throughout30 with respect to
either dominant-strategy implementation or Bayesian–Nash implementation.
That is, let the one-dimensional random variable Xℓi ≥ 0 denote the value
of good ℓ to buyer i, for ℓ = 1, 2 and i = 1, ..., n. Write X i = (X1i , X2i ) ∈ R2+
for the valuation vector of buyer i for both goods, and Xℓ = (Xℓi )i=1,...,n ∈ Rn+
for the vector of values of all buyers for good ℓ. “Independent buyers” means
that the random vectors X 1 , X 2 , ..., X n are independent; “independent goods”
means that the random vectors X1 and X2 are independent.31 Theorem 5
is proved in Appendix A.4, which also contains the precise notations and
statements; the proof is again a generalization of the Proof of Theorem 1 (ii)
for one buyer (Section 3.1).
30
I.e., for the two goods, as well as for each good separately.
Independent buyers together with independent goods means that the 2n random variables Xℓi are independent.
31
21
Remarks. (a) Dependent buyers and dominant-strategy implementation. In
the dominant-strategy case, our proof does not use the independence between the buyers’ valuations; thus GFOR(separate ) ≥ 1/2 holds under
dominant-strategy implementation for two independent goods and any number of buyers, whether independent or not; see Theorem 18 in Appendix
A.4.
(b) Dependent buyers and Bayesian–Nash implementation. In the Bayesian–
Nash case our proof does not extend when the buyers are not independent
(see Appendix A.4). However, for this case Crémer and McLean (1988)
show that, under a certain general “correlation-between-buyers” condition,
the seller can extract all the surplus from any single good: Rev(Xℓ ) =
E [max1≤i≤n Xℓi ] . Since the most that the seller can extract from the two
goods is, by IR,32 E [maxi X1i ] + E [maxi X2i ], it follows that in this case
Rev(X1 , X2 ) = Rev(X1 )+Rev(X2 ), and so GFOR(separate) = 1. We
do not know what GFOR(separate) is when the buyers are neither independent nor satisfy the Crémer–McLean condition.
3.4
Bundling
We come now to the other simple mechanism for two goods: bundled selling.
The bundled revenue is BRev(X) =Rev(X1 + X2 ), and this requires solving
only a one-dimensional problem (cf. (1)). Hart and Nisan (2012) provide
bounds on the fraction of optimal revenue that is guaranteed by bundling,
namely,33
2
e
2
·
≤ GFOR(bundled; 2 i.i.d. goods) ≤ ;
3 e+1
3
1
1
≤ GFOR(bundled; 2 independent goods) ≤ .
4
2
Interestingly, we have identified a class of two-good i.i.d. distributions for
which bundling is optimal (and so GFOR(bundled) = 1 in this case).
32
(see Appendix P
A.4P
for notations),Pbi (x) ≥ 0 implies si (x) ≤ q i (x) · xi and so
P Indeed
P
i
i
i
i
i
i
i s (x) ≤
i q (x) · x =
ℓ
i qℓ (x) xℓ ≤
ℓ maxi xℓ .
33
These bounds are obtained from those of Theorem 1 (i) and (ii) together with bounds
on the ratios between bundling and separate selling.
22
Theorem 6 Let a > 0 and let f be a density function with support included
in [a, ∞) that is differentiable and satisfies
3
xf ′ (x) + f (x) ≤ 0
2
(5)
for every x > a. Then bundling is optimal for two goods X = (X1 , X2 ) where
X1 and X2 are independent and identically distributed with density function
f , i.e.,
Rev(X) = BRev(X) = Rev(X1 + X2 ).
Theorem 6 is proved in Appendix A.2. Condition (5) is equivalent to
¡ 3/2
¢′
x f (x) ≤ 0, i.e., x3/2 f (x) is nonincreasing in x (the support of f is thus
either some finite interval [a, A] or the half-line [a, ∞)). When f (x) = cx−γ ,
(5) holds whenever γ ≥ 3/2. In particular, the ER distribution (where γ = 2)
satisfies (5), as well as any general Pareto distribution with index α ≥ 1/2.
3.5
More Than Two Goods
Consider now the case where there may be more than two goods, i.e., k ≥ 2.
It is well known, and not difficult to prove, that when the goods are
independent and their number k tends to infinity, the bundling revenue
approaches the optimal revenue (Armstrong 1999; Bakos and Brynjolfsson
1999). Moreover, the seller can extract essentially all the surplus by selling
the bundle of all the goods together. The logic is quite simple: the Law of
Large Numbers implies that there is almost no uncertainty as to the sum of
many independent random variables, and so the seller essentially knows this
sum—the value of the bundle—and may post something close to this value as
a bundle price (and this is the most the seller can obtain, as the revenue can
never exceed the expected value of the bundle, by IR and the nonnegativity
of the values).
Despite the apparent strength of this result, it does not provide any
guarantees for any fixed value of k. Indeed, for every (large enough) k we
have GFOR(bundled; k i.i.d. goods) ≤ 0.57 (see Hart and Nisan 2012,
which also contains additional results that yield lower and upper bounds on
23
GFOR(bundled; k independent goods)).
In the case of separate selling and more than two goods, our proof technique of Section 3.1 extends to a general decomposition result that yields
bounds that are logarithmic in k; specifically, there exist positive constants
c1 , c2 such that, for every34 k ≥ 2,
c1
c2
≤ GFOR(separate; k independent goods) ≤
.
2
log k
log k
4
Correlated Goods
In this section the two goods need no longer be independent, and we allow an
arbitrary dependence between their values. Our result in this case (Theorem
1 (iii)) is that selling the two goods separately cannot guarantee any positive
fraction of the optimal revenue—unlike the case of independent goods where
we have shown that this fraction is at least 50%. Indeed, we will prove here
that
Theorem 7 There exists a two-good valuation X with values in R2+ for which
the separate revenue is finite and the optimal revenue is infinite:
SRev(X) < ∞ and Rev(X) = ∞.
This immediately implies the result of Theorem 1 (iii) that GFOR(separate) =
0. Theorem 7 is proved in Section 4.1 below.
Any valuation X obtained in Theorem 7 is necessarily unbounded from
above, since otherwise the revenue would be bounded. This raises the question of whether the result holds also for bounded valuations. The answer is
easily seen to be yes, and so GFOR(separate) = 0 is true also for bounded
valuations.
Corollary 8 For every ε > 0 there exists a two-good valuation X with values
in [0, 1]2 such that
SRev(X) < ε · Rev(X).
34
See Hart and Nisan (2012). Following our work, the lower bound has been improved
to c1 / log k by Li and Yao (2013); see Section 1.1.
24
Proof. Apply Proposition 3 (iii) to the valuation X given by Theorem
7 to get N large enough so that Rev(X 1kXk≤N ) > (1/ε)· SRev(X) ≥
(1/ε)·SRev(X 1kXk≤N ). Rescaling yields the desired valuation: take X ′ :=
(1/N )X 1kXk≤N .
Given that separate selling cannot guarantee any positive fraction of the
optimal revenue, how about other simple mechanisms, e.g., bundled selling,
or even the larger class of deterministic mechanisms? In Section 4.2 we
show that neither one performs better: their GFOR is 0 as well. Moreover,
this negative result is extremely robust: GFOR equals 0 for any class of
mechanisms that are bounded in the number of menu entries. This result
is proved in Section 4.3, which includes additional results on the relation
between menu size and revenues.
