THE WORLD’S NEWSSTAND®
COMMUNICATIONS NETWORK ECONOMICS
Network Market Design Part I:
Bandwidth Markets
Rahul Jain, University of Southern California
ABSTRACT
Markets for network resources have become
increasingly important. Such resources include
bandwidth in wireline networks to wireless spectrum. It is clear that if the markets are not properly designed, they can function rather poorly,
even leading to market failure. This then leads
to suboptimal use of network resources. In this
article, we present the game theoretic framework behind the market design principles for
network resources. We present some of the
extant work on network market mechanisms in
the context of wireline networks in part I. Part II
[1] relates to network market design in the context of wireless systems. We also discuss some of
the key challenges for the future.
INTRODUCTION
The basic problem in communication networking
is how to effectively share resources, such as the
bandwidth of communication links or buffer
space in routers, or the use of a shared wireless
medium. From an engineering perspective,
resource sharing underlies the design of protocols used throughout the network stack.
Resource sharing is also at the heart of economics. Indeed, the role of a market is to allocate resources among competing agents. Often,
engineers view the economic considerations as
being separable from the design of the underlying technology used for resource sharing, and
focus mainly on the latter issue. For modern
communication infrastructure, which is owned
and operated by multiple, independent, profitmaking entities and handles a heterogeneous
mix of traffic in diverse environments, it is
becoming increasingly clear that such a separation is not tenable. This will become even more
apparent in next-generation networks, which are
envisioned to be even more complex hybrid
wireless/wireline networks with greater capacity,
reliability, and capability. Moreover, new architectures may evolve that tap user cooperation in
social networks to enhance network capabilities
and provide ubiquitous connectivity.
There are several challenges to such a vision.
Network capacity will remain constrained due to
the scarcity of resources such as spectrum, and
sometimes bandwidth as well; network utilization
will remain suboptimal due to the static and
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0163-6804/12/$25.00 © 2012 IEEE
often centralized allocation mechanisms used,
particularly with smarter selfish users; and service quality will remain variable and without
guarantees in the absence of adoption of network architectures for quality of service (QoS)
provisioning.
Network capacity can be enhanced by sophisticated information-theoretic and network coding methods, but those will not be enough
without incentivizing cooperation between
increasingly sophisticated and selfish users. Network utilization will improve with better decentralized algorithms, but what if these can be
manipulated by strategic users? A dependable
service quality on the Internet shall remain a distant dream until economic impediments to network service provider cooperation in
QoS-provisioning are removed.
The basic thesis of this article is that a systematic and foundational understanding which synthesizes the economic and technical approaches
for resource allocation will aid in meeting these
challenges by facilitating scalable, flexible protocols that account for user incentives.
FROM NUM TO NMD
Dynamic decentralized algorithms for network
flow optimization have a long history. Since
Kelly’s seminal work [2], network theorists
have used ideas from mathematical economics
and optimization to achieve distributed flow
control and optimization in large-scale networks. Such a theory has both informed Internet congestion control protocol design (e.g.,
TCP-Vegas) as well as yielded power control
algorithms for wireless networks. The rich
mathematical theory underlying this work is
often referred to as network utility maximization (NUM). Although NUM has its roots in
ideas from economics, it uses these ideas mainly as a metaphor to guide engineering design as
opposed to truly incorporating economic incentives. It is not robust to manipulation by
increasingly sophisticated users, who act strategically and often have economic interests at
stake. In fact, it was shown in [3] that the Kelly
mechanism can have arbitrarily bad efficiency
when users behave selfishly and strategically.
Thus, there is a need to go beyond the current
distributed optimization-based NUM framework to a game-theoretic and market economics-based network market design (NMD)
IEEE Communications Magazine • November 2012
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framework that takes incentive issues into
account in design of network resource allocation algorithms and protocols, which are robust
to strategic manipulation.
The theory of mechanism design in game theory [4, 5] indeed deals precisely with these problems but does not always provide solutions for
communication-network-specific problems. For
example, network resources are usually allocated
in bundles (e.g., bandwidth on links that constitute a route) and are often well approximated as
being divisible, meaning that they can be arbitrarily divided among users. Mechanism design
theory can in principle handle bundles and divisible resources, but the mechanisms it prescribes
quickly become too complex to implement both
in terms of their communication as well as their
computational costs. This is in stark contrast to
the scalability considerations that are essential
for networking protocols. Another consideration
is that most of this theory is for single-sided auctions (allocations by one auctioneer to multiple
buyer) as opposed to double-sided (exchange
between buyers and sellers facilitated by an auctioneer), which is more relevant in many network settings. Also, network resource allocation
problems are inherently dynamic where user
population, utilities, network capacities and even
topologies change over time while current design
solutions treat these as static, one-shot problems.
Partly, the reason for this is that dynamic mechanism design is itself under-developed and a very
hard problem. Finally, mechanism design focuses
on allocating a given set of assets to the users. In
many networking contexts, by making different
engineering choices, one can redefine the underlying assets in different ways, creating a new
dimension in market design.
In this article, we explain some of the basic
issues that need to be addressed in designing
network markets, and discuss some of the
main impossibility results as well as fundamental design guidelines. We then present
some of the work that has been done in the
last decade on network market design. Part I
presents this in the context of network bandwidth, while Part II will present it in the context of wireless spectrum. We conclude with
some of the open and challenging problems
for future research.
THE NUM FRAMEWORK FOR
DECENTRALIZED NETWORK
RESOURCE MANAGEMENT
Consider a network described by a graph G =
(V, E). A simple example is illustrated in Fig. 1.
Let there be n users or enterprise customers.
User i wants capacity on a bundle of links R i ,
which could constitute a route. Let the utility to
user i of obtaining x i bandwidth on links R i be
ui(xi). Let there be M network service providers
(NSPs) who own capacity on various links in the
network. For example, NSP j might own capacity
on a bundle of links Lj which could constitute a
route. Let the total capacity on link l be denoted
by Cl. Let the cost of provisioning yj capacity on
links Lj be cj(yj). The system objective is to
IEEE Communications Magazine • November 2012
Network operators
1,1’
Network operators
2,2’
v(x)
c(y)
A
B
C
Quantity
Service provider 1,
v1(x)
Service provider 2,
v2(x)
Figure 1. A network with network operators and service providers/users.
SYSTEM : max S ( x, y ) = ¨ vi ( xi ) ¨ c j ( y j )
i
j
(1)
subject to Flow Constraints:
¨
i :l ‘Ri
xi f
¨
j :l ‘L j
y j , l ‘ E , xi , y j v 0, i, j .
(2)
S(x, y) is called the social welfare function. Typically, the utility functions vi(x) are assumed to be
non-negative increasing concave functions, while
the cost functions are assumed to be non-negative
increasing convex functions. This makes the social
welfare maximization problem a convex optimization problem that can be solved easily using standard methods available in optimization theory.
The difficulty here is that both the utility and cost
functions are private information of the users and
network operators, respectively. Thus, this problem cannot be solved in a centralized manner.
Assume there is only one provider for each
link, and assume the capacities y j are given. It
was shown by Kelly [2] that the system problem
above can be decomposed into a user problem:
USERi : max x i v i (x i ) – S lŒR i l l x i , and a network
problem: NETWORKj: maxyj SlŒLj llxj – cj(yj), such
that they both together solve the SYSTEM problem. Here, lLj is shorthand for (ll : l ΠLj), and
these are the dual variables corresponding to the
flow constraints in the SYSTEM problem, and
interpreted as (shadow) prices for per-unit
capacity on the links. Moreover, Kelly also proposed a distributed algorithm (henceforth called
Kelly’s algorithm) that iteratively solves the
SYSTEM problem by having each user and each
operator solve their own problems.
