1. Combined Cooperation and Non-Cooperation for
Channel Allocation and Transmission Power
Control
Wi-5: What to do With the Wi-Fi Wild West
by
M.P.P. (Maran) van Heesch BSc (729120)
A thesis submitted in partial fulfillment of the requirements for the degree of
Master of Science in Econometrics and Mathematical Economics.
School of Economics and Management
Tilburg University
Supervisors
prof. dr. A.J.J. Talman (Tilburg Univrsity)
prof. dr. P.E.M. Borm (Tilburg University)
P.L.J. Wissink MSc (TNO)
drs. F.T.H.M. Berkers (TNO)
August 21, 2016
2. Abstract
In the last decade there has been an explosive growth in the use of wireless network services.
However, the spectrum which can be used for Wi-Fi is limited and unmanaged. As a conse-
quence, the Quality of Service (QoS) of Wi-Fi for all users may decrease as the number of users
grows. As to this date there is no solution for this problem.
In this thesis we game theoretically research a user’s incentive to join a technological aid for
spectrum management, known as the Wi-5 mechanism. The Wi-5 mechanism is a method that
aims to tackle the so called spectrum commons problem by managing the users’ channel selection
and transmission power with a controller.
To theoretically research the user’s incentive to join the Wi-5 mechanism, we propose a game
theoretic framework in which we combine non-cooperative and cooperative game theory. In par-
ticular, we consider the non-cooperative concept of Nash equilibria and the cooperative concept
of Nash bargaining. We describe three different scenarios: (i) the scenario in which no Wi-Fi
users join the Wi-5 mechanism, (ii) the scenario in which all users join the Wi-5 mechanism, and
(iii) the scenario in which there are both users that join the Wi-5 mechanism and users that do
not join. With the framework we can determine the ratio of users that joins the Wi-5 mechanism
in different scenarios.
A use case, named ‘the apartment building’, is used to initialize the framework. In the use
case we assume that a pre-specified number of apartment owners are Wi-Fi users, who continu-
ously transmit with their maximal transmission power in the default scenario, i.e., the scenario
in which not a single user joins the Wi-5 mechanism.
We illustrate the framework with two examples: a two person and a three person example.
We show that in the two user example, the users do not have the incentive to join the Wi-5
mechanism. This is because it would not be rational for one of the users, since his expected QoS
of Wi-Fi decreases when the expected QoS of the other user increases due to the transmission
powers selected by the controller. In the three user example there are two users that are willing
to join the Wi-5 mechanism since their expected QoS increases, albeit lowering the expected QoS
of the third user. This is because controller does not take the non-joining user into account when
determining the channel selection and the transmission power of the two joining users. Therefore
it might be beneficial for the non-joining user to join the Wi-5 mechanism, once the other two
users have joined. This could be researched by applying the framework, conditioned on the case
that the two users join the Wi-5 mechanism.
3. Acknowledgements
I would like to thank my thesis supervisors prof. dr. A.J.J. Talman and prof. dr. P.E.M. Borm
of the School of Economics and Management at Tilburg University. Thank you for making sure
that I was not too ambitious and for the continuous flow of ideas and commentary, it has led me
to develop the framework as presented in this thesis.
Next I would like to thank my daily supervisors P.L.J. Wissink MSc and drs. F.T.H.M. Berkens
at TNO. Thank you for making me feel welcome at TNO, teaching me how to make coffee, and
discussing examples in which the three of us hypothetically join or not join the Wi-5 mechanism.
Thirdly I would like to thank Mr. M. Djurica at TNO for his technical expertise. Thank
you for teaching me about the frequency spectrum and helping me initialize the use case.
Furthermore I would like to thank all mentioned afore and I.J. Blankers MSc for reviewing
my thesis. Your comments and input have lifted this report to a higher level and I could not
have done it without.
Last but certainly not least, I would like to thank my friends and family for their love and
support. Rick, thank you for keeping me sane the last four months. Martijn, thank you for
letting me crash at your place. Dad, thank you for pushing me to get the unit of SINR right.
Maran van Heesch
6. List of Figures
2.1 The USA frequency spectrum allocation chart. . . . . . . . . . . . . . . . . . . . 15
2.2 The 2.4 GHz frequency spectrum with channels. . . . . . . . . . . . . . . . . . . 16
3.1 Schematic overview of the game theory in the Wi-5 model. . . . . . . . . . . . . 18
5.1 Illustration of the use case, 2 appartments. . . . . . . . . . . . . . . . . . . . . . 29
5.2 Illustration of the use case, 6 appartments. . . . . . . . . . . . . . . . . . . . . . 30
5.3 Illustration of in-house communications. . . . . . . . . . . . . . . . . . . . . . . . 33
5.4 Relation between data speed (Mbps) and the Signal-to-Noise ratio (dB). . . . . . 34
5.5 Relation between the monthly contract fee (e) and target Signal-to-Noise ratio
(dB). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.1 Layout of the apartments of the two users, including the location of the transmit-
ters and receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2 Layout of the apartments of the three users, including the location of the trans-
mitters and receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.3 Utility levels in the non-cooperative scenario. . . . . . . . . . . . . . . . . . . . . 41
3
7. List of Tables
3.1 Two player example: stay at home or go to a party? . . . . . . . . . . . . . . . . 20
5.1 Relation between data speed (Mbps) and the Signal-to-Noise ratio (dB). . . . . . 33
6.1 Data initialization of the two users p1 and p2. . . . . . . . . . . . . . . . . . . . . 37
6.2 Transmission power and utility in the two-user non-cooperative scenario. . . . . . 37
6.3 Transmission power and utility in the 2-user cooperative scenario. . . . . . . . . 38
6.4 Data initialization of the three users p1, p2, and p3. . . . . . . . . . . . . . . . . 39
6.5 Transmission power and utility in the 3-user non-cooperative scenario. . . . . . . 40
6.6 Strategy choices per user, given the strategy of the other two users. . . . . . . . . 41
6.7 Transmission power and utility in the 3-user cooperative scenario. . . . . . . . . 42
6.8 Transmission power and utility in the 3-user mixed scenario, p2 is a non-joining
user. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4
8. Nomenclature
α The attenuation index used to determine the path loss.
∆(Si) The set of mixed strategies for player i ∈ N.
γ(mi) The target Signal-To-Inference-and-Noise ratio of player i ∈ N, dependent on mi.
×i∈N ∆(Si) The set of mixed strategy profiles.
A The alternative set in a bargaining game.
Bn
An n−person bargaining game.
C The set of frequency channels.
ci The frequency channel selected by player i ∈ N, ci ∈ C.
d The disagreement point in a bargaining game.
dj
Ri
The jth coordinate of the position of the receiver of player i ∈ N, j ∈ {1, 2, 3}.
dj
Ti
The jth coordinate of the position of the transmitter of player i ∈ N, j ∈ {1, 2, 3}.
Gij Path loss between the transmitter of player i ∈ N and the receiver of player j ∈ N.
I(c, c ) The inference function characterizing the interference between channel c ∈ C and
c ∈ C.
L The constant loss used to determine the path loss.
mi The contract fee paid by player i ∈ N to gain access to the frequency spectrum.
N The set of all players.
n The number of players.
Nk
c The set of cooperative players transmitting on channel k ∈ C.
Nc The set of cooperative players.
nc The number of cooperative players.
n0i The constant individual noise factor of player i ∈ N.
Nnc The set of non-cooperative players.
pi The transmission power at the transmitter of player i ∈ N.
5
9. pmax
i The maximal transmitting power at the transmitter of player i ∈ N.
S The set of pure strategy profiles.
S(n, k) Stirling number of the second kind.
Si The set of pure strategies of player i ∈ N.
si The current strategy of player i ∈ N.
s−i The current strategy of the players other then player i ∈ N.
Ui The expected utility function of player i ∈ N, in a mixed profile.
ui The utility function of player i ∈ N, in a pure profile.
6
10. Chapter 1
Introduction
1.1 Motivation
In the last decade there has been an explosive growth in the number of devices connected to
wireless networks. Examples are the growing use of laptops and tablets, as well as the increasing
number of smartphones. Experts estimate that the number of smartphone subscriptions will
increase up to 6.3 billion by 2021, which is almost double of the 3.2 billion smartphone subscrip-
tions in 2015[31]. This indicates that the number of users that wants to use the wireless services
will also grow further.
However, the spectrum which can be used for Wireless Fidelity (Wi-Fi) access is limited and
unmanaged. When more Wi-Fi users join the Wi-Fi spectrum, the overall Quality of Service
(QoS) of Wi-Fi for the users will decrease due to interference. This phenomenon is known as
the tragedy of the spectrum commons [18]. Therefore there is need for a more intelligent way of
spectrum management, in order to keep the QoS of Wi-Fi for the users as high as possible.
Currently there is no intelligent way of spectrum management implemented. A technical mech-
anism is under development, named the Wi-5 mechanism, as a way of intelligently managing
the spectrum. The Wi-5 mechanism is a method that automatically manages the use of the
spectrum for the users that agreed to cooperate under the predefined set of rules imposed by the
mechanism. The mechanism aims to tackle the tragedy of the spectrum commons problem in
areas with densely populated Wi-Fi users.
In this thesis we theoretically research whether there is an incentive for individual Wi-Fi users to
join the Wi-5 mechanism. In order to do so, we develop a game theoretic model that describes
the use of the Wi-5 mechanism using both non-cooperative and cooperative game theory, to
study the choices made by the Wi-Fi users. The Wi-Fi choices we consider are the frequency
channel (i.e., the exact ‘location’ in the available spectrum) and the transmission power (i.e., the
strength of the sending and receiving signal).
This thesis is written as part of the work within the EU-funded ‘What to do With the Wi-
Fi Wild West’ (Wi-5) project,1
in which the Wi-5 mechanism is developed. The aim of the
project is to develop a mechanism that can be readily integrated into existing solutions and
1The EU H2020 Wi-5 Project, part of the Horizon 2020 Framework Programme of the European Union,
http://www.wi5.eu/.
7
11. deployed to solve the problem of the spectrum commons. This thesis contributes by providing a
building block which can be used to theoretic proof that the Wi-5 mechanism improves the QoS
of Wi-Fi.
1.2 Aim
In order to cover all options of users joining the Wi-5 mechanism, we aim to find a model for
the following three scenarios: the non-cooperative, the cooperative, and the mixed scenario. In
the non-cooperative scenario we consider the scenario in which all Wi-Fi users do not (have the
option to) join the Wi-5 mechanism, i.e. the users are non-joining users. In the cooperative
scenario we consider the scenario where all Wi-Fi users are willing to join the Wi-5 mechanism,
i.e. all users are joining users. In this scenario we determine whether all users have the incentive
to join the Wi-5 mechanism. In the mixed scenario we consider the scenario in which each user
is either non-joining or joining. In this scenario we determine whether it is profitable for the
joining users to join the Wi-5 mechanism, given the behaviour of the non-joining users.
