![](media/image1.wmf){width="1.1416666666666666in" height="2.0in"}

> **Federal Communications Commission**

**Office of Strategic Planning and Policy Analysis**

**445 12th Street, SW**

**Washington, DC 20554**

#  {#section .unnumbered}

#  {#section-1 .unnumbered}

#  {#section-2 .unnumbered}

**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 Operations**[^1]

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

**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](#introduction)

2 User Selection Between Licensed and Unlicensed Service Options
[2](#user-selection-between-licensed-and-unlicensed-service-options)

2.1 Tragedy of the Commons Problem: Nash Prediction
[6](#tragedy-of-the-commons-problem-nash-prediction)

2.2 Alternative Congestion Etiquettes
[9](#alternative-congestion-etiquettes)

2.3 Summary Analysis of Alternative Congestion Etiquettes
[14](#__RefHeading___Toc191191015)

3 A Simple Competitive Model [17](#a-simple-competitive-model)

4 Simulations with Random Parameter Selection
[23](#simulations-with-random-parameter-selection)

5 Concluding Comments [25](#concluding-comments)

Appendix A: Detailed Data from Examples
[27](#appendix-a-detailed-data-from-examples)

# 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 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.

# 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.

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 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**

![](media/image2.wmf){width="6.0in" height="3.8048611111111112in"}

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 *P~A~* = 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 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
![](media/image3.wmf){width="1.125in" height="0.375in"}, and consumer
surplus is defined as ![](media/image4.wmf){width="1.9861111111111112in"
height="0.375in"}, where *q~i~* represents user *i*'s demand, *v~i~*
represents value per unit of demand for user *i*, and *P~A~* and *P~B~*
represent the prices for options A and B respectively.[^5]

Figure 2. Efficient Assignment of Spectrum in Example 1

![](media/image5.wmf){width="5.998611111111111in"
height="4.0256944444444445in"}

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 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.

## 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 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

  ----------- ------------ ------------- ------------- ------------ --------------
  User Choice                            Performance                
  Selection                              Measures                   

  **Option    **Option B** **Neither     **Total       **Consumer   **Total
  A**                      Option**      Surplus**     Surplus**    Efficiency**

  *1, 3*      *5, 6, 7, 8* *2, 4*        *743*         *483*        *96.5%*

  *1, 3*      *4, 5, 6, 7, *2*           *743*         *483*        *96.5%*
              8*                                                    

  *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, *4*           *386*         *126*        *50.1%*
              8*                                                    

  *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, *5*           *386*         *126*        *50.1%*
              8*                                                    

  *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, *6*           *386*         *126*        *50.1%*
              8*                                                    

  *1, 3*      *2, 4, 5, 6* *7, 8*        *567*         *307*        *73.6%*

  *1, 3*      *2, 4, 5, 6, *7*           *386*         *126*        *50.1%*
              8*                                                    

  *1, 3*      *2, 4, 5, 6, *8*           *386*         *126*        *50.1%*
              7*                                                    

  **1, 3**    **2, 4, 5,                 **386**       **126**      **50.1%**
              6, 7, 8**                                             
  ----------- ------------ ------------- ------------- ------------ --------------

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 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%.

## 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
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**

![](media/image6.wmf){width="6.001388888888889in"
height="4.043055555555555in"}

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 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 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, 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

![](media/image7.wmf){width="5.541666666666667in"
height="3.611111111111111in"}

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%.

## Summary Analysis of Alternative Congestion Etiquettes[^15]

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 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 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              72.8%        54.7%        93.1%[^18]   75.0%
  Randomization                                                

  WTP                   66.5%        76.9%        100%         100%
  --------------------- ------------ ------------ ------------ ------------

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              81.5%        76.4%        91.5%        75.0%
  Randomization                                                

  WTP                   25.9%        49.2%        78.1%        18.7%
  --------------------- ------------ ------------ ------------ ------------

# 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 *P~A~*. 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 *p~B~* ≥ *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.
*P~A~* \> *m k*). [^19] In contrast, Option B is assumed to be
competitively supplied, so that the price
![](media/image8.wmf){width="0.7777777777777778in"
height="0.5965277777777778in"} 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 *P~A~*) or to use spectrum provided
by a supplier of Option B (at an endogenously determined per unit price
*p~B~*), or to choose neither option. The subscription price *P~A~* 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 *p~B~* 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 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

