1290 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 5, OCTOBER 2011
Bilateral and Multilateral Exchanges for Peer-Assisted
Content Distribution
Christina Aperjis, Ramesh Johari, Member, IEEE, and Michael J. Freedman, Member, IEEE
Abstract—Users of the BitTorrent file-sharing protocol and its
variants are incentivized to contribute their upload capacity in a
bilateral manner: Downloading is possible in return for uploading
to the same user. An alternative is to use multilateral exchange to
match user demand for content to available supply at other users
in the system. We provide a formal comparison of peer-to-peer
system designs based on bilateral exchange with those that enable
multilateral exchange via a price-based market mechanism to
match supply and demand. First, we compare the two types of
exchange in terms of the equilibria that arise. A multilateral
equilibrium allocation is Pareto-efficient, while we demonstrate
that bilateral equilibrium allocations are not Pareto-efficient in
general. We show that Pareto efficiency represents the “gap”
between bilateral and multilateral equilibria: A bilateral equilibrium
allocation corresponds to a multilateral equilibrium
allocation if and only if it is Pareto-efficient. Our proof exploits the
fact that Pareto efficiency implies reversibility of an appropriately
constructed Markov chain. Second, we compare the two types of
exchange through the expected percentage of users that can trade
in a large system, assuming a fixed file popularity distribution.
Our theoretical results as well as analysis of a BitTorrent dataset
provide quantitative insight into regimes where bilateral exchange
may perform quite well even though it does not always give rise to
Pareto-efficient equilibrium allocations.
Index Terms—Asymptotic analysis, market equillibria, Markov
processes, peer-to-peer systems, random graphs.
I. INTRODUCTION
E
ARLY peer-to-peer systems did not provide any incentives
for participation, leading to extensive free riding:
Many peers were using the resources of other peers without
contributing their own [2], [18]. The peer-to-peer community
responded with mechanisms to prevent free riding by incentivizing
sharing on a bilateral exchange basis, as used by
BitTorrent [11] and its variants [32], [39], [35].
According to the BitTorrent protocol, each user splits his
available upload rate among users from which he gets the
highest download rates. As a result, an increase in the upload
rate to one user may increase the download rate from that
Manuscript received April 06, 2010; revised October 18, 2010; accepted January
01, 2011; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor
J. C. S. Lui. Date of publication March 10, 2011; date of current version
October 14, 2011. This work was supported in part by the National Science
Foundation under Grants 0428868 and 0904860.
C. Aperjis is with HP Labs, Palo Alto, CA 94304 USA (e-mail:
christina.aperjis@hp.com).
R. Johari is with Stanford University, Stanford, CA 94305 USA.
M. J. Freedman is with the Department of Computer Science, Princeton University,
Princeton, NJ 08540 USA.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNET.2011.2114898
particular user. However, it does not increase the download
rate from other users. Thus, two users are incentivized to
exchange only if each has content the other wants. This results
in a significant drawback of bilateral exchange: It breaks down
between users that do not have reciprocally desired files.
The difficulties of bilateral exchange (or barter) in an
economy have been long known, the most important being the
improbability of coincidence between persons wanting and
possessing. As discussed in [21], “there may be many people
wanting, and many possessing those things wanted; but to allow
of an act of barter, there must be a double coincidence, which
will rarely happen.” In modern economies, the aforementioned
difficulty is eliminated by the use of money. Money can enable
multilateral exchange by serving as a medium of exchange and
a common measure of value. Even though modern societies
take the use of money for granted, the same is not the case in
peer-to-peer systems.
Peer-to-peer systems could potentially also use market-based
multilateral exchange to match user demand for content to available
supply at other users in the system. This can be done by
using virtual currency and assigning a budget to each user that
decreases when downloading and increases when uploading.
Monetary incentives with a virtual currency have been previously
proposed to encourage contribution in peer-to-peer systems
[17], [38], [36], [6], [5]. However, such designs are usually
more complex than bilateral protocols and are not widespread.
The two system designs present a significant tradeoff: Bilateral
exchange without money is simple, while multilateral exchange
allows more users to trade. In this paper, we provide a
formal comparison of two peer-to-peer system designs: bilateral
barter systems such as BitTorrent and a market-based exchange
of content enabled by a price mechanism to match supply and
demand. Our main goal is to identify precisely what benefits a
currency-based system might offer and whether these benefits
are sufficient to actually warrant all the complexity of implementation
presented by such systems.
We start in Section II with a fundamental abstraction of content
exchange in systems like BitTorrent: exchange ratios. The
exchange ratio from one user to another gives the download
rate received per unit upload rate. Exchange ratios are a useful
formal tool because they allow us to define and study the equilibria
of bilateral exchange. In the model of bilateral exchange
we consider, each user optimizes with respect to exchange ratios.
In Section III, we define bilateral equilibrium as a rate
vector and a vector of exchange ratios, where all users have
simultaneously optimized given exchange ratios. We also define
multilateral equilibrium, where users optimize with respect
to prices; our definition of multilateral equilibrium is the same
as competitive equilibrium in economics [26]. In a multilateral
1063-6692/$26.00 © 2011 IEEEAPERJIS et al.: BILATERAL AND MULTILATERAL EXCHANGES FOR PEER-ASSISTED CONTENT DISTRIBUTION 1291
equilibrium, the presence of money enables a potentially wider
set of exchanges than is possible in bilateral equilibrium.
Our main results are the following.
1) Characterizing efficient equilibria. We compare bilateral
and multilateral peer-to-peer systems through the allocations
that arise at equilibria. A multilateral equilibrium
allocation is always Pareto-efficient, while bilateral equilibria
may be inefficient. Our main result is that a bilateral
equilibrium allocation is Pareto-efficient if and only if it is
a multilateral equilibrium allocation—in other words, efficient
bilateral equilibria must effectively yield “supporting
prices” as in a multilateral equilibrium. This result provides
formal justification of the efficiency benefits of multilateral
equilibria. The proof exploits an interesting connection between
equilibria and Markov chains: An important step of
the proof is to show that Pareto efficiency of a bilateral
equilibrium rate allocation implies reversibility of an appropriately
defined Markov chain, and that this chain has
an invariant distribution that corresponds to a price vector
of a multilateral equilibrium.