4.1
Proof for Correlated Goods
To prove Theorem 1 (iii)—more precisely, Theorem 7—we will construct a
two-dimensional random variable X such that from each one of its values
one obtains enough revenue so that the total revenue is infinite, while the
single-good revenues are kept finite. For this purpose we will construct a
sequence of allocations q (m) in [0, 1]2 (Proposition 10), and use it to generate
the distribution of values of X (Proposition 9). The proof is inspired by
Briest, Chawla, Kleinberg, and Weinberg (2010) who prove a similar result
for unit-demand buyers, and k ≥ 3 goods.35
Given a sequence q (0) , q (1) , q (2) , . . . of points in [0, 1]2 with q (0) = 0, define
for every m ≥ 1,
gap(m) := q (m) · q (m) − max q (j) · q (m) ;
0≤j<m
thus gap(m) ≤ q (m) ·q (m) ≤ 2 (since q (0) = 0 and q (m) ∈ [0, 1]2 ); it is convenient
to put gap(0) := 0.
35
The smaller the dimension, the harder it is to make such constructions; see also Remark
(b) in Section 4.3.
25
Proposition 9 Let (q (m) )m=0,1,... , where q (0) = 0, be a sequence of points in
[0, 1]2 with positive gaps, i.e., gap(m) > 0 for every m ≥ 1. Then there exists
a two-good valuation X = (X1 , X2 ) that takes values (x(m) )m=0,1,2,... in R2+
and satisfies
Rev(X1 ) ≤ 2,
Rev(X2 ) ≤ 2,
∞
X
Rev(X1 , X2 ) ≥
gap(m) .
and
m=1
P
Moreover, this revenue of m gap(m) is obtained by an IC and IR mechanism
£
¤
µ = (q, s) where q(x(m) ) = q (m) and s(x(m) ) = gap(m) /P X = x(m) for every
m ≥ 0.
(j)
m+1
Proof. For each m ≥ 1 let α(m) := (Πm
> 0. Since gap(m) ≤ 2
j=1 gap )/4
for every m it follows that α(1) ≤ 1/8 and α(m) ≤ α(m−1) /2, and so α(m) ≤
P
P
(m)
(m)
2−m−2 . Therefore ∞
≤ 1/4, and we put α(0) := 1− ∞
≥ 3/4
m=1 α
m=1 α
(m)
(m−1)
(thus α
≤α
/2 holds also for m = 0). Finally, for every m ≥ 0, put
x(m) := (1/α(m) ) q (m) ∈ R2+ and let X be the random variable that takes the
values x(m) with the probabilities α(m) , for all m ≥ 0.
We start with the single-good revenues, and show that Rev(Xℓ ) ≤ 2
for ℓ = 1, 2. By Myerson’s (1981) characterization (1), we need to evaluate
P [Xℓ ≥ t] for every t > 0. If 1/α(m) < t ≤ 1/α(m+1) for some m ≥ 0,
£
¤ P
(j)
(j)
then P [Xℓ ≥ t] ≤ P Xℓ > 1/α(m) ≤ j>m α(j) (since xℓ = (1/α(j) ) qℓ ≤
1/α(j) ≤ 1/α(m) for 0 ≤ j ≤ m), and this sum is at most 2α(m+1) (since
the α(j) decrease by a factor of at least 2); similarly, if 0 < t ≤ 1/α(0) then
P
P [Xℓ ≥ t] ≤ P [Xℓ > 0] ≤ j>0 α(j) ≤ 2α(1) . Thus t · P [Xℓ ≥ t] ≤ 2 holds for
all t > 0, which yields Rev(Xℓ ) ≤ 2 by (1).
Next, consider the two-good revenue. For every m ≥ 0 put s(m) :=
gap(m) /α(m) and define the two-dimensional mechanism µ = (q, s) on the
range {x(m) : m ≥ 0} of X by q(x(m) ) = q (m) and s(x(m) ) = s(m) for every
m ≥ 0 (the menu is thus {(q (m) , s(m) ) : m ≥ 0}). We claim that µ is IC and
IR. To prove IC, we will show that the buyer with valuation x(m) prefers
26
(q (m) , s(m) ) to any other (q (j) , s(j) ). Indeed: first, when 0 ≤ j < m,
b(x(m) ) = q (m) · x(m) − s(m) =
≥
1
α(m)
1
α(m)
(q (m) · q (m) − gap(m) )
q (j) · q (m) = q (j) · x(m) ≥ q (j) · x(m) − s(j)
(the first inequality follows from the definition of gap(m) , and the second
inequality from s(j) ≥ 0). In particular, when j = 0 we get b(x(m) ) ≥ 0 (since
q (0) = 0 and s(0) = 0), which proves IR. Second, when j > m,
q (j) · x(m) − s(j) =
≤
1
α(m)
2
α(m)
1
gap(j)
(j)
α
2
4
≤ (m) − (m) < 0 ≤ b(x(m) )
α
α
q (j) · q (m) −
−
4
α(j−1)
(for the first inequality we used q (j) · q (m) ≤ 2, as both vectors are in [0, 1]2 ,
and α(j) /α(j−1) = gap(j) /4, by definition of the α(m) ). Thus the mechanism
µ is IC and IR. Moreover, the revenue it extracts from X is R(µ; X) =
P∞
P
(m) (m)
(m)
s = ∞
.
m=0 α
m=1 gap
Remark. The bundled revenue from X satisfies BRev(X) = Rev(X1 +
X2 ) ≤ 4. This is proved in a similar way to Rev(Xℓ ) ≤ 2: for 2/α(m) < t ≤
¤
£
2/α(m+1) with m ≥ 0 we have P [X1 + X2 ≥ t] ≤ P X1 + X2 > 2/α(m) ≤
P
(j)
≤ 2α(m+1) , and for 0 < t ≤ 2/α(0) we have P [X1 + X2 ] ≤
j>m α
P
P [X1 + X2 > 0] ≤ j>0 α(j) ≤ 2α(1) , and thus t · P [X1 + X2 ≥ t] ≤ 4 for
every t > 0. This result will be used in the Proof of Theorem 11 in Section
4.3 below.
We now construct a sequence of points q (m) with gaps that are large
enough so that their sum is infinite.
Proposition 10 There exists an infinite sequence (q (m) )m=0,1,... of points in
[0, 1]2 with q (0) = 0, and a constant c > 0, such that gap(m) ≥ c m−6/7 for
every m ≥ 1.
Proof. The sequence of points (q (m) ) is composed of a sequence of “shells,”
numbered N = 1, 2, ..., each containing multiple nonnegative points that lie
27
in the unit ball {q : ||q||2 = q · q ≤ 1}, and thus in [0, 1]2 . The shells will be
getting closer and closer to each other, approaching, as N goes to infinity,
the boundary of the unit ball: specifically, all the points q (j) in the N -th shell
P
P
−3/2
−3/2
will be of length ||q (j) || = N
/ρ, where ρ := ∞
< ∞. Each
i=1 i
i=1 i
shell N will contain N 3/4 different points in it so that the angle between any
two of them is at least36 Ω(N −3/4 ).
Now for j < m we estimate q (j) ·q (m) = ||q (j) ||·||q (m) ||·cos(θ), where θ is the
angle between q (j) and q (m) . Let N denote the index of the shell containing
q (m) . Since j < m there are two possibilities: either q (j) is in the same shell
as q (m) or it is in a smaller shell. In the first case we have θ ≥ Ω(N −3/4 ),
and thus cos(θ) ≤ 1 − Ω(N −3/2 ) (since cos(x) = 1 − x2 /2 + x4 /24 − . . .);
because ||q (m) || = Θ(1) it follows that q (m) · q (m) − q (j) · q (m) ≥ Ω(N −3/2 ).