The model should contain the following main elements: possible (voluntary) cooperation be-
tween users, a spectrum manager (controller) that can prioritize between Wi-Fi users, and a way
to research the number of users that would benefit from cooperation. In particular, we want to
consider frequency channel selection and transmission power. Because in practice each Wi-Fi
user is able to determine the level of interference he receives, we can safely assume that all users
know the frequency channels and transmission powers of all other users in the spectrum. This
sort of information is also referred to as global information.
Section 1.3 discusses research on both non-cooperative game theory and cooperative game the-
ory that could be of interest to model the afore mentioned scenarios. We are interested in
non-cooperative games because they could be used to model the current scenario, in which Wi-
Fi users do not join the Wi-5 mechanism and therefore do not cooperate in their channel selection
and transmission power selection. We consider strategic games because the selected channel and
transmission power chosen by an individual Wi-Fi user affects all other Wi-Fi users. We are
interested in cooperative games because users that are willing to join the Wi-5 mechanism can
be regarded as users that cooperate in a game theoretic sense. We research bargaining games,
which we may use to determine the pay-off received by the users who join the Wi-5 mechanism,
and coalition games, which we may use to determine the users who have an incentive to join the
Wi-5 mechanism.
1.3 Literature study
In the last two decades there has been a lot of research to spectrum allocation in wireless networks
and transmission power control using game theoretic models. Previous works consider four cat-
egories of game theory, namely non-cooperative games, cooperative games, auction games, and
stochastic games.
In this section we discuss the literature that is most relevant to modelling the Wi-5 mecha-
nism that is studied in this thesis. We differentiate between research using non-cooperative
and cooperative game theory. Both [11] and [22] give an overview of the most relevant literature
on modelling communication networks (channel allocation and power control) using game theory.
8
12. In this thesis we do not consider auction games, i.e., games which model auction markets. This
is because the spectrum is unlicensed, and therefore the Wi-Fi users do not need to pay to obtain
access and it is therefore not possible to auction the frequency channels in this setting. Stochastic
games are sequential games which are in a random state at the beginning of each game, which
can be used to model dynamic environments. E.g., in [19], a finite state Markov chain is used
to model the choice of transmission power of Wi-Fi users. We do not consider stochastic games
because they are beyond the scope of this thesis, although they are interesting for future research.
Non-cooperative games
Non-cooperative games are games in which players make strategic decisions independently of
each other. The most used solution concept of non-cooperative games is the Nash equilibrium,
first introduced in [27]. The Nash equilibrium is a strategy profile in which none of the players
have the incentive to deviate and choose a different strategy.
One type of non-cooperative games are strategic games, i.e., games in which the utility of players
does not only depend on the strategy of the player but also on the strategy of the other players.
There are various types of strategic games, for example potential games, first introduced in [24].
Potential games are strategic games in which existence of a Nash equilibrium is guaranteed. Be-
cause of this guarantee, potential games have become popular building blocks to model wireless
networks. Standard functions [35] are also used in modelling wireless networks, since they also
ensure convergence to a Nash equilibrium.
The concept of potential games is first used in the context of wireless radio networks in [5]
and has been a popular tool used in other radio network models since. In [14], a framework is
proposed for transmission power management using potential games in which the radios (or access
points) can choose any power level. The utility function of the radios depends on the Signal-to-
Interference-and-Noise ratio (SINR) and costs associated with choosing a certain power level.
In [9], a channel allocation model is proposed, based on no-regret learning and a potential game,
partially using the framework proposed in [14]. In this model two types of players are defined,
non-cooperative and cooperative players. The differentiation between the players is made by
using two different utility functions. The non-cooperative players only consider the interference
encountered when transmitting on a channel, whereas the cooperative players also consider the
interference they cause to other players. In the scenario with only non-cooperative users and
no-regret learning it is shown that the scenario converges to a channel allocation equilibrium. In
the scenario with only cooperative users it is shown that there exists a pure channel allocation
equilibrium, using the potential game structure. In a later work [2], the authors include trans-
mission power control in the model. This is using a target Signal-to-Interference ratio (SIR), and
the transmission power is chosen such that the target SIR is exactly reached in each of the chan-
nel allocations. A cooperative utility function is considered, and Nash equilibria are computed.
This can be seen as an enforced cooperation between the players, due to the fact that players are
considered to make independent decisions in the non-cooperative game structure. It is shown
that, although only considering channel allocation or transmission power control management
improve the QoS of Wi-Fi, combined management leads to an even further improved QoS of
Wi-Fi.
In [35], an iterative framework is proposed using a standard function and one frequency channel.
That is, using a function that describes the interference each individual player must overcome
to have an acceptable QoS of Wi-Fi, which depends on the transmission power of all players.
9
13. In [35], there are multiple possible standard functions described and various constraints on the
transmission power are considered. In every modelling choice that can be made, a transmission
power that provides an acceptable QoS of Wi-Fi for the players is found if it exists.
We already mentioned the cost element in the utility function in [14]. In [15], the effect of
adding a pricing element to the utility of selfish users is investigated, in the scenario with only
one channel. The authors compare Nash equilibria in the case that the utility of the users is the
SIR, to the equilibria in the case that the utility function is the difference between the SIR and
a linear pricing function that depends on the transmission power. In the latter case it is shown
that there may exist multiple Nash equilibria, for which the transmission power vectors yield
higher net utilities than any other equilibrium power vector.
In [7], a dynamic, hierarchical, non-cooperative game is proposed. In this game (i) a spectrum
manager maximizes the spectrum efficiency through pricing, (ii) service providers maximize their
revenue by deploying services over their licensed spectrum bands, and (iii) Wi-Fi users make a
trade off between QoS of Wi-Fi and spectrum cost through transmission power control, modelled
using a standard function. It is shown that the Nash equilibrium that solves this game exists
and is unique.
Repeated games are games which are solved by solving the underlying strategic game sequentially
(infinitely) many times keeping track of previous solution strategies. The pay-off function for a
user is the discounted average of immediate pay-offs from each round of the repeated game. Re-
peated games are used to model spectrum access, for example in [20] and [12]. In [20], a dynamic
framework is proposed for spectrum access control in which two types of users are considered. It
is shown that the less prioritized users can obtain optimal access with only local information.
Cooperative games
Cooperative games are games in which players jointly decide on a possible pay-off distribution
among the players. Two classes of cooperative games are used in modelling wireless networks,
namely bargaining games and coalition games. Bargaining games are games in which all in-
dividual players have the opportunity to reach an agreement but have their own choice if the
agreement is not made. A coalition game is a game which describes how all subsets of players
can cooperate and improve their pay-off. Coalition games can be used to decide on optimal col-
laboration strategies [25]. We discuss both these classes, because of their potential in modelling
the Wi-5 mechanism.
Bargaining games are cooperative games in which all players can decide to cooperate, considering
a disagreement point. They can achieve some degree of fairness among the players in multiple
ways, by using different solution concepts. Examples of solution concepts for bargaining games
are the Egalitarian bargaining solution [21] which maximizes the minimum of surplus utilities
and the Nash bargaining solution [28] which maximizes the product of surplus utilities. The
Egalitarian bargaining solution imposes absolute fairness on the players whereas the Nash bar-
gaining solution imposes relative fairness.
The disagreement point represents the pay-off of the players in the case that the players de-
cide not to cooperate. The disagreement point can be defined in various ways. For example, in
[16], a bargaining model is proposed which schedules the communication (Wi-Fi usage) of the
Wi-Fi users in advance, concerning frequency channel and time slots. The users do not transmit
continuously on a channel and it might be better for a user to wait before transmitting. The
10
14. game is solved, after a learning process, using Nash bargaining in which the disagreement point is
the vector of utilities achieved by the learning process. It is shown that this model could be used
to reduce the overhead of continuous sensing the available spectrum. In [1], a bargaining model
is proposed for throughput maximization in which there are two types of users. The pay-off set
is the set of feasible throughput values of the players. To prioritize one type of user over the
other, two values are used as the disagreement point. The prioritized type of users receive the
higher disagreement value, and the other users receive the lower value. In [30], a bargaining
model for channel selection is proposed in which the disagreement point is chosen as the threat
made by heterogeneous individual users. The pay-off set is based on the SINR. In [6], bargaining
is used to deal with the case in which users only know the frequency channel and transmission
power of close by other users, i.e. only local information is available. The approach allows Wi-
Fi users to self organize in bargaining groups and approximate their optimal channel assignment.
Coalition games, with or without transferable pay-off, are used to study user cooperation and
design optimal collaboration strategies. For example, in [23], a coalition game is proposed to
model cooperation among users in a single frequency channel. Users that cooperate form a coali-
tion and jointly decode received signals. A coalition views incoming signals from users not in the
coalition as interference. The value a coalition is assigned is defined as the maximal data rate
that can be achieved in the coalition. In the game it is assumed that the data rate is transferable
and the Nash bargaining rule and a proportional fair solution are proposed to allocate the data
rate. It is shown that the grand coalition, i.e., the coalition with all users, is stable. That is,
none of the users have the incentive to leave the grand coalition and form other coalitions.
1.4 General approach
Because Wi-Fi uses an unlicensed spectrum, it is not possible to force users to join the Wi-5
mechanism. Therefore some users may not be willing to join the Wi-5 mechanism. In order to
model the scenario in which there exist users that are willing to join the Wi-5 mechanism and
other users that are not willing to join, we propose to combine non-cooperative and coopera-
tive game theory. However, according to the best of our knowledge, a model that uses both
non-cooperative and cooperative game theory does not yet exist in the current literature. In [9],
there is a deviation between non-cooperative and cooperative users, only it is assumed that all
users act selfishly and make strategic decisions independently. Due to the controller in the Wi-5
mechanism, the joining users will not act selfishly and therefore the model in [9] is not suitable to
model the Wi-5 mechanism. Therefore we construct a new model, the Wi-5 model, to model the
Wi-5 mechanism that includes all elements mentioned in Section 1.2 and facilitates the option
of considering both joining and non-joining users.
In the discussed literature there are ideas and elements that we found are very interesting for
modelling the Wi-5 mechanism. We use the idea of transmission power control in the Wi-5 model,
as is proposed in [2], but we consider a different power control criterion. We do not choose the
transmission powers such that the target SINR of the joining users is exactly reached, but we
maximize the minimal transmission power and guarantee joining users that they have at least
their target SINR. We also use the idea of a pricing element [15] to prioritize between the users
that join the Wi-5 mechanism in the transmission power control mechanism. Furthermore we
use the idea to use the threat made by the individual users as disagreement point in the Nash
bargaining rule [30] in the cooperative scenario and the mixed scenario.
