Federal Communications Commission
Office of Strategic Planning and Policy Analysis
445 12th Street, SW
Washington, DC 20554
OSP Working Paper Series
42 Modeling the Efficiency of Spectrum Designated to
Licensed Service and
Unlicensed Operations
February 2008
Mark M. Bykowsky
Mark A. Olson
William W. Sharkey
Modeling the Efficiency of Spectrum Designated
to Licensed Service and Unlicensed Operations1
Mark M. Bykowsky2
Office of Strategic Planning and Policy Analysis
Federal Communications Commission
Washington, D.C.
William W. Sharkey
Wireline Competition Bureau
Federal Communications Commission
Washington, D.C.
and
Mark A. Olson
George Mason University
Arlington, Virginia
February 2008
OPP Working Paper No. 42
1 This paper represents an extended theoretical discussion of issues which arise in OSP Working Paper #41
entitled “Enhancing Spectrum’s Value Through Market-informed Congestion Etiquettes” co-authored by
Kenneth Carter in addition to the three current authors.
2 The authors would like to thank Weiren Wang, Charles Needy, and Thomas Spavins for very helpful
comments on an earlier draft. Mark Olson participated in this project under a contract with the FCC. This
document is available on the FCC’s World Wide Web site at http://www.fcc.gov/osp/workingp.html
.
The FCC Office of Strategic Planning and Policy Analysis’s Working Paper Series
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Abstract
Obtaining the economically efficient use of spectrum requires that the price signals that
guide its allocation and use be competitive. One approach involves establishing a non-
exclusive licensing regime that provides for open and free access to spectrum. However,
myopic, self-interested behavior on the part of users can give rise to excessive wireless
spectrum congestion. This paper examines the Nash Equilibria that exist in an
environment that exhibits characteristics present in the naturally occurring environment.
The various Nash Equilibria generate total surplus levels that are far below the levels
achieved under the efficient allocation of spectrum. This paper also examines the
performance properties of congestion etiquettes that utilize various types of user
information to address the congestion problem. Termed “market-informed” etiquettes,
Nash Equilibrium theory predicts that these etiquettes are superior to the currently and
widely used congestion etiquette, despite the fact that such etiquettes introduce a
surprising strategic effect that may substantially reduce total welfare.
TABLE OF CONTENTS
1 Introduction ................................................................................................................1
2 User Selection Between Licensed and Unlicensed Service Options .........................2
2.1 Tragedy of the Commons Problem: Nash Prediction ............................................... 6
2.2 Alternative Congestion Etiquettes .............................................................................. 9
2.3 Summary Analysis of Alternative Congestion Etiquettes....................................... 14
3 A Simple Competitive Model ....................................................................................17
4 Simulations with Random Parameter Selection......................................................23
5 Concluding Comments .............................................................................................25
Appendix A: Detailed Data from Examples ....................................................................27
- 1 -
1 Introduction
Recent advances in radio technology have made it possible to transmit data, voice,
and video in digital form over high-speed wireless networks. In response to high
demand, service providers and equipment manufacturers are constantly attempting to
improve the transmission capacity of the available spectrum in an effort to avoid wireless
congestion. For its part, the Federal Communications Commission (FCC) has the ability
to reduce congestion by allocating additional spectrum for the provision of such services.
To do so, the FCC must first determine whether to allocate spectrum to either licensed or
unlicensed use. It is generally agreed that, to date, the allocation of spectrum to
unlicensed use has significantly enhanced consumer welfare. However, the issue of
whether to allocate additional spectrum to unlicensed use is both a contentious and
complex one.
One complicating factor is that allocating additional spectrum to unlicensed use may,
under certain conditions, reduce consumer welfare. The economic problem is
straightforward. Markets work best when the choices that economic actors make reflect
the cost that their decisions impose on society. Economic agents typically incur the cost
of their decisions by paying an appropriately informed market price. Due to the free
entry condition inherent in unlicensed use (as traditionally conceived), there is no
assurance that consumers will take into account the negative effect of their spectrum
consumption decisions on the value other consumers place on using the same spectrum.
Thus, rational users may “over-consume” freely available spectrum compared to the level
that would promote both consumers’ and society’s interests. This welfare reducing
outcome is referred to as the “Tragedy of the Commons.”
One solution involves treating spectrum allocated to unlicensed use as a “common
pool resource” and utilizing a congestion etiquette to allocate spectrum to competing
users. Most existing etiquettes are based solely on engineering principles, as opposed to
market principles. This study defines four new potential congestion etiquettes, and
examines the Nash equilibrium outcomes of each when spectrum users are allowed to
choose between licensed and unlicensed options. We show that while each of the
etiquettes improves the performance of the congestion-prone service, by reducing or
eliminating the Tragedy of the Commons problem, some of the etiquettes also have an
- 2 -
unanticipated secondary effect of distorting user choices between the two services. In
many cases, the secondary distortion outweighs the primary beneficial effect, and
aggregate total surplus is lower under an etiquette than under ordinary Nash equilibrium.
These results demonstrate that modest changes to the spectrum assignment process can
potentially increase the value that society receives from allocating spectrum to unlicensed
use, but that great care must be taken in choosing a specific mechanism in which
unlicensed spectrum is allocated.
Our results have implications for potential future public policies regarding the
designation of spectrum to licensed use versus unlicensed operations through an auction
mechanism. Some of these issues are addressed in a recently released working paper by
the authors.3 More broadly, this work is also relevant to the ongoing discussion of
broadband infrastructure development and competition policy for local
telecommunications. In particular, an understanding of, and potential resolution of the
Tragedy of the Commons problem for usage of wireless spectrum will help policy makers
evaluate the viability of a third wireless competitor (in addition to cable and wire line
telecommunications providers) in the market for “last mile” broadband connections.
2 User Selection Between Licensed and Unlicensed Service Options
The Tragedy of the Commons refers to a situation in which myopic, self-interested
behavior leads to the excessive use of a common pool resource. Because of this
excessive use, the welfare of individual users is not maximized, and further, society fails
to obtain the most efficient use of its scarce resources. In some situations, users of a
common pool resource can avoid the Tragedy of the Commons problem by tacit
coordination in their usage decisions. For example, bus riders in many cities voluntarily
agree to wait in line for the next bus. In other cases (e.g. commercial fishing limits) both
voluntary industry agreements and explicit governmental regulations have been used to
address the tragedy of the commons problem. While such coordination has been
successful in these instances, the radio spectrum market has several characteristics that
make user coordination particularly difficult.
3 Bykowsky, M., Olson, M., and Sharkey W. (2008) “A Market-based Approach to Establishing Licensing
Rules: Licensed Versus Unlicensed Use of Spectrum,” OSP Working Paper #43.
- 3 -
One market characteristic is the wide variation in uses to which users employ
spectrum. Some users employ spectrum to download streaming video, while other users
employ spectrum to send and receive text messages. The level of satisfaction a user
obtains from employing spectrum depends importantly on the amount of unused capacity
available to satisfy his or her desired application. Equally important, the level of
available capacity needed to provide satisfactory streaming video service is different (in
this case substantially greater) than the level needed to provide satisfactory text
messaging service. Such differences in the amount of spectrum required by different
applications complicate the ability of users to coordinate their uses. Therefore, in order
to avoid the Tragedy of the Commons problem, users must not only recognize that their
spectrum use negatively affects other users, but they must also take into account the
specific demands of other users. As an example, the economic cost of adding one
additional user who wishes to download streaming video may be negligible if all current
users are employing spectrum for text messaging, but it may be very high if existing users
are also downloading streaming video applications, or are otherwise intolerant of
spectrum congestion.