Since the introduction of this idea to networking, it has been very well developed, and a
whole framework, NUM, has been developed
that allows decentralized resource allocation in
large-scale communication, transportation, and
power networks. However, with non-cooperative
and strategic users and network service providers
(NSPs), distributed NUM algorithms can fail to
achieve predicted performance, and are often
highly inefficient. For example, Kelly’s mechanism for network allocation can have social welfare that is arbitrarily close to zero with selfish
strategic users [3]. Thus, a different framework
that is more robust, and takes cooperation incentives and economic interests of users and NSPs
into account is needed for such settings. We call
it the NMD framework based as it is on market
economics, auctions, and game theory.
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MECHANISM DESIGN:
IN A NUTSHELL
Mechanism design (MD) in game theory provides a framework in which design of markets
can be considered in a systematic manner. It
indeed offers the basis for the NMD theory, but
there are challenges specific to communication
networks. For example, auction theory deals with
mostly indivisible goods, whereas network
resources such as bandwidth and spectrum are
often regarded as divisible goods. If standard
mechanisms such as the Vickrey-Clarke-Groves
(VCG) mechanism are used, this introduces
informational problems since they require infinite-length bid messages to yield incentive-compatible outcomes. Furthermore, bundles of goods
are mostly considered in combinatorial auction
contexts, which usually have computational feasibility issues due to the integer programming
problems involved. But above all, network
resource allocation frequently involves exchange
between agents (some being buyers, others sellers), and thus, mechanisms (e.g., VCG) that are
not budget-balanced (i.e., payments made by the
buyers equal payments made to the sellers) are
untenable. And last but not least, network environments are inherently dynamic, so we need to
go beyond the one-shot mechanisms available in
state-of-the-art theory. Below, we first present
the MD framework, and some of the most useful
and celebrated results in the static setting.
Consider n agents with the set of outcomes X.
Agent i’s utility is u i(x, q i) with type q i Œ Q i. A
social choice function (s.c.f.) f(q 1 , …, q n ) Œ X
maps agent types to outcomes. This can be interpreted to be the outcome that is “socially desirable” given agent types or utility functions. An
example is f(q 1, …, q n) Œ arg maxx S i ui(xi, q i),
where x = (x 1, …, x n) Œ X. An s.c.f. f Q Æ X is
called ex post efficient if for no q is there an x Œ
X such that ui(xi, qi) ui(fi(q), qi) for all i, with a
strict inequality for some i.
The question now is what s.c.f.s can be implemented (in a distributed manner) when agents’
types are private information? And how?
We define a mechanism G = (S1, …, Sn, g(.))
as a collection of strategy sets and an outcome
function g(s1, ..., sn) ΠX. A mechanism G implements a s.c.f. f if there is an equilibrium strategy
profile s* = (s *1 (q 1 ), …, s*(q
n
n )) of the game
induced by G such that g(s*1 (q 1 ), ..., s*(q
n n )) =
f(q1, ..., qn), "qi ΠQi, "i. That is, the game corresponding to G is such that the equilibrium outcome is what would be regarded as the outcome
for the social good (i.e., the s.c.f. f).
Now, we say that a strategy profile s* is a
Nash equilibrium if for each player i, ui(g(s*,
i s*
–i),
qi) ui(g(si¢, s*–i), qi) for every strategy si¢ Œ Si of
player i. That is, given other players play strategies s*–i = (s1*, …, s*i–1, s*i+1, …, s*n), player i’s best
response is to pick s*.
i Note that s*
i does depend
on s*–i, in general. However, it is quite possible
that a player has one best strategy to play no
matter what strategies others play. Such a strategy is called a dominant strategy for player i. If a
Nash equilibrium consists of a dominant strategy
for every player, we call it a dominant-strategy
equilibrium.
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If the equilibrium strategy profile of the
mechanism G is a dominant strategy (Nash)
equilibrium, it is called a dominant strategy (or
Nash) implementation of the s.c.f. f. The question
now is are all s.c.f.s implementable in some way?
Do we need to consider all possible mechanisms,
which is a rather huge space? Fortunately, the
revelation principle provides a very helpful
answer to the latter question.
REVELATION PRINCIPLE
Suppose there is a mechanism G that implements the s.c.f. f in dominant strategies. Then
the direct mechanism (wherein Si = Qi, g = f) is
incentive-compatible; that is, truthfully reporting
types is a dominant strategy equilibrium.
This has great implications: If an s.c.f. is
implementable at all (in dominant strategies), it
suffices to consider the direct mechanism wherein each player is just asked to report his/her true
type. Thus, it suffices to consider direct mechanisms alone for dominant strategy implementation.
We now discuss some desirable properties a
mechanism should have:
• A mechanism is ex post individual-rational
(IR) if the payoff at any equilibrium is nonnegative for all agents.
• A mechanism is said to be efficient if it maximizes social welfare.
• A mechanism is said to be budget-balanced
if the sum of payments of agents is zero.
• A mechanism is said to be incentive compatible (IC) if at equilibrium it is a dominant strategy for all agents to bid truthfully.
We now present two common auction mechanisms.
FIRST-PRICE AUCTIONS
Consider n bidders for a good. X is the set of
allocations of good to each of the bidders. Each
player has a utility function ui(xi, qi) = xiqi. The
types q i have U[0, 1] distribution, and this is
common knowledge. The s.c.f. is f(q1, …, qn) =
maxxŒX S i ui(xi, qi), where the outcome space X
is the good being assigned to each of the bidders. The auction rule is that bidders submit
sealed bids; the highest bidder wins and pays
his/her bid. In this case, it is easy to show that
there is a Bayesian-Nash equilibrium such that
each bidder bids exactly half of what s/he is willing to pay for the good. Thus, the good will go
to the bidder who has the highest value for it,
although he does not reveal his/her true valuation for the good.
SECOND-PRICE AUCTIONS
Now, consider the following auction rule: The
highest bidder wins but pays the second highest
price [6]. It is easy to show that in this case a
dominant-strategy equilibrium is for each bidder
to bid exactly his/her true value for the good.
Thus, in this case not only is the equilibrium
outcome efficient (i.e., the bidder who values the
good the most gets it), but also each bidder
truthfully reveals their true valuation. Moreover,
this is a dominant strategy for each player, independent of what the others know or do. Thus,
this is a strong implementation of the s.c.f. f
defined above.
IEEE Communications Magazine • November 2012
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We note that the celebrated VCG mechanism, which is a generalization of the secondprice auction, is IC, ex post IR, and efficient but
not budget-balanced. This raises the question of
whether there are mechanisms that have all four
properties. The answer turns out to be negative.
HURWICZ IMPOSSIBILITY THEOREM (1975)
There is no mechanism (for simple exchange with
quasi-linear preferences) that is IC, ex post IR,
efficient, and budget-balanced [7].
We say that a mechanism is Bayesian IC
(BIC) if truth revelation is a Bayesian-Nash equilibrium of the associated incomplete information
game. The dAGVA (expected externality) mechanism (d’Aspremont and Gerard-Varet, 1979,
Arrow, 1979), a variation of the VCG mechanism for the incomplete information setting, is ex
ante IR, BIC, efficient, and strongly budget-balanced when agents have quasi-linear preferences. The Myerson-Satterthwaite Impossibility
Theorem (1983) [8] states that it is impossible to
achieve efficiency, budget balance, and interim
individual rationality in a BIC mechanism even
with quasi-linear utilities.
We refer the reader to [4, 5] for a more thorough treatment of the theory of mechanism
design.