11
15. We restricted the scope of this thesis to model the Wi-5 mechanism as a one-shot game. Therefore
we do not consider repeated games. We do not use the utility function of the users to differentiate
between non-joining and joining users as in [9]. This is because we want the strategic decisions of
the joining players to be made by a controller, whereas in [9] it is implied that joining users make
their strategic decisions independently. We also do not use the concept of local bargaining ap-
plied in [6] because we assume global information. Furthermore we do not use a coalition game to
model the Wi-5 mechanism. This is because we assume that individual users can join or not join
the Wi-5 mechanism. Therefore it is not possible for coalitions of users to decide to separately
join the mechanism. Lastly we find that the spectrum manager modelled in hierarchical model in
[7] an interesting idea, but it does not fit to model the Wi-5 mechanism. This is because the Wi-
5 controller does not have a utility of its own, and should not be modelled as a player in the game.
As solution concept for the non-cooperative scenario we use a Nash equilibrium. We use the
Nash bargaining rule as solution concept for the cooperative scenario due to the relative fairness
the Nash bargaining rule imposes on the joining users. In the mixed scenario we combine a Nash
equilibrium with the Nash bargaining rule.
The Wi-5 modes provides a one-shot solution for each of the three scenarios. Considering the
outcome of all scenarios we are able to determine the fraction of Wi-Fi users that is willing to
join the Wi-5 mechanism. Note that for a given set of users multiple mixed scenarios exist, since
various sets of users may be willing to join the Wi-5 mechanism.
1.5 Contribution
The main contribution of this thesis to current literature is a game theoretic framework, in which
we propose a new approach in combing non-cooperative and cooperative game theory. We use
this theoretical framework to model spectrum management, frequency channel allocation and
transmission power control, considering the Wi-5 mechanism. We use the framework to research
a Wi-Fi user’s incentive to join the Wi-5 mechanism (cooperate), considering the case in which
all users do not join the Wi-5 mechanism. The framework therefore contributes by providing a
theoretical basis for investigating the willingness to cooperate in tragedy of the spectrum com-
mons solution mechanisms.
The framework may potentially be used in numerous other kind of applications to research
if players have an incentive to cooperate, and to determine the number of players that will ben-
efit from cooperation.
In this thesis we make significant first steps as to theoretically showing that the Wi-5 mech-
anism is able increase the QoS of Wi-Fi using transmission power control. The conclusions we
draw from the examples discussed in Chapter 6 are in line with conclusions drawn from empirical
simulations performed by Dr. P.L. Kempker and A.S. Popescu at TNO. The empirical results,
as well as the game theoretic framework and more extensive examples, are to be published in De-
liverable 2.2 ‘Wi-Fi optimisation solutions roadmap’ of the Wi-5 project, which is to be released
in December 2016.
12
16. 1.6 Outline
In Chapter 2 we give background information on Wi-Fi and describe the technical details that
we use to initialize the Wi-5 model. Chapter 3 describes the mathematical theory used in this
thesis to theoretically research the incentive of Wi-Fi users to joint the Wi-5 mechanism. We
introduce strategic games and the concept of (mixed) Nash equilibrium in Section 3.2, and bar-
gaining games and the Nash bargaining solution in Section 3.3.
In Chapter 4 we introduce the Wi-5 model and describe the three scenarios in Sections 4.3,
4.4, and 4.5. In Chapter 5 we introduce the use case ‘the apartment building’. We initialize
the utility function, the transmission power control mechanism, and the parameters of the Wi-5
model in the use case in Section 5.3.
In Chapter 6 we illustrate the Wi-5 model, considering the use case, with a two person and
a three person example. We discuss the results of these examples in Sections 6.2.4 and 6.3.5.
Finally, in Chapter 7 we provide conclusions of our work and discuss what further research can
be done to improve upon the conclusions of this thesis.
13
17. Chapter 2
Wi-Fi and the Wi-5 mechanism
2.1 Introduction
A Wi-Fi user is an individual that uses the Wi-Fi spectrum. The Wi-Fi spectrum allows elec-
tronic devices to connect to a wireless local area network. Devices which can use Wi-Fi technology
include mobile phones, laptops, tablets, and televisions. To get access to the Wi-Fi spectrum, a
Wi-Fi user sets up a Wi-Fi connection. In order to do this, a Wi-Fi user needs an access point
(AP), which we will refer to as a transmitter node, and a receiver node. Both nodes need to be
able to connect to the wireless network. The transmitter node can transmit a Wi-Fi signal with
a certain transmission power, which can be picked up by the receiver node, creating the Wi-Fi
connection.
In this chapter we describe the background information and the technical details on Wi-Fi that
we use to define our model in Chapter 4. We discuss the Wi-Fi spectrum, the regulations con-
cerning the spectrum, and the Signal-to-Noise-and-Interference ratio in Section 2.2.
In Section 2.3 we introduce the Wi-5 mechanism, developed in the ‘What to do With the Wi-Fi
Wild West’ (Wi-5) project.2
The Wi-5 mechanism is a technical mechanism, which is a way of
intelligently managing the spectrum to keep the QoS of Wi-Fi on a high level when the number
of Wi-Fi users increases.
2.2 The unlisenced Wi-Fi spectrum
Wi-Fi connections are set up using radio frequencies, which are part of the total radio frequency
spectrum. The total radio frequency spectrum, the electromagnetic spectrum, is a finite spec-
trum. This means that there are only a finite number of frequencies available. It is not an option
to ‘create’ new frequencies, since this is physically not possible.
The radio frequency spectrum is divided in bands, radio frequency bands, and since the spec-
trum is finite the number of bands is finite. The use of these radio frequency bands is in most
countries regulated by the government. If a part of the spectrum has a specific purpose according
to the regulations, this spectrum is called licensed. Furthermore, the spectrum regulations are
harmonized between governments due to technical and economic reasons.
2See footnote 1.
14
18. Figure 2.1: The USA frequency spectrum allocation chart.
The unlicensed spectrum, all the spectrum which is not licensed, can be used for Wi-Fi. Figure
2.1 illustrates the radio frequency band regulations of the United States, including all allocations
of the bands to various services. The unlicensed spectrum in the United States is equal to the
unlicensed spectrum in the other countries of the world. As indicated in Figure 2.1 with the
circles, the spectrum which is allocated as unlicensed spectrum, is very limited. There are three
parts of the spectrum that are unlicensed, which correspond to the 2.4 GHz, 5 GHz, and 60 GHz
part of the spectrum. The higher the amount of GHz, the smaller the reach of the Wi-Fi signal
will be. To compare, the 2.4 GHz signal can cover an entire house whereas the 5 GHz signal will
only cover a single room.
Each section of the frequency spectrum can be divided in different channels, where a chan-
nel is simply a small range of frequencies. To illustrate the existence of multiple channels we
consider the 2.4 GHz unlicensed spectrum. Figure 2.2 illustrates the unlicensed frequency spec-
trum and the channels that are present in this part of the spectrum.
In total there are 14 channels on the 2.4 GHz spectrum. Maximally four of these channels
are non-overlapping, namely channels 1, 6, 11, and 14. Non-overlapping channels are channels
that do not share any frequency with each other. If two Wi-Fi users have set up a Wi-Fi connec-
tion on the same channel or on two different overlapping channels, they interfere with each other.
This means that the quality of their Wi-Fi connection is lower than if they would have set the
connections up on two non-overlapping channels,assuming that their transmission nodes trans-
mit with the same transmission power. In general it is such that the more interference a Wi-Fi
15
19. Figure 2.2: The 2.4 GHz frequency spectrum with channels.
user experiences, the lower the QoS of his Wi-Fi connection. A standard interference measure
in practice and in literature, e.g. used in [2], [8] and [9], is the Signal-to-Interference-and-Noise
ratio (SINR), expressed in decibels (dB).
In all countries, except for Japan, it is illegal to use channel 14. This means that in the 2.4
GHz spectrum there are maximally three non-overlapping channels available for Wi-Fi usage.
Channels 1, 6, and 11 are the default non-overlapping channels used in practice.
2.3 The Wi-5 mechanism
Due to the increasing number of Wi-Fi users and the fact that the unlicensed spectrum that
can be used for Wi-Fi is limited, the QoS of Wi-Fi for all the users may decrease because of the
increasing level of interference. This phenomenon is known as tragedy of the spectrum commons,
a special case of the concept of tragedy of the commons. The concept of tragedy of the commons,
the idea that multiple actors are selfishly competing for a share of limited resources would be
worse for their own utility than considering collective actions, was first introduced by Hardin [17].
Due to the increasing Wi-Fi demand, there has been interest in the development of an intel-
ligent channel allocation and transmission power control mechanism that is beneficial for all the
participating Wi-Fi users. TNO (Netherlands Organisation for Applied Scientific Research), a
Dutch research organisation, participates in the European Wi-5 project to contribute designing
such an intelligent mechanism, the Wi-5 mechanism. The idea behind the Wi-5 mechanism is to
tune the channels and transmission powers of all users to improve the overall Wi-Fi connection
quality.
In practice, if a Wi-Fi user accepts the Wi-5 mechanism, he is guaranteed a prespecified QoS of
Wi-Fi and he allows a controller to choose his frequency channel and transmission power. The
controller is able to change the selected frequency channel and transmission power of a user in
real time, i.e. without the user experiencing any inconvenience from this and immediately after
a user that does not join the Wi-5 mechanism changes his frequency channel or transmission
power. Furthermore the controller is able to change the frequency channel and the transmission
power of all users simultaneously.
16
20. The goal of the Wi-5 mechanism, realised by the controller, is to find a channel allocation (with
corresponding transmission power) for all the users that join, such that the quality of their Wi-
Fi connections is better than their Wi-Fi connection quality in the current scenario without the
mechanism. Due to the technical abilities of the controller it is possible that the found channel
allocation is a mixed allocation in which some channel allocations are chosen with certain prob-
abilities.
We note that it is legally not allowed to obligate users to join the Wi-5 mechanism, i.e. the
Wi-5 mechanism imposes a legally non-binding structure. Therefore, when we model the Wi-5
mechanism we need to take into account that it is possible that some users do not want to
participate with the Wi-5 mechanism.
17
21. Chapter 3
Mathematical preliminaries
3.1 Introduction
In this thesis we consider both non-cooperative and cooperative game theory. In a non-cooperative
game, individuals are only interested in their individual pay-off or utility when choosing their
strategy and there are no legally binding contracts possible between players. In a cooperative
game, individuals have the potential to work together via a binding contract in order to obtain
a certain utility.