+--------+--------+--------+------------+---------+-------+---------+
| Para   |        | Equil  |            |         | P     |         |
| meters |        | ibrium |            |         | erfor |         |
|        |        | User   |            |         | mance |         |
|        |        | Sele   |            |         |       |         |
|        |        | ctions |            |         |       |         |
+--------+--------+--------+------------+---------+-------+---------+
| **Capa | **P    | **     | **Option   | **Ne    | **To  | **Con   |
| city** | rice** | Option | B**        | ither** | tal** | sumer** |
|        |        | A**    |            |         |       |         |
| **     | **     |        |            | **O     | *     | **Su    |
| Option | Option |        |            | ption** | *Surp | rplus** |
| B**    | B**    |        |            |         | lus** |         |
+--------+--------+--------+------------+---------+-------+---------+
| *100*  | *1.69* | *1,3*  | *          | *2,4*   | *743* | *383*   |
|        |        |        | 4,5,6,7,8* |         |       |         |
+--------+--------+--------+------------+---------+-------+---------+
| *16    |        | *1,3*  |            | *2,4,5  | *386* | *126*   |
| 6.667* |        |        |            | ,6,7,8* |       |         |
+--------+--------+--------+------------+---------+-------+---------+
| *16    | *1.81* | *1*    | *3,        | *2*     | *855* | *5      |
| 6.667* |        |        | 4,5,6,7,8* |         |       | 58.333* |
+--------+--------+--------+------------+---------+-------+---------+
| **166  | **     | *      | **2,4      |         | **    | **55    |
| .667** | 1.67** | *1,3** | ,5,6,7,8** |         | 980** | 3.333** |
+--------+--------+--------+------------+---------+-------+---------+
| *16    | *1.68* | *3*    | *1,        | *2*     | *855* | *5      |
| 6.667* |        |        | 4,5,6,7,8* |         |       | 58.333* |
+--------+--------+--------+------------+---------+-------+---------+
| *16    | *1.67* |        | *1,        | *2,4*   | *743* | *5      |
| 6.667* |        |        | 3,5,6,7,8* |         |       | 76.333* |
+--------+--------+--------+------------+---------+-------+---------+
| *195*  |        | *1,3*  |            | *2,4,5  | *386* | *126*   |
|        |        |        |            | ,6,7,8* |       |         |
+--------+--------+--------+------------+---------+-------+---------+
| *      | **     | **1**  | **2,3,4    |         | **    | **655** |
| *195** | 1.67** |        | ,5,6,7,8** |         | 980** |         |
+--------+--------+--------+------------+---------+-------+---------+
| *195*  | *1.68* |        | *1,3,      | *2*     | *855* | *660*   |
|        |        |        | 4,5,6,7,8* |         |       |         |
+--------+--------+--------+------------+---------+-------+---------+
| *195*  | *1.81* | *3*    | *1,        | *4*     | *868* | *543*   |
|        |        |        | 2,5,6,7,8* |         |       |         |
+--------+--------+--------+------------+---------+-------+---------+
| *20    |        | *1,3*  |            | *2,4,5  | *386* | *126*   |
| 6.667* |        |        |            | ,6,7,8* |       |         |
+--------+--------+--------+------------+---------+-------+---------+
| **206  | **     | **1**  | **2,3,4    |         | **    | **64    |
| .667** | 1.77** |        | ,5,6,7,8** |         | 980** | 3.333** |
+--------+--------+--------+------------+---------+-------+---------+
| *20    | *1.78* |        | *1,3,      | *2*     | *855* | *6      |
| 6.667* |        |        | 4,5,6,7,8* |         |       | 48.333* |
+--------+--------+--------+------------+---------+-------+---------+
| **206  | **     | **3**  | **1,2,4    |         | **    | **64    |
| .667** | 1.67** |        | ,5,6,7,8** |         | 980** | 3.333** |
+--------+--------+--------+------------+---------+-------+---------+
| *235*  |        | *1,3*  |            | *2,4,5  | *386* | *126*   |
|        |        |        |            | ,6,7,8* |       |         |
+--------+--------+--------+------------+---------+-------+---------+
| *      | **     |        | **1,2,3,4  |         | **    | **745** |
| *235** | 1.67** |        | ,5,6,7,8** |         | 980** |         |
+--------+--------+--------+------------+---------+-------+---------+

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]

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.