A simple example can be used to describe the unique and complicated nature of the
Tragedy of the Commons problem involving spectrum usage. Assume that there are
eight different spectrum users, each of which is defined by a set of characteristics. Each
user varies in the minimum amount of bandwidth that he or she needs in order to satisfy
their service needs, as well as the value they place on having their service needs fully
satisfied. Because of differences in service needs, users also vary in the extent to which
they can tolerate spectrum congestion. For example, a user who requires a very high
quality of service (e.g. video streaming or other applications that require a high and
reliable data transmission rate) might find quality unacceptable whenever aggregate
demand exceeds 60% of system capacity. Users who require a lower quality of service
(e.g. email) would find quality acceptable if aggregate demand exceeds a higher
percentage of system capacity (e.g. 80%). In what follows we measure a user’s demand
for service quality by his or her “congestion limit,” expressed as the maximum amount of
spectrum congestion the user can tolerate before the value he or she places on employing
- 4 -
spectrum falls from some desired value to zero.4 Figure 1 presents hypothetical data for a
set of spectrum users, including the value that each user places on spectrum (expressed
on a per megabit basis), the minimum amount of spectrum required by the user, and each
user’s quality of service requirement as measured by his/her congestion tolerance limit.
Figure 1. Spectrum User Characteristics in Example 1
Willingness to Pay
per MB
Quantity Demanded (MB)
5
10
#3
10¢
17 MB
60%
#1
9¢
24 MB
60%
#4
7¢
16 MB
60%
#5
7¢
18 MB
80%
7¢
13MB
80%
#8
#6 #7
5¢
17 MB
80%
141
5
11 MB
80%
#2
5¢
25 MB
60%
$2.16 $1.25 $1.70 $1.12 $1.26 $0.55 $0.85 $0.91
60% Congestion Tolerance
80% Congestion Tolerance
Key
¢ Per bit
Megabits
Congestion
$ Total
216 125 170 112 126 55 85 91
Value/MB
Total Value
Total Valuation = 980
Suppose that each user in the above example has the option to choose between a
reliable and non-congestible transmission system that requires a fixed payment PA = 130
(Option A), and a transmission system (Option B) that is “free,” but where the value to
each user depends on the system congestion level. The capacity of the free service is
assumed to be equal to 100 units. The identification of the efficient assignment of
spectrum depends on the amount of spectrum demanded by the users, the value that users
4 A user’s demand for quality is, in fact, a demand for a number of data transmission characteristics
including file transfer rate, latency tolerance, and allowable level of jitter. These service attributes tend to
degrade quite rapidly when total user demand exceeds a given percentage of total system capacity, which
may vary by user. While an “all or nothing” congestion limit is an obvious simplification, it captures all of
the strategic considerations of user selection, which is the main focus of this paper, and avoids the need to
model the full complexity of choices between services with differing quality characteristics.
- 5 -
place on this spectrum, and the minimum quality of service that each user requires in
order to obtain that value.
The unique efficient assignment of spectrum in this example involves assigning
spectrum to users as shown in Figure 2. In Figure 2 and later tables, total surplus is
defined as åå
ÎÎ
+
Bi
ii
Ai
ii qvqv , and consumer surplus is defined as
( ) ( )åå
ÎÎ
-+-
Bi
Bii
Ai
Aii PqvPqv , where qi represents user i’s demand, vi represents value
per unit of demand for user i, and PA and PB represent the prices for options A and B
respectively.5
Figure 2. Efficient Assignment of Spectrum in Example 1
Quantity Demanded (MB)
#3
10¢
17 MB
60%
#1
9¢
24 MB
60%
#4
7¢
16 MB
60%
#5
7¢
18 MB
80%
7¢
13MB
80%
#8
#6
#7
5¢
17 MB
80%
60% Capacity
Congestion
Limit5
11 MB
80%
Option A Option B
Consumer Surplus = 384
5
10
5
10
Excluded
5841
$2.16 $1.70 $1.12 $1.26 $0.55
$0.85
$0.91
#2
5¢
25 MB
60%
$1.25
Willingness to Pay
per MB
216 170 112 126 55
85
91
Consumer Surplus = 126
Provider Surplus = 260
Total Surplus = 770
125
Notice that in the efficient assignment, users 2 and 7 are excluded from spectrum
available via both Options A and B. Under Option A, both users 2 and 7 would receive
5 By default, the price of Option B is equal to zero. Under some etiquettes, to be discussed later, a positive
price might be assigned. In the following section, a per unit price, PB , may be assigned to users of Option
B. In the above notation, producer surplus is defined as PA nA + PB nB, where nA and nB represent the
number of subscribers choosing options A and B respectively.
- 6 -
negative surplus, since the value they place on service is less than the subscription fee of
130. Moreover, including either user 2 or user 7 under the free service (Option B) would
require displacing at least one of the existing users of that service, and one can verify that
any such change would result in a reduction in total surplus.
2.1 Tragedy of the Commons Problem: Nash Prediction
Suppose users simultaneously and independently choose whether to select the
subscription service (Option A), the free service (Option B), or neither service. A Nash
equilibrium is a particular joint selection such that no individual user could increase his
or her payoff by selecting an alternative strategy, assuming that all other users maintain
their equilibrium strategy choices. We can now see immediately that the efficient
assignment in Example 1 is not a Nash equilibrium. Suppose that user 7 contemplates
joining the free service. Such a change would increase the total demand for this service
from 58 to 75 units, but since user 7 expects to receive acceptable quality of service as
long as total demand is less than 80 units, this alternative selection would increase user
7’s surplus from zero to 85 valuation units.6 Under Nash equilibrium assumptions, user 7
would therefore choose this alternative. User 7’s deviation from the efficient allocation
unfortunately reduces total surplus. With a total demand of 75 for Option B, user 4 no
longer receives a satisfactory quality of service, and under our assumptions, her entire
surplus of 112 valuation units is assumed to be forfeited.
While the efficient assignment cannot be sustained as a Nash equilibrium, there are
alternative joint user selections that satisfy the Nash equilibrium conditions. The
magnitude of the Tragedy of the Commons problem in Example 1 is measured by the
difference between the total surplus under the efficient assignment and total surplus under
the Nash equilibrium condition. The value of this measure, as predicted by Nash
Equilibrium Theory, depends on the joint user selection. One common element of each
Nash assignment (in this example) is that both users 1 and 3 are assigned to the
subscription service (Option A). To see this, suppose that users 1 and 3 select Option A,
users 5, 6, 7, and 8 select Option B, and that users 2 and 4 choose neither option. These
6 User 7 has a congestion tolerance of 80%. Given that the free service’s capacity is 100 Mb, user 7 would
obtain full value from selecting Option B
- 7 -
selections satisfy the Nash equilibrium conditions. If either user 1 or user 3 attempts to
switch to the free service in order to avoid paying the subscription fee, demand for
Option B would rise above either of their congestion limits, and they would forfeit the
value received by selecting Option A. Each of the users choosing Option B would clearly
see a reduction in surplus by switching to Option A, since they would be required to pay
the subscription fee of 130, under which they receive negative surplus. Finally, users 2
and 4 receive zero surplus by choosing neither option, but would do no better by
choosing either Option A or Option B.7
Table 1. Nash Equilibrium Assignments for Example 1
Total surplus in the above equilibrium selection is equal to 743 valuation units. In
fact, this is the highest total surplus obtainable in any of the Nash equilibria, although it is
lower than the surplus of 770 obtained under the efficient assignment. Thus, if this
equilibrium was obtained, the social cost of the Tragedy of the Commons problem would
7 All of the assertions made about specific Nash equilibrium outcomes can be verified by simple
computations. Results presented in later tables are based on computer algorithms which are able to
exhaustively repeat these computations for all possible joint strategies.