In this chapter we describe the game theoretic components that we use to model the Wi-5
mechanism (Section 2.3). We use this model, the Wi-5 model (Chapter 4), to research a player’s
incentive to join the Wi-5 mechanism. In Section 3.2 we introduce strategic games in non-
cooperative game theory with the mixed Nash equilibrium concept. In Section 3.3 we introduce
bargaining games in cooperative game theory with the Nash bargaining rule as solution concept.
In Figure 3.1 we provide a schematic overview of the game theory that is used in this thesis.
Game theory
Non-cooperative games
Strategic games
The mixed Nash equilibrium
Cooperative games
Bargaining games
The Nash Bargaining rule
Figure 3.1: Schematic overview of the game theory in the Wi-5 model.
3.2 Non-cooperative game theory
Non-cooperative games are games in which players are only interested in their individual utility
and make their strategic decisions independently. In non-cooperative games, it is assumed that
legal contracts, binding the players to agreed upon strategic decisions, do not exist.
Strategic games are non-cooperative games in which all players make their strategic decisions
18
22. simultaneously. In strategic games the utility of a player depends on his own strategy and the
strategies of all other players. Therefore a player should consider the viewpoints of all other
players while selecting an appropriate strategy.
Strategic games are of the following form:
N, {Si}i∈N , {ui}i∈N ,
where N defines the finite set of players, N = {1, . . . , n}. The set of pure strategies of player
i ∈ N, is given by Si, and S = ×i∈N Si is the set of pure strategy profiles. A pure strategy
provides a complete definition of how a player will act in the game, so if si ∈ Si is the pure
strategy selected by player i ∈ N, then the player acts according to strategy si. The set of
probability distributions over Si is defined by ∆(Si). An element si ∈ ∆(Si) is called a mixed
strategy of player i ∈ N, and ×i∈N ∆(Si) is the set of mixed strategy profiles. A strategic game
is called finite if Si is finite for all i ∈ N.
The utility function of player i ∈ N, ui : S → R, imposes a preference relation on the set
of pure strategy profiles. The expected utility function of player i ∈ N, Ui : ×i∈N ∆(Si) → R,
defines the expected utility of player i. Note that Ui is multilinear, that is for si, ti ∈ ∆(Si),
s−i ∈ ×j∈N,j=i∆(Sj) and λ ∈ [0, 1] we have that Ui(λsi + (1 − λ)ti, s−i) = λUi(si, s−i) + (1 −
λ)Ui(ti, s−i). In the case that the strategic game is finite we have that for all s ∈ ×i∈N ∆(Si):
Ui(s) =
|S|
k=1
Pr[sk
|s]ui(sk
),
where S = {s1
, . . . , s|S|
}. Here Pr[sk
|s] is the probability that sk
is the chosen pure profile, given
mixed profile s. Using the fact that the players decide on their strategy independently of each
other, we have that Pr[sk|s] can be computed as
Pr[sk
|s] =
i∈N
Pr[sk
i |si],
where Pr[sk
i |si] is the probability that player i ∈ N, chooses the pure strategy sk
i , given his mixed
strategy si.
There exist various solution concepts to determine equilibrium strategies for the players in strate-
gic games. We focus on the mixed Nash equilibrium (NE), first introduced in [26], a one-shot
solution concept in which no player has the incentive to choose a different mixed strategy since
he cannot obtain a higher expected utility by deviating from the mixed NE.
Definition 3.1. Consider the strategic game N, {Si}i∈N , {ui}i∈N . A mixed Nash equilibrium
of the strategic game is a profile s∗
∈ ×i∈N ∆(Si) such that for every player i ∈ N it holds that
Ui(s∗
i , s∗
−i) ≥ Ui(si, s∗
−i), (3.1)
for all si ∈ ∆(Si).
The following theorem by Osborne and Rubinstein [29] is on the existence of a mixed Nash
equilibrium for finite strategic games.
Theorem 3.2. Every finite strategic game N, {Si}i∈N , {ui}i∈N has at least one mixed Nash
equilibrium.
19
23. It is also possible that in a mixed Nash equilibrium all players select a pure strategy. In the case
an equilibrium is reached at a pure strategy profile, it is called a pure Nash equilibrium.
We note that there exist games in which multiple Nash equilibria exist, for example in the
2-player game described in Example 3.3.
Example 3.3. In this example, two players need to decide if they want to spend the evening at
home, watching tv, or go to a party. The utility levels of player 1 is described in the first entry
of the tuples, the utility levels of player 2 in the second entry.
home party
home (10,5) (0,2)
party (0,0) (5,10)
Table 3.1: Two player example: stay at home or go to a party?
We see that there exist two pure Nash equilibria, namely in the case that both players stay at
home and watch tv together, or if both players go to a party. Player 1 prefers that they both
stay at home, whereas player 2 prefers that they both go to the party, but both players prefer to
not engage in an activity alone. There also exists a mixed Nash equilibrium, namely in the case
that player 1 stays home with probability 8/13 and player 2 stays home with probability 1/3.
The utility profile in the mixed Nash equilibrium is (3.33,4.41). We computed these equilibria
considering the expected utility of the players, and to set the first derivative equal to zero.
In the case that in a game multiple equilibria exist, the equilibria satisfying Pareto efficiency
(Definition 3.4) are widely used in economics and social sciences. This is because a utility profile
is called Pareto optimal if no player can increase his utility by changing his strategy without
decreasing the utility of at least one of the other players.
Definition 3.4. Let U ⊂ Rn
be a set. Then u ∈ U is Pareto efficient if there is no u ∈ U for
which ui > ui for all i ∈ {1, . . . , n}. u ∈ U us strongly Pareto efficient if there is no u ∈ U for
which ui ≥ ui for all i ∈ {1, . . . , n} and ui > ui for some i ∈ {1, . . . , n}. The Pareto frontier is
defined as the set of all u ∈ U that are Pareto efficient.
In Example 3.3 there are three equilibria: (i) (10,5), (ii) (5,10), and (iii) (3.33,4.41). Both
equilibria (i) and (ii) are Pareto efficient, and they define the Pareto frontier in this example.
3.3 Cooperative game theory
Cooperative games are games in which players can agree to a legal contract in which an alter-
native is selected that is beneficial to all players that agreed on the contract. An alternative
represents a pay-off or utility vector (or profile) for all players. In this thesis we focus on bar-
gaining games. Bargaining games are games in which either all players agree on an alternative
and sign a binding contract or there exists no alternative on which all players agree. In the latter
case all players disagree, no binding contract is signed, and a disagreement point is chosen as an
alternative.
An n-person bargaining game is of the following form:
Bn
= (A, d).
20
24. The set A ⊂ Rn
defines the set of alternatives that can be achieved by the players. The vector
d ∈ Rn
is the disagreement point, which is chosen in the case that the players cannot agree on
any outcome in A. Usually, the following assumptions are imposed on bargaining games [4]:
· A = ∅, A is closed and convex.
· A is comprehensive, i.e. (a ∈ A, a ≤ a ⇒ a ∈ A).
· The set {a ∈ A|a ≥ d} is bounded.
· There is an alternative a ∈ A such that ai > di for all i ∈ N.
(3.2)
There exist various solution concepts, i.e. bargaining rules, for bargaining games. Examples
are the strategic model proposed by Rubinstein [32] and the axiomatic model proposed by Nash
[28]. Bargaining rules can satisfy an assortment of axioms, a few of which we mention. Let
f : Bn
→ Rn
, f(A, d) ∈ A, be a bargaining rule.
Individual rationality f satisfies individual rationality if fi(A, d) ≥ di for all i ∈ N.
This means that the players will only agree on an alternative other then the disagreement
point if they obtain at least their disagreement pay-off.
Symmetry f satisfies symmetry if f(A, d) is such that π(f(A, d)) = f(π(A), π(d)), where π :
Rn
→ Rn
is any permutation function.
Symmetry implies that the order of the players does not affect the outcome of the bargaining
rule.
Invariance f satisfies invariance (with respect to affine transformations) if for every T : Rn
→
Rn
of the form T(x) = ax + b with a ∈ R+, and b ∈ Rn
, it holds that T(f(A, d)) =
f(T(A), T(d)).
This implies that rescaling the disagreement values, and the alternative set, changes the
outcome of the bargaining rule in corresponding way.
Independence of irrelevant alternatives f satisfies independence of irrelevant alternatives
if for all A1, A2 ⊂ Rn
such that A1 ⊂ A2 and f(A2, d) ∈ A1 it holds that f(A1, d) =
f(A2, d).
This implies that if players would bargain over a smaller alternative set, in which the
solution of the original bargaining game is available, the same alternative is selected as in
the original bargaining game.
In this thesis we focus on the Nash bargaining rule, proposed in [28]. This is because the Nash
bargaining rule is the only bargaining rule that provides a Pareto optimal solution (Definition
3.4) and satisfies all the above mentioned axioms (Theorem 3.6).
Definition 3.5. The Nash bargaining rule N assigns to each bargaining game Bn
= (A, d)
satisfying (3.2) the unique alternative N(A, d) ∈ A such that
N(A, d) = arg max
a∈A
n
i=1
(ai − di),
with ai ≥ di for i ∈ {1, . . . , n}.
The following theorem in [4, Theorem 9.2] gives a full characterisation of the Nash bargaining
rule.
Theorem 3.6. The Nash bargaining rule is the unique bargaining rule that satisfies Pareto
efficiency, symmetry, invariance, and independence of irrelevant alternatives.
21
25. Chapter 4
The Wi-5 model
4.1 Introduction
When given the option to join the Wi-5 mechanism, the questions Wi-Fi users will ask them-
selves are ‘Should I join the Wi-Fi mechanism or should I not join the Wi-5 mechanism?’ and
‘Which of the other users should join the Wi-5 mechanism in order for me to benefit from the
Wi-5 mechanism?’. To help answer these questions theoretically, we propose the Wi-5 model.
With the Wi-5 model, we can research whether it is rational for a fixed subset of users, that is
willing to join the Wi-5 mechanism, to join or not join the Wi-5 mechanism. In case it is not
rational for at least one of the users to join the Wi-5 mechanism, we assume that none of the
Wi-Fi users in the subset will join the Wi-5 mechanism. It is possible to use the Wi-5 model to
research for which subsets of Wi-Fi users it would be beneficial to cooperate, and we are able to
determine a largest set of users that has an incentive to join the Wi-5 mechanism.
In this chapter we describe the Wi-5 model, which consists of three scenarios. Together, these
three scenarios describe all possible scenarios of users willing or not willing to join the Wi-5
mechanism, given a fixed set of users. We initialize the game theoretic parameters and functions
used in the Wi-5 model in Section 4.2. We discuss the case that all Wi-Fi users do not (have the
option to) join or are not willing to join the Wi-5 mechanism in Section 4.3. In Section 4.4 we
discuss the scenario in which all users are willing to join the Wi-5 mechanism. The outcome of
this scenario is either all users join the Wi-5 mechanism, or none of the users joins. In Section
4.5 we discuss the scenario in which there are both users willing to join the Wi-5 mechanism and
users that are not willing to join. The question in this scenario remains the same, namely if the
users willing to join the Wi-5 mechanism have an incentive to join or not, but now the users not
willing to cooperate need to be taken into consideration.