In the second case we have ||q (m) || − ||q (j) || ≥ N −3/2 /ρ , and so, again using
||q (m) || = Θ(1), it follows that q (m) · q (m) − q (j) · q (m) ≥ Ω(N −3/2 ). Thus for
any point q (m) in the N -th shell we have gap(m) ≥ Ω(N −3/2 ). Since the first
P
3/4
N shells together contain N
= Θ(N 7/4 ) points, we have m = Θ(N 7/4 )
i=1 i
and thus gap(m) ≥ Ω(N −3/2 ) = Ω(m−6/7 ).
We can now prove our result.
Proof of Theorem 7.
Take the sequence of points (q (m) ) constructed
in Proposition 10 and apply Proposition 9, to get the two-good valuation
X = (X1 , X2 ), for
³Pwhich Rev(X
´ 1 )+Rev(X2 ) ≤ 2 + 2 = 4 and Rev(X) ≥
P∞
∞
(j)
−6/7
≥Ω
= ∞.
j=1 gap
j=1 j
4.2
Simple Mechanisms and Menu Size
Theorem 7 says that separate selling can be arbitrarily far from optimal.
How about other simple mechanisms, such as bundled, or deterministic?
It turns out that they are just as “bad,” i.e., they cannot guarantee any
positive fraction of the optimal revenue. The proof of Theorem 7 in the
36
We use the following standard notations for functions f and g defined on the natural
numbers: f (n) ≤ O(g(n)) and f (n) ≥ Ω(g(n)) if there is c > 0 such that f (n) ≤ cg(n),
respectively f (n) ≥ cg(n), for all n ≥ 1; and f (n) = Θ(g(n)) if f (n) = O(g(n)) and
f (n) = Ω(g(n)) both hold.
28
previous section suggests that the key to all these negative results is the
number of outcomes available to the buyer, i.e., the number of entries on
the “menu” of the mechanism. More precisely, we have constructed there a
mechanism that has a large (even infinite) number of menu entries, together
with an appropriate distribution for which a significant amount of revenue is
extracted by each menu entry—while no simple mechanism, which necessarily
has a small menu, can do the same. The large menu thus appears to be the
crucial attribute of the high-revenue mechanisms: it enables the sophisticated
screening between different buyer types required for high-revenue extraction.
This suggests focusing on the menu size as a complexity measure
of mechanisms, and studying the revenue extraction capabilities of mechanisms that are limited in their menu size.
The menu size of a mechanism is the number of possible non-zero outcomes of the mechanism (we do not count the “zero” outcome where the
allocation is (0, ..., 0) and the payment is 0, which without loss of generality
we may assume is always available, as it corresponds to the “outside option”
that yields IR). Formally, we define the menu size of a mechanism µ = (q, s)
as the number of elements of its menu,37 i.e.,
MS(µ) := |{(q(x), s(x)) 6= ((0, ..., 0), 0) : x ∈ Rk+ }|.
For an IC mechanism (where q(x) = q(x′ ) implies s(x) = s(x′ )), this is the
same as the number of non-zero allocations: MS(µ) = |{q(x) 6= (0, ..., 0) :
x ∈ Rk+ }|. For example, posted-price mechanisms for one good have menu size
1, and so does the optimal bundled mechanism for two goods (which posts a
price for the bundle); optimal separate selling for two goods has menu size
3 (since the possible non-zero allocations are (1, 0), (0, 1), and (1, 1)), and in
fact any deterministic mechanism has the same menu size 3 (when there are
k goods the menu size becomes 2k − 1); finally, general mechanisms that use
lotteries may have infinite menu size.
For each finite m ≥ 1, let38 MRev[m] (X) denote the maximal revenue
37
38
We write |A| for the number of elements of the set A (finite or infinite).
In Hart and Nisan (2013) this was denoted [m]-Rev.
29
that can be obtained from X using mechanisms with menu size at most m;
i.e., MRev[m] (X) := supµ R(µ; X) where the supremum is taken over all IC
and IR mechanisms µ ∈ M with menu size at most m, i.e., MS(µ) ≤ m.
While for one good Rev(X) = MRev[1] (X) (by Myerson’s 1981 result (1)),
this is not true for more than one good: the revenue may increase as we
allow the menu size to increase. For two goods, BRev(X) ≤ MRev[1] (X)
(Proposition 12 below will show that in fact we have equality here), and
SRev(X) ≤ DRev(X) ≤ MRev[3] (X).
Our main result is that no finite menu size can guarantee a positive fraction of the optimal revenue: there exists a two-good valuation from which
one can extract an infinite revenue, but that revenue cannot be obtained from
any finite menu size (specifically, we take the valuation X of Proposition 9).
Again, as such a valuation is necessarily unbounded, we also provide a similar result for bounded valuations: for every menu size m there are bounded
valuations for which every mechanism of size at most m can extract no more
than an arbitrarily small proportion of the optimal revenue.
Theorem 11 For every m ≥ 1,
GFOR(menu-size ≤ m) = 0.
Moreover, there exists a two-good valuation X with values in R2+ such that
Rev(X) = ∞ and MRev[m] (X) < ∞ for all m ≥ 1;
and, for every m ≥ 1 and ε > 0, there exists a two-good valuation X with
values in [0, 1]2 such that
MRev[m] (X) < ε · Rev(X).
Theorem 11 is proved in the next section.
30
4.3
Menu Size and Revenue
We start with two immediate properties of MRev: first, menu size 1 yields
the bundled revenue, and second, the revenue can increase at most linearly
with menu size.
Proposition 12 MRev[1] (X) =BRev(X).
Proof. Let (g, t) ∈ [0, 1]2 × R be the non-zero menu entry of an IC and IR
mechanism µ with menu size MS(µ) = 1. Replacing it with ((1, 1), t) yields an
IC and IR mechanism that is bundled and extracts at least as much revenue,
since every buyer valuation x that preferred (g, t) to the zero outcome will
also prefer ((1, 1), t) to it. The converse inequality follows from the fact that
the bundled revenue is obtained by posting a price for the bundle, which has
menu size 1.
Proposition 13 MRev[m] (X) ≤ m·MRev[1] (X) for every m ≥ 1.
Proof. Let (g, t) ∈ [0, 1]2 × R be an entry in the menu of an IC and IR
mechanism µ = (q, s); the revenue from it is t · P [(q(X), s(X)) = (g, t)] .
Keeping only this entry and dropping all the other non-zero entries can only
increase the revenue obtained from (g, t), since any valuation x that chose it
will continue to do so. Therefore this revenue is at most the revenue from
menu size 1, i.e., t · P [(q(X), s(X)) = (g, t)] ≤ MRev[1] (X); now sum over
the non-zero menu entries of µ.
We can now prove Theorem 11.
Proof of Theorem 11. Take the sequence (q (m) ) of Proposition 10 and
the resulting random variable X of Proposition 9, for which Rev(X) = ∞.
Since BRev(X) = Rev(X1 +X2 ) ≤ 4 (see the Remark following the Proof of
Proposition 9), it follows from Propositions 12 and 13 that MRev[m] (X) ≤
m·BRev(X) ≤ 4m. The “moreover” statement follows in the same way that
Corollary 8 follows from Theorem 7.