NETWORK AUCTION DESIGN:
DIVISIBLE BANDWIDTH
Consider a network with L links, each with divisible bandwidth; that is, bandwidth can be allocated in any fraction. Let link l have capacity C l .
There are n buyers, and buyer i wants a bundle
of links Ri that constitute a route. If buyer i gets
bandwidth xi on route Ri, it derives a utility vi(xi).
If it makes a payment Pi, its net payoff is ui(xi) =
vi(xi) – Pi. The goal is to determine an allocation
x** that maximizes the social welfare S(x) = Si
vi(xi) subject to the flow constraints: S i:lŒRi xi £
Cl for all links l.
The difficulty in determining the allocation
x** is that the utility functions are private information. And buyers may not reveal them. If the
utility functions are not known, how can we still
determine the optimal allocation x**, if at all?
Well, one way is to elicit some information from
the buyers by conducting an auction.
Suppose buyer i reports a bid signal b i for
route R i as indicative of his utility function v i.
The auction system would then take the bids bi
of all the buyers, and determine an allocation
x*(b) and payments P i (b i , b –i ). Of course, the
buyers are selfish and will act strategically. They
will pick a bid bi to maximize their payoff ui(b)
= v i (x i (b)) – P i (b). See Fig. 2 for a graphical
illustration. So, unless the right allocation function x*(b) and payment functions Pi are picked,
and the SYSTEM and buyer objectives are
aligned, there is no reason to believe that the
auction allocation x* will be the same as the
optimal allocation x**. A larger question is
whether this is even possible. It turns out that
this is indeed so!
For simplicity, consider a single-link network
with capacity C. One way to allocate divisible
bandwidth via an auction is what is called Kelly’s
IEEE Communications Magazine • November 2012
Buyer i (bi, Ri )
(true vi )
Auction system
compute allocation
and prices
Buyer i (xi*, Pi )
Figure 2. An auction system for network bandwidth.
mechanism: Each player submits a bid, b i. The
allocation for user i then is xi = Cbi/Si bi, that is,
the capacity C is allocated proportionally (w.r.t.
their bids) among the users. This can be seen as
a solution of the SYSTEM problem with the surrogate utility functions ui(xi) = bi log xi. A perunit price is computed as m = Si bi/C. Thus, user
i pays mxi = bi. If the users are non-strategic (or
price-takers), it can be shown that this mechanism is indeed efficient, in the sense of achieving
the optima of the SYSTEM problem above. Unfortunately, when the users are strategic, the mechanism has equilibria wherein the social welfare is
arbitrarily small [3].
To address this problem, the VCG-Kelly
mechanism was proposed [3]. Users submit onedimensional bids bi. The mechanism determines
an allocation x* that solves the SYSTEM problem
with surrogate utility functions ui(xi) = bi log xi.
Bidder i makes a “VCG payment” Pi(b) = max
Sj i bj log xj – Sj i bj log x*.
j It can be shown that
if there are at least two special buyers (with
u¢(0+) = +•) on each link who make positive
bids, the Nash equilibrium point exists, and is
unique and efficient. A similar idea has been
developed in [9] in more generality. One of the
shortcomings of this mechanism may be that it
seems impractical. In most auctions and markets,
the bidders are rather used to not only specifying how much they would pay, but how much
they want as well.
A mechanism that has a philosophy very similar to the VCG-Kelly mechanism, and yet
addresses the above concerns, is the following.
In the network second price (NSP) mechanism [10], buyers submit bids bi = (bi, di), which
is interpreted as willingness to pay $b ii per unit
for up to di units of bandwidth on route Ri. The
mechanism then determines an allocation x* that
maximizes “social welfare” Si bixi subject to the
flow constraints, and ensuring that xi Π[0, di] for
all buyers i. Buyer i receives x i * bandwidth on
route Ri and pays Pi(bi, b–i) = S j i bj (xj–i* – x*j),
where x–i* is the auction allocation when bidder i
does not participate. We illustrate how this
works through a simple example.
EXAMPLE 1
Suppose C = 6 units of capacity on a single-link
network. There are three players. Player R bids
($3/unit, 3 units), Player B ($2/unit, 2.5 units),
and Player G ($1/unit, 4 units). Then it can easily be checked that the allocation is x*
R = 3, x*
B =
2.5, and x*
G = 0.5, and the payments would be PR
= $3, PB = $2.5, and PG = 0. This is illustrated
in Fig. 3.
In fact, it can be shown that there is a Nash
equilibrium b* at which the corresponding allocation x* is efficient: S(x*) = S(x**) [10]. We
81
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Bids
1
3
3
2
2
3
2
Allocation
Payments
1
2
2
$0
$2.5
2
1
3
1
$3
3
1
3
Figure 3. Example 1: Allocation and payments with NSP.
Seattle
Boston
Chicago
AOL: v1(b,R)
NY
MCI: c2(b,L)
Comcast: v2(b,R)
Earthlink: v3(b,R)
SF
AT&T: c1(b,L)
DC
Sprint: c3(b,L)
Houston
Figure 4. A network with bandwidth owners and ISPs.
Buyer i (bi , Ri )
(true vi )
Seller j (aj , Lj )
(true cj )
Auction system
compute allocation
and prices
Buyer i (xi0, Pi c )
Seller j (yj0, -Tj )
Figure 5. A market system for bandwidth exchange.
note, however, that there are many Nash equilibria, not all efficient. Although some of the inefficient Nash equilibria can be eliminated through
reserve prices (i.e., by having each winner pay a
reserve price plus the designed payment). Thus,
the mechanism proposed above may be considered a weak Nash implementation, is ex post
individual rational, but does not have the budget-balance property.
NETWORK MARKET DESIGN:
INDIVISIBLE BANDWIDTH
One difficulty with the mechanisms we considered in the previous section is that often bandwidth is only be available in discrete units (e.g.,
in increments of 1 Gb/s). That is, it is an indivisible good. Another difficulty is that, often we
have a market setting, in which the bandwidth
exchange involves both multiple buyers and multiple sellers, facilitated by an “auctioneer” or a
market maker. In this case, it becomes crucial
that there be budget balance; that is, the payments made by the winning buyers equal the
payments made by the winning sellers, and the
mechanisms proposed earlier become irrelevant
including any VCG variants.
As before, consider that buyer i wants bandwidth on route Ri. If s/he gets xi units, and pays
82
Pi, his/her payoff is uib(xi, Pi) = vixi – Pi, where vi
is the valuation that buyer i has for each unit of
bandwidth on route R i up to some d i units. A
seller j has s j units of bandwidth to sell on link
Lj. If it sells yj units and receives a payment Tj,
its payoff is usj (yj, Tj) = Tj – cjyj, where cj is what
it costs the seller to sell a unit of bandwidth. See
Fig. 4 for a simple illustration. Such costs may
arise because while the seller may have sunk in
some fixed infrastructure cost, s/he would still
incur some maintenance costs, and this could
depend on how much bandwidth in the pipe is
actively used.
The goal in designing a network (bandwidth)
market then would be to determine an exchange
(x**, y**) that maximizes the social welfare S(x,
y) = Si vixi – Sj cjyj subject to the flow constraints:
Si: l Œ Ri xi £ Sj:l=Lj yj for each link l. Note that
while buyers want routes, so bundles of links,
sellers sell bandwidth on individual links. And if
they own bandwidth on multiple links, they
could offer them separately but need not bundle
them. This makes the problem tractable.