4.2 Game theoretic elements
In this section we initialize the building blocks for the Wi-5 model, needed to define the appro-
priate strategic games (Section 3.2) and bargaining games (Section 3.3). Throughout this thesis
we use the following notation:
N The finite set of Wi-Fi users, N = {1, . . . , n}.
dj
Ri
jth coordinate of the position of the receiver node of user i ∈ N, j ∈ {1, 2, 3}.
22
26. dj
Ti
jth coordinate of the position of the transmitter node of user i ∈ N, j ∈ {1, 2, 3}.
mi Monthly contract fee of user i ∈ N.
C The finite set of channels.
ci The channel selected by user i ∈ N, ci ∈ C.
pi The transmission power of user i ∈ N, pi ∈ (0, pmax
i ].
pmax
i The maximal transmission power of user i ∈ N, pmax
i > 0.
ui The utility function of user i ∈ N.
The players We consider a fixed set of Wi-Fi users, N, to be the players in the Wi-5 model
and the users own a receiver and transmitter node (Section 2.1). The location of the nodes is
given by 3D coordinates and two nodes cannot have the exact same coordinates. Each user i ∈ N
has an individual internet plan, with a monthly contract fee mi in euros. This amount cannot
be freely changed by the user and is therefore a fixed parameter in the model.
There is a fixed finite set of frequency channels, the same for each user, a user can choose
from. This is due to the fact that the licensed spectrum used for Wi-Fi is available to all Wi-Fi
users (Section 2.2). Furthermore, the transmission power of the users is bounded from above
with maximal transmission power pmax
i due to the Code of Federal regulations title 47, part 15.
We assume that we know in advance which of the Wi-Fi users are willing to join the Wi-5
mechanism (joining users) and which users are not willing to join the Wi-5 mechanism (non-
joining users).
The utility function We consider the Signal-to-Interference-and-Noise Ratio (SINR), intro-
duced in Section 2.2, as the utility function. We use the following notations to define the
SINR[10]:
Gij The path loss between the transmitter node of user i ∈ N and the receiver node
of user j ∈ N, where Gii is the path loss of user i ∈ N. Path loss is defined as the
reduction in power density (attenuation) of the transmission power as it propagates
through space. Gij depends on the distance between the considered transmitter
and receiver, which is determined using their locations.
I(c, c ) The function characterizing the interference between channel c ∈ C and c ∈ C.
n0i
The individual noise factor of user i ∈ N which is an additional noise term other
then the interference experienced.
Definition 4.1. The Signal-to-Interference-and-Noise ratio (SINR) for user i ∈ N, is the func-
tion SINRi given as follows:
SINRi(c1, p1, c2, p2 . . . , cn, pn) = 10 log10
piGii
j∈N,j=i pjGijI(ci, cj) + n0i
.
We obtain for user i ∈ N:
ui(c1, p1, . . . , cn, pn) = SINRi(c1, p1, . . . , cn, pn).
The SINRi of user i, depends not only on his selected channel and transmission power but also on
the selected channels and transmission powers of all other users. This implies, for the non-joining
users, that we consider a strategic game.
23
27. 4.3 The non-cooperative scenario
We name the scenario in which none of the Wi-Fi users (is able to) join the Wi-5 mechanism the
non-cooperative scenario. In this scenario, all Wi-Fi users are non-joining users. We model this
scenario as a strategic game, and aim to find an equilibrium utility profile of the users.
The strategy Each non-joining user is able to make two choices, namely (i) on which chan-
nel to transmit and (ii) with which transmission power. Note that in the case that pi can be
chosen arbitrarily between 0 and pmax
i , then the strategy set of non-joining user i ∈ N is not finite.
Considering the SINR, we find that it is in a non-joining user’s best interest to transmit at
maximal transmission power. This is because the user’s SINR increases in the case that his
transmission power increases. Therefore the user’s SINR is optimal, considering the individual
transmission power choices the user can make, in the case that his transmission power is max-
imal. So we assume that a non-joining user transmits at maximal transmission power and this
implies that in the case that N consists only of non-joining users, there are |C|n
pure strategies
profiles in total.
The equilibrium utility profile We determine an equilibrium strategy profile for the non-
joining users to compute their expected equilibrium utility profile. We need the expected equi-
librium utility profile to investigate if non-joining users might obtain a higher expected utility in
the case that they would join the Wi-5 mechanism.
An equilibrium strategy profile for the non-cooperative scenario can be determined by com-
puting a mixed Nash equilibrium (Definition 3.1). The equilibria can be computed using the
utility profiles in each of the pure strategy profiles of the users. A mixed Nash equilibrium
always exists (Theorem 3.2). In the case that there exist multiple equilibrium utility profiles,
select an equilibrium that is Pareto efficient.
4.4 The cooperative scenario
We name the scenario in which all Wi-Fi users are willing to join the Wi-5 mechanism the
cooperative scenario. In this scenario, all Wi-Fi users are joining users. We model this scenario
as an n−person bargaining game, Bn
= (A, d) (Section 3.3), such that we can investigate if the
joining users have an incentive to join the Wi-5 mechanism.
The alternative set For joining users we define the alternative set based on their utility
levels, considering the channel selection and transmission power of all the users, joining and
non-joining. In each of the channel allocations, the Wi-5 controller determines the transmission
power of the joining users, using the intelligent transmission power control mechanism (Section
2.3), and the non-joining users choose their maximal transmission power (Section 4.3). The Wi-5
controller considers the monthly contract fee of the joining users in order to prioritize the users
while determining the transmission powers. The alternative set of the joining users is the convex
hull of these utility levels.
We note that it is possible that the utility levels in this set of alternatives are different from
the utilities obtained by the joining users if they would act as non-joining users. This is because
the transmission powers may differ.
24
28. The bargaining solution We want to select an individually rational alternative from the al-
ternative set A for the joining users, that is, an alternative in which all users obtain a higher (or
equal) expected utility compared to the disagreement point (Section 3.3). If individually rational
alternatives exist, the users agree to join the Wi-5 mechanism and all sign a binding contract
that binds the users to one of these alternatives. If an individually rational alternative does not
exist, the joining users decide to not cooperate and act as non-joining users. In the latter case,
each of the users needs to determine a strategy independently, i.e. a frequency channel, and
transmits with his maximal transmission power.
We let the disagreement point d ∈ Rn
be the utility profile obtained by choosing a mixed
Nash equilibrium strategy found by assuming that the joining users act as non-joining users, as
described in Section 4.3. This is a feasible choice for d because when the joining users cannot
agree on an alternative, they act as non-joining users. It is possible that d /∈ A, because in
the non-cooperative scenario the users transmit with maximal transmission power and in the
cooperative scenario they may not.
In case that {a ∈ A|a ≥ d} = ∅, there exists no individually rational alternative and the
joining users will not join the Wi-5 mechanism. The users act as non-joining users.
In case that {a ∈ A|a ≥ d} = ∅, we use the Nash bargaining rule (Definition 3.5) to find a
solution for the defined bargaining game. In the case that the solution is an alternative in A
other than d, the joining users cooperate and are bounded by a contract to join the solution of the
bargaining game. In the case that the solution of the bargaining game is the disagreement point
d, the joining users are indifferent to cooperation. Whether the users join the Wi-5 mechanism
or not depends on the use case.
4.5 The mixed scenario
Since it is legally not possible to obligate Wi-Fi users to join the Wi-5 mechanism, there is always
the possibility that there are users not willing to join the Wi-5 mechanism. We name the scenario
in which there are both joining and non-joining users the mixed scenario.
Let Nc be the set of users willing to join the Wi-5 mechanism, |Nc| = nc, and Nnc = NNc the
set of users not willing to join the Wi-5 mechanism. The Wi-Fi users in Nc only join the Wi-5
mechanism if the expected utility of every joining user increases. We model this scenario as an
nc−person bargaining game, BNc
= (Ac, dc), Section 3.3, such that we can research if the joining
users should join the Wi-5 mechanism or not.
We let the disagreement point dc ∈ Rnc
be the sub-vector of d ∈ Rn
with indices in Nc, with
d as in the non-cooperative scenario described in Section 4.3. This is a feasible choice for the
disagreement point because the joining users act as non-joining users if they decide to not join
the Wi-5 mechanism.
We let the alternative set Ac of the joining users be the convex hull of the utility levels of
the pure channel selections of the joining users, with transmission powers determined by the
Wi-5 controller, considering the equilibrium strategy of the non-joining users (Section 3.2) in
each of these channel selections. The following steps describe how to construct Ac:
Step 1. Compute for all pure channel selections the transmission powers of the joining users.
25
29. Determine the utility levels of the non-joining users in each of the pure channel selections,
considering the transmission powers of the joining users. Use these utility levels to compute
a mixed Nash equilibrium for the non-joining users. Store the corresponding (mixed)
strategies.
Step 2. Compute the utility levels for the joining users in each of the pure channel allocations
with corresponding transmission powers, considering the equilibrium strategies of the non-
joining users. Determine the alternative set Ac as the convex hull of these pure alternatives.
In case that {a ∈ Ac|a ≥ dc} = ∅, there exists no individually rational alternative and the joining
users will not joint the Wi-5 mechanism. All users act as non-joining users.
In case that {a ∈ Ac|a ≥ dc} = ∅, we use the Nash bargaining rule (Definition 3.5) to find
a solution for the defined bargaining game. In the case that the solution is an alternative in Ac
other then dc, the joining users cooperate and join the Wi-5 mechanism. In the case that the
solution of the bargaining game is the disagreement point dc, the joining users are indifferent to
cooperation. Whether the users cooperate or not depends on the use case.
26
30. Chapter 5
Use case ‘The apartment building’
5.1 Introduction
The use case that we consider in this thesis is called ‘the apartment building’. This use case is
one of the use cases defined in Deliverable 2.3 ‘Wi-5 use cases and requirements’ of the Wi-5
project [3].
In this use case we consider an apartment building in which there are various apartments, each
owned by a single rational Wi-Fi user who owns a transmitter node and a receiver node. We
consider the scenario where all users are constantly transmitting at their maximal power. So,
we consider an area with relatively many access points where all users constantly interfere with
each other. Furthermore we assume that all available channels are non-overlapping channels.
Therefore each pair of users either fully interferes with each other, i.e. transmit on the same
channel, or they do not interfere with each other.