2) Quantifying the efficiency gap between bilateral and multilateral
exchange. From a practical standpoint, the preceding
insight is somewhat unsatisfying because it does not
quantify the benefits of multilateral exchange. Although all
efficient equilibria are multilateral equilibria, if the potential
loss of efficiency in bilateral equilibrium is small, then
it may be an acceptable tradeoff in return for a significantly
simpler system design. To address this issue, we perform
a quantitative comparison of bilateral and multilateral exchange
by quantifying how rarely a double coincidence of
wants occurs under different assumptions on the popularity
of files in the system.
a) Theoretical analysis. We first perform an asymptotic
analysis assuming that file popularity follows a power
law. We find that asymptotically all users are able to
trade bilaterally when the file popularity is very concentrated
(i.e., when the popularity distribution has a
relatively light tail). On the other hand, multilateral
exchange may perform significantly better than bilateral
exchange when the file popularity is not concentrated
(i.e., when the distribution has a heavy tail). Importantly,
we also find that increasing the number of
files that each user shares or wants by a small amount
can significantly improve the performance of bilateral
exchange.
b) Empirical validation. We complement our theoretical
analysis by studying file popularity from a large Bit-
Torrent dataset [33]. We find that on this dataset, bilateral
exchange may in general exhibit significant inefficiency
relative to multilateral exchange. However,
consistent with our theoretical observation, the gap
between bilateral and multilateral exchange can be
narrowed significantly if each user shares or is interested
in a sufficiently large number of files. For example,
for systems of the size in the dataset, over 96%
of users can trade bilaterally if each user shares at least
10 files. The last result is informative: It suggests that
taking small steps to increase the number of bilateral
matches possible can actually significantly eliminate
almost all the advantage of multilateral exchange.
The remainder of the paper is organized as follows.
Section IV characterizes efficient equilibria. Section V provides
both a theoretical and an empirical quantification of the
efficiency gap between bilateral and multilateral exchange.
Section VI discusses the related literature, and Section VII
concludes the paper.
II. EXCHANGE RATIOS IN BILATERAL PROTOCOLS
Many peer-to-peer protocols enable exchange on a bilateral
basis between users: A user uploads to a user if and only if
user uploads to user in return. Of course, such an exchange
is only possible if each user has something the other wants;
this is known as “double coincidence of wants” in economics.
The foremost examples of such a protocol are BitTorrent and its
variants. While such protocols are traditionally studied solely
through the rates that users obtain, this section provides an interpretation
of these protocols through exchange ratios. As exchange
ratios can be interpreted in terms of prices, these ratios
allow us to compare bilateral barter-based peer-to-peer systems
with multilateral price-based peer-to-peer systems.
Let denote the rate sent from user to user at a given
point in time in a bilateral peer-to-peer protocol. We define the
exchange ratio between user and user as the ratio
; this is the download rate received by from per unit of
rate uploaded to . By definition, . Clearly, a rational
user would prefer to download from users with which he has
higher exchange ratios.
The exchange ratio has a natural interpretation in terms of
prices. In particular, assume that users charge each other for
content in a common monetary unit, but that all transactions are
settlement-free, i.e., no money ever changes hands. In this case,
if user charged user a price per unit rate, the exchange of
content between users and must satisfy
We refer to as the bilateral price from to . Note that the
preceding condition thus shows the exchange ratio is equivalent
to the ratio of bilateral prices:
(as long as the
prices and rates are nonzero).
What is the exchange ratio for BitTorrent? A user splits his
upload capacity equally among those users in his active set from
which he gets the highest download rates. Let be the size of
the active set. Suppose all rates that user receives from
users are fixed, and let be the th highest rate that
receives. Let be the upload capacity of user . Then,
depends on . In particular
if
otherwise.
Thus, for BitTorrent, the exchange ratio is
if
user is in the active set, and zero otherwise. Note that the exchange
ratios and may be different for two users
in ’s active set.
The exchange ratio decreases with as long as user
is in user ’s active set (in which case is constant). Hence, a1292 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 5, OCTOBER 2011
strategic user would prefer to choose as small as possible
while remaining in ’s active set. This behavior is exactly the
approach taken by the BitTyrant [32] variation on BitTorrent. In
fact, if all users follow this policy, then for all users
in ’s active set. Note that in this case,
. Thus,
user has the same exchange ratio to all users with which he
bilaterally exchanges content.
The preceding discussion highlights the fact that the rates in a
bilateral peer-to-peer system can be interpreted via exchange ratios.
Thus far, we have assumed that transfer rates are given, and
exchange ratios are computed from these rates. In Section III,
we turn this relationship around: We explicitly consider an abstraction
of bilateral peer-to-peer systems where users react to
given exchange ratios, and we compare the resulting outcomes
to price-based multilateral exchange.
III. BILATERAL AND MULTILATERAL EQUILIBRIA
In this section, we define bilateral equilibrium (BE) and
multilateral equilibrium (ME), i.e., the market equilibria corresponding
to bilateral and multilateral exchange. In the formal
model we consider, a set of users shares a set of files .
User has a subset of the files and is interested in
downloading files in
. Throughout, we use
to denote the rate at which user uploads file to user .We
then let be the rate at which user downloads
file . We denote the vector of download rates for user by
. Finally, let be the total
upload rate of user . We measure the desirability of a rate
vector to user by a utility function , according to
the following assumption. This assumption remains in force
throughout the paper.
Assumption 1: The preference relation of a user on the set
of feasible rate vectors is represented by a continuous strictly
concave utility function , which is strictly increasing
in each download rate for all and strictly decreasing
in the upload rate . We further assume that has finite
derivative everywhere with respect to all .
Note that utility functions in our model depend on instantaneous
transfer rates rather than the number of bytes exchanged.
This is consistent with the “snapshot” view that our model of
peer-to-peer file sharing adopts: Informally, it is as if we are analyzing
efficiency of the system at a fixed moment in time. This
is also why our model keeps fixed both the set of files available
for upload and the set of files desired for download at a given
user; these sets remain constant on the timescale we are considering.
An important open direction related to this work concerns
the analysis of dynamic efficiency, where these sets might
change over time.