31
Our construction shows that the revenue may strictly increase with the
menu size. A more precise estimate is the following:39
Proposition 14 There exists a two-good valuation X with 0 <MRev[1] (X) <
∞ and a constant c > 0 such that, for all m ≥ 1,
MRev[m] (X) ≥ c m1/7 · MRev[1] (X).
Proof. Take the mechanism with menu {(q (j) , s(j) ) : 0 ≤ j ≤ m} and
the random variable X of Propositions 9 and 10: the expected revenue
P
(j)
is m
≥ Ω(m1/7 ) (by the “moreover” statement in Proposition 9),
j=0 gap
whereas, by the Remark following the Proof of Proposition 9, MRev[1] (X) =
BRev(X) ≤ 4.
Remarks. We collect here a number of extensions; see Hart and Nisan
(2013) for further details.
(a) Multiple buyers. All the results that GFOR = 0 (i.e., Theorems 1
(iii), 7, and 11) clearly extend to any number n of buyers, as one can always
add n − 1 buyers with zero (or close to zero) valuations.40
(b) More than two goods. All the results that GFOR = 0 also extend to
any number k ≥ 2 of goods, as one can always add goods with value zero (or
close to zero). Moreover, everything in this section (such as the propositions
above) holds for any number of goods k ≥ 2, as the definitions and proofs
apply to any k.
(c) Deterministic mechanisms. Deterministic mechanisms—where, for
every valuation x and every good ℓ, the allocation qℓ (x) is either 0 or 1—
have menu size at most 3 when there are two goods, and at most 2k − 1
when there are k goods (this is the number of possible vectors q(x) 6= 0).
39
The increase is at a polynomial rate in m, and we do not think that the constant of
1/7 we obtain is tight. For larger values of k the construction in Briest, Chawla, Kleinberg,
and Weinberg (2010) implies a somewhat better polynomial dependence in m. For m that
is at most exponential in k, the proof of Theorem C in Hart and Nisan (2013) actually
shows that the growth can be almost linear.
40
In the case of multiple buyers there are several different possible notions of menu-size
complexity, which are briefly discussed in Appendix 3 of Hart and Nisan (2013).
32
Proposition 13 implies that41
DRev(X) ≤ MRev[m] (X) ≤ m · MRev[1] (X) for m = 2k − 1.
In Hart and Nisan (2013, Theorem C and Lemma 5.2) we show that for every
k ≥ 2 there exists a k-good valuation X (with values in [0, 1]k ) such that42
′
DRev(X) ≥ m′ · MRev[1] (X) ≥ MRev[m ] (X) for m′ =
2k − 1
.
2k
Thus, deterministic mechanisms yield essentially the same revenue as any
class of mechanisms with similar menu size, namely, exponential in the number of goods k. This suggests that it is the limited menu size—rather than
not using lotteries—that is the main bottleneck of deterministic mechanisms.
(d) Additive menu size. Our complexity measure so far has been extremely simple, just counting the total number of possible outcomes. In
particular, the mechanism that sells each good separately (in k completely
separate one-good posted-price mechanisms) has 2k −1 menu-size complexity
(since the buyer may acquire any subset of the goods), even though intuitively
we would consider it a simpler mechanism; after all, it is determined by k
parameters only (the posted prices of the k goods). Moreover, the separate
revenue satisfies SRev(X) ≤ k·MRev[1] (X) (since for every good ℓ we have
P
Rev(Xℓ ) ≤ Rev( kj=1 Xj ) = BRev(X) = MRev[1] (X); this improves upon
k
SRev(X) ≤ MRev[2 −1] (X) ≤ (2k − 1)· MRev[1] (X)). We thus introduce
in Hart and Nisan (2013, Section 7) a stronger and more refined notion of
mechanism complexity, which allows “additive menus” where the buyer may
choose not just a single menu entry but rather any subset of menu entries,
provided that no good is allocated more than once in total. The resulting
“additive menu-size” complexity measure equals k for separate selling of k
goods, and moreover satisfies the same properties as the original menu-size
41
For the special case of k = 2 goods, Hart and Nisan (2013, Appendix 2) provide a more precise estimation tehcnique that yields the tight bound DRev(X) ≤
(5/2)·MRev[1] (X). In the independent case, Hart and Nisan (2012) shows that
DRev(X) ≤ (c/k)·MRev[1] (X).
42
Use Proposition 13.
33
complexity (in particular, Propositions 12 and 13).
(e) Bounded values. Consider the case where the values of the goods are
bounded from below, i.e., away from 0, as well as from above; that is, the
values of all goods lie in some interval [a, A] with 0 < a ≤ A < ∞. In this
case our constructions do not apply. Moreover, one can show that in this case
mechanisms of bounded menu size yield good approximations of the optimal
revenue; see Hart and Nisan (2013, Theorem E and Section 8).
(f ) Exact comparisons to bundling. In Hart and Nisan (2013, Appendix 2)
we provide a technique that yields precise bounds for various revenues relative
to the bundled revenue BRev=MRev[1] ; for example, it yields DRev(X) ≤
(5/2)·BRev(X) for any two-good valuation X.
5
Summary and Open Problems
To summarize the results of this paper, taking simple mechanisms to mean
“mechanisms with menu size at most m for some m ≥ 1,” we have:
Simple mechanisms guarantee 73%, 50%, and 0% of the
optimal revenue for two goods that are, respectively, independent and identically distributed, independent, and
correlated.43
A number of interesting problems remain open. As attested by the long
time that has passed since Myerson’s (1981) work in the one-good case, characterizing the optimal mechanisms in the multiple-goods case—even when
there are just two goods—is an extremely difficult problem.44 While the
general problem appears very complex, one may well be able to obtain results for certain useful classes of valuations and mechanisms. Following are
some specific questions arising from our study:
43
In the independent case this holds for m ≥ 3 (so that separate selling is included); for
m = 1, 2 the best bounds we have obtained are 48% for i.i.d. and 25% for independent
(see Section 3.4 and Proposition 12). We do not know better bounds on GFOR for other
simple mechanisms (see Open Problem 5 below).
44
Recall footnote 9 on conceptual complexity.
34
1. Characterize distributions where separate selling is optimal (cf. the
parallel result for bundling of Theorem 6).
2. Provide bounds for GFOR(deterministic), the fraction of optimal
revenue that is guaranteed by mechanisms that do not use randomizations. In addition, characterize distributions where deterministic
mechanisms are optimal.45
3. Tighten the bounds on GFOR(separate). While the gap in the i.i.d.
case (78% vs. 73%) is quite small, we do not know what the right
value is; also, is GFOR in the independent case in fact lower than in
the i.i.d. case?
4. Evaluate GFOR(separate) for Bayesian–Nash implementation when
there are multiple buyers that are neither independent nor satisfy the
Crémer–McLean condition (see Remark (b) in Section 3.3).
5. Find simple mechanisms different than separate selling that can guarantee a larger fraction of the optimal revenue.
6. In the case of independent (even identically distributed) goods, does
GFOR(menu-size ≤ m) approach 1 as m increases? That is, whether
for every ε > 0 there is m such that MRev[m] (X) ≥ (1 − ε)·Rev(X)
for all X (with m depending on ε but not on X).
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A
A.1
Appendix
Proof for I.I.D. Goods
In this appendix we prove Theorem 1 (i), which says that selling two i.i.d.
goods separately yields at least e/(e + 1) of the optimal revenue. It follows
38
a similar line of argument as the proof of Theorem 1 (ii) in Section 3.1, but
is more intricate as we make all the estimates much tighter.