We relate the Wi-5 model to the use case in Section 5.2 and we initialize the use case spe-
cific functions and parameters of the Wi-5 model in Section 5.3. In Chapter 6 we illustrate the
use case and the Wi-5 mechanism using the Wi-5 model in a two person and a three person
example.
5.2 The Wi-5 mechanism in the use case
The Wi-5 mechanism will be carried out as follows in the apartment building. First the apart-
ment owners can choose to either join or not join the Wi-5 mechanism. Once this choice is
made, the joining owners are asked to provide data about their monthly contract fee. With this
data, the public information about the position of the transmitters and receivers and the current
channel allocation and transmission power of all users, a new channel allocation and transmission
power is determined by the controller for the joining users.
Users that join the Wi-5 mechanism are guaranteed to reach a certain QoS of Wi-Fi (Section
2.3), called the target SINR. The target SINR of a user i ∈ N, is based on the monthly contract
fee of the user and denoted by γ(mi). The controller takes these target SINRs into consideration
when he determines the transmission power of the joining users in each of the possible channel
selections. We initialize the target SINR in Section 5.3.1.
27
31. The controller aims to find a channel selection and transmission power for all Wi-Fi users such
that the solution is feasible, that is such that all users have QoS of Wi-Fi of at least the target
SINR. So for a feasible solution we need for all users i ∈ Nc that join the Wi-5 mechanism:
ui(c1, p1, . . . , cn, pn) ≥ γ(mi).
In Figure 5.1 we illustrate the use case in which there are two apartments. SPi illustrates the
supplier of network access to user i, and APi illustrates the access point (transmitter node) of
user i. Figure 5.1a illustrates a non-cooperative scenario with interference and maximal trans-
mission power and Figure 5.1b illustrates a cooperative scenario with the transmission power
controlled by the controller.
In Figure 5.2 we illustrate the use case in which there are six apartments. We illustrate a
situation with interference and maximal transmission power in Figure 5.2a and a situation with
less interference and controlled transmission power using the Wi-5 mechanism, with user 3 the
only non-joining user, in Figure 5.2b.
5.3 Initialization
In this section we initialize the interference function and the path loss used to define the utility
function, the transmission power control mechanism, and the parameters introduced in Section
4.2 to model this use case.
The utility funtion As defined in Section 4.2, we use the SINR (Definition 4.1) as the utility
function in the Wi-5 model. In this use case we compute Gij using (5.1), see [10], where dij is
the Euclidean distance between the transmitter node of user i ∈ N and the receiver node of user
j ∈ N in km, L is a constant loss, and α is a power density index (generally valued between 2
and 6)[13]:
Gij =
L
dα
ij
, i ∈ N, j ∈ N. (5.1)
Since in the use case we only consider non-overlapping channels, we define the function charac-
terizing the interference between c ∈ C and c ∈ C by:
I(c, c ) =
1 if c = c ,
0 otherwise.
(5.2)
Note that there is either ‘full’ interference experienced between two users, if they transmit on
the same channel, or there is no interference experienced, if they transmit on different channels.
To deal with overlapping channels one could consider an interference function which is not valued
either zero or one, but is valued between zero and one depending on the intensity of the overlap
between channels. For example, as illustrated in Figure 2.2, two users transmitting on channel
1 and channel 2 interfere more with each other than if they would transmit on channel 1 and
channel 4.
We note that, with the interference function as defined for the use case, the SINR does not
take into account the specific channel on which a Wi-Fi user transmits, but it considers the other
28
32. (a) A non-cooperative scenario, with interference.
(b) A cooperative scenario, without interference.
Figure 5.1: Illustration of the use case, 2 appartments.
29
33. (a) A non-cooperative scenario, with interference.
(b) A cooperative scenario, with less interference.
Figure 5.2: Illustration of the use case, 6 appartments.
30
34. users that transmit on the same channel. For example, let there be three users and two channels,
then the scenarios that users 1 and 3 transmit on channel ch1 and user 2 transmits on channel
ch2 and the scenario that user 2 transmits on channel ch1 and users 1 and 3 transmit on channel
ch2 lead to the same utility levels. To compute the number of unique utility profiles we use the
Stirling numbers of the second kind, introduced by Stirling in [34].
Definition 5.1. (Stirling numbers of the second kind) The Stirling number of the second kind
S(n, k) computes the number of partitions of n labelled objects into k non-empty subsets, where
S(n, k) =
1
k!
k
j=0
(−1)k−j k
j
jn
,
with S(n, n) = 1, S(n, 1) = 1, and S(n, 0) = 0.
We use the Stirling number of the second kind, S(n, k), to compute the number of ways n different
Wi-Fi users can be divided over k identical channels. To compute the number of unique utility
profiles in Rn
we sum the Stirling numbers of the second kind:
|C|
k=1 S(n, k). We do this to also
consider the scenarios in which there are frequency channels available on which no Wi-Fi user
transmits.
The transmission power control mechanism Transmission power control is an important
tool in the Wi-5 mechanism to decrease interference. This is because if the transmission power of
the transmitter of a user is decreased, the other users who are transmitting on the same channel
experience less interference.
There are multiple ways to adopt a transmission power control mechanism in the Wi-5 con-
troller. One way is to choose the transmission power of the users that join the Wi-5 mechanism
such that the target SINR (defined in Section 5.2) is exactly reached for all the users. This
concept is considered in [2].
The Telecom providers currently consider the concept in which all users are equal, this is to
avoid customer complaints about unfair treatment. For this use case we consider a mechanism,
in which we prioritize between the users based on their monthly contract fee. To prioritize be-
tween the users we assign the users weights wi (5.4), such that the sum of the weights is equal
to one.
Let Nk
c ⊂ Nc be the set of joining users that have a Wi-Fi connection on channel k ∈ C. Let
(pi)i∈Nk
c
be the transmission powers of the users in Nk
c and let ˜ui((pi)i∈Nk
c
) = ui(c1, p1, . . . , cn, pn)
be the utility function of the user i ∈ Nk
c . The utility function ˜ui is well defined since the joining
users in NcNk
c transmit on a different channel and therefore do not cause any interference to
the users in Nk
c and the it is assumed that the non-joining users who transmit on channel k
transmit at maximal transmission power.
We solve the following optimization problem to determine the transmission power of the users
in Nk
c , ∀k ∈ C:
max
(pi)i∈Nk
c
min
i∈Nk
c
wi ˜ui((pi)i∈Nk
c
) − di, (5.3)
such that for all i ∈ Nk
c : ˜ui((pi)i∈Nk
c
) ≥ γ(mi),
pi ∈ (0, pmax
i ],
31
35. where in (5.3), given the channel selection and transmission power of all users in the model, di
is the expected equilibrium utility of user i ∈ Nk
c in the non-cooperative scenario (Section 4.3).
We subtract the disagreement value to make sure that as much users as possible benefit from
cooperating. We want to guarantee the Wi-5 mechanism to select a feasible system, hence the
boundary condition ˜ui ≥ γ(mi) for all i ∈ Nk
c , for all k ∈ C. If a different channel allocation is
considered for the users, a different transmission power vector may be found.
The weights wi of a joining user i ∈ Nk
c , k ∈ C, are computed by:
wi =
bk
mi
for all i ∈ Nk
c , (5.4)
with bk a constant such that i∈Nk
c
wi = 1. Note that since we determine the weights of users
as in (5.4), we provide users with a high monthly fee with a lower weight. We do this because
we maximize the minimum in (5.3), and in this case the utility of users with a lower weight will
increase more.
If the system is not a feasible system, the transmission powers which lead to the smallest differ-
ence between the utility and the target utility are selected.
5.3.1 Parameter initialization
In this section we initialize the following model parameters:
pmax
i The maximal transmission power of user i ∈ N, in mW.
γ(mi) The target SINR of user i ∈ N, in dB.
n0i
The individual noise factor of user i ∈ N, in mW.
L The constant loss in (5.1).
α The power density index in (5.1).
The maximal transmitting power pmax
i of user i ∈ N, depends on the spectrum band the user is
transmitting on. In this use case we consider only the 2.4 GHz spectrum band. The maximal
transmitting power in Europe is determined by the Code of Federal regulations title 47, part 15
(47 CFR 15). If user i ∈ N, transmits on the 2.4 GHz spectrum, then pmax
i = 100mW.
A Telecom provider provides data at a data speed in megabits per second (Mbps) to the ac-
cess point of a user, the data speed depending on the monthly contract fee. The target SINR of
a user depends on the data speed and therefore the target SINR depends on the user’s monthly
contract fee. A user’s transmission power, if he adopts the Wi-5 mechanism, depends on the
target SINR, as can be seen in Section 5.2.
However, a Wi-Fi user may also have some in-house connections, e.g. a personal cloud (Network
Attached Storage (NAS)) for which he also needs transmission power. The incoming data and
the in-house connections are illustrated in Figure 5.3. Network access is provided by the network
supplier SP. The Defense Switched Network (DSN) line illustrates the actual network and is
connected to the access point of the user.
32
36. Figure 5.3: Illustration of in-house communications.
For a user to be able to use his in-house connections and obtain a sufficiently high data speed
we increase the SINR compared to the situation in which we only consider the data speed
provided by the Telecom provider. We do this by adding 2 Mbps3
to the data speed provided
by the Telecom provider and determine the target SINR for a user from the increased data speed.
It is not straightforward to compute the target SINR from a given data speed. There is no
standard formula for this computation, since it depends on numerous external factors. We will
consider data provided in [33], which can be found in Table 5.1, to convert data speed to a tar-
get Signal-to-Noise Ratio (SNR). The SNR excludes the interference experienced by the Wi-Fi
users, but to correct for this we subtract 100 dB4
from the target SNR to obtain the target SINR.
The data from Table 5.1 is illustrated in Figure 5.4 for the 2.4 GHz and the 5 GHz spectrum.
2.4 GHz 5 GHz 2.4 and 5 GHz
Data speed (Mbps) 1 2 5.5 11 6 9 12 18 24 36 48 54
SNR (dB) 4 6 8 10 4 5 7 9 12 16 20 21
Table 5.1: Relation between data speed (Mbps) and the Signal-to-Noise ratio (dB).
3Personal communication with Mr. M. Djurica, Senior Research Scientist at TNO.
4See footnote 3.
33
37. Figure 5.4: Relation between data speed (Mbps) and the Signal-to-Noise ratio (dB).
If a user in the Netherlands would choose Vodafone as the provider, he has the choice of the fol-
lowing data speeds for the following monthly costs:5
20 Mbps for e23.50 and 50 Mbps for e32.50.
Given the data from [33] and Vodafone, and the interference correction, the target SINR of
a user using the 2.4 GHz spectrum can be defined as follows. We use a polynomial of degree four
to describe the Mbps to SNR conversion and a polynomial of degree two to determine the euro
to Mbps conversion, satisfying that if the monthly contract fee is e0 then 0 Mbps is received.