Each user is assumed to have a constraint on the available
upload rate. Let denote this upper bound for user . We assume
that users do not face any constraint on their download
rate; this is consistent with most end-users’ asymmetric connections
today, where upload capacity is far exceeded by download
capacity. Furthermore, for the purposes of this paper, we also assume
that there are no constraints in the middle of the network,
though our prior work suggests a natural approach for including
such constraints [5].
Let
be the set of feasible rate vectors. In particular, this ensures
that: 1) all rates are nonnegative; 2) users only upload files they
possess; and 3) each user does not violate his upload capacity
constraint.
In Sections III-A and III-B, we look at two different models
of equilibrium, corresponding to exchange in bilateral and multilateral
systems, respectively. In bilateral exchange, we assume
that users maximize their utility given the exchange ratios they
see to other users. Thus, in a bilateral equilibrium, the exchange
ratios must be chosen to exactly balance the rates each user considers
optimal—this is informally the condition that “supply
equals demand.” In multilateral exchange, on the other hand,
we assume that users can earn currency by uploading to others,
and they can spend that currency in downloading from any users
they wish. In this model, users maximize their utility given the
current prices in the system, and the prices must be set to exactly
balance the rates each user considers optimal. Thus, in both
types of exchange, “supply equals demand” at an equilibrium or,
equivalently, the market clears.
A. Bilateral Equilibrium
We start by considering users’ behavior in bilateral schemes,
given a vector of exchange ratios . User solves
the Bilateral Peer Optimization problem given in Fig. 1. 1
In this optimization problem, in addition to the definition of
upload and download rates, each user faces one constraint for
each potential peer with which might exchange content: the
restriction that
ensures that the rate at
which downloads from is exactly the exchange ratio times
the rate at which uploads to .
We can now define bilateral equilibrium.
Definition 1: The rate allocation and the exchange ratios
, with for all , constitute a
BE if, for each user solves the Bilateral Peer Optimization
problem given exchange ratios .
Definition 1 requires that: 1) all users have optimized with
respect to the exchange ratios; and 2) the market clears. Even
though the market clearing condition is not explicitly stated, it
is implicitly required since the same vector is an optimal
solution of the Bilateral Peer Optimization problems of all users.
The following proposition shows that a BE exists.
Proposition 1: A BE exists.
B. Multilateral Equilibrium
By contrast, in a multilateral price-based exchange, the
system maintains one price per user, and users optimize with
1 Note that we allow users to bilaterally exchange content over multiple files.
This is partly supported by swarming systems like BitTorrent through bundles
[28]. BitTorrent also has a mixed bundling mechanism, which could be adapted
to allow trade across files in a bundle (but currently is not flexible).1294 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 5, OCTOBER 2011
A user serves requests sequentially without preemption and updates
its price according to the mismatch between requests received
and available capacity. Simulations suggest that such
price dynamics converge for a variety of topologies in the case
of multilateral exchange [5]. The same approach can be applied
to BE by considering exchange ratios instead.
IV. EFFICIENCY OF EQUILIBRIA
This section rigorously analyzes the efficiency properties of
bilateral and multilateral exchange. We assume users explicitly
react to exchange ratios or prices, and we compare the schemes
through their resulting equilibria. In order to proceed, we first
formally define Pareto efficiency.
Definition 3: Given rate allocations
, let
be the corresponding download rates, and let be the
corresponding upload rates. Then, Pareto-dominates if
for all , with strict inequality for at least
one .
A rate allocation is Pareto-efficient if it is not Paretodominated
by any other rate allocation .
Thus, a rate allocation is Pareto-efficient if there is no way to
increase the utility of some user without decreasing the utility
of some other user. An ME allocation is always Pareto-efficient;
this is the content of the first fundamental theorem of welfare
economics [26]. For completeness, we include the result here.
Proposition 3: If the rate allocation and the user prices
with for all constitute an ME, then
the allocation is Pareto-efficient.
BE may not be Pareto-efficient. Inefficiencies may arise because
trade does not occur at a BE, while users do trade at an
ME of the same system. Moreover, even when all users trade at a
BE, the allocation may not be Pareto-efficient, as the following
example shows.
Example 1: Consider a system with users and files, for
. Each user has file and wants files and . The
utility of user is
, i.e., user wants the files of both user and
user , but derives a higher utility from the file of user .
We first consider a symmetric BE with exchange ratios
and
. The equilibrium rates are
and
, and the download rates are
and . The utility of each user is
. On the other hand, prices for all , and
rates
constitute an ME. The utility
of each user is
, i.e., significantly larger than
the utility of a user at the BE. This demonstrates that the BE
allocation is not Pareto-efficient.
The previous example shows that BE may not be Pareto-efficient.
By changing the utility function of a user in this example,
we next provide an example of a BE rate allocation that
is Pareto-efficient.
Example 2: Consider a system with users and files, for
. Each user has file and wants files and . The
utility of user is
.
We consider a symmetric BE with exchange ratios
and . The equilibrium rates are and
. The BE rate allocation is Pareto-efficient. In particular,
it corresponds to an ME: Prices for all , and
rates
constitute an ME.
Thus, BE may be inefficient, while ME always has Pareto-efficient
allocations (Proposition 3). In Example 2, the BE rate allocation
is Pareto-efficient and corresponds to an ME. Our main
result is that a BE allocation is Pareto-efficient if and only if it is
an ME allocation. In particular, if a BE allocation is Pareto-efficient,
then there exist “supporting prices,” i.e., prices such that
the BE rate allocation is optimal for the Multilateral Peer Optimization
problem of each user. Informally, Pareto efficiency
represents the “gap” between BE and ME.
Proposition 4: Assume that for every user and any fixed
as . Let be a BE. The rate
allocation is Pareto-efficient if and only if there exists a price
vector such that and constitute an ME.
Proposition 4 assumes that as for
every user and every fixed . This assumption ensures that
the total upload rate of a user is strictly smaller than his upload
capacity at the BE. This is a reasonable assumption for a peer-topeer
setting since we do not expect users to use all their upload
capacity. We note that if the total upload rate of a user is equal
to his upload capacity, then there may exist Pareto-efficient BE
that do not correspond to ME simply because users have already
“maxed out” their available upload capacity.