User Choice Selection Performance Measures
Option A Option B Neither Option Total Surplus Consumer Surplus Total Efficiency
1, 3 5, 6, 7, 8 2, 4 743 483 96.5%
1, 3 4, 5, 6, 7, 8 2 743 483 96.5%
1, 3 2, 6, 7, 8 4, 5 617 357 80.1%
1, 3 2, 5, 7, 8 4, 6 688 428 89.4%
1, 3 2, 5, 6, 8 4, 7 658 398 85.5%
1, 3 2, 5, 6, 7 4, 8 652 392 84.7%
1, 3 2, 5, 6, 7, 8 4 386 126 50.1%
1, 3 2, 4, 7, 8 5, 6 562 302 73.0%
1, 3 2, 4, 6, 8 5, 7 532 272 69.1%
1, 3 2, 4, 6, 7 5, 8 526 266 68.3%
1, 3 2, 4, 6, 7, 8 5 386 126 50.1%
1, 3 2, 4, 5, 8 6, 7 603 343 78.3%
1, 3 2, 4, 5, 7 6, 8 597 337 77.5%
1, 3 2, 4, 5, 7, 8 6 386 126 50.1%
1, 3 2, 4, 5, 6 7, 8 567 307 73.6%
1, 3 2, 4, 5, 6, 8 7 386 126 50.1%
1, 3 2, 4, 5, 6, 7 8 386 126 50.1%
1, 3 2, 4, 5, 6, 7, 8 386 126 50.1%
- 8 -
be very small. However, as noted earlier, there are many joint user selections that satisfy
the Nash equilibrium conditions, many of which generate much lower levels of total
surplus. The full set of Nash equilibria and the associated total efficiency levels for
Example 1 are shown in Table 1.8
While there a large number of outcomes predicted under the assumption that users
follow Nash equilibrium behavior, there are strong theoretical grounds for assuming that
the outcomes that generate the greatest amount of social loss are the ones that are the
most likely to occur. To see this, note that users who select Option B are guaranteed to
receive at least a zero payoff, and there is a possibility that they could receive a positive
payoff if other users deviate (contrary to Nash assumptions) from the assumed
equilibrium. Those who select “neither option” receive a zero payoff with certainty. In
the language of non-cooperative game theory, in this case a strategy in which a user
chooses “neither option” is “weakly dominated” by the strategy in which that user
chooses Option B. The only equilibrium in which no user selects a weakly dominated
strategy is shown in the last row of Table 1. This outcome is the worst of all Nash
equilibrium outcomes, measured in terms of both total and consumers’ surplus, and its
total efficiency is equal to 50.1%.
8 In some cases all of the users who select Option B receive zero surplus due to the congestion caused by
their collective choices. However, these outcomes satisfy the Nash equilibrium conditions since no
individual user can improve his or her payoff by selecting an alternative strategy. For example, when users
1 and 3 select Option A, users 2, 5, 6, and 7 choose Option B, and user 4 chooses neither option, all users
except 1 and 3 receive zero surplus.
- 9 -
2.2 Alternative Congestion Etiquettes
In this section we consider a number of service etiquettes that can potentially lead to
superior market outcomes. We first consider an etiquette that addresses the congestion
problem by changing the payoffs of users and, in so doing, their service choice. This
etiquette is based on the observation that the Tragedy of the Commons problem is made
worse by the fact that the strategy of selecting Option B weakly dominates the strategy of
selecting neither Option A nor B. One method of changing user incentives, herein called
the “Pay to Connect” etiquette, is to charge users a small non-refundable fee if they select
Option B. Under pay to connect, a user who selects Option B, but who also expects that
his/her decision would raise the demand for spectrum available under Option B to a level
above his or her own congestion limit, would expect to receive negative surplus, and
would therefore prefer to choose to sit on the sidelines and earn a zero payoff.
Given a fee for selecting Option B, the last row of Table 1 would no longer represent
a Nash equilibrium. Similarly, any equilibrium from Table 1 in which at least one user
receives zero surplus while choosing Option B cannot be an equilibrium under the Pay to
Connect option. In fact, in Example 1, the only Nash equilibrium strategy selection that
survives the imposition of a small but positive fee for Option B is the highest surplus
outcome in Table 1. In this outcome total efficiency is equal to 96.5%.
We next consider two etiquettes that employ a rationing technique to more
efficiently allocate spectrum to those users that select Option B. As before, users first
decide whether to select Option A or Option B. For those who select Option B, a random
number generator is used to assign a priority level to each such user. If n users select
Option B, then there are n! permutations of users, and each permutation is assumed to be
chosen with equal probability.9 Users are then assigned spectrum under Option B until
the total demand exceeds a pre-determined level, which we assume to be equal to the
highest congestion limit of all users who are expected to make use of the system.10 Herein
9 When all 8 users choose Option B, there are 40,320 possible permutations.
10 In an alternative version of this randomization etiquette, users could be served only until total demand
exceeds the lowest congestion limit of expected users. There are clear trade-offs involved in selecting the
appropriate system capacity, and neither rule is optimal in all situations. With a high limit, some high
priority users with low congestion limits would be granted service under the etiquette, but the service
would be worthless to them if the total demand exceeded their personal congestion limit. On the other
hand, there can be situations in which some users would be served when system capacity is set at the high
congestion limit but would be rejected under the lower limit.
- 10 -
referred to as “Simple Randomization,” this etiquette addresses the tragedy of the
commons problem by guaranteeing that some users (those with high priority and high
congestion limits) receive some value from their assigned spectrum under Option B. A
particular implementation of the simple randomization etiquette for Example 1 is
illustrated in Figure 3 below. In this example, all 8 users are assumed to select Option B,
and the random priority assignment ranks users in the order {5, 8, 1, 2, 4, 3, 7, and 6}.
Figure 3. Simple Randomization in Example 1
$2.16 $1.25 $1.70$1.12$1.26 $0.55$0.85$0.91
Quantity Demanded (Mbps)
5
10
#3
10¢
17 Mbps
60%
#1
9¢
24 Mbps
60%
#4
7¢
16 Mbps
60%
#5
7¢
18 Mbps
80%
7¢
13Mbps
80%
#8
#6#7
5¢
17 Mbps
80%
141
#2
5¢
25 Mbps
60%
60% Capacity
Congestion
Limit
80% Capacity
Congestion
Limit
Willingness to Pay
per Mbps
80%
11 Mbps
216 125 170112126 558591
5
Under the simple randomization etiquette, users 5, 8, 1, and 2 are provided spectrum,
while all lower priority users are excluded. Including any additional user would raise
total demand above the system capacity of 80, which is equal to the maximum congestion
limit. In this case, users 1 and 2 would receive no value from their spectrum assignment,
since the total demand would exceed their personal congestion limits, and the resulting
total efficiency would be equal to 16.3%. Alternative user selections between Options A
and B will result in different outcomes under the Simple Randomization etiquette. When
users take all such random outcomes into account there is a unique Nash equilibrium
- 11 -
whose outcome is shown in Appendix A. In this equilibrium, total efficiency is equal to
83.6%.