Proof of Theorem 1 (i). Let X = (Y, Z) where Y and Z are i.i.d. nonnegative one-dimensional random variables, and let r := Rev(Y ) = Rev(Z) =
supt≥0 t · G(t) be the revenue from each good separately, where G(t) :=
P [Y ≥ t]. We want to prove that Rev(Y, Z) ≤ ((e+1)/e)(Rev(Y )+Rev(Z)) =
2(1 + 1/e)r.
Take a two-good IC and IR mechanism µ = (q, s) with buyer payoff
function b (i.e., b(x) = q(x) · x − s(x) for all x). Without loss of generality
assume that it also satisfies NPT, i.e., s(x) ≥ 0 for all x (Proposition 3 (iii));
thus in particular b(0) = s(0) = 0 (since they are both nonnegative and
b(0) + s(0) = q(0) · 0 = 0). Since X is symmetric we will also assume that
µ is symmetric, i.e., q1 (y, z) = q2 (z, y) and s(y, z) = s(z, y) for all (y, z)—
and so also b(y, z) = b(z, y)—for all y, z ≥ 0 (indeed, µ can be replaced by
its “symmetrization” µ̄ = (q̄, s̄) given by q̄1 (y, z) = q̄2 (z, y) = (q1 (y, z) +
q2 (z, y))/2 and s̄(y, z) = s̄(z, y) = (s(y, z) + s(z, y))/2, which also satisfies
IC, IR, and NPT, and yields the same revenue E [s̄(Y, Z)] = E [s(Y, Z)] .
Proposition 2 implies that for almost every (y, z) ∈ R2+ we have q1 (y, z) =
q2 (z, y) = by (y, z) = bz (z, y), where by and bz are the derivatives of b with
respect to its first and second coordinates, respectively (these exist almost
everywhere since b is a convex function).
For every t ≥ 0 define Φ(t) := b(t, t)/2 and ϕ(t) := q1 (t, t) = q2 (t, t); then
Φ is a convex function, ϕ(t) is a subgradient of Φ at t (use Proposition 2),
Ru
and so Φ(u) = 0 ϕ(t) dt (formally, use Corollary 24.2.1 in Rockafellar 1970;
recall that Φ(0) = b(0, 0) = 0).
We will consider the two regions Y ≥ Z and Z ≥ Y separately; by
symmetry, the expected revenue in the two regions is the same, and so it
suffices to show that46
µ
¶
1
E [s(Y, Z)1Y ≥Z ] ≤ 1 +
r.
e
46
E [s(Y, Z) 1Y <Z ] ≤ E [s(Y, Z) 1Y ≤Z ] since s ≥ 0.
39
For each fixed z ≥ 0 define a mechanism µ̂z = (q̂ z , ŝz ) for the first good
by q̂ z (y) := q1 (y, z) and ŝz (y) := s(y, z) − q2 (y, z) · z for every y ≥ 0; the
buyer’s payoff remains the same: b̂z (y) = b(y, z). The mechanism µ̂z is
IC and IR for y, since µ is IC and IR for (y, z). Let Yz denote the random variable Y conditional on the event Y ≥ z, and consider the revenue
R(µ̂z ; Yz ) = E [ŝz (Yz )] = E [ŝz (Y )|Y ≥ z] of µ̂z from Yz . Since Yz ≥ z we
have b̂z (Yz ) = b(Yz , z) ≥ b(z, z) = 2Φ(z) (because b is nondecreasing) and
q̂ z (Yz ) = q1 (Yz , z) ≥ q1 (z, z) = ϕ(z) (because b is a convex function, and so
its subgradient q is monotonic, which in particular implies that q1 (y, z) is
nondecreasing in y). Applying Lemma 15 below to Yz yields
E [ŝz (Y )|Y ≥ z] ≤ (1 − ϕ(z))Rev(Yz ) + zϕ(z) − 2Φ(z).
(6)
Since P [Yz ≥ t] = P [Y ≥ t] /P [Y ≥ z] = G(t)/G(z) for all t ≥ z, we get from
(1)
Rev(Yz ) = sup u · P [Yz ≥ u] = sup u ·
u≥0
u≥z
G(u)
G(u)
r
≤ sup u ·
=
.
G(z)
G(z)
G(z)
u≥0
Multiply (6) by P [Y ≥ z] = G(z) to get
E [ŝz (Y )1Y ≥z ] ≤ (1 − ϕ(z))r + (zϕ(z) − 2Φ(z))G(z),
and then take expectation over the values z of Z:
£
¤
E ŝZ (Y )1Y ≥Z ≤ r − r E [ϕ(Z)] + E [(Zϕ(Z) − 2Φ(Z)1Y ≥Z ] .
(7)
Now s(y, z) = ŝz (y)+q2 (y, z) z ≤ ŝz (y)+q2 (y, y) z = ŝz (y)+zϕ(y) (use z ≥ 0
and the monotonicity of q2 (y, z) in z, again from the convexity of b), which
together with (7) yields
£
¤
E [s(Y, Z)1Y ≥Z ] = E ŝZ (Y )1Y ≥Z + E [Zϕ(Y )1Y ≥Z ]
≤ r − r E [ϕ(Z)] + E [W 1Y ≥Z ] ,
40
(8)
where we put
W := Zϕ(Y ) + Zϕ(Z) − 2Φ(Z).
(9)
We want to bound from above the right-hand side of (8). This expression
Ru
is affine in ϕ (recall that Φ(u) = 0 ϕ(t) dt), which is a real nondecreasing
function with values in [0, 1]. Since any such function lies in the closed convex
hull of the extreme functions47 ϕ(v) = 1[v,∞) for all v ≥ 0, it suffices to bound
(8) for these extreme functions.
Ru
Consider such an extreme ϕ = 1[v,∞) for v ≥ 0; then Φ(u) = 0 ϕ(t) dt =
[u − v]+ . Substituting in (9) yields
Thus


 2Z − 2(Z − v) = 2v,
W =
Z − 0 = Z,


0,
if Y ≥ Z ≥ v,
if Y ≥ v > Z,
if v > Y ≥ Z.
E [W 1Y ≥Z ] = 2v P [Y ≥ Z ≥ v] + E [Z 1Y ≥v>Z ]
= v P [Y ≥ v] P [Z ≥ v] + P [Y ≥ v] E [Z 1v>Z ]
= P [Y ≥ v] E [v 1Z≥v ] + P [Y ≥ v] E [Z 1v>Z ]
= P [Y ≥ v] E [min{Z, v}] = G(v) E [min{Z, v}]
(we have used the fact that Y, Z are i.i.d.—and so P [Y ≥ v] P [Z ≥ v] =
P [Y ≥ v, Z ≥ v] = P [Y ≥ Z ≥ v] + P [Z ≥ Y ≥ v] = 2P [Y ≥ Z ≥ v]—and
¤
£
min{Z, v} = v 1Z≥v + Z 1v>Z ). Together with E [ϕ(Z)] = E 1Z∈[v,∞) =
P [Z ≥ v] = G(v), (8) becomes
r − rG(v) + G(v) E [min{Z, v}] = r + G(v) (E [min{Z, v}] − r) .
(10)
Let ρ(v) denote the expression in (10). When v ≤ r we have ρ(v) ≤ r since
47
This is one way of proving (1).