**Table 5. Maximum Total Surplus and Efficiency Results**

in the Competitive Model with K = 100

+-------+--------+------------+-------------+------------+------------+
|       |        | Example 1  | Example 2   | Example 3  | Example 4  |
+-------+--------+------------+-------------+------------+------------+
| MC    | Max TS | 770        | 1114        | 1114       | 561        |
| ~B~=1 |        |            |             |            |            |
|       | Eq TS  | 743        | 1018        | 1114       | 476        |
|       |        | (96.5%)    | (91.4%)     | (100%)     | (84.8%)    |
+-------+--------+------------+-------------+------------+------------+
| MC    | Max TS | 770        | 1114        | 1114       | 561        |
| ~B~=2 |        |            |             |            |            |
|       | Eq TS  | 743        | 1018        | 1114       | 476        |
|       |        | (96.5%)    | (91.4%)     | (100%)     | (84.8%)    |
+-------+--------+------------+-------------+------------+------------+
| MC    | Max TS | 624        | 1018        | 1114       | 561        |
| ~B~=4 |        |            |             |            |            |
|       | Eq TS  | 386        | 857 (84.2%) | 1114       | 0 (0%)     |
|       |        | (61.9%)    |             | (100%)     |            |
+-------+--------+------------+-------------+------------+------------+

**Table 6. Maximum Consumer Surplus and Efficiency Results**

in the Competitive Model with K = 100

+-------+--------+------------+------------+------------+------------+
|       |        | Example 1  | Example 2  | Example 3  | Example 4  |
+-------+--------+------------+------------+------------+------------+
| MC    | Max CS | 428        | 528        | 814        | 461        |
| ~B~=1 |        |            |            |            |            |
|       | Eq CS  | 383        | 528 (100%) | 814 (100%) | 376        |
|       |        | (89.5%)    |            |            | (81.6%)    |
+-------+--------+------------+------------+------------+------------+
| MC    | Max CS | 328        | 428        | 714        | 361        |
| ~B~=2 |        |            |            |            |            |
|       | Eq CS  | 283        | 428 (100%) | 714 (100%) | 276        |
|       |        | (86.3%)    |            |            | (76.5%)    |
+-------+--------+------------+------------+------------+------------+
| MC    | Max CS | 128        | 228        | 714        | 161        |
| ~B~=4 |        |            |            |            |            |
|       | Eq CS  | 126        | 207        | 714 (100%) | 0 (0%)     |
|       |        | (98.4%)    | (90.8%)    |            |            |
+-------+--------+------------+------------+------------+------------+

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 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

  ---------------- ------------ ------------ ------------ ------------------- ------------- ------------
  Examples         Parameters                             Equilibrium User    Performance   
                                                          Selections[^24]                   

                   ***P^A^***   ***K~B~***   ***p^B^***                       **Total       **Consumer
                                                                              Surplus**     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      50[^25]      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
  ---------------- ------------ ------------ ------------ ------------------- ------------- ------------

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 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.

# 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 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 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.

# 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 "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.

# Appendix A: Detailed Data from Examples {#appendix-a-detailed-data-from-examples .unnumbered}

**Table 9: Parameters for Example 1**

+------+------------------+----------------+----------+--------------+
| P    |                  |                |          |              |
| rice |                  |                |          |              |
| of   |                  |                |          |              |
| Op   |                  |                |          |              |
| tion |                  |                |          |              |
| A:   |                  |                |          |              |
| 130  |                  |                |          |              |
| Capa |                  |                |          |              |
| city |                  |                |          |              |
| of   |                  |                |          |              |
| Op   |                  |                |          |              |
| tion |                  |                |          |              |
| B:   |                  |                |          |              |
| 100  |                  |                |          |              |
|      |                  |                |          |              |
| T    |                  |                |          |              |
| otal |                  |                |          |              |
| Sur  |                  |                |          |              |
| plus |                  |                |          |              |
| Max: |                  |                |          |              |
| 770  |                  |                |          |              |
| Cons |                  |                |          |              |
| umer |                  |                |          |              |
| Sur  |                  |                |          |              |
| plus |                  |                |          |              |
| Max: |                  |                |          |              |
| 528  |                  |                |          |              |
+------+------------------+----------------+----------+--------------+
| User | Value per Unit   | Unit Demand    | Surplus  | Congestion   |
|      |                  |                |          |              |
| \    | (v~i~)           | (q~i~)         | (v~i~    | Limit        |
| (i\) |                  |                | q~i~)    |              |
+------+------------------+----------------+----------+--------------+
| 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-Existence**[^28]