In an alternative version of the randomization etiquette, to be called “Informed
Randomization”, users are first asked to report their personal congestion limits to a
system manager.11 As in simple randomization, a random number generator is used to
assign a priority level to each user who selects Option B. Service is now assigned,
however, by taking account of the congestion limits of each potential user. If the demand
of the highest priority user is less than that user’s congestion limit, that user is guaranteed
service. The etiquette then considers each additional user’s demand and personal
congestion limit in order of decreasing priority. The etiquette ensures that total demand
always remains less than or equal to the minimum congestion limit of all users who are
guaranteed service. With each new user, the relevant system congestion limit is set equal
to the minimum limit of the current user being considered, and the limits of all higher
ranking users already guaranteed service. If the total demand, including the current user,
is less than or equal to the congestion limit, that user is granted service. Otherwise, that
user is not served and the next user is considered. The etiquette continues to examine
users until all have been considered, or until demand is exactly equal to the system limit
as determined by the currently served users.
Under the Informed Randomization Etiquette, users 2, 4, 5, 6, 7, and 8 would expect
to receive positive surplus under Option B, since there is positive probability that they
will be assigned a high priority rank. They will therefore choose Option B in preference
to choosing neither Option A nor Option B. In addition, user 3 in this example has an
incentive to select Option B over Option A, because the expected surplus to him or her
under the randomization protocol is higher than could be achieved with certainty by
11 The system manager would most likely be embodied in a hardware devise, and user reports about
personal congestion limits would correspond to priority bits under existing internet protocols. The
sensitivity of the randomization (and later) etiquettes to personal congestion limits raises the question of
whether users will have the incentive to report their limits truthfully. In fact, “truth telling” is in this case is
a weakly dominant strategy (in game theoretic terminology) since a user who selects Option B can never
increase his or her payoff by falsely reporting a congestion limit. To see this, note that a high priority user
with a low (e.g. 60%) congestion limit would never wish to report a higher limit. If the user would have
been excluded under a truthful (60%) report, than, while a false report (e.g. 80%) might result in providing
service, the service would be of no value to the user. Similarly, a user with a high congestion limit would
never wish to report a lower limit, since in some cases this would result in that user being denied service,
while in no cases would a lower total demand increase the quality of service under the present assumptions.
- 12 -
choosing Option A. For user 1, the certain surplus under Option A is higher than that
under the randomization etiquette, whether or not user 3 chooses Option B. The resulting
total efficiency is equal to 72.8%.
While both versions of the randomization etiquette guarantee that those users who
select Option B, and who have a sufficiently high priority level, receive positive surplus,
these protocols do not necessarily create the correct incentives for users to make optimal
selections between Options A and B. In particular, by raising a user’s expected return
from selecting Option B, the randomization etiquettes may induce high valued users (who
can afford to select Option A) to select Option B. While this choice is privately welfare
maximizing for the user, it can potentially reduce total surplus.
Finally, we consider an etiquette that utilizes reported information on a user’s
congestion tolerance and willingness to pay (WTP) to more efficiently allocate spectrum
to those users who select Option B. In one version of a WTP etiquette (hereafter called
the “Lump Sum WTP Etiquette”), users who select Option B would be asked to report
both their personal congestion limit and their willingness to pay to receive or send
information. The reported willingness to pay values would then be used to define a
priority ranking of users, and the etiquette would proceed in a similar manner as the
randomization etiquette (but without the random component). In one important
difference, however, users under the WTP etiquette would be required to actually pay a
price to receive service. Under this etiquette, the price is determined by the reported
willingness to pay of the marginal user who is excluded from service by the etiquette.
Only users who are guaranteed service under the etiquette are required to pay this price.12
As an example, suppose that in Example 1, all users bid “truthfully” in the following
sense: users 1 and 3 bid 130, which is the price they would have paid if they had selected
Option A, and all remaining users bid their actual value, which in each case is less than
130.13 Users 1 and 3 would then be tied for the highest priority of service, and they
would be followed in order by users 5, 2, 4, 8, 7 and 6. Under the WTP etiquette, users 1,
12 This pricing rule is based on standard “second price” auction theory. In the present context, there is
some ambiguity about the definition of the marginal user. The present version of the etiquette assumes that
the extra-marginal user is defined as the highest priority user who is excluded from service, and such that
no lower priority user is assigned service.
13 In general, truthful bidding of this form is not guaranteed. While truthfully revealing a personal
congestion limit is still a weakly dominant strategy, there can be situations in which users may prefer to
misrepresent their willingness to pay in an effort to lower the market price.
- 13 -
3 and 5 would be assigned spectrum under Option B, and each would pay a price equal to
125, which is the highest rejected bid. No other users will be assigned spectrum, since
the addition of any other user would violate the requirement that total demand remain less
than or equal to the minimum congestion limit of all users who are guaranteed service.
Figure 4: Willingness to Pay in Example 1
Quantity Demanded (MB)
10¢
17 MB
60%
9¢
24 MB
60%
141
60% Capacity
Congestion
Limit
80% Capacity
Congestion
Limit
$2.16$1.70
#5
18 MB
80%
#8
Lump Sum Bids
100
#3
60%
#1
60%
#4
#6
#7
17 MB
80% 11 MB
80%
130
130
16 MB
60%
112
126
55
85
13 MB
80%
91
#2
25 MB
60%
125
150
50
The ability of users to express their willingness to pay introduces some surprising and
important strategic effects. For example, users with high valuations can now predict with
greater certainty whether or not they will be served if they choose Option B. No user
who can afford to pay the subscription price for Option A can gain by choosing Option B
and making a bid higher than the subscription price for Option A (for which satisfactory
service is guaranteed).14 Assuming that users place “truthful” bids as described above, all
users select Option B in the set of undominated Nash equilibria for Example 1. In this
equilibrium both total surplus and consumer surplus are lower than surplus under both of
the randomization etiquettes and under and Pay to Connect. Total efficiency under the
WTP etiquette for Example 1 is equal to 66.5%.
14 A bid for service under Option B that is higher than the price of Option A would differ from a bid equal
to the price of Option A only if the bid of the marginal user under Option A was higher than the price of
Option A. In such a case, the bidder would be better off choosing Option A instead of Option B.
- 14 -
2.3 Summary Analysis of Alternative Congestion Etiquettes15
In this section we will summarize the above analysis as it applies to Example 1, as
well as three additional examples, which are reported in detail in Appendix A. In each
example, there are eight different users. Demand for bandwidth, the value of completed
service, and the demand for service quality as defined by a personal congestion limit
varies among the users. The examples differ primarily in terms of the set of users who
can potentially afford to purchase the subscription service under Option A. In Example
1, with a price for Option A equal to 130, two out of eight users can afford Option A as
illustrated in Figure 1. In Example 2, the price of option A remains equal to 130, but user
demands and values of service are changed in such a way that five out of eight users can
afford Option A. Examples 3 and 4 maintain the demand and value of service
assumptions of Example 2, but adjust the price of Option A with the result that eight out
of eight users can afford Option A in Example 3, while zero out of 8 can afford Option A
in Example 4.
The examined etiquettes fall into two categories. One class of etiquette attempts to
solve the congestion problem by changing the payoffs that some users receive by
selecting Option B. In the Pay to Connect etiquette, users pay a small fee for selecting
Option B, and have an incentive to refrain from selecting this option when it is expected
that their decision would lead to an outcome that is both individually non-rational and
globally inefficient. Thus, the Pay to Connect option can lead to superior choices
between Options A and B, but at the same time, it does not lead to efficient spectrum
allocation among the users who finally choose Option B.16
Another class of etiquettes attempts to more efficiently allocate spectrum among
those users that choose Option B. Both the Simple and Informed Randomization
etiquettes work by explicitly denying service to low priority users when total demand
threatens to exceed a pre-defined system congestion limit. However, randomization has a
potentially perverse effect on user choices between Options A and B. On the one hand, it
gives every user who cannot afford to choose the subscription service (Option A) a
15 Data for three additional examples are provided in an Appendix.
16 In general, Pay to Connect does not result in a unique equilibrium, or guarantee that the highest
equilibrium surplus is attained.