41
E [min{Z, v}] ≤ r. When v > r we have
E [min{Z, v}] =
=
Z
∞
Z0 v
0
Z
v
P [Z ≥ u] du
P [min{Z, v} ≥ u] du =
Z r
Z v 0
³v ´
r
G(u) du ≤
1 du +
du = r + r log
,
r
0
r u
where we have used G(u) ≤ min{r/u, 1} (which follows from r = supu≥0 uG(u)).
Therefore in this case
µ
¶
³v ´
´
log w
r³
r + r log
−r =r 1+
,
ρ(v) ≤ r +
v
r
w
where w := v/r > 1. Since maxw≥1 w−1 log w = 1/e (attained at w = e), it
follows that ρ(v) ≤ (1 + 1/e)r for all v > r, hence for all v ≥ 0 (because for
v ≤ r we have seen above that ρ(v) ≤ r). Recalling (8) and (10) therefore
yields E [s(Y, Z)1Y ≥Z ] ≤ (1 + 1/e)r, and so E [s(Y, Z)] ≤ 2(1 + 1/e)r and the
proof is complete.
Lemma 15 Let X be a one-good valuation whose support is included in
[x0 , ∞) for some x0 ≥ 0, and let b0 ≥ 0 and 0 ≤ q0 ≤ 1 be given. Then
for every IC and IR mechanism µ = (q, s) with buyer payoff function b, such
that q(x) ≥ q0 and b(x) ≥ b0 for all48 x ≥ x0 , the revenue is
R(µ; X) = E [s(X)] ≤ (1 − q0 )Rev(X) + q0 x0 − b0 .
(11)
Proof. If q0 = 1 then q(x) = 1 for all x, and so s(x) = s(x0 ) for all x by
IC; now b0 ≤ q(x0 ) · x0 − s(x0 ) = x0 − s(x0 ) yields s(x0 ) ≤ x0 − b0 , and so
(11) holds since E [s(X)] = s(x0 ). If q0 < 1, then, given any IC mechanism
(q, s) satisfying q(x) ≥ q0 and q(x) · x − s(x) ≥ b0 for all x, define a new
mechanism (q̂, ŝ) by q̂(x) := λ(q(x) − q0 ) and ŝ(x) := λ(s(x) + b0 − q0 x0 ),
where λ := 1/(1 − q0 ) > 0 (thus 0 ≤ q̂(x) ≤ 1). It is immediate to verify that
(q̂, ŝ) is an IC and IR mechanism (since [q̂(x) · x − ŝ(x)] − [q̂(x̃) · x − ŝ(x̃)] =
λ ([q(x) · x − s(x)] − [q(x̃) · x − s(x̃)]) ≥ 0, and q̂(x) · x − ŝ(x) ≥ q̂(x0 ) · x −
48
This is easily seen to be equivalent to q(x0 ) ≥ q0 and b(x0 ) ≥ b0 , as IC implies that b
and q are monotonic functions.
42
ŝ(x0 ) ≥ q̂(x0 ) · x0 − ŝ(x0 ) = λ([q(x0 ) · x0 − s(x0 )] − b0 ) = λ(b(x0 ) − b0 ) ≥ 0).
Therefore Rev(X) ≥ E [ŝ(X)] = λ(E [s(X)] + b0 − q0 x0 ), which yields (11).
A.2
Distributions Where Bundling Is Optimal
In this appendix we prove Theorem 6 which is stated in Section 3.4: for
two i.i.d. goods, if the one-good distribution f satisfies condition (5), then
bundling is optimal.
Proof of Theorem 6. Let X = (Y, Z) where Y, Z are i.i.d. with cumulative
distribution F and probability density function f that satisfies (5). We will
show that for every IC and IR mechanism µ there is a bundled mechanism
µ̂ that yields at least as much revenue, i.e., R(µ̂; X) ≥ R(µ; X); this proves
that Rev(X) =BRev(X).
Let µ = (q, s) be an IC and IR mechanism with buyer payoff function b;
as in the proof of Theorem 1 (i) in Appendix A.1, we assume without loss
of generality that the mechanism is symmetric and satisfies NPT. Since q(x)
equals the gradient of b at x for almost every x (see Proposition 2) and the
distribution is continuous, we have
R(µ; X) =
Z
∞
Z
∞
Z
∞
a
=
a
=
Z
∞
Z
∞
Z
∞
a
a
s(y, z)f (y)f (z) dy dz
(yby (y, z) + zbz (y, z) − b(y, z)) f (y)f (z) dy dz
(2yby (y, z) − b(y, z)) f (y)f (z) dy dz.
a
a
For each M > a, let rM (b) denote the same double integral but with upper
limits M instead of ∞, i.e.,
rM (b) :=
Z
a
M
Z
M
(2yby (y, z) − b(y, z)) f (y) dy f (z) dz;
(12)
a
then R(µ; X) = limM →∞ rM (b) (recall that the integrand s(y, z) is nonnegative).
43
For each z, integrate by parts the 2yby (y, z)f (y) term:
Z
M
2by (y, z)yf (y) dy =
a
[2b(y, z)yf (y)]M
a
−
Z
M
2b(y, z) (f (y) + yf ′ (y)) dy
a
= 2b(M, z)M f (M ) − 2b(a, z)af (a)
Z M
2b(y, z) (f (y) + yf ′ (y)) dy.
−
a
Substituting this in (12) yields
Z
M
rM (b) = 2M f (M )
b(M, z)f (z) dz
a
µ
¶
Z MZ M
3
′
b(y, z) − f (y) − yf (y) f (z) dy dz
+2
2
a
a
Z M
−2af (a)
b(a, z)f (z) dz.
(13)
a
Define b̂(y, z) := b(y + z − a, a) = b(a, y + z − a) for every (y, z) with
y, z ≥ a; then b̂ is a symmetric convex function on the quadrant [a, ∞)2 , it
coincides with b on the boundaries y = a and z = a, and is at least as large
as b everywhere: indeed, the convexity of b yields
z−a
y−a
b(y + z − a, a) +
b(a, y + z − a) (14)
y + z − 2a
y + z − 2a
= b(y + z − a, a) = b̂(y, z)
b(y, z) ≤
for every (y, z) ∈ [a, ∞)2 . Replacing b with b̂ can only increase the first and
second terms of (13), since all the coefficients of b there are nonnegative (use
(5)), whereas it does not affect the third term. Therefore rM (b) ≤ rM (b̂) for
all M .
Define q̂(y, z) := (q1 (y + z − a, a), q1 (y + z − a, a)) ∈ [0, 1]2 and ŝ(y, z) :=
q̂(y, z) · (y, z) − b̂(y, z); then q̂(y, z) is a subgradient of49 b̂ at (y, z), and so
µ̂ = (q̂, ŝ) is an IC and IR mechanism (by Proposition 2). Since R(µ̂; X) =
limM →∞ rM (b̂) and rM (b) ≤ rM (b̂) for all M, we get R(µ; X) ≤ R(µ̂; X).
Finally, µ̂ is a bundled mechanism, since q̂ and ŝ are functions of y + z (the
49
At points of differentiability this is b̂y (y, z) = b̂z (y, z) = by (y + z − a, a).
44
corresponding one-good mechanism for T := Y + Z is (q̃, s̃) with q̃(t) :=
q̂(t − a, a) and s̃(t) := ŝ(t − a, a)). This completes the proof.