  ---------- ------------- --------------- ------------- ------------------
  **Option   **Option B**  **Neither       **Total       **Consumer
  A**                      Option**        surplus**     surplus**

  1, 3       2, 4, 5, 6,                   386           126
             7, 8                                        
  ---------- ------------- --------------- ------------- ------------------

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,                 643.367       383.367
             8                                            
  ---------- -------------- --------------- ------------- -----------------

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,                 560.25        430.25
            7, 8                                         
  --------- --------------- -------------- ------------- -----------------

Table 14. Undominated Nash Equilibrium Assignments for Example 1

Assuming Lump-Sum Willingness to Pay Etiquette

  ---------- ---------------- -------------- ------------- -----------------
  **Option   **Option B**     **Neither      **Total       **Consumer
  A**                         Option**       surplus**     surplus**

             1, 2, 3, 4, 5,                  512           137
             6, 7, 8                                       
  ---------- ---------------- -------------- ------------- -----------------

**Table 15: Parameters for Example 2**

+------+------------------+----------------+----------+--------------+
| P    |                  |                |          |              |
| rice |                  |                |          |              |
| of   |                  |                |          |              |
| Op   |                  |                |          |              |
| tion |                  |                |          |              |
| A:   |                  |                |          |              |
| 130  |                  |                |          |              |
| Capa |                  |                |          |              |
| city |                  |                |          |              |
| of   |                  |                |          |              |
| Op   |                  |                |          |              |
| tion |                  |                |          |              |
| B:   |                  |                |          |              |
| 100  |                  |                |          |              |
|      |                  |                |          |              |
| T    |                  |                |          |              |
| otal |                  |                |          |              |
| Sur  |                  |                |          |              |
| plus |                  |                |          |              |
| Max: |                  |                |          |              |
| 1114 |                  |                |          |              |
| Cons |                  |                |          |              |
| umer |                  |                |          |              |
| Sur  |                  |                |          |              |
| plus |                  |                |          |              |
| Max: |                  |                |          |              |
| 628  |                  |                |          |              |
+------+------------------+----------------+----------+--------------+
| User | Value per Unit   | Unit Demand    | Surplus  | Congestion   |
|      |                  |                |          |              |
| \    | (v~i~)           | (q~i~)         | (v~i~    | Limit        |
| (i\) |                  |                | q~i~)    |              |
+------+------------------+----------------+----------+--------------+
| 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-Existence**[^29]

  ---------- ----------- --------------- -------------- ------------------
  **Option   **Option    **Neither       **Total        **Consumer
  A**        B**         Option**        surplus**      surplus**

  1, 2, 4    3, 5, 6, 7,                 1018           628
             8                                          
  ---------- ----------- --------------- -------------- ------------------

**Table 17: Nash Equilibrium Assignment for Example 2**

**Assuming Pay to Connect Etiquette (Price of B = 1)**

  ---------- ---------- ---------------- -------------- -------------------
  **Option   **Option   **Neither        **Total        **Consumer
  A**        B**        Option**         surplus**      surplus**

  1, 2, 4    5, 6, 7, 8 3                1018           624
  ---------- ---------- ---------------- -------------- -------------------

**Table 18: Nash Equilibrium Assignment for Example 2**

**Assuming Simple Randomization Etiquette**

  ---------- ----------- --------------- -------------- ------------------
  **Option   **Option    **Neither       **Total        **Consumer
  A**        B**         Option**        surplus**      surplus**

  1, 2, 4    3, 5, 6, 7,                 1018           628
             8                                          
  ---------- ----------- --------------- -------------- ------------------

**Table 19: Nash Equilibrium Assignment for Example 2**

**Assuming Informed Randomization Etiquette**

  ---------- --------------- -------------- ------------- -----------------
  **Option   **Option B**    **Neither      **Total       **Consumer
  A**                        Option**       surplus**     surplus**

  4          1, 2, 3, 5, 6,                 609.593       479.593
             7, 8                                         
  ---------- --------------- -------------- ------------- -----------------