In a market mechanism, the buyers can be
asked to report (bi, Ri, di), which is interpreted to
mean that the buyer is willing to pay up to bi per
unit for up to d i units on route R i. Meanwhile,
the sellers can be asked to report (aj, Lj, sj) which
is interpreted to mean that the seller wants at
least aj per unit and can sell up to sj units on link
Lj. This is graphically illustrated in Fig. 5. Note
that unless the market allocation (x*(b, a), y*(b,
a)) and the payment functions Pi and Tj are properly designed, the SYSTEM and player objectives
are not going to be aligned to result in the social
welfare maximizing outcome (x**, y**).
In the combinatorial Seller’s Bid Double
Auction (c-SeBiDA) that was proposed in [11],
the allocation (x*, y*) can be determined by
maximizing the “trading surplus” S ibixi – S jajyj
subject to the flow constraints above and xi Π{0,
…, di} and yj Œ {0, …, sj}. Once the allocation is
determined, we can determine a price on each
link as pl = sup {aj: yj* > 0, l = Lj}, that is, the
highest ask price aj among all “matched” sellers
on the link l. Now, a “matched” buyer pays the
sum of prices pl on the links in his/her route Ri
times the allocation xi*, and a “matched” seller
receives a payment equal to price p L j times
quantity qj. We now illustrate through an example how the mechanism works.
EXAMPLE 2
Consider a single link network with three buyers
with valuations $3.1, 2.1, and 1.1, respectively, for
one unit, and three sellers with costs $1, 2, and 3,
respectively, for one unit. If each of them bids
their true valuation or cost, the optimal “trading
surplus” subject to flow constraints would be 2.2
when buyers 1 and 2 are matched, and sellers 1
and 2 are matched, but buyer 3 and seller 3 are
not. In this case, the price will be p = 2, the highest ask price among the matched sellers. This is
graphically represented in the first part of Fig. 6.
Note that what the c-SeBiDA mechanism
does not do is try to match each buyer with a
seller such that each such matching has a positive
surplus. If this were the case, buyer 1 could be
matched with seller 3, buyer 2 with seller 2 and
buyer 3 with seller 1, and surplus in each match-
IEEE Communications Magazine • November 2012
THE WORLD’S NEWSSTAND®
ing would be positive. But this does not maximize
the “trading surplus,” or the social welfare.
For the c-SeBiDA market mechanism, it has
been shown in [11] that:
• The mechanism is budget-balanced and ex
post IR.
• On a single link, every Nash equilibrium
(NE) allocation (in weakly rationalizable
strategies) is efficient. Moreover, there is an
NE at which all players except the highest
matched seller bid truthfully (almost IC).
• In the network case, it was established that
if an NE exists with a trade for every link,
every such NE allocation is efficient.
The case of incomplete information has also
been addressed. It was shown that the market
mechanism is asymptotically Bayesian-IC and
efficient, ex post individual rational, and budget
balanced. Recall from the Myerson-Satterthwaite
theorem [8] that we cannot have a mechanism
which has all four non-asymptotic properties. So,
in some sense, this is the best we can hope for.
Network market design with indivisible bandwidth is an important problem to run bandwidth
markets efficiently. However, there is a paucity
of work on it. The only known related work is
[12] which proposes a double auction mechanism
that only has the weak budget balance property
(i.e., the auctioneer’s expected payoff is nonnegative).
OPEN PROBLEMS AND
FUTURE DIRECTIONS
From the above discussion, it should be clear
that auction and market design solutions for
network resources exist for a wide variety of
scenarios. Single-sided auction designs for
divisible goods have been fairly well explored.
Designing suitable double-sided auctions, or
markets, has proved to be rather challenging
though. Part of the reason is Hurwicz’ negative
result: It is impossible to have all four properties. We must make a compromise on one, and
it proves rather difficult to trade off incentive
compatibility and efficiency while ensuring
budget balance and individual rationality.
Designing markets for indivisible goods proves
to be even more challenging. Not only the
computational issues become important, but
the paucity of appropriate mathematical tools
to deal with non-smooth problems makes the
analysis rather difficult. There is thus scope to
introduce new design methodologies and mathematical tools for such network market design
problems.
The other issue we have ignored is that markets usually operate dynamically. When markets
are dynamic, participants not only glean valuable
information from past plays, but they may also
have the option of waiting until the next round.
This belongs to the realm of dynamic mechanism
design, a theory yet underdeveloped, but receiving increasing attention. It should be obvious
that in dynamic mechanisms and markets, strategic learning and experimentation issues also
become important, yielding new models such as
multi-armed bandit games that have yet to be
understood.
IEEE Communications Magazine • November 2012
b1=3.1
b1=3.1
a3=3
a3=3
b2=2.1
b2=2.1
a2=2
c2=2
b3=1.1
b3=1.1
a1=1
a1=1
(a)
a2=2.1
(b)
Figure 6. Example 2: C-SeBiDA outcome and equilibrium.
Designing markets for spectrum opens up a
whole new host of issues: What is a good here?
What do property rights mean? And, in the context
of cognitive radio systems, what is the right framework for spectrum contract design? Some of these
issues are addressed in the companion article [1].
REFERENCES
[1] R. Berry, “Network Market Design II: Spectrum Markets,” IEEE Commun. Mag., this issue.
[2] F. P. Kelly, “Charging and Rate Control for Elastic Traffic,”
Euro. Trans. Telecommun., vol. 8, 1997, pp. 33–37.
[3] S. Yang and B. Hajek, “VCG-Kelly Mechanisms for Divisible Goods: Adapting VCG Mechanisms to One-Dimensional Signals,” IEEE JSAC, vol. 25, no. 6, 2007, pp.
1237–43.
[4] V. Krishna, Auction Theory, 2nd ed., Academic Press,
2009.
[5] A. Mas-Colell, M. D. Whinston, and J. R. Green, Microeconomic Theory, Oxford University Press, 1995.
[6] W. Vickrey, “Counterspeculation, Auctions And Competitive Sealed Tenders,” J. Finance, vol. 16, no. 1, 1961,
p. 837.
[7] L. Hurwicz, “On Informationally Decentralized Systems,”
C. McGuire and R. Radner, Eds., Decision and Organization: A Volume in Honor of Jacob Marschak, NorthHolland, 1975.
[8] R. B. Myerson and M. A. Satterthwaite, “Efficient Mechanisms for Bilateral Trading,” J. Economic Theory, vol.
28, 1983, pp. 265–81.
[9] R. Johari and J. N. Tsitsiklis, “Efficiency of Scalar-Parameterized Mechanisms,” Operations Research, 2010.
[10] R. Jain and J. Walrand, “An Efficient Nash-Implementation Mechanism for Divisible Resource Allocation,”
Automatica, vol. 46, no. 8, Aug. 2010, pp. 1276–83.
[11] R. Jain and P. P. Varaiya, “An Asymptotically Efficient
Mechanism for Combinatorial Network Markets,” submitted to Operations Research, Nov. 2011.
[12] L. Y. Chu and Z.-J. Max Shen, “Truthful Double Auction Mechanisms,” Operations Research, vol. 56, no. 1,
2008, pp. 102–20.
BIOGRAPHY
R AHUL J AIN ([email protected])
___________ is an assistant professor
and the K. C. Dahlberg Early Career Chair in the Electrical
Engineering Department at the University of Southern California. He received his Ph.D. in EECS and an M.A. in statistics from the University of California, Berkeley, and his
B.Tech from IIT Kanpur. He is a winner of numerous
awards including the NSF CAREER award, an IBM Faculty
award, and the ONR Young Investigator award. His
research interests span wireless communications, network
economics and game theory, queueing theory, power systems, and stochastic control theory.