Remark. The function b̂ that we have constructed in the proof is the maximal convex function that coincides with b on the boundary (indeed, any
such function will necessarily satisfy the inequality (14)). Consider now the
nonsymmetric version of this problem (for simplicity we state it for a = 0),
where b(t, 0) need not equal b(0, t). It turns out that the construction of the
maximal b̂ uses the function s (where s(x) = ∇b(x) · x − b(x) with ∇b(x) a
(sub)gradient of b at x). Specifically, one can show that for any two boundary points (y, 0) and (0, z) with the same value of s, i.e., s(y, 0) = s(0, z),
and every 0 ≤ λ ≤ 1, we have b̂(λy, (1 − λ)z) = λb(y, 0) + (1 − λ)b(0, z)
(moreover, this implies that the resulting ŝ(x) = ∇b̂(x) · x − b̂(x) is constant
there: ŝ(λy, (1 − λ)z) = s(y, 0) = s(0, z)). Geometrically, this says that b̂ is
affine along “equi-s” lines. What may come as a surprise here is that the
problem of the maximal convex function with given boundary values, which
has nothing to do with mechanism design, is solved in terms of the payment
function s of mechanism design.
A.3
ER Goods
In this appendix we consider two i.i.d.-ER goods X1 , X2 (the ER cumulative
distribution function is F (x) = 1 − 1/x, and the probability density function
is f (x) = 1/x2 , for x ≥ 1; see Section 2.4). Theorem 6 of Section 3.4 implies
that the optimal revenue from X = (X1 , X2 ) is obtained by bundling, and
so we evaluate the bundling revenue Rev(X1 + X2 ) (using (1)).
Lemma 16 Let X1 , X2 be i.i.d.-ER. Then50
P [X1 + X2 ≥ t] =
2
2
log(t − 1) +
2
t
t
for every t ≥ 2, and P [X1 + X2 ≥ t] = 1 for every t ≤ 2.
50
log denotes natural logarithm.
45
Proof. For t ≤ 2 we have P [X1 + X2 ≥ t] = 1 since Xℓ ≥ 1. For t ≥ 2 we
have
Z
P [X1 + X2 ≥ t] =
f (x) (1 − F (t − x)) dx
Z t−1
Z ∞
1 1
1
=
dx
+
dx
2
x2 t − x
1
t−1 x
·
¸t−1
1
1
1
1
=
+
log x − 2 log (t − x) −
2
t
t
tx 1
t−1
1
1
1
2
+ +
,
= 2 log (t − 1) −
t
t(t − 1) t t − 1
2
2
= 2 log (t − 1) + .
t
t
Corollary 17 Let X = (X1 , X2 ) where X1 , X2 are i.i.d.-ER. Then
Rev(X) = BRev(X) = 2(w + 1) ≈ 2.557... ,
where w = W (1/e) is the solution of the equation wew = 1/e.
Proof. Lemma 16 yields t · P [X1 + X2 ≥ t] = (2/t) log(t − 1) + 2 (for t ≥ 2),
and this attains its maximum of 2w + 2 at t = 1 + 1/w; using (1) yields
BRev. The equality to Rev is obtained by applying Theorem 6.
Remark. Take two i.i.d. goods X1 and X2 , without loss of generality
normalized so that Rev(X1 ) = Rev(X2 ) = 1. Then each Xℓ is first-order
stochastically dominated by ER, and so X = (X1 , X2 ) is first-order stochastically dominated by Y = (Y1 , Y2 ) where Y1 , Y2 are i.i.d.-ER. However, we cannot deduce from this that Rev(X) ≤ Rev(Y ) = 2(w + 1) (which would have
given a better bound—and with a simpler proof—for GFOR(separate)).
Indeed, unlike the case of one good, the revenue need not increase with the
valuation when there are two (or more) goods; see Hart and Reny (2012) for
this nonmonotonicity of the optimal revenue.
46
A.4
Multiple Buyers
The generalization from one buyer to n ≥ 1 buyers is as follows (recall
Section 2.1). One seller is selling k goods to n buyers; these goods have
no value nor cost for the seller. For each buyer i = 1, ..., n and each good
ℓ = 1, ..., k, buyer i’s value for good ℓ is given by a nonnegative random
variable Xℓi ; put X i = (Xℓi )ℓ=1,..,k for the Rk+ -valued valuation vector of buyer
i, and Xℓ = (Xℓi )i=1,...,n for the Rn+ -valued vector of values of good ℓ; put
also X = (Xℓi )i=1,...,n;ℓ=1,...,k , and let F be the joint distribution of all the
kn values. The distribution F is commonly known; in addition, each buyer
i knows the realization of his valuation X i . Finally, the seller and all the
buyers are risk-neutral and have quasi-linear utilities, and the valuation to
each buyer of a set of goods is additive.
A (direct) mechanism µ = (q i , si )i=1,...,n consists of an allocation function
k
i
kn
q i : Rkn
+ → [0, 1] and a payment function s : R+ → R for each buyer
Pn
i = 1, .., n, where
i=1 qℓ (x) ≤ 1 for every good ℓ = 1, .., k; the payoff
of buyer i is bi (x) = q i (x) · xi − si (x), and that of the seller is S(x) :=
Pn i
i=1 s (x). Two standard equilibrium notions are used for the n-person game
among the buyers (after the mechanism µ is given): “dominant strategy” (ds)
and “Bayesian–Nash” (bn), which yield the so-called ex-post and interim
equilibria, respectively. The corresponding conditions are:
• In the ds case: incentive compatibility (IC-DS) requires that
£
¤
bi (x) = q i (x) · xi − si (x) = max q i (x̃i , x−i ) · xi − si (x̃i , x−i )
x̃i ∈Rk+
for every i = 1, ..., n and x ∈ Rkn
+ ; individual rationality (IR-DS)
i
requires that b (x) ≥ 0 for every i and x ∈ Rkn
+ .
• In the bn case: incentive compatibility (IC-BN) requires that51
£
¤
£
¤
b̄i (xi ) := E bi (X)|X i = xi = max E q i (x̃i , X −i ) · xi − si (x̃i , X −i )|X i = xi
x̃i ∈Rk+
51
The conditions in the Bayesian–Nash case depend on the mechanism µ and (the distribution of) the valuations X, whereas in the dominant-strategy case they depend only
on the mechanism µ.
47
for every i = 1, ..., n and xi ∈ Rk+ ; individual rationality (IR-BN)
requires that b̄i (xi ) ≥ 0 for every i and xi ∈ Rk+ .
P
Let R(µ; X) := E [S(X)] ≡ E [ ni=1 si (X)] denote the seller’s expected
revenue from the mechanism µ for the valuations X. Let RevDS (X) stand for
the maximal revenue obtained from k goods and n buyers with valuations X,
using dominant-strategy implementation, i.e., mechanisms that satisfy IC-DS
and IR-DS; let RevBN (X) stand for the maximal revenue using Bayesian–
Nash implementation, i.e., mechanisms that satisfy IC-BN and IR-BN (in
the one-buyer case, i.e., when n = 1, these two concepts clearly coincide).
Remarks. (a) Bayesian–Nash implementation for independent buyers. In
the Bayesian–Nash case, when the buyers’ valuation vectors X 1 , X 2 , ..., X n
are independent, only the expectations, conditional on each buyer’s own valuation, q̄ i (xi ) := E [q i (X)|X i = xi ] = E [q i (xi , X −i )] and s̄i (xi ) := E [si (X)|X i = xi ] =
E [si (xi , X −i )] , of allocations and payments matter: replacing q i (x) with
q̄ i (xi ) and si (x) with s̄i (xi ) throughout affects neither the IC-BN and IR-BN
constraints nor the revenue.52 We can thus assume without loss of generality
in this case that the mechanism is given in “reduced form,” where q i and si
depend only on xi (rather than on the entire x), for all i.