**Table 20: Undominated Nash Equilibrium Assignments for Example 2**

**Assuming Lump-Sum Willingness to Pay Etiquette**[^30]

  ---------- ------------- --------------- ------------- ------------------
  **Option   **Option B**  **Neither       **Total       **Consumer
  A**                      Option**        surplus**     surplus**

  1, 2       3, 4, 5, 6,                   857           309
             7, 8                                        

  1, 4       2, 3, 5, 6,                   857           309
             7, 8                                        

  1, 6       2, 3, 4, 5,                   857           309
             7, 8                                        

  1, 7       2, 3, 4, 5,                   857           309
             6, 8                                        

  2, 4       1, 3, 5, 6,                   857           309
             7, 8                                        

  2, 6       1, 3, 4, 5,                   857           309
             7, 8                                        

  2, 7       1, 3, 4, 5,                   857           309
             6, 8                                        

  4, 6       1, 2, 3, 5,                   857           309
             7, 8                                        

  4, 7       1, 2, 3, 5,                   857           309
             6, 8                                        

  6, 7       1, 2, 3, 4,                   857           309
             5, 8                                        
  ---------- ------------- --------------- ------------- ------------------

**Table 21: Parameters for Example 3**

+------+------------------+----------------+----------+--------------+
| P    |                  |                |          |              |
| rice |                  |                |          |              |
| of   |                  |                |          |              |
| Op   |                  |                |          |              |
| tion |                  |                |          |              |
| A:   |                  |                |          |              |
| 50   |                  |                |          |              |
| Capa |                  |                |          |              |
| city |                  |                |          |              |
| of   |                  |                |          |              |
| Op   |                  |                |          |              |
| tion |                  |                |          |              |
| B:   |                  |                |          |              |
| 100  |                  |                |          |              |
|      |                  |                |          |              |
| T    |                  |                |          |              |
| otal |                  |                |          |              |
| Sur  |                  |                |          |              |
| plus |                  |                |          |              |
| Max: |                  |                |          |              |
| 1114 |                  |                |          |              |
| Cons |                  |                |          |              |
| umer |                  |                |          |              |
| Sur  |                  |                |          |              |
| plus |                  |                |          |              |
| Max: |                  |                |          |              |
| 914  |                  |                |          |              |
+------+------------------+----------------+----------+--------------+
| User | Value per Unit   | Unit Demand    | Surplus  | Congestion   |
|      |                  |                |          |              |
| \    | (v~i~)           | (q~i~)         | (v~i~    | Limit        |
| (i\) |                  |                | q~i~)    |              |
+------+------------------+----------------+----------+--------------+
| 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-Existence**[^31]

  ---------- ---------- ---------------- -------------- -------------------
  **Option   **Option   **Neither        **Total        **Consumer
  A**        B**        Option**         surplus**      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   **Option   **Neither        **Total        **Consumer
  A**        B**        Option**         surplus**      surplus**

  1, 2, 3, 4 5, 6, 7, 8                  1114           910
  ---------- ---------- ---------------- -------------- -------------------

**Table 24: Nash Equilibrium Assignments for Example 3**

**Assuming Simple Randomization Etiquette**

  ---------- ---------- ---------------- -------------- -------------------
  **Option   **Option   **Neither        **Total        **Consumer
  A**        B**        Option**         surplus**      surplus**

  1, 2, 3, 4 5, 6, 7, 8                  1114           914
  ---------- ---------- ---------------- -------------- -------------------

**Table 25: Nash Equilibrium Assignments for Example 3**

**Assuming Informed Randomization Etiquette**

  ---------- ---------- ---------------- -------------- -------------------
  **Option   **Option   **Neither        **Total        **Consumer
  A**        B**        Option**         surplus**      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 Etiquette**[^32]

  ---------------- ---------- -------------- ------------- -----------------
  **Option A**     **Option   **Neither      **Total       **Consumer
                   B**        Option**       surplus**     surplus**