We provide an overview of the proof of Proposition 4, which
demonstrates an interesting connection between equilibria
and Markov chains; the details of the proof are provided in
the Appendix. From a BE rate allocation , we construct a
transition rate matrix for a continuous-time Markov chain,
such that if , and .We
first observe that implies that the multilateral budget
constraint is satisfied with price vector . Therefore, for any
invariant distribution is feasible for the Multilateral Peer
Optimization problem of every user when prices are equal to
. We then show that if is also Pareto-efficient, there exists
an invariant distribution of , say , such that is an optimal
solution of the Multilateral Peer Optimization problem of each
user when the prices are equal to . We conclude that and
constitute an ME.
A key step of the proof is to show that Pareto efficiency of
implies reversibility of . This is proven by contradiction: If
is not reversible, then we can find a cycle of users that can
change their rates only along successive pairs of users on the
cycle and, in doing so, make all their utilities strictly higher.
Now let be an invariant distribution of with all entries
positive. If the matrix is reversible, then for all
pairs of users and that trade at the BE. We conclude that if
is Pareto-efficient, then solves the Multilateral Peer Optimization
problem for each user given prices if the user is
restricted to trade with peers he trades with at the BE. Much
of the complexity in the proof is to show that this result holds
even if user is not restricted to trade only with those users it
transacts with at the BE.
The matrix corresponding to the BE allocation of Example 1
is not reversible, which implies that the BE allocation is not
Pareto-efficient. On the other hand, the matrix corresponding to
the BE allocation of Example 2 is reversible, and the BE allocation
is Pareto-efficient and corresponds to an ME allocation.APERJIS et al.: BILATERAL AND MULTILATERAL EXCHANGES FOR PEER-ASSISTED CONTENT DISTRIBUTION 1295
In closing, we note that if the rate matrix corresponding to an
ME is reversible, then the ME allocation is a BE allocation and
can be realized through bilateral trade. In particular, from an ME
rate allocation , we construct a transition rate matrix for a
continuous-time Markov chain as described. If is reversible,
then there exists an invariant distribution with all entries positive
such that for all . Setting for
all , we conclude that is a BE.
V. ABILITY TO TRADE
Propositions 1 and 2 show that both BE and ME exist under
general conditions (in particular if Assumption 1 holds). However,
not all users are guaranteed to trade at these equilibria. For
instance, if user does not have reciprocally desired files with
any other user (that is, for any user , either or
), at a BE is sufficiently low for all that have
files desired by so that does not trade. In this section, we
compare bilateral and multilateral exchange through the corresponding
percentages of users that can trade. Though distinct
from Pareto efficiency, this metric provides quantitative insight
into the comparison of the two types of exchange. In particular,
we expect that systems that perform well will also generally encourage
high levels of participation. We characterize regimes
where bilateral exchange performs very well with respect to this
metric, and for which, as a result, it may not be worth the effort
to use multilateral exchange.
In Section V-A, we introduce the framework we use to study
the percentage of users that can trade bilaterally and multilaterally.
Our analysis is based on a random model, where we assume
file popularity follows a power law. In Section V-B, we
carry out an asymptotic theoretical analysis as the number of
users and files grows large. In Section V-C, we complement our
theoretical analysis by studying file popularity from a large Bit-
Torrent dataset. Here, we find that the ability to trade bilaterally
improves significantly if each user shares or is interested in a
sufficiently large number of files.
A. Framework
1) Definitions: We start by formally defining the quantities
we compare. For a given peer-to-peer system, we define the
system profile to consist of the specification of which files each
user desires and possesses, i.e., .
We say that user can trade bilaterally under if there exists
some user such that and , that is, and
have reciprocally desired files. Given a system profile , let
be the percentage of users that cannot trade bilaterally.
Similarly, we say that user can trade multilaterally under
if there exist users
such that
for and . In other
words, user is able to trade multilaterally if and only if there
exists a cycle of users starting (and ending) at such that each
user possesses a file that is desired by the next user in the cycle.
Clearly, if user can trade bilaterally under , then he can also
trade multilaterally under . Let be the percentage of
users that cannot trade multilaterally.
Note that whether or not a user actually trades in equilibrium
depends on the specific utility functions chosen. For example,
if the marginal utility for downloading is sufficiently low,
the optimal decision for a user may be to not download at all,
even if he can. However, the definition above ensures that if
a user cannot trade bilaterally (resp., multilaterally), then this
user never trades in a BE (resp., ME), regardless of the utility
functions.
2) Random Model: We assume that the system profile is
chosen according to some distribution that depends on the popularity
of different files, and that the sets and are chosen
independently for each user . We denote by the popularity
of the th file and assume that the probability that the th file
is desired or possessed by a user is proportional to . We assume
that each does not depend on the number of files in
the system. On the other hand, the probability that the th file is
chosen clearly depends on the number of files in the system
since it is .
We are interested in comparing the expected proportions of
users that cannot trade bilaterally and multilaterally—that is,
the expected values of and —for given file
popularities.
B. Asymptotic Analysis
This section theoretically compares the two types of exchange
through the expected percentages of users that cannot trade.
We focus on large systems and consider the asymptotic regime
where the number of files and users in the system becomes large.
We assume the files that users possess and desire are drawn
independently from a Zipf file-popularity distribution that is
identical for each user. Our motivation to study this distribution
comes from the fact that Zipf’s law has been observed in many
settings and has been suggested as a good model for file popularity
(e.g., [1], [4], and [10]). 5 Zipf’s law states that the popularity
of the th largest occurrence is proportional to a power of
its inverse rank. We adjust this definition to our setting.
Definition 4: File popularity has a Zipf distribution with exponent
if the th most popular file has popularity .
Note that corresponds to the uniform distribution. On
the other hand, the distribution becomes more concentrated as
increases.
Recall that we are interested in the expected percentage of
users that cannot trade. This is a function of the number of
users , the number of files , and the Zipf exponent . Let
and
be the expected percentages
of users that cannot trade bilaterally and multilaterally, respectively.
In particular, (resp., ) is the
expected value of (resp., ) over system profiles.
We consider a sequence of peer-to-peer systems indexed by
. The th system has users and files, where
is a nondecreasing function of . The function represents
how the number of files scales with the number of users.
For simplicity, we suppress the dependence of on .We
study an asymptotic regime where .
Since the number of users that cannot trade bilaterally is
always greater than or equal to the number of users that cannot
trade multilaterally, we have .