- 15 -
positive incentive to choose Option B, rather than choose neither option. It can also give
some users an incentive to switch from Option A to Option B, in cases where their
expected payoff in Option B is higher than the certain payoff in Option A. The latter
effect is more pronounced under the informed version of the etiquette than under the
simple version. As with the Pay to Connect etiquette, an etiquette based on the
randomized ranking of users does not ensure that the spectrum available under Option B
is assigned to those users that value such spectrum the most.
The Willingness to Pay etiquette has both potential advantages and potential
disadvantages compared to the randomization etiquettes. User willingness to pay can
potentially allocate service to the highest value users who choose Option B. However,
high value users (users who would be willing to pay the subscription price for Option A)
can reduce the uncertainty regarding whether they will obtain service via Option B by
submitting a bid that is equal to the cost of accessing spectrum via Option A. As a result
of this reduction in uncertainty, more high value users now have an incentive to select
Option B, and these decisions can potentially reduce total surplus by generating more
potential congestion under option B.
The results of all four examples, presented in detail in Appendix A, are summarized
in Tables 2 and 3 below. In these tables, efficiency is defined as the surplus obtained
under the corresponding etiquette, as a fraction of maximum possible surplus, expressed
in percentage terms. Each row reports only the equilibrium surplus associated with un-
dominated strategies under the corresponding etiquette. Examples 3 and 4 were
deliberately chosen as extreme cases, where the co-existence equilibrium achieves full
efficiency in Example 3 and zero efficiency in Example 4. These results are explained as
follows. In Example 3, all users can afford to purchase Option A, so that in equilibrium,
no user would willingly choose service under Option B while expecting that enough other
users would do so to raise total demand above his or her personal congestion limit. Since
Option A is available in this case, that option dominates a choice of Option B. In
Example 4, the situation is reversed. No users can afford to purchase Option A, and
therefore, in an undominated strategy equilibrium, all users must choose Option B. Any
user for whom total demand for Option B divided by total system capacity for Option B
is greater than his or her personal congestion limit can do no better than a zero payoff in
- 16 -
this case. In Example 4, all users happen to fall into this category under the assumed
parameters.
In each of the examples, two of the four pro-active etiquettes (pay to connect and
simple randomization) achieve higher total surplus than that achieved under simple co-
existence.17 Informed randomization achieves higher total surplus in only two of the four
examples, while the willingness to pay etiquette achieves equal or higher surplus in three
of the four examples. In terms of consumer surplus, the results are similar. Simple
randomization achieves outcomes that are equal to or superior to the co-existence
outcome. Pay to connect achieves higher consumer surplus in two of the four examples
and marginally smaller surplus in the remaining two cases. The informed randomization
and willingness to pay etiquettes achieve higher consumer surplus in two of the four
examples and lower surplus in the remaining two cases.
Table 2. Summary of Total Surplus Efficiency Results
Etiquette Example 1 Example 2 Example 3 Example 4
Co-existence 50.1% 91.4% 100% 0%
Pay to Connect 96.5% 91.4% 100% 84.8%
Simple Randomization 83.6% 91.4% 100% 43.6%
Informed Randomization 72.8% 54.7% 93.1%18 75.0%
WTP 66.5% 76.9% 100% 100%
17 Note, however, that some co-existence equilibria involving dominated strategies (where one or more
users select neither option and receive a zero payoff with certainty) can achieve higher total surplus than
some of these etiquettes.
18 This result and the corresponding entry in the next table represent an average of gross surplus due to four
equilibria as presented in Appendix A.
- 17 -
Table 3. Summary of Consumers’ Surplus Efficiency Results
Etiquette Example 1 Example 2 Example 3 Example 4
Co-existence 23.9% 100% 100% 0%
Pay to Connect 90.7% 99.4% 99.6% 84.1%
Simple Randomization 72.6% 100% 100% 43.6%
Informed Randomization 81.5% 76.4% 91.5% 75.0%
WTP 25.9% 49.2% 78.1% 18.7%
3 A Simple Competitive Model
In this section we introduce a simple supply side to the analysis in order to provide
further insights into the question of spectrum allocation between licensed and unlicensed
uses. In contrast to previous sections, where the subscription service (Option A) was
assumed to be offered at a fixed price, and a second service (Option B) was assumed to
be offered at fixed capacity (either on a free basis or with consumer fees based on certain
congestion etiquettes), this section will consider the possible market behavior in cases in
which spectrum, like many other production inputs, is freely available at a constant unit
cost. While this assumption is unrealistic, given the way that spectrum is currently
allocated, it may provide some insights into the sources of market behavior (and possible
market failure) that may underlie spectrum allocation which this paper is concerned with.
Throughout this section, we allow both the capacity of Option B and the assumed pricing
behavior by a competitive supplier of that option to vary.
In order to model the supply side, we now assume that service providers for Options
A and B are required to acquire spectrum at a constant unit cost equal to m per megabit.
Under Option A, the service provider is assumed to provide guaranteed access of k units
of capacity to each subscriber at a lump sum price PA. Under Option B, the service
provider is assumed to provide K units of total capacity, at a cost of m K, that can be sold
to users at a per-unit price pB ? m. Subscription service under Option A is assumed to be
provided by a single monopoly supplier, who may be able to earn positive profits (i.e. PA
- 18 -
> m k). 19 In contrast, Option B is assumed to be competitively supplied, so that the price
å
Î
=
Bi
i
B q
mKp satisfies a zero profit constraint. Note that, in general, p
B > m since K must
be larger than the total demand of users who select Option B in order to satisfy their
congestion limits.
As in the previous section, we continue to assume that spectrum users individually
and simultaneously decide whether to subscribe to service under Option A (with lump
sum price PA) or to use spectrum provided by a supplier of Option B (at an endogenously
determined per unit price pB), or to choose neither option. The subscription price PA is
assumed to be independent of the number of users who select that option, and service
under this option is assumed to be congestion free. In contrast, the per-unit price pB of
Option B is assumed to depend on the set of users who ultimately select that option, and
the capacity of Option B is assumed to depend on the service providers expectation about
those users expected to select it.20 Under these assumptions, both optimal and Nash
equilibrium outcomes can be computed in exactly the same manner as in the previous
section. As above, we can summarize these outcomes in terms of gross and net
efficiency, which measure respectively the percentage of total and consumers’ surplus as
a percentage of the maximum corresponding surplus.
Table 4 presents a detailed description of the possible Nash equilibrium outcomes that
might occur under the competitive scenario under the assumptions of Example 1.21 In
that example, the lump sum price of Option B is equal to 130, and the service demands
and values are described in Figure 1. Now, however, the capacity of Option B is
assumed to be variable, and as described above, users who select that option are assumed
19 Given our previous assumption that service under Option A is congestion free, we also assume (and all of
our examples will verify) that the price P is sufficient to recover the capacity cost (m qi) of each subscriber
of Option A, or alternatively that the aggregate revenues of all subscribers are sufficient to recover total
capacity costs. In addition, we assume that each subscriber to Option A is guaranteed a dedicated capacity
of service, so that congestion costs are not relevant to any user who selects that option.
20 This is a subtle point, which may become more clear after the detailed discussion of Example 1 is
presented. Essentially, our analysis assumes that capacity of Option B is consciously chosen in order to
maximize total gross surplus, which is in turn based on equilibrium user selections given any particular
level of capacity. Since there may be multiple equilibria associated with any given capacity choice, there
can be no guarantee that the optimal gross surplus is attained in every possible Nash equilibrium, although
we will demonstrate that for each “optimal” capacity level, at least one equilibrium must attain this level of
total surplus.