83
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COMMUNICATIONS NETWORK ECONOMICS
Network Market Design Part II:
Spectrum Markets
Randall A. Berry, Northwestern University
ABSTRACT
Market-based approaches have promising
potential for allocating network resources. Part I
of this article introduced the game theoretic
underpinnings of market design and argued for
the need to jointly consider market design with
the underlying engineering issues in communication networks. Example research questions in
this area were reviewed for wireline networks. In
this part, we turn to network market design for
wireless systems and in particular for the flexible
sharing of wireless spectrum. We use this as a
vehicle for discussing challenges that arise in the
design of markets in the presence of externalities.
INTRODUCTION
Resource sharing is the basic issue underlying
both the design of communication protocols as
well as the design of economic markets. The first
part of this article made the case that these two
approaches to resource sharing are not separable and espoused the need for a theory of network market design, which combines both
economic and technical considerations. Examples of such an approach were discussed in the
context of designing markets for bandwidth in a
wireline network. In this part, we turn to wireless networks and discuss several other issues
that such a theory would need to address. In
particular, we focus on market-based approaches
for spectrum sharing.
Currently, most licensed spectrum for wireless services is allocated on very coarse scales in
both time and space. For example in the United
States, many licenses are allocated for 10-year
timeframes and typically cover a large portion of
the country. It has been widely argued that this
approach has to led to underutilization of spectrum; this in turn has led to an array of techniques being proposed to enable more efficient
use of limited spectrum resources (e.g., [1] provides an overview of this area). These techniques
include various market structures for trading
and/or leasing spectrum on finer temporal and
spatial scales (e.g., see [2]). Here, we use such
markets as a way to discuss a number of issues
related to network market design, which may
also be relevant in other settings.
We highlight a few key differences between
wireless spectrum markets and the bandwidth
84
0163-6804/12/$25.00 © 2012 IEEE
markets considered in Part I. First, in a bandwidth market, to the first order it is fairly clear
how to define the set of assets being allocated.
For example, on a single link, the assets are
units of bandwidth whose sum is no greater than
the link’s capacity. Of course, there are secondary considerations. For example, as discussed
in Part I, there is still an issue that in practice
bandwidth is not infinitely divisible, so there is a
choice to be made on how this asset is divided
up into indivisible bundles. If temporal dynamics
are taken into account, there is also an issue of
deciding on the timescale at which allocations
occur. However, in the context of wireless spectrum, even such a first order approximation is
not clear. For example, one approach to allocating a band of spectrum to a group of users is to
require that all users transmit using spread spectrum over the entire band, treating interference
from others as noise. With such an approach,
the allocation of spectrum might correspond to
determining the allowed transmission power of
each user or each set of users. Another approach
is to use frequency-division multiplexing (FDM)
and allocate exclusive use of frequency bands to
different users. These two approaches lead to
very different definitions of the set of assets
being traded. These are engineering choices, but
the resulting choice also affects any market that
emerges. Jointly considering such effects is a key
dimension of network market design, which
clearly requires both engineering and economic
insights.
A second basic property of wireless spectrum is that two users utilizing the same frequency band at nearby locations mutually
interfere with each other. Hence, an agent’s
value for a given “spectrum asset” may depend
in part on the allocation of other assets to
other agents. Such effects are known in economics as externalities. When spectrum is allocated on coarse geographic scales, such
interference externalities are a relatively minor
problem, in part because the boundaries can be
drawn through sparsely populated areas and in
part because the boundaries comprise a much
smaller portion of the total area. However, if
spectrum was allocated on a finer spatial scale,
these issues become more relevant. It is well
known in economics that the presence of such
externalities can greatly complicate market
design. We begin in the next section, with a
more complete discussion of such externalities.
IEEE Communications Magazine • November 2012
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We follow this with a discussion of several
approaches for dealing with interference externalities in spectrum markets.
Transmitters
IEEE Communications Magazine • November 2012
Receivers
h11
EXTERNALITIES AND TRAGEDIES
In the model for bandwidth allocation considered in Part I, a user’s utility is simply a function
of the amount of bandwidth she is allocated, and
in particular does not depend on the bandwidth
allocated to other users. In the case of wireless
spectrum, this may not be the case. Due to interference, the utility an agent derives from an allocation may decrease if a “nearby” agent increases
his allocation. Economists refer to such effects
as externalities. This interference effect is a negative externality, since increasing the allocation to
one agent has a negative effect on the performance of all other agents. Networking problems
may also exhibit positive externalities when the
actions of one agent lead to higher utility of
another. An example of this is a peer-to-peer file
sharing system in which users bring resources
that help other users as well as themselves. Here
we focus on negative externalities.
More formally, as in Part I, consider a general resource allocation problem of selecting an
outcome x = (x1, …, xN) for N agents from a set
of feasible outcomes X, where x i represents
agent i’s share of the resource. Loosely, without
externalities, each agent i’s utility for a given
allocation will depend only on xi, that is, it will
be a function ui(xi) and not depend on the values
of x j, for j i. On the other hand, when externalities are present, agents will have utilities that
depend on both xi and xj for j i, that is, these
will be functions of the form ui(x1, …, xN). This
is not a precise definition. In particular, note
that the choice of labels for each outcome is
arbitrary, and by simply changing the labels, one
can change the dependence of the utilities on
these labels. As an extreme example, for any
problem, we can simply label each outcome with
the corresponding utility received by each agent,
so we would have u i (x i ) = x i and thus would
never see any dependence on xj in user i’s utility.
A more precise definition of externalities is
somewhat subtle. To see this, note that even in
the case of allocating the bandwidth of a single
link, one agent’s bandwidth allocation reduces
the amount of bandwidth available and thus will
have an effect on the utility that can be received
by other agents. Is this an externality? One
approach to answer this is to define an externality not simply in terms of the underlying resource,
but in terms of a market for that resource. Given
a market, an externality is defined as an effect
that one agent causes others that is not accounted for in the market [3]. For example, there are
no externalities in a market based on the Kelly
mechanism discussed in Part I for allocating the
bandwidth of a single link, since the dependency
of the users via the common capacity constraint
is represented via the per-unit price, m. Although
this definition is more precise, the term externality is more often used in the former sense to
simply reflect the dependence of one agent’s
utility function on the resources obtained by
another, under some “natural” parameterization
of the resources.
n0
p1
h12
h21
n0
p2
h22
..
.
n0
..
.
pM
Figure 1. Channel model for M transmitter/receiver pairs sharing spectrum via
power allocation.
To further illustrate the role of a market in
determining an externality, consider an “open”
market for sharing a single link with capacity C
among N agents. In this market, each agent simply requests some amount of bandwidth and is
allocated their request, without paying any
charge, provided the sum of their requests is less
than C; otherwise, they receive nothing. As in
Part I, each agent i receives a utility u i(x i) that
depends on the amount of bandwidth x i she is
allocated. Now additionally assume that the
agents receive a disutility per unit flow that is
proportional to the total traffic on the link, divided by C, which models some form of congestion
on the link. Agent i’s utility is now given by
ui ( xi ) 1
xi
C
(¨ x ),
j j
where the second term represents a negative
externality not accounted for in this market. To
see the effect of this, further assume that ui(xi)
= x i for each agent i. It can be shown that the
game defined by this market structure has a
unique Nash equilibrium in which each agent
requests
xi =
C
1+ N
units of bandwidth. For comparison, the efficient
allocation in this setting is to give each agent
xi =
C
.