  1, 2, 3, 4, 5,                             1114          714
  6, 7, 8                                                  

  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**

+------+------------------+----------------+----------+--------------+
| P    |                  |                |          |              |
| rice |                  |                |          |              |
| of   |                  |                |          |              |
| Op   |                  |                |          |              |
| tion |                  |                |          |              |
| A:   |                  |                |          |              |
| 250  |                  |                |          |              |
| Capa |                  |                |          |              |
| city |                  |                |          |              |
| of   |                  |                |          |              |
| Op   |                  |                |          |              |
| tion |                  |                |          |              |
| B:   |                  |                |          |              |
| 100  |                  |                |          |              |
|      |                  |                |          |              |
| T    |                  |                |          |              |
| otal |                  |                |          |              |
| Sur  |                  |                |          |              |
| plus |                  |                |          |              |
| Max: |                  |                |          |              |
| 561  |                  |                |          |              |
| Cons |                  |                |          |              |
| umer |                  |                |          |              |
| Sur  |                  |                |          |              |
| plus |                  |                |          |              |
| Max: |                  |                |          |              |
| 561  |                  |                |          |              |
+------+------------------+----------------+----------+--------------+
| User | Value per Unit   | Unit Demand    | Surplus  | Congestion   |
|      |                  |                |          |              |
| \    | (v~i~)           | (q~i~)         | (v~i~    | Limit        |
| (i\) |                  |                | q~i~)    |              |
+------+------------------+----------------+----------+--------------+
| 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 28: Undominated Nash Equilibria Assignment for Example 4**

**Assuming Co-Existence**[^33]

  ---------- --------------- -------------- ------------- -----------------
  **Option   **Option B**    **Neither      **Total       **Consumer
  A**                        Option**       surplus**     surplus**

             1, 2, 3, 4, 6,                 0             0
             7, 8                                         
  ---------- --------------- -------------- ------------- -----------------

**Table 29: Nash Equilibrium Assignment for Example 4**

**Assuming Pay to Connect Etiquette (Price of B = 1)**

  ---------- ---------- ---------------- -------------- -------------------
  **Option   **Option   **Neither        **Total        **Consumer
  A**        B**        Option**         surplus**      surplus**

             5, 6, 7, 8 1, 2, 3, 4       476            472
  ---------- ---------- ---------------- -------------- -------------------

**Table 30: Nash Equilibrium Assignments for Example 4**

**Assuming Simple Randomization Etiquette**

  ---------- ---------------- -------------- ------------- -----------------
  **Option   **Option B**     **Neither      **Total       **Consumer
  A**                         Option**       surplus**     surplus**

             1, 2, 3, 4, 5,                  244.589       244.589
             6, 7, 8                                       
  ---------- ---------------- -------------- ------------- -----------------

**Table 31: Nash Equilibrium Assignments for Example 4**

**Assuming Informed Randomization Etiquette**

  ---------- ---------------- -------------- ------------- -----------------
  **Option   **Option B**     **Neither      **Total       **Consumer
  A**                         Option**       surplus**     surplus**

             1, 2, 3, 4, 5,                  420.975       420.975
             6, 7, 8                                       
  ---------- ---------------- -------------- ------------- -----------------

**Table 32: Undominated Nash Equilibrium Assignments for Example 4**

**Assuming Lump-Sum Willingness to Pay Etiquette**[^34]

  ---------- ---------------- -------------- ------------- -----------------
  **Option   **Option B**     **Neither      **Total       **Consumer
  A**                         Option**       surplus**     surplus**

             1, 2, 3, 4, 5,                  561           105
             6, 7, 8                                       
  ---------- ---------------- -------------- ------------- -----------------

#  {#section-3 .unnumbered}

[^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>

    .

[^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*.

[^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]: 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, *P~B~* , may
    be assigned to users of Option B. In the above notation, producer
    surplus is defined as *P~A~ n~A~ + P~B~ n~B~,* where *n~A~* and
    *n~B~* represent the number of subscribers choosing options *A* and
    *B* respectively*.*

[^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]: 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.

[^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]: 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.

[^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]: 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.

[^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.

[^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.

[^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.

[^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 q~i~*) 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.

[^22]: Recall, however, that these etiquettes do not account for any
    costs associated with provision of Option B.

[^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.

[^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.

[^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.

[^28]: There are 18 Nash equilibria altogether. Total surplus ranges
    from 386 to 743. Consumer surplus ranges from 126 to 483.

[^29]: There are two Nash equilibria in total. Both equilibria achieve
    the same total and consumer surplus values as reported in this
    table.

[^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.

[^32]: There are 95 Nash equilibria involving undominated strategies. In
    each case total and consumer surplus are equal to 1114 and 714
    respectively.

[^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.