21 None of the equilibria in Table 4 involve the use of weakly dominated strategies.
- 19 -
to be charged a per-unit price which guarantees that the service provider of that option
will earn zero economic profits. In Table 4, we assume that the marginal cost of
spectrum capacity is equal to 1.
Table 4: Nash Equilibria in Example 1 with Variable Capacity and a Zero Profit
Constraint for Option B with Marginal Cost = 1
Parameters Equilibrium User Selections Performance
Capacity
Option B
Price
Option B Option A Option B
Neither
Option
Total
Surplus
Consumer
Surplus
100 1.69 1,3 4,5,6,7,8 2,4 743 383
166.667 1,3 2,4,5,6,7,8 386 126
166.667 1.81 1 3,4,5,6,7,8 2 855 558.333
166.667 1.67 1,3 2,4,5,6,7,8 980 553.333
166.667 1.68 3 1,4,5,6,7,8 2 855 558.333
166.667 1.67 1,3,5,6,7,8 2,4 743 576.333
195 1,3 2,4,5,6,7,8 386 126
195 1.67 1 2,3,4,5,6,7,8 980 655
195 1.68 1,3,4,5,6,7,8 2 855 660
195 1.81 3 1,2,5,6,7,8 4 868 543
206.667 1,3 2,4,5,6,7,8 386 126
206.667 1.77 1 2,3,4,5,6,7,8 980 643.333
206.667 1.78 1,3,4,5,6,7,8 2 855 648.333
206.667 1.67 3 1,2,4,5,6,7,8 980 643.333
235 1,3 2,4,5,6,7,8 386 126
235 1.67 1,2,3,4,5,6,7,8 980 745
The first row of the table assumes that capacity of Option B is equal to 100, which is
the same assumption maintained in the previous section. The results from this row are
therefore directly comparable to the results for the various congestion etiquettes
described above. These results show that when a per-unit price for Option B is set in
order to satisfy a zero profit constraint for Option B, the resulting total surplus
corresponds exactly to total surplus attained under the Pay to Connect etiquette. These
surplus levels are generally, though not universally, higher than total surplus resulting
form the other etiquettes. For consumer surplus, the results lie between those attained by
the congestion etiquettes.22
22 Recall, however, that these etiquettes do not account for any costs associated with provision of Option B.
- 20 -
Rows 2 through 15 of Table 4 assume that capacity of Option B is chosen in order to
maximize total surplus. One can verify that when marginal cost of spectrum capacity is
equal to 1, the maximum possible total surplus in this example is equal to 980, and an
inspection of the total surplus column in the table shows that this level of surplus is
attained at four different levels of capacity. For each level of capacity, however, there are
multiple equilibria based on user choices, and so the maximum possible surplus cannot be
guaranteed without further coordination. While the maximum total surplus is equal to
980, there are Nash equilibria in which total surplus can be as low as 386. These
inefficient equilibria occur because the capacity for Option B is assumed to be chosen
under the assumption that a particular set of users will ultimately choose that option.
However, under the current pricing assumption, there can be additional equilibria in
which fewer, or even none, of these users can (in equilibrium) rationally choose Option B
given the expected choices of other users. These inefficient equilibria occur in part
because of our maintained assumption that prices are set after user selection.
For consumer surplus, a similar analysis is possible. The last row of Table 4
describes the outcome when capacity of Option B is set equal to 235, and all users in
equilibrium select that option. Both total surplus and consumer surplus attain their
maximum values in this equilibrium. At this capacity level, however, there remains an
alternative, and much inferior equilibrium, in which two users select Option A and the
remaining users choose neither option.
Tables 5 and 6 present the results corresponding to a fixed capacity for Option B
equal to 100 for each of the four examples considered above, and for varying levels of
marginal cost. In both tables, each cell reports both the maximum possible surplus
attainable and the best possible surplus attained in equilibrium, both in total and as a
percentage of the maximum. A comparison of Tables 2 and 3 with Tables 5 and 6 shows
that when marginal cost is equal to 2 or less, the gross efficiency results exactly match
the results for the pay to connect etiquette. More generally, both the total surplus and
consumer surplus results in Tables 5 and 6 closely follow the corresponding results of the
pay to connect etiquette in Tables 2 and 3.
- 21 -
Table 5. Maximum Total Surplus and Efficiency Results
in the Competitive Model with K = 100
Example 1 Example 2 Example 3 Example 4
MCB=1
Max TS
Eq TS
770
743 (96.5%)
1114
1018 (91.4%)
1114
1114 (100%)
561
476 (84.8%)
MCB=2
Max TS
Eq TS
770
743 (96.5%)
1114
1018 (91.4%)
1114
1114 (100%)
561
476 (84.8%)
MCB=4
Max TS
Eq TS
624
386 (61.9%)
1018
857 (84.2%)
1114
1114 (100%)
561
0 (0%)
Table 6. Maximum Consumer Surplus and Efficiency Results
in the Competitive Model with K = 100
Example 1 Example 2 Example 3 Example 4
MCB=1
Max CS
Eq CS
428
383 (89.5%)
528
528 (100%)
814
814 (100%)
461
376 (81.6%)
MCB=2
Max CS
Eq CS
328
283 (86.3%)
428
428 (100%)
714
714 (100%)
361
276 (76.5%)
MCB=4
Max CS
Eq CS
128
126 (98.4%)
228
207 (90.8%)
714
714 (100%)
161
0 (0%)
As Table 4 illustrates, there are a large number of Nash equilibria possible in the
competitive model, both because different levels of system capacity for Option B might
be chosen, and because multiple equilbria for any given level of capacity are possible.
For any given level of capacity, however, some of the equilibria require some users to
refrain from choosing either Option A or Option B. As long as it is reasonable to assume
that users who select Option B and are subsequently denied satisfactory service do not
pay the competitive price for Option B, these equilibria are weakly dominated for exactly
- 22 -
the same reason as discussed in the previous section.23 If all weakly dominated equilibria
are removed, only the five rows in Table 4 highlighted in bold remain as Nash equilibria.
Table 7 presents summary results for Example 1 and each of the remaining examples
for the weakly undominated equilibrium corresponding to the highest capacity choice that
maximizes consumer surplus.
Table 7: Nash Equilibrium Outcomes in the Competitive Model with Optimal
Capacity for Option B
A comparison of Tables 5, 6 and 7 allows us to quantify the impact of adjusting
capacity for Option B to optimal levels, rather than maintaining it at an arbitrary level of
100 units. In Example 1, with marginal cost is equal to 1, the highest possible total
surplus with capacity for Option B equal to 100 is equal to 770. Table 7 reveals that both
the equilibrium surplus and the maximum total surplus increase to 980 when capacity is
23 Rather than choosing neither option, and accept a zero payoff with certainty, any user would weakly
prefer to choose Option B even if in equilibrium such a user would expect to receive a zero payoff.
24 In this column the optimal strategy is represented by a vector describing the equilibrium strategy choice
of each user, where 0 corresponds to neither option, 1 corresponds to Option A and 2 corresponds to Option
B.
25 Note that when marginal cost for Option B (and by assumption also for Option A) is equal to 4, high
demand users who select Option A can only be provided their necessary capacity if a sufficient number of
lower demand customers also subscribe to this option. For example, the cost of serving user 2 is equal to
100 while in this example, the lump sum price of service is equal to 50. In the equilibrium shown, the
service provider for Option A is nevertheless guaranteed to earn positive profits.