The following propositions imply that in a large system,
5 Gummadi et al. find that peer-to-peer file popularity follows a flattened Zipflink
distribution [16]. However, the Zipf distribution is still the closest approximation
for which analytical work is possible.1296 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 5, OCTOBER 2011
may be significant when ,
but is always negligible when .
Proposition 5: If , then as
for any nondecreasing .
Since
, we conclude that if
files are chosen according to a Zipf distribution with ,
then both and as
. Thus, when , bilateral exchange performs very well
asymptotically for any scaling of and . This result holds
regardless of the number of files that users possess or desire.
For details, see the proof of Proposition 5 in the Appendix. We
note that the result of Proposition 5 can be generalized to all
popularity distributions for which
; the Zipf distribution
with is of course a special case.
This is an interesting result: Even though bilateral exchange
significantly restricts trade compared to multilateral exchange,
almost all users can trade in expectation under both types of
exchange when the system is large and file popularity follows
a Zipf distribution with exponent . The intuition behind
this result is that when is large, the popularity distribution is
more concentrated, i.e., the most popular files are chosen with
relatively high probability. As a result, for any user , both
and probably consist of one of the most popular files, and it
is more likely that there exists a user such that and have
reciprocally desired files.
When , the asymptotic behavior can be quite different,
as the following proposition shows.
Proposition 6: Assume , and for
all . As , we have the following.
i) If , then
.
ii) If , then .
iii) If , then .
iv) If , then .
The case where scales slower than but faster than
is of particular interest. In this case, according to
Proposition 6,
and
as . That is, when the system is large, almost all users
can trade multilaterally, but a constant proportion of users
cannot trade bilaterally. Thus, for this case, multilateral exchange
performs significantly better than bilateral exchange in
terms of the ability of users to trade. We note that if , in
this regime
; that is, almost all users can
trade multilaterally, but cannot trade bilaterally.
By contrast, when and scales faster than ,
both bilateral and multilateral exchange perform well. On the
other hand, if scales slower than , then neither type of exchange
performs well. We note that Proposition 6 has a small
gap since it does not say how well multilateral exchange performs
when scales faster than yet slower than .
Fig. 2 shows the percentages of users (from simulations)
that cannot trade bilaterally when popularities follow a Zipf
distribution for various values of the exponent , assuming
that users desire and possess one file. (We assume the system
consists of 7323 files, as this is the number of files in the dataset
considered in Section V-C.) We observe that bilateral exchange
does not perform well when the exponent is small, which
agrees with our theoretical results. As the exponent increases,
the performance of bilateral exchange improves. When the
Fig. 2. Percentages of users (from simulations) that cannot trade bilaterally
when file popularities follow a Zipf distribution with exponent
. Users desire and possess one file, i.e.,
for all users . There are 7323 files in the system, and the
number of users is shown on the horizontal axis.
exponent is greater than one, bilateral exchange performs
reasonably well.
In [7], we show that if , then the conclusions of
Proposition 6 hold even if users possess and desire multiple
files. In fact, the proof of Proposition 7 in [7] shows that when
each user possesses files and desires files, there exist
constants and such that
for all sufficiently large and . Thus, to first order, the exponent
scales like
. This suggests that small increases
in the number of files that agents are willing to trade
can lead to significant improvements in system performance. Indeed,
we observe precisely this phenomenon in the Section V-C
analysis.
C. Data Analysis
This section quantitatively compares bilateral and multilateral
exchange using data on BitTorrent peer-to-peer file-sharing
collected by Piatek et al. [33]. We find that a significant percentage
of users cannot trade bilaterally when each user is
sharing one file. However, the percentage becomes negligible
as users share more files. We conclude by discussing this
finding’s implications for the design of peer-to-peer content
exchange systems.
The dataset consists of 1 364 734 downloads, 679 523 users,
and 7323 files. 6 We use the number of downloads of each file
in the dataset to estimate the popularities of different files. We
thus abstract from the details of the specific BitTorrent trace
and only use the information on the preferences of the users
for different files in order to compare bilateral and multilateral
exchange through simulations.
The estimated popularities are shown in Fig. 3. As before,
we assume that the probability that a given file is selected is
6 We equate files to torrents, and neglect bundles, as a first-order approximation.APERJIS et al.: BILATERAL AND MULTILATERAL EXCHANGES FOR PEER-ASSISTED CONTENT DISTRIBUTION 1297
Fig. 3. Popularity of each unique file ( for all files )—that is, the number
of times file was downloaded—shown in decreasing popularity on a log–log
scale.
Fig. 4. Percentages of users (from simulations) that cannot trade bilaterally,
trilaterally, and multilaterally when users desire and possess one file, i.e.,
for all . The horizontal axis shows the number of users in the system.
proportional to its popularity. We then use these probabilities to
generate system profiles and compute the percentages of users
that cannot trade bilaterally and multilaterally. We assume that
there are 7323 files with the given distribution and vary the
number of users in the system. 7
The algorithm we use to compute is exact. For every
user , we check whether there is some user such that and
have reciprocally desired files. On the other hand, computing
the exact value of for a large system appears computationally
intractable. Therefore, we use an approximation: We recursively
remove users that either possess files not desired by others
or desire files not possessed by others since such users cannot
trade multilaterally. Simulations for small numbers of users suggest
that this algorithm provides a very good approximation for
. 8
Bilateral and Multilateral Trade: We first assume that each
user possesses and desires exactly one file, i.e.,
for every . Fig. 4 shows the percentages of users that
cannot trade bilaterally and multilaterally from simulations for
various numbers of users in the system. We observe that a significant
majority of users cannot trade bilaterally, while nearly
all users can trade multilaterally. Finally, as the number of users
increases, the percentages of users that can trade increase for
both bilateral and multilateral exchange.
Trading Trilaterally: Fig. 4 also shows the percentage of
users that cannot trade in triangles, i.e., triples , where
uploads to uploads to , and uploads to . 9 We observe
that a very large percentage of users is able to trade in triangles
when there are at least 600 000 users in the system. 10
Fig. 5. Percentages of users (from simulations) that cannot trade bilaterally
when each user desires one file
and possesses multiple files. The
legend shows for each line. The horizontal axis shows the number of users
in the system. In the top graph, all users possess the same number of files
. In the bottom graph, we consider the case where different
users possess different numbers of files, where this number is drawn from the
dataset distribution (denoted by ).