2N
The difference between these two allocations is
because the market does not provide a way to
account for the negative costs an agent’s allocation has on other agents’ utilities, leading agents
to over-consume. To see the effect of this, note
that the total welfare obtained by the equilibrium allocation is
We =
NC
(1 + N )2
,
85
THE WORLD’S NEWSSTAND®
0.25
CDF
0.2
0.15
0.1
0.05
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Efficiency
0.85
0.9
0.95
1
Figure 2. CDF of the efficiency of the second-price sequential auction from [8]
for randomly placed nodes; in all cases the efficiency exceeded the worst case
bound in [8], and there was no efficiency loss in 80 percent of the realizations.
which goes to zero as N increases. With the efficient allocation, the total welfare is C/4, independent of the number of agents. One common
measure of the degradation of welfare in such a
market compared to the optimal allocation is
given by the market’s efficiency, which is the
ratio of the equilibrium allocation to the optimal
allocation. For this example, the efficiency is
4N
(1 + N )2
,
which approaches 0 percent as the size of the
market grows, a situation sometimes referred to
as a tragedy of the commons.
PRICES
One classical “solution” to the previous tragedy
is to use a price to “internalize the externality “.
In this model, the externality each user contributes to the overall welfare is given by
1
xi ¨ j |i x j .
C
Note that when every other user has an efficient
allocation, the marginal change in this term for
each user i is given by
1
1
.
2 2N
It follows that if we instead charge each user a
price of
1
1
p= 2 2N
p=
per unit bandwidth, the resulting market would
have a unique equilibrium in which the users
receive the efficient allocation. This price is also
known as a Pigovian tax after the economist
Alfred Pigou, who proposed such a solution to
86
externality problems [4]. Note that in this problem all agents were charged a common Pigovian
tax of p. This sufficed due to the symmetry of
the problem. Without this symmetry, the
marginal change in the welfare of the other
users due to an change in user i’s allocation may
depend on i, so N different Pigovian taxes may
be required. Actually determining the optimal
Pigovian tax required knowledge of the utilities
of each user. Of course, the main motivation for
using a market to allocate resources is that this
information is not known a priori.
To illustrate this difficulty in the context of
spectrum, consider a model for sharing a band
of spectrum among M agents in a given area.
For simplicity assume that each agent corresponds to a single transmitter/receiver pair and
that spectrum is shared by specifying the power,
P i , that each agent uses to transmit over the
common band. Interference from other agents is
then treated as additional noise. Each agent i
receives a utility ui(gi) that is an increasing function of their signal-to-interference-plus-noise
ratio (SINR) gi, which is given by
Li =
hii Pi
,
n0 + ¨ j |i h ji Pj
where h ij denotes the channel gain from transmitter i to receiver j and n 0 is the noise power
(Fig. 1). Here, each user generates a negative
externality due to the interference. If users
ignore this externality and simply choose their
own power Pi to maximize their own utility ui(gi),
they would all choose to use the maximum power
possible, since their own utility is increasing in
their own power. This can result in a total welfare that is much smaller than the optimal. What
would Pigovian taxes look like here? They can
be expressed in terms of interference prices as
introduced in [5]. The interference price p j of
user j is the marginal decrease in that user’s utility due to an increase in the total interference at
that user (i.e., an increase in S j i h jiP j). Given
the set of interference prices, the Pigovian tax
charged to user i (per unit transmit power) is
given by the sum of the interference prices from
each user j i weighted by the cross channel
gains hij. In general, each user will have a different interference price, which depends in part on
that user’s utility and SINR, and these prices will
be weighted differently for each user, resulting
in a different tax for each user i. Determining
these taxes appears to require knowledge of all
channel gains as well as the utilities of each
agent. However, in [5] it is observed that each
agent can compute their own interference price
given only local knowledge. Based on this observation, a distributed algorithm is developed in
which the agents iteratively exchange interference prices and update powers. With certain
restrictions on the class of utilities this is shown
to converge to the socially optimal power allocation. Like the Network Utility Maximization
(NUM) framework described in Part I, the interference pricing algorithm from [5] uses these
prices mainly as a metaphor to develop a distributed algorithm and is not truly modeling economic incentives. In particular, the agents have
an incentive to announce inflated interference
IEEE Communications Magazine • November 2012
THE WORLD’S NEWSSTAND®
prices, since this would reduce their interference
and not cost them anything.
The theory of mechanism design discussed in
Part 1 of this article provides a framework for
dealing with such incentive issues. Indeed, the
Vickery-Clarke-Groves (VCG) mechanism can
be used in settings with externalities to again
provide an incentive compatible and efficient
allocation. Recall, for a setting without externalities, i.e., one in which each user’s utility only
depends on his allocation x i , this mechanism
requires agents to essentially submit their utility
function u i (x i ) and the mechanism is then
required to solve an optimization problem corresponding to maximizing the total welfare to
determine the allocation and N related problems
to determine the payments. In a setting with
externalities, the required bids are again the utility functions, only now these must be given as a
function of the joint allocation to all agents, i.e.,
agents must specify ui(x1, …, xn). For the power
allocation problem described above, this corresponds to reporting not only one’s utility ui(gi) as
a function of the SINR, but also the relevant
channel gains involved in determining the SINR
as a function of the powers allocated to each
agent. In practice, this could be an excessive
amount of information to report and may not
even be known accurately (e.g., cross channel
gains could be difficult to measure). Once again
the mechanism is required to solve N + 1 optimization problems: one for the allocation and N
for the payments. These tend to be more difficult to solve in the presence of externalities. For
example in this power allocation problem, these
optimization problems may be non-convex due
to the interference. We also note that the VCG
payment can be viewed as the total “externality”
that an agent imposes on the welfare of all other
agents. This can be contrasted with the Pigovian
tax, which gives the marginal change in the
externality.
Given the aforementioned difficulties with
the VCG approach, it is natural to seek a simpler procedure for pricing such interference
externalities, such as the Kelly mechanism discussed in Part 1 for bandwidth allocation. What
would be a Kelly-like mechanism for this power
allocation problem? In the case of bandwidth,
the Kelly mechanism is a market-clearing mechanism, i.e., it sets a per unit price so that supply
(the links capacity) meets demand (represented
by the bids). For this wireless model, the “supply” is less clear. One way to think about this is
that each receiver has a supply of interference it
is willing to tolerate. This suggests having nodes
bid for the supply of interference at each node,
and then again set prices as in the Kelly mechanism so that supply meets demand. Such an
approach was studied in [6, 7] for pricing the
received interference at only a single “measurement point” with a fixed supply of interference.
In such a setting even if user’s are “price taking,” they are still strategically coupled due to
the interference. It is shown in [6, 7] that the
resulting games for both price anticipating and
price-taking users have Nash equilibria. However, in general these equilibria are not efficient,
even for price taking users, and the efficiency
may go to zero. In this case, not only is this mar-
IEEE Communications Magazine • November 2012
1
2
3
5
4
Figure 3. Example of an interference graph G for 5 spectrum assets. One independent set is shown in orange.
ket clearing mechanism not accounting for “price
anticipating users,” but the resulting single price
is not accurately reflecting the externalities
among the agents. As we have discussed, truly
capturing the externalities requires a different
interference price at each receiver. Why not simply run a Kelly-like mechanism for the interference at each of these? There are several
difficulties with such an approach. First, how
much interference a node is willing to supply
may depend on how much power it can use (due
to the available interference at other nodes).
Second, how much interference a node is willing
to buy at one node would depend on how much
it may buy at other nodes, since its total power is
determined by the smallest “interference allocation” it receives. Finally, implementing such an
approach would require the seller to have knowledge of the received interference at each receiver, which may not be feasible in practice.