Furthermore, we use that a SNR of 50 dB provides an optimal service.6
Figure 5.5 illustrates
the relation between the target SNR and the monthly contract fee.
Definition 5.2. (Target Signal-to-Interference-and-Noise Ratio) For a user with monthly con-
tract fee m ≥ 0 in euros, his target Signal-to-Interference-and-Noise Ratio (target SINR) γ(m)
in decibels, is given as follows:
γ(m) = − 100 + max{−0.00001 · 24
+ 0.00016 · 23
− 0.0622 · 22
+ 1.1025 · 2 + 3.4785,
min{−0.00001(z + 2)4
+ 0.00016(z + 2)3
− 0.0622(z + 2)2
+ 1.1025(z + 2) + 3.4785, 40}},
where
z = 0.0764m2
− 0.9438m − 0.00000000000003.
5See https://www.vodafone.nl/shop/vodafone-thuis/internet/?&channel=1_0_SEA_GOO&cmpid=00390c_
|vt_nb_pr_internet_algemeen_high_00390|internet_abonnement_thuis|internet%20abonnement%20thuis||.
6See http://www.wireless-nets.com/resources/tutorials/define_SNR_values.html.
34
38. Since the polynomial that describes the conversion from euros to Mbps has a minimum at 6.18,
we consider the maximum over the target SNR with a monthly contract fee of e0 and the target
SNR with the actual contract fee. We do this because it is not logical that a user who pays
a contract fee between e6.18 and e12.36 would have a lower target SNR than a user with a
contract fee lower than e6.18.
Figure 5.5: Relation between the monthly contract fee (e) and target Signal-to-Noise ratio (dB).
To be consistent with [10] we let n0i = n0 = 2 ∗ 10−13
be the constant individual noise factor of
user i, i ∈ N, in milliwatts (mW). To compute the path loss we use L = 10−11
and α = 2, as
suggested in [10].
35
39. Chapter 6
Illustration of the Wi-5 model
6.1 Introduction
In this chapter we illustrate the Wi-5 model using a two person and a three person example. In
both of these examples we consider the use case ‘the apartment building’ (Chapter 5). We use
MATLAB to compute the transmission powers for the joining users, the utilities, and the Nash
bargaining solutions.
In Section 6.2 we consider a two user example, initialized in Section 6.2.1. We consider the
non-cooperative scenario in Section 6.2.2 and the cooperative scenario in Section 6.2.3. We show
that in this two user example the two users will not cooperate, as is explained in Section 6.2.4.
Furthermore we also discuss that in a general two user case, the two users will not join the Wi-5
mechanism.
In Section 6.3.1 we initialize a three user example. In Section 6.3.2 we consider the non-
cooperative scenario, in Section 6.3.3 we consider the cooperative scenario, and in Section 6.3.4
we consider the mixed scenario. We show that the three users are not willing to join the Wi-5
mechanism, but that two of the three users will join. In Section 6.3.5 we discuss that conditioning
on other scenarios may affect the user’s incentive to join and we discuss three user examples in
general.
6.2 Two person example
6.2.1 The example initialization
In this section we define the two user example in Table 6.1 and sketch the layout of the apartments
in Figure 6.1. We compute the equilibrium strategies in each of the two scenarios: the non-
cooperative and cooperative scenario. Note that in this example, a mixed scenario does not
exist. This is because only scenarios with at least three users contain a mixed scenario, since
cooperation exists only between two or more users and there should be at least one non-joining
player.
36
40. Transmitter coordinates (km) Receiver coordinates (km) Monthly fee (e)
User 1 (p1) 0.008 0.000 0.002 0.005 0.004 0.0015 20
User 2 (p2) 0.012 0.000 0.002 0.017 0.004 0.0015 35
Table 6.1: Data initialization of the two users p1 and p2.
To determine the coordinates of the transmitters and receivers we take the lower, left corner of the
apartment of user p1 to have coordinates (0,0,0). The transmitters are placed in the cupboard,
which is close to the entrance door. The receiver is placed in the middle of the apartment. We
correlate the monthly contract fee of the users to the size of their apartment, a bigger apartment
corresponds to a higher monthly fee.
Figure 6.1: Layout of the apartments of the two users, including the location of the transmitters
and receivers.
Furthermore we assume that there is one frequency channel, ch1. If we would assume that there
are two or three non-overlapping channels, this example would be trivial. This is because in that
case both users can transmit on a different channel, this will be the chosen strategy and each
user will transmit with maximal transmission power.
6.2.2 The non-cooperative scenario
As in Section 4.3, both users transmit at their maximal transmission power in this non-cooperative
scenario. The utilities of the two users in the one possible channel allocation can be found in
Table 6.2.
Channel selection Transmission power (mW) Utility (SINR) (dB)
p1 p2 p1 & p2 p1 p2
ch1 ch1 100 0.0147 -4.3681
Table 6.2: Transmission power and utility in the two-user non-cooperative scenario.
Since there is only one possible channel allocation, we find that in the non-cooperative scenario
the users have the following utilities:
p1 : 0.0147,
p2 : −4.3681.
(6.1)
37
41. It is in this scenario logical that user p1 obtains a higher SINR. This is because the distance
between R1 and T1 is smaller than the distance between R2 and T2, and this distance is the
only parameter differentiating p1 from p2 in this non-cooperative scenario.
6.2.3 The cooperative scenario
In this scenario the users join the Wi-5 mechanism, and they transmit with the transmission
power computed by the controller, as described in Section 5.3. To compute the transmission
powers we set the minimal transmission power to 0.01 mW, otherwise it is possible that a user
gets a transmission power of 0 mW. The transmission powers and utilities of the two users can
be found in Table 6.3.
Channel selection Transmission power (mW) Utility (SINR) (dB)
p1 p2 p1 p2 p1 p2
ch1 ch1 17.9542 9.4549 2.7997 -7.1532
Table 6.3: Transmission power and utility in the 2-user cooperative scenario.
To check whether the transmission powers are computed correctly, we compute the value of
the objective function (5.3) for both of the users and check if the minimum value is maxi-
mized. In this example, the minimal objective value is maximized in the case user p1 and p2
have the same objective value. Otherwise, when the values are not equal, it is possible to in-
crease the transmission power of the user with the lower objective value such that it increases
and the values of the two users level out. It is possible to increase the transmission power of
both the users, since they do not transmit with maximal transmission power. We have that
c
20 ·2.7997−0.0147 = c
35 ·(−7.1532)+4.3681 = 1.766, with c = 12 8
11 . So the transmission powers
are computed properly.
Since there is only one possible channel allocation, we find that in the cooperative scenario
the users have the following utilities:
p1 : 2.7997,
p2 : −7.1532.
(6.2)
Note that even though user p2 has a higher monthly contract fee than user p1, his utility remains
lower than the utility of user p1. This can be explained by using the low disagreement value of
user p2, which provides him with a disadvantage over p1.
6.2.4 Discussion
Comparing the utility levels in the non-cooperative scenario and the cooperative scenario, (6.1)
and (6.2), respectively, we conclude that users p1 and p2 decide not to join the Wi-5 mechanism.
This is because it is not rational for user p2 to join the Wi-5 mechanism, since his utility is lower
in the cooperative scenario than in the non-cooperative scenario.
Considering two user examples in general, we find that the two users do not have an incen-
tive to join the Wi-5 mechanism. This is because in the case that the SINR of one of the users
increases when the users join the Wi-5 mechanism, his transmission power increases compared
to the transmission power of the other user. This will always decrease the SINR of the other
user, which makes it not rational for him to cooperate. In the case that the two users have
38
42. the same monthly contract fee and the distances between the four transmitter-receiver pairs are
the same, the two users are indifferent to joining the Wi-5 mechanism. This is because if the
users join the Wi-5 mechanism, their utility does not change since the controller will not lower
their transmission power. Since if the transmission power is lowered, the nominator in the SINR
decreases relatively faster than the denominator due to the added individual noise.
6.3 Three person example
6.3.1 The example initialization
In this section we define the three-user example in Table 6.4 and sketch the layout of the apart-
ments in Figure 6.2. We compute the equilibrium strategies in each of the three scenarios: the
non-cooperative, cooperative, and mixed scenario.
Transmitter coordinates (km) Receiver coordinates (km) Monthly fee (e)
User 1 (p1) 0.007 0.007 0.002 0.0045 0.012 0.0015 60
User 2 (p2) 0.006 0.003 0.002 0.011 0.004 0.0015 35
User 3 (p3) 0.004 0.0015 0.002 0.002 0.0035 0.0015 20
Table 6.4: Data initialization of the three users p1, p2, and p3.
To determine the coordinates of the transmitters and receivers we take the lower, left corner of the
apartment of user p3 to have coordinates (0,0,0). The transmitters are placed in the cupboard,
which is close to the entrance door. The receiver is placed in the middle of the apartment. We
correlate the monthly contract fee of the users to the size of their apartment, a bigger apartment
corresponds to a higher monthly fee.
Figure 6.2: Layout of the apartments of the three users, including the location of the transmitters
and receivers.
39
43. Furthermore we assume that there are two non-overlapping frequency channels, ch1 and ch2. If
we would assume that there are three non-overlapping channels, this example would be trivial.
This is because in that case all three users can transmit on a different channel, this will be the
chosen strategy and each user will transmit with maximal transmission power.
6.3.2 The non-cooperative scenario
As in Section 4.3, all users transmit at their maximal transmission power in this non-cooperative
scenario. The utilities of the three users in each of the eight channel allocations can be found in
Table 6.5.
Channel selection Transmission power (mW) Utility (SINR) (dB)
p1 p2 p3 p1, p2 & p3 p1 p2 p3
ch1 ch1 ch1 100 1.7936 -1.7972 1.4267
ch1 ch1 ch2 100 4.2338 -0.1687 87.8252
ch1 ch2 ch1 100 5.4603 82.7984 6.5758
ch1 ch2 ch2 100 82.0066 3.2516 3.0103
ch2 ch1 ch1 100 82.0066 3.2516 3.0103
ch2 ch1 ch2 100 5.4603 82.7984 6.5758
ch2 ch2 ch1 100 4.2338 -0.1687 87.8252
ch2 ch2 ch2 100 1.7936 -1.7972 1.4267
Table 6.5: Transmission power and utility in the 3-user non-cooperative scenario.
To find an equilibrium strategy profile in this non-cooperative example, we compute a mixed
Nash equilibrium. As described in Section 4.3, we consider the utility profiles to compute the
equilibrium. We compute a Nash equilibrium in the following paragraph. This leads to the
following expected utilities:
p1 : 5.4603,
p2 : 82.7984,
p3 : 6.5758.