7 We have also considered the case that the number of users is fixed and the
number of files varies; as expected, performance degrades as increases.
8 For instance, suppose there are 1000 users and 200 files in the system whose
popularities are equal to the popularities of the 200 most popular files of the
dataset. In 100 simulations, 972 users can trade multilaterally on average, while
our heuristic finds that 976 users can trade multilaterally on average (99.6%
accuracy).
9 We estimate by sampling at least a few thousand users and using an
exact algorithm to compute the proportion of the sampled users that cannot trade
in triangles.
10Analytically, it can be shown that if we allow trade in triangles, then
performance under uniform file popularity (i.e., ) is good as long as
as (see [7, Appendix, Prop. 8]). This is a
significant improvement on the corresponding result for bilateral equilibrium.
Uploading (or Downloading) Multiple Files: We next assume
each user desires one file and possesses multiple files. As the
number of files that each user has increases, the number of possible
trades increases, and as a result the percentage of users
that can trade bilaterally increases. In Fig. 5 (top graph), we
show the percentages of users that cannot trade bilaterally when
each user desires one file and possesses multiple files
. In these experiments, all users possess
the same number of files, i.e., for all (the
case of is already shown in Fig. 4). Note that, by symmetry,
we get exactly the same graph if users possess one file
and desire multiple files .1298 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 5, OCTOBER 2011
From these simulations, we observe a significant decrease in
the percentage of users that cannot trade when increases
from 1 to 20 (resp., when and increases from 1 to
20). We can illustrate this by considering the minimum required
number of users in the system so that at most 10% are not able
to trade: 1 000 000 users are required when each user has five
files, but only 50 000 users are needed when each user possesses
20 files.
Distribution of Across Users: Our simulations up to now
have assumed that all users in the system possess the same
number of files, i.e., for all . We next assume
that the number of files that users possess varies across different
users, inferring this distribution for from the dataset. 11 We
are interested in whether the percentage of users that can trade
bilaterally increases as the variance of the distribution of
increases (assuming that the mean remains the same). At first,
it may seem plausible that users with very large would be
able to accommodate a lot of trades, and as a result, should
increase as the ’s become more dispersed. However, this is
not the case, as we discuss next.
The dataset shows that most users, in fact, possess only a few
files. Only 32% of users possess more than a single file, and only
2% of users possess more than 10 files. There are, however, a
few users that have more than 400 files. Since the mean value
of in the dataset is 2.0084, we are interested in whether
increases compared to the case that for all . This
comparison is shown in Fig. 5 (bottom graph).
We observe that when is drawn from the dataset distribution
, the percentages of users that cannot trade bilaterally
are between the cases of and , even though
the expected value of is slightly greater than 2. This
occurs because, even though some users have a large number
of files and are thus more likely to be able to trade bilaterally
when the distribution of is more dispersed, the percentage
of users that only have one file also increases. Moreover, since
each user desires one file, the probability that a user that has one
file is matched with a user with multiple files does not significantly
increase.
VI. RELATED WORK
In this paper, we have provided a formal comparison of
peer-to-peer system designs, and have studied the advantages
and disadvantages of bilateral and multilateral exchange.
Menasché et al. investigate direct and indirect reciprocity in
peer-to-peer systems [27], which correspond to bilateral and
multilateral exchange in our model. They upper bound the efficiency
loss of direct reciprocity assuming that users are willing
to download files they do not desire for bartering purposes.
They also consider dynamic scenarios in simulations. On the
other hand, we compare bilateral and multilateral exchange
through equilibrium outcomes and through the expected percentage
of users that can trade, and we do not assume that users
download files they do not desire. DeFigueiredo et al. also
consider direct and indirect reciprocity [12], but do not focus
on comparing the two.
11 We assume that the number of files that a user possesses is equal to the
number of files he downloads in the dataset; note that this may be a bit optimistic,
as it ignores the possibility of deletions over time from a user’s set of shared files.
The “gap” between bilateral and multilateral exchange in
terms of both efficiency and complexity has motivated the
study of incentive mechanisms that lie between the two types
of exchange in terms of both metrics. Through trace-driven
analysis and measurements of a deployment on PlanetLab,
Piatek et al. find that allowing trades to pass through one
intermediary improves performance and incentives relative to
BitTorrent [33]. Liu et al. study a similar mechanism assuming
that peers belong to an underlying social network [25]. Finally,
the performance implications of bundling have been considered
[28].
Our work is also related to the study of equilibria in
economies where not all trades are allowed. Kakade et al.
introduce a graph-theoretic generalization of classical
Arrow–Debreu economics, in which an undirected graph
specifies which consumers or economies are permitted to engage
in direct trade [22]. However, the inefficiencies of bilateral
exchange do not arise in their model. The monetary economics
literature has long studied how money reduces the double
coincidence problem. The implementation of a competitive
equilibrium is a central theme in this literature. The superiority
of monetary exchange has been studied [37], and dynamics of
bilateral trading processes have been considered [29], [13]. The
transactions role of money is surveyed in [40].
Finally, as discussed in the Introduction, we note that a
number of studies consider incentives in peer-to-peer systems
(e.g., [3], [9], [11], [14], [15], [24], [33], [38], [40]). Our paper
contributes to this broad line of literature.
VII. CONCLUSION
This paper provides a formal comparison of two peer-to-peer
system designs: bilateral barter systems such as BitTorrent, and
a market-based exchange of content enabled by a price mechanism
to match supply and demand. Our results demonstrate that
even though bilateral equilibria are not Pareto-efficient in general,
bilateral exchange may perform very well in terms of the
expected percentage of users that can trade for certain file probability
distributions. Moreover, our data analysis shows a significant
increase in the percentage of users that can trade bilaterally
when each user shares or desires multiple files. Informally,
the fact that BitTorrent breaks files into chunks greatly increases
the number of matches possible, leading to performance gains
similar to those described in Section VI. More generally, our
insight suggests that bilateral incentives in BitTorrent could be
made even stronger if the protocol considered exchanges across
different files, rather than restricting exchange in a single swarm
to chunks of the same file. 12
We conclude by noting that our paper has considered a static
“snapshot” view of a file-sharing system. This is complemented
by our earlier work in [5], where we considered a system design
for multilateral exchange in a dynamic setting. In a dynamic
system, the number of simultaneous matches possible may be
quite small, even if all content possessed by the users is available
for upload. In such systems, money plays another important
role: It can act as a store of value (e.g., see [21]) over time, allowing
a user to upload now and earn the ability to download
12 One can think of this as a more flexible version of mixed bundling, adapted
to allow trade across files in a bundle.APERJIS et al.: BILATERAL AND MULTILATERAL EXCHANGES FOR PEER-ASSISTED CONTENT DISTRIBUTION 1299
later. Our design in [5] leverages this advantage by designing
a system where pricing mechanisms serve only as algorithmic
devices to ensure efficient exchange; to keep the system simple,
we never expose prices directly to the end-user. 13 However, in
our system design, currency functions as a store of value and allows
dynamic trades over time. Quantifying this advantage (in
the sense of Section V) remains an important direction for future
work.