(b) Non-Positive Transfers (NPT) and subdomain. A mechanism µ =
(q i , si )i=1,...,n for n ≥ 1 buyers and k ≥ 1 goods satisfies NPT if si (x) ≥ 0
for every i and every x. The results (i)–(v) of Proposition 3 in Section 2.3
extend to multiple buyers.
In the dominant-strategy case, we consider separately each i = 1, ..., n
k(n−1)
, and obtain, in particular, that NPT is equivalent to
and each x−i ∈ R+
i
−i
s (0, x ) = 0 for all i and all x−i , that NPT can be assumed without loss of
generality when maximizing revenue, and that the subdomain property holds:
P
E [ i si (X) 1X∈A ] ≤ RevDS (X 1X∈A ) ≤ RevDS (X) for every A ⊂ Rkn
+ .
In the Bayesian–Nash case, when the buyers are independent and each
payment si depends only on xi (see Remark (a) above), we obtain in par52
P
However, the feasibility conditions i qℓi ≤ 1 on the allocations cannot be directly
expressed in terms of the q̄ i (this is known as the “implementability” condition; see Border
1991 for a necessary and sufficient condition for implementability, and Hart and Reny 2011
for a simple restatement and proof).
48
ticular that NPT is equivalent to s̄i (0) = 0 for all i, that NPT can be assumed
P
without loss of generality when maximizing revenue, and that E [ i si (X) 1X∈A ] =
P
E [ i s̄i (X i ) 1X∈A ] ≤ RevBN (X 1X∈A ) ≤ RevBN (X) for every53 A ⊂ Rkn
+ .
We state the result separately for the two kinds of implementation, since
in the dominant-strategy case the result is stronger: the requirement that
the buyers’ valuations are independent is not needed (i.e., while there is
independence between the two goods—X1i and X2j are independent for any
two buyers i, j = 1, ..., n—we allow, for each good ℓ = 1, 2, the buyers’ values
Xℓ1 , Xℓ2 , ..., Xℓn to be arbitrarily correlated). The two theorems below imply
Theorem 5.
Theorem 18 In the case of n buyers, two goods, and dominant-strategy implementation: if the valuation vectors of the two goods X1 = (X1i )i=1,...,n and
X2 = (X2i )i=1,...,n are independent, then GFOR(separate) ≥ 1/2, i.e.,
1
RevDS (X1 ) + RevDS (X2 ) ≥ RevDS (X1 , X2 ).
2
Theorem 19 In the case of n independent buyers, two goods, and Bayesian–
Nash implementation: if the valuation vectors of the two goods X1 = (X1i )i=1,...,n
and X2 = (X2i )i=1,...,n are independent, then GFOR(separate) ≥ 1/2, i.e.,
1
RevBN (X1 ) + RevBN (X2 ) ≥ RevBN (X1 , X2 ).
2
Proof of Theorems 18 and 19. Put Y = X1 = (X1i )i=1,...,n and Z =
X2 = (X2i )i=1,...,n for the valuation vectors of good 1 and good 2, respectively;
thus Y and Z are Rn+ -valued random variables and X = (Y, Z). Let µ =
(q i , si )i=1,...,n be a mechanism for the two goods that satisfies either IC-DS
and IR-DS, or IC-BN and IR-BN; in the BN-case, we assume in addition that
53
NPT need not hold in the Bayesian–Nash case when the buyers’ valuations are not
independent. In this case, optimal mechanisms may make use of negative payments
si (x) (i.e., positive transfers); cf. Crémer and McLean (1988). In addition, the requirement that si depends only on xi is needed because the restriction to a set A of
values
X may introduce
£P of
¤
£P dependencies
¤ between the coordinates of X 1X∈A , and then
i
i
i
E
s
(X)
1
=
E
s̄
(X
)
1
need not hold.
X∈A
X∈A
i
i
49
it is in reduced form: q i and si depend only on xi (cf. Remark (a) above).54
We split the total expected revenue from µ into two parts, according to
which one of55 Y (1) = maxi Y i and Z (1) = maxi Z i is higher, and show that
£
¤
E S(Y, Z)1Y (1) ≥Z (1) ≤ 2Rev(Y ) and
E [S(Y, Z)1Z (1) >Y (1) ] ≤ 2Rev(Z);
(15)
(16)
adding the two inequalities yields our result.
To prove (15), for every fixed vector of values z ∈ Rn+ of the n buyers
for the second good define a mechanism (q̂, ŝ) ≡ (q̂ z , ŝz ) for the first good
by q̂ i (y) := q1i (y, z) and ŝi (y) := si (y, z) − q2i (y, z) z i for every y ∈ Rn+ and
P
i = 1, ..., n, and put Ŝ(y) := i ŝi (y). The mechanism (q̂, ŝ) is IC and IR
for y, since (q, s) was IC and IR for (y, z) (for IC: only constraints (ỹ i , z i ) vs.
P
(y i , z i ), matter; for IR, b̂i (y) = bi (y, z) ≥ 0). Then S(y, z) = i si (y, z) =
P i
P i i
P i
(1)
ŝ
(y)+
z
q
(y,
z)
≤
= Ŝ(y)+z (1) (the inequality obtains
2
i
i
i ŝ (y)+z
P
because 0 ≤ z i ≤ z (1) and i q2i ≤ 1). Since Y is independent of Z we get
¤
£
£
¤
E S(Y, Z)1Y (1) ≥Z (1) | Z = z = E S(Y, z)1Y (1) ≥z(1)
i
h
£
¤
≤ E Ŝ(Y )1Y (1) ≥z(1) + E z (1) 1Y (1) ≥z(1) .
The first term is the revenue from a subdomain of values of y, and so is at
most the maximal revenue Rev(Y ) by Remark (b) above (in the BN case,
ŝi depends only on y i since si and q i depend only on xi ); as for the second
term,
£
¤
¤
£
E z (1) 1Y (1) ≥z(1) = z (1) P Y (1) ≥ z (1) ≤ Rev(Y ),
(17)
since posting a price of z (1) and giving the good y to a buyer i with y i ≥ z (1) ,
54
Formally, put here q i (x) := q̄ i (xi ) and si (x) := s̄i (xi ); this allows the proof to apply
mutatis mutandis to both implementations, dominant-strategy and Bayesian–Nash.
55 (1)
a := maxi=1,...,n ai denotes the maximal coordinate of a vector a = (ai )i=1,...,n ∈ Rn .
50
if any, constitutes an IC and IR mechanism for56 y. Thus
¤
£
E S(Y, Z)1Y (1) ≥Z (1) | Z = z ≤ 2Rev(Y )
for every value z of Z; taking expectation over z yields (15).
The same argument also proves that E [S(Y, Z)1Y (1) >Z (1) ] ≤ 2Rev(Y )
(the only place where the strict inequality Y (1) > z (1) matters is in (17),
which clearly holds also in this case). Interchanging Y and Z yields (16).
56
As in the Proof of Theorem 1 (ii) in Section 3.1, we are not using the characterization
of optimal mechanisms in the one-good case (Myerson 1981), but only that posting a price
is IC and IR.
51