Parameters Performance
Examples PA K
B pB
Equilibrium User
Selections24 Total Surplus Consumer Surplus
Ex. 1, MC=1 130 235 1.67 {2,2,2,2,2,2,2,2} 980 745
Ex. 1, MC=2 130 235 3.33 {2,2,2,2,2,2,2,2} 980 510
Ex. 1, MC=4 130 73.8 5.0 {1,0,1,0,2,2,2,2} 743 188
Ex. 2, MC=1 130 230 1.67 {2,2,2,2,2,2,2,2} 1114 884
Ex. 2, MC=2 130 230 3.33 {2,2,2,2,2,2,2,2} 1114 654
Ex. 2, MC=4 130 80 5.0 {1,1,0,1,2,2,2,2} 1018 308
Ex. 3, MC=1 50 230 1.67 {2,2,2,2,2,2,2,2} 1114 884
Ex. 3, MC=2 50 80 2.5 {1,1,1,1,2,2,2,2} 1114 754
Ex. 3, MC=4 5025 0 {1,1,1,1,1,1,1,1} 1114 714
Ex. 4, MC=1 250 230 1.67 {2,2,2,2,2,2,2,2} 1114 884
Ex. 4, MC=2 250 230 3.33 {2,2,2,2,2,2,2,2} 1114 654
Ex. 4, MC=4 250 180 6.67 {2,2,0,2,2,2,2,0} 948 228
- 23 -
set optimally at 235 units instead of 100. This represents a gain of 27% over the
maximum surplus at K = 100, and a gain of 32% over the equilibrium surplus attained in
that case. When marginal cost is equal to 4, the equilibrium and maximum surplus at the
optimal capacity of 73.8 both fall to 743. This represents a gain of 19% over the
maximum surplus at K = 100, and a gain of 93% over the equilibrium surplus in that
case.
4 Simulations with Random Parameter Selection
In this section, we present a table showing the results of a set of simulations with
random parameter selection for each of the outcomes discussed in sections 3 and 4.
While supportive of the conclusions that we have already drawn in those sections, these
results are important in that they provide a useful comparison of outcomes under both
congestion etiquettes and competitive pricing (under the assumptions defined in Section
5). Since parameters for the computations below must be selected with some constraints
in mind, the following results are not truly random, except within the specifications of
those constraints. Nevertheless, these results may be considered to be more general than
those previously reported, since they do not depend on the specific parameter sets chosen
for individual examples.
Table 8 shows the results of the random simulations. Each entry in the table shows
the relevant computation averaged over 100 randomly chosen parameter sets involving 6
(instead of 8) users. The random input values are chosen as follows: user values are
random integers between 2 and 10; user demands are random integers between 1 and 25;
user congestion factors are random numbers between 0.5 and 0.8; the capacity of Option
B is equal to the total demand of all users multiplied by a random number between 0.3
and 1.0; and the price of Option A is a random number chosen between the minimum and
maximum total surplus for each user.26 Marginal cost in the competitive scenarios is
26 Recall from the discussion in Section 3 that undominated Nash equilibria under co-existence are
guaranteed to achieve 100% total surplus efficiency when all users can afford Option A. Furthermore,
expected total efficiency is necessarily small when no users can afford Option A. Actual efficiency in any
particular random simulation depends on the specific parameters which define system capacity of Option B,
the total demands of users, and their congestion limits.
- 24 -
assumed to be equal to 1. In cases where multiple equilibria exist, the reported result is
an average of all equilibria.27
Table 8: Outcomes of Random Simulations
The results shown in Table 8 may be briefly summarized as follows. When the
capacity of Option B is fixed at its randomly chosen value as described above, the Pay to
Connect Etiquette achieves superior outcomes to co-existence and all other etiquettes in
terms of both total efficiency and consumer surplus efficiency. The remaining three
congestion etiquettes (simple randomization, informed randomization and lump sum
willingness to pay) do worse in terms of total and consumer surplus, but both of the
randomization etiquettes achieve better outcomes than simple co-existence in terms of
consumer surplus.
Results for the competitive outcomes, when capacity of Option B is set at its
randomly chosen default level, show that in the competitive equilibrium, total efficiency
27 Note, however, that in the case of competitive equilibria with optimal capacity for Option B, the capacity
is chosen in order to maximize total surplus, as explained in the previous section. Also, in this case, when
there are multiple equilibria, Proposition 1 guarantees that at least one equilibrium achieves full efficiency
in terms of total surplus.
Average TS Average CS TS Efficiency CS Efficiency
Surplus Max
(Original KB) 379.87 251.31
Co-existence 316.79 171.58 83.4% 68.3%
Pay to Connect 346.06 225.85 91.1% 89.9%
Simple Rand 289.25 195.08 76.1% 77.6%
Informed Rand 281.43 202.36 74.1% 80.5%
WTP 299.06 142.08 78.7% 56.5%
Competitive
Nash Eq with
Original KB
343.63 180.95 90.6% 72.0%
Surplus Max
(Optimal KB) 425.82 287.81
Competitive
Nash Eq with
Optimal KB
363.33 212.99 85.3% 74.0%
- 25 -
is greater than the corresponding values both of the randomization etiquettes and for
WTP. Total surplus efficiency is somewhat less than under the pay to connect etiquette.
With optimally chosen capacity for Option B, both total surplus and consumer surplus are
higher than each of the corresponding outcomes attained with the default capacity level.
Since efficiency is in this case measured with respect to the maximum possible surplus
(with optimal capacity instead of default capacity), the corresponding efficiencies
reported are not, however, comparable to other values in the table.
5 Concluding Comments
The demand for wireless services can, at times, place a substantial burden on the
transmission and, thus, the quality of service capabilities of the available spectrum. One
approach to the resource congestion problem is to attempt to serve all users, thereby
degrading service for all. Another response is to exclude some users in an effort to
guarantee an acceptable quality of service for some. In this paper, we have examined the
performance characteristics of a number of service etiquettes that might be utilized to
reduce or eliminate congestion for users of unlicensed spectrum. We have demonstrated
through examples that each of the etiquettes succeeds in this task, either by employing
direct rationing of user demands or by using market pricing principles to allocate users to
available spectrum. However, the same etiquettes that improve reliability in the
congestible service option also have the effect of shifting demand for wireless use from
traditional licensed use, with guaranteed service quality, to unlicensed use. Even when
service etiquettes guarantee acceptable quality of service for some users, the net effect
may be to lower total surplus.
Our analysis of each of the service etiquettes was done under the assumption that the
total capacity of the congestible service was exogenously determined, and the prices (if
any) charged to users of that service were not required to recover the cost of that service.
In section 4 of the paper, we relaxed both of these assumptions. First, we incorporated a
simple supply side to the model by imposing a zero profit constraint on the provider of
the congestible service option, which we assumed to be supplied at constant marginal
cost. If the capacity of the congestible service is fixed at the same level as assumed in the
previous sections, we showed that when marginal cost is sufficiently small, the
- 26 -
“competitive” model results in outcomes identical to those under the pay to connect
etiquette. In many cases, both total surplus and consumers’ surplus are higher when
consumers are required to bear the full cost of service for both licensed and unlicensed
spectrum than in the model with free unlicensed capacity. When the capacity of the
congestible service is allowed to vary, while the zero profit constraint is maintained, we
were able to show that when capacity is chosen correctly, there exists a Nash equilibrium
that maximizes total surplus, resulting in fully efficient user selections between the two
service options.