APPENDIX
Note: Throughout the Appendix, we write if all components
of are positive.
Proof of Proposition 4: Define
, the total
rate that user sends to user . We define the matrix such that
if , and . By construction, is
a transition rate matrix of a continuous-time Markov chain with
no transient subclasses since implies that (by
the definition of BE). In what follows, we consider the communicating
classes of : If , then users and are in the
same communicating class. For the purposes of this proof, let
be the set of users with which trades under , i.e.,
. Note that is a subset of
the communicating class containing . Finally, a transition rate
matrix is called reversible if for every cycle of distinct nodes
, there holds:
; i.e., the product of transition rates
is the same in each direction around the cycle. If a matrix is
reversible, and is a strictly positive invariant distribution, i.e.,
and , then the detailed balance conditions hold:
For all
. See [23] for details.
Let be an invariant distribution of , i.e., . We observe
that implies that the budget constraint in the Multilateral
Peer Optimization problem is satisfied with prices .
Therefore, for any invariant distribution is feasible for
the Multilateral Peer Optimization problem of every user when
prices are equal to . We show that if is Pareto-efficient, then
for some invariant distribution of and constitute an
ME. In particular, we show that for each user solves the
Multilateral Peer Optimization problem under .
This is done in three steps. First, we show that if is Paretoefficient,
then corresponds to a reversible Markov chain. This
implies that if is an invariant distribution of with all components
strictly positive, then whenever , and
as a result solves the Multilateral Peer Optimization problem
of user given prices if user is restricted to trade with users in
(Step 1). We then show that if user is restricted to trade
with users in the same communicating class under prices ,
then is an optimal solution of the Multilateral Peer Optimization
problem (Step 2). Step 2 completes the proof if consists
of one communicating class. Finally, we show that if there are
multiple communicating classes, there exists an invariant distribution
(derived as a convex combination of the invariant
distributions corresponding to the communicating classes) such
that is an optimal solution of the Multilateral Peer Optimization
problem of each user (Step 3). We show each of these steps
by demonstrating that if the desired conclusion of the step does
13 See recent work in [34] for a similar approach in peer-to-peer distributed
backup.
not hold, then there exists a rate vector that Pareto-improves
—violating the assumption that is Pareto-efficient.
Before beginning the proof, we derive a simple condition
that allows us to test whether a new rate vector improves
user ’s utility. Suppose solves the Bilateral Peer Optimization
problem of user under . Let and
be the corresponding download and upload rates for user .
Consider a rate allocation where and are
the corresponding download and upload rates for user . Fix
a file , and assume that for all files .
If and are sufficiently small, we can use
Taylor’s approximation to conclude that user is strictly better
off under if
Let . If user is getting file from
user at the BE, it must be that . Substituting in the
other constraints of the Bilateral Peer Optimization problem of
user (given in Fig. 1), we conclude that .
Thus, user wishes to maximize . The
first-order optimality conditions (which are necessary and sufficient
since the objective is concave) and the fact that
yield that
whenever . (Here, we use the fact that
as to ignore the constraint .) Combining
(2) with (1), we see that user strictly prefers to if
assuming that and are sufficiently small. Thus,
if , and we increase the download rate of file to user
as well as the total upload rate of user such that the previous
condition holds, then user is strictly better off.
Step 1. If is Pareto-efficient, then is reversible. Furthermore,
if is an invariant distribution of , then solves
the Multilateral Optimization Problem of user given prices
if user is restricted to trade only with the users in . Let
be a strictly positive invariant distribution of , i.e.,
and . If is reversible, then the detailed balance equations
hold for every , i.e., . We note that
if , then it must be that , so the detailed balance
equation trivially holds for and . On the other hand, the budget
constraint of the Bilateral Peer Optimization problem of user
implies that . We conclude that is reversible if
and only if whenever .
We show that Pareto efficiency of implies reversibility of
. Assume that is not reversible. Then, for
some with . Since is an invariant distribution,
, and thus . On the other
hand, the budget constraint of the Bilateral Peer Optimization
(1)
(2)
(3)1300 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 5, OCTOBER 2011
problem of user implies that . Summing over
and substituting, we conclude that
We first assume that as . Then,
is the sum of a finite number of terms, each
of which approaches 0 as . Thus,
as .
Now, assume that as , and let
If for some with , the previous equation
implies that there exists some user such that
and . Without loss of generality, we relabel
to be , and to be . Then, .
Applying this reasoning recursively, we can find a sequence of
users such that and
for all .
We show how the utility of each user in
can increase while the rate allocation to users outside remains
the same. In particular, we increase and by for all
. We note that users’ upload capacity constraints do not
bind (i.e., remain inactive) at the BE, a consequence of the assumption
that as . Therefore, it is feasible
to slightly increase the upload rates of all users. Applying
(3), user is better off if
We observe that as . Since
is finite. We have
Let
Since , it follows that . Then,
it is possible to make all users in the set better off by, e.g.,
choosing and small enough, and setting
, for all .
We conclude that if is the rate allocation of a BE and is
Pareto-efficient, then is reversible. Furthermore, if is
an invariant distribution of , then whenever
. This means that solves the Multilateral Peer Optimization
problem of user given prices if he is restricted to trade with
users in . The remainder of the proof shows that there
exists an invariant distribution such that is optimal for the
Multilateral Peer Optimization problem under . Due to space
limitations it is provided in [7].
