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. Bykowsky

2

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 

presents staff analysis and research in various states. These papers are intended to 

stimulate discussion and critical comment within the FCC, as well as outside the 

agency, on issues in communications policy. Titles may include preliminary work 

and progress reports, as well as completed research. The analyses and conclusions 

in the Working Paper Series are those of the authors and do not necessarily reflect 

the view of other members of the Office of Strategic Planning and Policy Analysis, 

other Commission Staff, or any Commissioner. Given the preliminary character of 

some titles, it is advisable to check with the authors before quoting or referencing 

these working papers in other publications.
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 nonexclusive 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.4Figure 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

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%

216 125 170 112 126 55 85 91

60% Congestion Tolerance

80% Congestion Tolerance

Key

¢ Per bit

Megabits

Congestion

$ Total

Key

¢ Per bit

Megabits

Congestion

Key

Value/MB

Total Value

Total Valuation = 980

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

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%

216 125 170 112 126 55 85 91

60% Congestion Tolerance

80% Congestion Tolerance

Key

¢ Per bit

Megabits

Congestion

$ Total

Key

¢ Per bit

Megabits

Congestion

Key

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 å å Î Î+

i B

i i

i A

viqiv q , and consumer surplus is defined as 

å( ) å( )

Î Î

- + -

i B

i i B

i A

viqi PAv q P , 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 

Limit 5

11 MB

80%

Option A Option B

Consumer Surplus = 384

5

10

5

10

Excluded

41 58

$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 

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 

Limit 5

11 MB

80%

Option A Option B

5

10

5

10

Excluded

41 58

216 170 112 126 55

85

91

Consumer Surplus = 126

Provider Surplus = 260

Total Surplus = 770

#2

5¢

25 MB

60%

125

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 

Limit 5

11 MB

80%

Option A Option B

Consumer Surplus = 384

5

10

5

10

Excluded

41 58

$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 

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 

Limit 5

11 MB

80%

Option A Option B

5

10

5

10

Excluded

41 58

216 170 112 126 55

85

91

Consumer Surplus = 126

Provider Surplus = 260

Total Surplus = 770

#2

5¢

25 MB

60%

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

$1.26 $0.91 $2.16 $1.25 $1.12 $1.70 $0.85 $0.55

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

$1.26 $0.91 $2.16 $1.25 $1.12 $1.70 $0.85 $0.55

80% Capacity

Congestion

Limit

Willingness to Pay 

per Mbps

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%

$1.26 $0.91 $2.16 $1.25 $1.12 $1.70 $0.85 $0.55

Willingness to Pay 

per Mbps

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

80%

#2

5¢

25 Mbps

60%

$1.26 $0.91 $2.16 $1.25 $1.12 $1.70 $0.85 $0.55

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

11 Mbps

#2

5

25 Mbps

60%

60% Capacity

Congestion

Limit

126 91 216 125 112 170 85 55

80% Capacity

Congestion

Limit 5

$1.26 $0.91 $2.16 $1.25 $1.12 $1.70 $0.85 $0.55

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

$1.26 $0.91 $2.16 $1.25 $1.12 $1.70 $0.85 $0.55

80% Capacity

Congestion

Limit

Willingness to Pay 

per Mbps

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%

$1.26 $0.91 $2.16 $1.25 $1.12 $1.70 $0.85 $0.55

Willingness to Pay 

per Mbps

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

80%

#2

5¢

25 Mbps

60%

$1.26 $0.91 $2.16 $1.25 $1.12 $1.70 $0.85 $0.55

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

11 Mbps

#2

5

25 Mbps

60%

60% Capacity

Congestion

Limit

126 91 216 125 112 170 85 55

80% Capacity

Congestion

Limit 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

$1.70 $2.16

#5

18 MB

80%

#8

Lump Sum Bids

100

10¢

17 MB

60%

#3

60%

9¢

24 MB

60%

#1

60%

#4

#6

#7

17 MB

80%

141

11 MB

80%

60% Capacity 

Congestion 

Limit

80% Capacity 

Congestion 

Limit

130

130

16 MB

60%

112

126

55

85

13 MB

80%

91

#2

25 MB

60%

125

150

50

Quantity Demanded (MB)

10¢

17 MB

60%

9¢

24 MB

60%

141

60% Capacity 

Congestion 

Limit

80% Capacity 

Congestion 

Limit

$1.70 $2.16

#5

18 MB

80%

#8

Lump Sum Bids

100

10¢

17 MB

60%

#3

60%

9¢

24 MB

60%

#1

60%

#4

#6

#7

17 MB

80%

141

11 MB

80%

60% Capacity 

Congestion 

Limit

80% Capacity 

Congestion 

Limit

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 undominated 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 coexistence.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 

åÎ

=

i B

i

B

q

mK

p satisfies a zero profit constraint. Note that, in general, pB > 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 NeitherOption

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 P

A KB 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.