Instead of pricing the received interference at
each receiver, an alternative market for this scenario is to view the supply as the potential transmission power available at each node i. Each
unit of this power can be allocated to either
node i, allowing her to increase her transmission
power, or to another agent j, which prevents i
from using this power and thus reduces the
interference at node j. A market for such a scenario is considered in [8], in which discrete units
of power at one user were allocated to two users
sequentially using a second-price auction for
each unit. It was shown that such a mechanism
can have an efficiency as small as 1/n, where n is
the number of discrete units of power. As n
increases, this shows that the efficiency can go to
zero. However, this is a worst-case result over
the set of possible utilities for two users. Assuming each user has a utility that is proportional to
its rate, and users are randomly placed in a given
area, numerical results [8] indicate that in average cases, there may be little if any efficiency
loss in many cases (e.g., Fig. 2). Extending the
analysis in [8] to more than two users appears to
be difficult, in part because whenever one agent
j i is allocated a unit of agent i’s transmission
power, it reduces the interference for all agents
87
THE WORLD’S NEWSSTAND®
Designing
approximation
schemes for general
combinatoric
auctions that
preserve the VCG’s
incentive properties
is a topic that has
received much
attention in the
algorithmic game
theory literature.
88
other than i, which gives these agents an incentive to “free-ride” on each other.
ASSET DESIGN
Another “solution” to the externality problem is
to design spectrum assets to limit or remove
externalities. To illustrate this, we return to the
power allocation problem introduced above.
Instead of sharing the band of spectrum by having each user spread their signal over the entire
band, we could instead subdivide the band into
orthogonal frequency bands (or assign users
orthogonal time slots). With such a change, the
spectrum allocation problem becomes equivalent
to a bandwidth allocation problem as in Part I,
so there are no longer any externalities present.
This leads to simpler market design, but from an
engineering perspective it may not be ideal. In
particular, we know that if the transmitter/receiver pairs are far enough apart, the earlier “power
sharing” model will better exploit frequency
reuse and result in better spectrum utilization.
This illustrates an under-explored trade-off
between designing assets so as to have simpler
market mechanisms vs. a more efficient allocation. As another example of this trade-off, one
can obtain an even simpler market design by
simply allocating the entire spectrum band to
one agent (e.g., via a second price auction), but
again, if each agent only requires a small amount
of the available spectrum, this could be very
inefficient.
A practical approach to navigating this tradeoff is to instead allow agents to use the same frequency band when they are far enough apart,
but require nearby users to use different bands.
When users sharing the same band are far
enough apart, the externalities will become
insignificant and can be safely ignored. For
example, suppose a spectrum asset is defined as
the right to transmit in a given area with a fixed
power mask in a given frequency band. We can
then represent a set of C such assets as nodes in
an interference graph, G, so that two assets are
connected by an edge if they significantly interfere with each other (Fig. 3). If we wish to allocate assets without significant interference, this
corresponds to only allocating assets that belong
to an independent set in G (here we focus on
simply allocating a single frequency band; with
multiple bands, a different independent set
could be used for different frequency bands). If
we simply fix one independent set for all time
and only allocate these assets, the assets would
have no externalities. However, the “best” choice
of assets, i.e., the independent set with the most
value may differ from time to time depending on
the agent’s demands. An alternative approach
would be to dynamically determine the independent set via a market mechanism. Given the true
value of each asset, the efficient allocation is to
find a maximum weight independent set in G ,
where the weights correspond to the values.
Unfortunately, this is an NP-hard problem; thus,
implementing such an allocation via a VCG
mechanism may be computationally challenging.
Of course, the importance of such computational
considerations depends on the size of problem
and the time scale at which allocations are per-
formed. If this complexity is a concern, instead
of exactly solving for the maximum independent
set, one could instead employ an approximation
algorithm; however, in such cases the incentive
properties of the VCG mechanism may no
longer hold.
Designing approximation schemes for general
combinatoric auctions that preserve the VCG’s
incentive properties is a topic that has received
much attention in the algorithmic game theory
literature (e.g., [9]). In the context of spectrum
markets, such questions have also received some
attention; for example, [10] develops a design
called VERITAS. VERITAS applies to the preceding setting in the special case where each bidder wants a single asset from C (more generally,
it applies to settings with multiple frequency
bands). Given an agent’s value for its asset,
VERITAS finds an independent set in time less
than O(n3). Moreover, this mechanism is truthful; that is, agents have no incentive to not bid
their true value. The key to showing this is establishing that the allocation rule is monotonic,
meaning that if bidder i wins an assets when bidding bi, it will continue winning if it raises its bid
to any value greater than bi, assuming all other
agents keep their bids fixed. For a setting such
as this in which each agent desires a single asset,
such monotonicity is known to be necessary and
sufficient for a mechanism to be truthful [11].
The cost for the computational tractability of
this mechanism is that it is no longer guaranteed
to find the optimal allocation (if it did, it would
be finding a maximum weight independent set in
polynomial time). No approximation bound for
VERITAS is given in [10], but the mechanism is
shown to perform well in numerical results.
Another example of a truthful, computationally
tractable mechanism is given in [12], which also
allows for agents to share the available assets.
Again, in this work agents value a single asset
(or share of an asset), so monotonicity suffices
to show truthfulness. In the more general setting
where agents may desire multiple spectrum
assets, it becomes more difficult to characterize
monotonicity and to design truthful tractable
mechanisms. Relatively little has been done in
this space in the context of spectrum markets.
When representing spectrum asset in terms of
an interference graph, a fundamental underlying
question is how to determine the level of interference required for link to be present in this
graph. This level could vary greatly with the
intended application and technology used by a
bidding agent, making it difficult to predetermine. An alternative to this, introduced in [13] is
to not allocate only independent sets but instead
allow every set of asset to be allocated. In such a
setting, suppose that an agent wanted an asset i
but only if there was no interference from a
neighboring asset j. The agent could achieve this
by simply purchasing both assets i and j and simply not using j. On the other hand, another agent
who wanted asset i but could tolerate interference from j would only need to purchase asset i.
With such an approach, spectrum assets have
complementarities, meaning that the value of a
bundle of assets may be greater than the sum of
the value of the individual assets. Such an
approach will be more efficient than restricting
IEEE Communications Magazine • November 2012
THE WORLD’S NEWSSTAND®
allocations to be independent sets. A model for
such a setting is studied in [13], for which it is
shown that the underlying allocation problems
are still NP-hard, unless the interference graph
is restricted to be a line or ring. Little work
appears to have been done to develop incentive
compatible approximation algorithms for such a
setting.
In the previous section we considered allocation spectrum in terms of assigning powers to
transmitting nodes, while in this section we
focused on models in which power is fixed and
one agent has the right to use this in a given spatial region. We briefly mention an approach
from [13] that combines these two ideas. Namely, a market is considered in which agents bid for
both a spatial region and the power emitted
from that region, which effectively determines
the boundary of the region. For a simplified
model, it is shown that adding this flexibility
actually makes the resulting allocation problem
computationally tractable. More precisely, this
model applies to a setting in which an agent’s
value for an asset is proportional to the “radius”
of the spatial region served. Under this assumption, the optimal boundaries for a given assignment can be found by solving a linear program.
Other ways of defining spectrum assets are also
possible. For example, the definition could
require nodes to follow a protocol like carrier
sense multiple access (CSMA), which results in a
different form of interference externality. Ideas
of “cognitive sensing” could also be incorporated
into the definition, perhaps leading to both primary and secondary spectrum assets. Finally,
more advanced physical layer techniques like the
use of multiple antennas could play a role. However, defining a spectrum asset in a way that is
tied to a specific technology also has a risk: it
may make it difficult for alternative competing
technologies to emerge. A clear picture of the
costs and benefits of such approaches has yet to
emerge.