Proof of Proposition 5: We first show the result when
for all users. Let
Since is finite and , it suffices to show that
as . We observe that for any fixed
The first term approaches zero as ; the second term does
not depend on , and approaches zero as . Thus, first
taking the limit as , then taking the limit as ,
the result follows.
We have shown the result for the case that
for
all users . When users desire or possess more files, the chance
of bilateral exchange increases, so again
as
.
The following result is used in the proof of Proposition 6.
Lemma 1: If and , then
(4)
The probability that and is equal to
Proof of Lemma 1: Let
Peer cannot trade bilaterally if there exists no user such that
and . The event and
occurs with probability
(since there are
users to choose from). Since is chosen independently
for each user , the expected percentage of users that cannot
trade bilaterally when there are files and users satisfies
(5)
It suffices to show that for . We observe that
and
for . This completes the proof.
Proof of Proposition 6: We follow the same notation as in
the proof of Proposition 5. In particular, we define as in
(4) and conclude that is given by (5).APERJIS et al.: BILATERAL AND MULTILATERAL EXCHANGES FOR PEER-ASSISTED CONTENT DISTRIBUTION 1301
We observe that
We next show ii). By Lemma 1
as
We now show iii). We observe that a user cannot trade multilaterally
if he has a file that no user wants. Thus
Thus
We first show i). Let
have
In the previous inequalities, we set
the upper bound in (6).
Define
(6)
.We
(7)
and use
A similar argument to the argument used in i) shows that
Finally, we show iv). Our proof exploits a connection to
classical Erdös–Rényi random graphs. Throughout, we assume
without loss of generality that
. Let
denote a graph drawn uniformly at random from
the possible directed graphs on nodes with
edges. We let
denote a random multigraph on
nodes with edges, where each edge is independently
placed from to with probability . Typically, will be a
probability distribution. However, in the subsequent analysis it
is convenient if we allow the possibility .
If , then we assume that each edge is not placed at all
with probability .
Now consider a random system with users and files,
where each user desires and has one file. For each user , draw
an edge from to if user wants file and has file . This
is exactly the random multigraph
described in the
preceding paragraph, with
Using the lower bound in (6) and the fact that
as , we have as . Also
observe that since the cardinality of is at least ,
we have
It follows from (7) that
Since
was arbitrary, the result follows.
where is defined as in (4).
Furthermore, observe that if
is strongly connected,
then all users can trade multilaterally. This follows
because if a user has and wants , then there is an edge from
to in . If is strongly connected,
then there must exist a path from to as well. This path
identifies a collection of users that, together with user , form a
cycle. Thus, user can trade multilaterally. It suffices to show
that with probability approaching 1 as
is strongly connected. (Convergence in probability implies
convergence in expectation, as is bounded.)
(8)1302 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 5, OCTOBER 2011
It is known that if , then
is connected as [31],
where we use “connected” to mean strongly connected. 14
In Lemma 2, we use this threshold to establish the same
threshold for a special class of
multigraphs, where
for all , with .
To complete the proof, fix such that , and observe
that from (6) we have for fixed and
parameter . Let denote the time until
we observe distinct items. Then
Since as , we conclude
Let , and let .
It follows that
is connected
is connected
Since the right-hand side approaches 1 as by Lemma 2,
we conclude that the left-hand side approaches 1 as well. Thus,
the probability that all users can trade multilaterally approaches
1.
The following lemma uses the same definitions as the preceding
proof.
Lemma 2: Suppose as , and
for all , where . Then
is connected as .
Proof: The random multigraph
differs in two
ways from the random graph . First, we may sample
the same edge twice (this is why
is a multigraph).
Second, with probability , a given edge may not be placed at
all. Informally, neither of these effects change the order scaling
of the number of edges needed to ensure connectivity. We now
formally justify this intuition.
Where clear from context, to compress notation, we let
and denote and , respectively. Let
denote the simple graph obtained from by replacing
any multiedges by a single edge. Observe that conditional
on having edges, has the same distribution
as . Thus it suffices to show that almost surely, the
number of edges in satisfies .
Since
, we can choose a sequence
such that and . (For example,
for each , choose such that for
all . For such that , let
.) Since and , it follows that
as . It suffices to show that
contains at least edges as .
Consider the following procedure for sampling with replacement
from
items. Each time we draw an item, we
record its number with probability ; with probability ,
we discard the observation. If we have recorded distinct items,
the time until we record the next distinct item is geometric with
14 This result is analogous to the same result for undirected Erdös–Rényi
random graphs and can be proven using similar counting arguments for
threshold behavior of those graphs; see, e.g., [19] and [20].)
Using Markov’s inequality
as . In other words, if we sample items with replacement
from a bin of items as described above, then we
obtain distinct items with probability approaching 1 as
. It follows that contains at least edges as
, as required.
ACKNOWLEDGMENT
The authors are grateful to M. Piatek for providing access to
the BitTorrent dataset [33] analyzed in Section V.
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Christina Aperjis received the B.S. degree in electrical
and computer engineering from the National
Technical University of Athens, Athens, Greece, in
2003, the M.S. degree in computer science from Columbia
University, New York, NY, in 2004, and the
Ph.D. degree in operations research from Stanford
University, Stanford, CA, in 2009.
She is currently a Researcher with the Social Computing
Group, HP Labs, Palo Alto, CA.
Ramesh Johari (M’05) received the A.B. degree in
mathematics from Harvard University, Cambridge,
MA, in 1998, the Certificate of Advanced Study
in mathematics from the University of Cambridge,
Cambridge, U.K., in 1999, and the Ph.D. in electrical
engineering and computer science from the Massachusetts
Institute of Technology (MIT), Cambridge,
in 2004.
He is currently an Assistant Professor of management
science and engineering and, by courtesy, computer
science and electrical engineering with Stanford
University, Stanford, CA.
Michael J. Freedman (S’99–M’09) received the
B.S. and M.Eng. degrees in electrical engineering
and computer science from the Massachusetts
Institute of Technology (MIT), Cambridge, in 2001
and 2002, respectively, and the Ph.D. degree in
computer science from the Courant Institute, New
York University, New York, in 2007, during which
he spent two years at Stanford University, Stanford,
CA.
He is currently an Assistant Professor of computer
science with Princeton University, Princeton, NJ.