27
Appendix A: Detailed Data from Examples
Table 9: Parameters for Example 1
Price of Option A: 130 Capacity of Option B: 100
Total Surplus Max: 770 Consumer Surplus Max: 528
User
(i)
Value per Unit
(vi)
Unit Demand
(qi)
Surplus
(vi qi)
Congestion
Limit
1 9 24 216 60
2 5 25 125 60
3 10 17 170 60
4 7 16 112 60
5 7 18 126 80
6 5 11 55 80
7 5 17 85 80
8 7 13 91 80
Table 10: Undominated Nash Equilibrium Assignments for Example 1
Assuming Co-Existence28
Option A Option B Neither Option Total surplus Consumer surplus
1, 3 2, 4, 5, 6, 7, 8 386 126
Table 11: Nash Equilibrium Assignment for Example 1
Assuming Pay to Connect Etiquette (Price of B = 1)
Option A Option B Neither Option Total Surplus Consumer Surplus
1, 3 5, 6, 7, 8 2, 4 743 479
Table 12: Nash Equilibrium Assignment for Example 1
Assuming the Simple Randomization Etiquette
Option A Option B Neither Option Total surplus Consumer surplus
1, 3 2, 4, 5, 6, 7, 8 643.367 383.367
Table 13: Nash Equilibrium Assignment for Example 1
Assuming Congestion Informed Randomization Etiquette
Option A Option B Neither Option Total Surplus Consumer Surplus
1 2, 3, 4, 5, 6, 7, 8 560.25 430.25
28 There are 18 Nash equilibria altogether. Total surplus ranges from 386 to 743. Consumer surplus ranges from
126 to 483.
28
Table 14. Undominated Nash Equilibrium Assignments for Example 1
Assuming Lump-Sum Willingness to Pay Etiquette
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4, 5, 6, 7, 8 512 137
Table 15: Parameters for Example 2
Price of Option A: 130 Capacity of Option B: 100
Total Surplus Max: 1114 Consumer Surplus Max: 628
User
(i)
Value per Unit
(vi)
Unit Demand
(qi)
Surplus
(vi qi)
Congestion
Limit
1 8 19 152 60
2 10 17 170 60
3 6 16 96 60
4 10 22 220 60
5 7 13 91 80
6 8 18 144 80
7 9 19 171 80
8 5 14 70 80
Table 16: Undominated Nash Equilibrium Assignments for Example 2
Assuming Co-Existence29
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 4 3, 5, 6, 7, 8 1018 628
Table 17: Nash Equilibrium Assignment for Example 2
Assuming Pay to Connect Etiquette (Price of B = 1)
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 4 5, 6, 7, 8 3 1018 624
Table 18: Nash Equilibrium Assignment for Example 2
Assuming Simple Randomization Etiquette
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 4 3, 5, 6, 7, 8 1018 628
Table 19: Nash Equilibrium Assignment for Example 2
Assuming Informed Randomization Etiquette
Option A Option B Neither Option Total surplus Consumer surplus
4 1, 2, 3, 5, 6, 7, 8 609.593 479.593
29 There are two Nash equilibria in total. Both equilibria achieve the same total and consumer surplus values as
reported in this table.
29
Table 20: Undominated Nash Equilibrium Assignments for Example 2
Assuming Lump-Sum Willingness to Pay Etiquette30
Option A Option B Neither Option Total surplus Consumer surplus
1, 2 3, 4, 5, 6, 7, 8 857 309
1, 4 2, 3, 5, 6, 7, 8 857 309
1, 6 2, 3, 4, 5, 7, 8 857 309
1, 7 2, 3, 4, 5, 6, 8 857 309
2, 4 1, 3, 5, 6, 7, 8 857 309
2, 6 1, 3, 4, 5, 7, 8 857 309
2, 7 1, 3, 4, 5, 6, 8 857 309
4, 6 1, 2, 3, 5, 7, 8 857 309
4, 7 1, 2, 3, 5, 6, 8 857 309
6, 7 1, 2, 3, 4, 5, 8 857 309
Table 21: Parameters for Example 3
Price of Option A: 50 Capacity of Option B: 100
Total Surplus Max: 1114 Consumer Surplus Max: 914
User
(i)
Value per Unit
(vi)
Unit Demand
(qi)
Surplus
(vi qi)
Congestion
Limit
1 8 19 152 60
2 10 17 170 60
3 6 16 96 60
4 10 22 220 60
5 7 13 91 80
6 8 18 144 80
7 9 19 171 80
8 5 14 70 80
Table 22: Undominated Nash Equilibrium Assignment for Example 3
Assuming Co-Existence31
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4 5, 6, 7, 8 1114 914
Table 23: Nash Equilibrium Assignment for Example 3
Assuming Pay to Connect Etiquette (Price of B = 1)
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4 5, 6, 7, 8 1114 910
30 There are 32 equilibria in all. Table 13 shows the 10 equilibria involving weakly un-dominated strategies. In the
full set of Nash equilibria, total surplus is equal to 857 in each equilibrium, and consumer surplus ranges between
207 and 387.
31 There are no additional equilibria involving weakly dominated strategies in this example.
30
Table 24: Nash Equilibrium Assignments for Example 3
Assuming Simple Randomization Etiquette
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4 5, 6, 7, 8 1114 914
Table 25: Nash Equilibrium Assignments for Example 3
Assuming Informed Randomization Etiquette
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 4, 6 3, 5, 7, 8 1007 807
1, 2, 4, 7 3, 5, 6, 8 1013.75 813.75
1, 4, 6, 7 2, 3, 5, 8 1114.0 914.0
2, 4, 6, 7 1, 3, 5, 8 1011.75 811.75
Table 26: Selected Nash Equilibria Assignments for Example 3
Assuming Willingness to Pay Etiquette32
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4, 5, 6, 7, 8 1114 714
1, 5, 6, 7,8 2, 3, 4 1114 714
4, 5, 6, 7, 8 1, 2, 3 1114 714
… … … … …
Table 27: Parameters for Example 4
Price of Option A: 250 Capacity of Option B: 100
Total Surplus Max: 561 Consumer Surplus Max: 561
User
(i)
Value per Unit
(vi)
Unit Demand
(qi)
Surplus
(vi qi)
Congestion
Limit
1 8 19 152 60
2 10 17 170 60
3 6 16 96 60
4 10 22 220 60
5 7 13 91 80
6 8 18 144 80
7 9 19 171 80
8 5 14 70 80
32 There are 95 Nash equilibria involving undominated strategies. In each case total and consumer surplus are equal
to 1114 and 714 respectively.
31
Table 28: Undominated Nash Equilibria Assignment for Example 4
Assuming Co-Existence33
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4, 6, 7, 8 0 0
Table 29: Nash Equilibrium Assignment for Example 4
Assuming Pay to Connect Etiquette (Price of B = 1)
Option A Option B Neither Option Total surplus Consumer surplus
5, 6, 7, 8 1, 2, 3, 4 476 472
Table 30: Nash Equilibrium Assignments for Example 4
Assuming Simple Randomization Etiquette
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4, 5, 6, 7, 8 244.589 244.589
Table 31: Nash Equilibrium Assignments for Example 4
Assuming Informed Randomization Etiquette
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4, 5, 6, 7, 8 420.975 420.975
Table 32: Undominated Nash Equilibrium Assignments for Example 4
Assuming Lump-Sum Willingness to Pay Etiquette34
Option A Option B Neither Option Total surplus Consumer surplus
1, 2, 3, 4, 5, 6, 7, 8 561 105
33 There are 149 Nash equilibria altogether. Total and consumer surplus both lie between 0 and 476 in all equilibria.
34 There are 32 equilibria in all. Table 13 shows the unique equilibrium involving weakly un-dominated strategies.
In the full set of Nash equilibria, total surplus is equal to 561 in each equilibrium, and consumer surplus ranges
between 105 and 351.