So far we have focused on a market for trading “raw spectrum” (i.e., access to the physical
medium). Another question related to asset
design is whether this is the right layer at which
to run a dynamic market. If a new entrant wishes
to offer a wireless service, it is not clear that it is
best served by acquiring raw spectrum and
deploying its own infrastructure. It could instead
contract with existing “infrastructure providers”
to offer its service over their infrastructure. Such
infrastructure providers could have a relatively
static supply of raw spectrum on which they
build a flexible infrastructure, which is used to
provide transport services to various higher-layer
agents. The resulting market would then be for
these transport services, and the market design
issues would be closer to those for wired bandwidth discussed in part I. To a limited extent,
this type of market exists today, for example,
when cellular service providers sell access to
their networks for various mobile virtual network operators (MVNOs). Enabling such a market on a much more extensive scale could help
facilitate more efficient spectrum usage without
requiring trading of raw spectrum. An advantage
of this approach is that if infrastructure providers own spectrum at neighboring locations,
IEEE Communications Magazine • November 2012
they can manage the interference externalities
themselves, making the market design issues
simpler. Potential disadvantages are that such a
market structure might not provide as much
opportunity for new physical layer services to
emerge, and the transport services offered by the
infrastructure providers may not be well suited
to all applications. Again, more work is needed
to better understand and model such trade-offs.
BARGAINING
The final approach for dealing with externalities
that we will discuss is via bargaining. To motivate
this, consider allocating power to two neighboring bands of spectrum owned by two different
wireless service providers. The two providers
interfere with each other, creating an externality.
We can model this by assuming that provider i’s
utility depends on both the power he is allocated
and the power allocated to the other provider.
Under a bargaining approach, instead of
attempting to price this externality or design
assets in such a way that the externality does not
arise, we simply allocate the bands to both providers and let them bargain with each other to
determine the needed power allocation. For
example, suppose that provider 1 sees a gain of
v 1 from provider 2 reducing her power, and
provider 2 only loses v 2 < v 1 from doing this.
Then if provider 1 offers provider 2 a payment
of p, where v 2 < p < v 1 to do this, provider 2
should be willing to accept this, and the overall
welfare will improve. This is an example of what
is known as the Coase theorem, which states that
if trade for an externality can occur, bargaining
will lead to an efficient outcome independent of
the initial allocation. This result is attributed to
the economist Ronald Coase, who in fact developed it while studying the allocation of wireless
spectrum in the 1950s [14].
A prerequisite for the Coase theorem to
apply is the existence of well defined and
enforceable property rights for the assets being
traded. In other words, agents must be able to
clearly value the assets and ensure that the
counter party follows through on their side of
the bargain. In the context of spectrum ensuring
well defined property rights may be subtle, as
illustrated by the recent dispute over LightSquared’s plans to offer nationwide fourth-generation Long Term Evolution (4G LTE) service.
At the heart of this dispute is the FCC requirement that services deployed in one band do not
cause “harmful interference” to services in adjacent bands. This does not provide a well-defined
property right, as harmful interference depends
in part on the type of receivers deployed in the
adjacent bands. The key point here is that the
definition of a property right has to clearly
define the underlying externality in order for
agents to able to bargain over it. In this spirit,
there has been a line of work by the legal and
policy communities on defining property rights
to enable spectrum markets (e.g., [15]). Once
again, this is an issue that is not cleanly separable from technical considerations, although the
technical community has provided little input.
The Coase theorem may seem to suggest a
very simple market structure for well defined
Under a bargaining
approach, instead of
attempting to price
this externality or
design assets in such
a way that the externality does not arise,
we simply allocate
the bands to both
providers and let
them bargain with
each other to determine the needed
power allocation.
89
THE WORLD’S NEWSSTAND®
Even with a well
designed market,
there may still be
opportunities for
agents to bargain
after receiving their
allocations. Incorporating such considerations in the design
of dynamic spectrum
markets is another
area that has not
been fully
developed.
assets; simply allocate the assets arbitrarily with
well defined property rights and let the agents
bargain. The problem with this conclusion is that
in practice there are a number of frictions, which
may impede efficient bargaining. For example,
there may be costs and delays involved in bargaining. These delays may be due to the time it
takes to agree on a bargain as well as the search
time needed to find a counter party to bargain
with. Moreover, these tend to increase with the
number of parties with whom one needs to bargain. In the case of spectrum markets, this suggests that bargaining may be more difficult when
spectrum is allocated on finer temporal and spatial scales; finer temporal scales restrict the time
for bargaining, and finer spatial scales may
increase the number of relevant counter parties.
Another friction in bargaining is due to imperfect
information, that is, when agents do not exactly
know each other’s valuations. Returning to the
previous example, suppose that v1 = 3 and v2 =
2 so that provider 1 should be willing pay some
price p slightly more than 2, which will result in
provider 2 lowering his power and the overall
welfare improving. Now suppose that provider 1
only knows that v2 is distributed uniformly on the
interval [0, 3]; in this case provider 1’s expected
benefit is maximized if she makes provider 2 an
offer of 1.5, which would not be accepted, even
though the overall welfare would improve if
trade occurred. Of course, this is just one possible way the two agents can bargain; however, a
result due to the economists Myerson and Satterthwaite [16] shows that the underlying problem is fundamental: in the presence of
incomplete information, there is no bargaining
procedure that can always guarantee the efficient
outcome. This discussion suggests that simply
relying on bargaining alone may not be sufficient;
markets can be used to reduce these frictions and
provide incentives for agents to reveal private
information. Of course, even with a well designed
market, there may still be opportunities for
agents to bargain after receiving their allocations.
Incorporating such considerations in the design
of dynamic spectrum markets is another area
that has not been fully developed.
CONCLUSIONS
From both parts of this article it should be clear
that network market design contains a number
of challenging and under-explored topics. In this
part we have focused on a number of such issues
for spectrum sharing markets that must operate
in the presence of interference externalities. In
this part, we have focused mainly on single-sided
markets in which one “spectrum manager” is
selling access to a set of service providers. As
discussed in Part I, market design becomes more
complicated for doubled-sided exchanges in
which there are both multiple buyers and sellers.
Such a setting could also arise in spectrum markets, for example, when multiple license holders
seek to lease their spectrum to multiple secondary service providers.
As in Part I, we have largely ignored temporal
dynamics, which raises a number of additional
questions. For example, the price of spectrum
assets will change based on time-varying demand.
90
More elastic users can attempt to exploit this by
deferring their usage to cheaper times; however,
their ability to do this will be based in part on
how well they can predict future usage, which is
determined by the actions of other strategic
agents. Less elastic users may want to guarantee
some level of service in the face of uncertain
future price fluctuations. Perhaps a “spectrum
futures” market could be used to meet this
demand. Analyzing such settings is challenging, in
part because the underlying economic theory for
dynamic environments is much less developed.
ACKNOWLEDGEMENTS
The author would like to acknowledge Junjik
Bae, Eyal Beigman, Jianwei Huang, Michael
Honig Thanh Ngyuen, Vijay Subramanian,
Rakesh Vohra, and Hang Zhou; many of ideas
in this article emerged from collaborations with
these researchers. This work was supported in
part by the National Science Foundation under
grants CNS-0519935, CNS-0905407, and CNS1147786.
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BIOGRAPHY
RANDALL A. BERRY ([email protected])
_________________ received
a Ph.D. in electrical engineering and computer science
from the Massachusetts Institute of Technology in 2000.
Subsequently, he joined Northwestern University, where he
is currently a professor in the Department of Electrical
Engineering and Computer Science. His research interests
include network economics and wireless communications.
IEEE Communications Magazine • November 2012