(6.3)
Determining a Nash equilibrium To compute a mixed Nash equilibrium, in this case a pure
equilibrium, we consider the four different utility profiles, the number which can be computed
using the Stirling numbers of the second kind:
S(3, 1) + S(3, 2) = 1 +
1
2
(0 − 2 + 8) = 4.
We consider the utilities in Figure 6.3.
40
44. p3
p1
ch1 ch2
p2
ch1 (1.79,-1.79,1.43) (82.00,3.25,3.010)
ch2 (5.46,82.80,6.58) (4.23,-0.17,87.83)
p1
ch1 ch2
p2
ch1 (4.23,-0.17,87.83) (5.46,82.80,6.58)
ch2 (82.00,3.25,3.010) (1.79,-1.79,1.43)
ch1 ch2
Figure 6.3: Utility levels in the non-cooperative scenario.
In Table 6.6 we describe the strategy decisions a user, p1, p2 or p3, makes if he knows the
strategy decisions of the other two users. We underline these decisions in Figure 6.3.
Using Table 6.3, we see that there exist two pure Nash strategies, namely
p1 : ch1
p2 : ch2
p3 : ch1
, and
p1 : ch2
p2 : ch1
p3 : ch2
.
Both of these strategies lead to the utility levels as in (6.3). These utility levels are the equilibrium
utility levels in the non-cooperative scenario.
Strategy decision of p1 and p2: p3 will transmit on channel:
p1 p2 p3
ch1 ch1 ch2
ch1 ch2 ch1
ch2 ch1 ch2
ch2 ch2 ch1
Strategy decision of p1 and p3: p2 will transmit on channel:
p1 p3 p2
ch1 ch1 ch2
ch1 ch2 ch2
ch2 ch1 ch1
ch2 ch2 ch1
Strategy decision of p2 and p3: p1 will transmit on channel:
p2 p3 p1
ch1 ch1 ch2
ch1 ch2 ch2
ch2 ch1 ch1
ch2 ch2 ch1
Table 6.6: Strategy choices per user, given the strategy of the other two users.
6.3.3 The cooperative scenario
In the cooperative scenario the users join the Wi-5 mechanism, and they transmit with the
transmission power computed by the controller, as described in Section 5.3. To compute the
transmission powers we set the minimal transmission power to 0.01 mW, otherwise it is possible
41
45. that a user gets a transmission power of 0 mW. The transmission powers and utilities of the
three users in each of the channel allocations can be found in Table 6.7.
Channel selection Transmission power (mW) Utility (SINR) (dB)
p1 p2 p3 p1 p2 p3 p1 p2 p3
ch1 ch1 ch1 0.01 100.00 0.03 -35.7673 35.8915 -31.8749
ch1 ch1 ch2 0.01 100.00 100.00 -35.7662 39.8311 87.8252
ch1 ch2 ch1 0.89 100.00 0.51 7.9116 82.7984 4.1245
ch1 ch2 ch2 100.00 100.00 0.01 82.0066 43.2512 -36.9897
ch2 ch1 ch1 100.00 100.00 0.01 82.0066 43.2512 -36.9897
ch2 ch1 ch2 0.89 100.00 0.51 7.9116 82.7984 4.1245
ch2 ch2 ch1 0.01 100.00 100.00 -35.7662 39.8311 87.8252
ch2 ch2 ch2 0.01 100.00 0.03 -35.7673 35.8915 -31.8749
Table 6.7: Transmission power and utility in the 3-user cooperative scenario.
In Table 6.7, we see that user p2 will always transmit with maximal transmission power. This
can be explained by considering the equilibrium utilities obtained in the non-cooperative scenario
(6.3). The equilibrium utility of user 2 is a lot higher than the utility of the other two users,
which is used in the objective function (5.3). Therefore we see that the utilities of users trans-
mitting on the same channel as user p2 (Table 6.5 and 6.7) are smaller in the cooperative scenario.
In this scenario we bargain over the convex set Conv({(−35.7673, 35.8915, −31.8749), (−35.7662,
39.8311, 87.8252), (7.9116, 82.7984, 4.1245), (82.0066, 43.2512, −36.9897)}) with disagreement point
(6.3). The solution of the Nash bargaining rule is the disagreement point (6.3) and the three
users will not cooperate. This can be explained by considering the utilities in Table 6.7. The only
alternative user p2 will agree upon is the alternative in which he transmits alone on a channel
and obtains the utility 82.7984. The utilities of the other two users are in this case: p1: 7.9116
and p3: 4.1245. It is not rational for user p3 to accept this alternative, since in the non-cooperate
scenario he obtains a utility of 6.5758. Therefore the solution of the Nash bargaining rule is the
disagreement point (6.3).
6.3.4 The mixed scenario
We have seen in Section 6.3.3 that the three users are not willing to cooperate. In this section we
consider the scenario in which users p1 and p3 are willing to join, and p2 is not willing to join.
We consider this scenario since p1 and p3 have the most to gain from cooperation, considering
that in the equilibrium strategies in the non-cooperative scenario they transmit on the same
channel. To compute the transmission powers of p1 and p3 we set their minimal transmission
power to 0.01 mW, otherwise it is possible that a user gets a transmission power of 0 mW. The
transmission powers and utilities of the three users can be found in Table 6.8.
42
46. Channel selection Transmission power (mW) Utility (SINR) (dB)
p1 p2 p3 p1 p2 p3 p1 p2 p3
ch1 ch1 ch1 78.10 100.00 100.00 0.7203 1.7274 -1.0888
ch1 ch1 ch2 100.00 100.00 100.00 4.2338 -0.1687 87.8252
ch1 ch2 ch1 0.89 100.00 0.51 7.9116 82.7984 4.1245
ch1 ch2 ch2 100.00 100.00 100.00 82.0066 3.2516 3.0103
ch2 ch1 ch1 100.00 100.00 100.00 82.0066 3.2516 3.0103
ch2 ch1 ch2 0.89 100.00 0.51 7.9116 82.7984 4.1245
ch2 ch2 ch1 100.00 100.00 100.00 4.2338 -0.1687 87.8252
ch2 ch2 ch2 78.10 100.00 100.00 0.7203 1.7274 -1.0888
Table 6.8: Transmission power and utility in the 3-user mixed scenario, p2 is a non-joining user.
We see in Table 6.8 that when p1 and p3 transmit on a different channel, their transmission
powers and utilities are equal as in the non-cooperative scenario (see Table 6.5). This is because
if a user transmits alone on a channel, he will transmit with maximal transmission power. Since
in this mixed scenario user p2 is not considered by the Wi-5 controller, he assigns user p1 and p3
maximal transmission power. Furthermore, in the case that user p1 and p3 transmit on the same
channel, and user p2 transmits on the other channel, we see that their utilities are the same as
in the cooperative scenario (see Table 6.7). This is because all three users act the same in both
of the scenarios.
In this scenario we bargain for the users p1 and p3 over the convex set Conv({(0.7203, −1.0888),
(4.2338, 87.8252), (7.9116, 4.1245), (82.0066, 3.0103)}) with disagreement point (5.4603, 6.5758).
The Nash bargaining solution is:
p1 : 42.0987,
p3 : 46.5317.
(6.4)
Comparing (6.4) to the disagreement point, we conclude that it is rational for both users p1 and
p3 to cooperate since their expected utilities increase. Therefore they will join the Wi-5 mecha-
nism. The mixed channel allocation for users p1 and p3 corresponding to the utility profile (6.4)
is: with probability 0.49 user p1 transmits alone on a channel and with probability 0.51 user p3
transmits alone on a channel. This means for user p2, that his expected utility in this scenario
is 0.49 · 3.2516 + 0.51 · −0.1687 = 1.4965. Recall that we can compute the expected utility of
user p2 in this way because the Wi-5 controller is able to respond in real time (immediate) to a
change in the strategy of user p2.
We find that in this mixed scenario, in which users p1 and p3 will join the Wi-5 mechanism, that
the expected utilities of the users are as follows:
p1 : 42.0987,
p2 : 1.4965,
p3 : 46.5317.
(6.5)
6.3.5 Discussion
In Section 6.3.3 we show that the three users do not all have an incentive to join the Wi-5
mechanism. This is because it would not be rational for at least one of the users. In the case
that an expected utility profile is rational for user p2, then the expected utility profile is not
43
47. rational for user p3 and vice versa. In Section 6.3.4 we see that users p1 and p3 do have an in-
centive to join the Wi-5 mechanism. But this will negatively effect the expected utility of user p2.
Because the expected utility of p2 decreases in the case that p1 and p3 join the Wi-5 mech-
anism, it could be rational for user p2 to join the Wi-5 mechanism the next opportunity he gets.
To research if p2 is willing to join the Wi-5 mechanism, we can compute the Nash bargaining
solution as in the cooperative scenario conditioned on the fact that users p1 and p3 have al-
ready joined the Wi-5 mechanism. To do this we use (6.5) as the disagreement point. The Nash
bargaining solution in this conditioned cooperative scenario is:
p1 : 42.6017,
p2 : 2.0434,
p3 : 47.0814.
(6.6)
The expected utility of all users increases compared to (6.5), which could potentially indicate
that all users join the Wi-5 mechanism.
Considering a three user example in general, we find that the three users do not all have an
incentive to join the Wi-5 mechanism conditioned on the non-cooperative scenario in which none
of the users join the Wi-5 mechanism. This is, due to the same reasoning as in Section 6.2.4,
because one user always obtains a lower expected utility if one or two of the other users obtain
a higher utility.
In the case that two users in a general three user example have the incentive to join the Wi-5
mechanism, the third user could have an incentive to join the Wi-5 mechanism conditioned on
the fact that the other two users join the Wi-5 mechanism. It is possible to investigate this using
the cooperative scenario, conditioning on the expected utility profile in the situation that the
two users join the Wi-5 mechanism. Conditioning on scenarios other then the non-cooperative
scenario can be seen as iteratively using the Wi-5 model.
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48. Chapter 7
Conclusion and future work
7.1 Conclusion
In this thesis we propose a game theoretic framework in which we combine the non-cooperative
Nash equilibrium and the cooperative Nash bargaining solution. The framework can be used
to model a player’s incentive to cooperate in various settings. In this thesis we use this frame-
work to research whether Wi-Fi users have an incentive to coordinate their channel selection and
transmission power by joining a controller-operated spectrum management mechanism, named
the Wi-5 mechanism. We name the framework, initialized to model spectrum management in
the Wi-5 mechanism, the Wi-5 model.
The aim in this thesis is to find a building block that can be used to compute the ratio of
Wi-Fi users that have an incentive to join the Wi-5 mechanism. The Wi-5 model can be used
to compute the number of users that are willing to cooperate versus the number of users that is
not willing to cooperate. We consider a use case, named ‘the apartment building’. The use case
defines a small area with densely populated Wi-Fi users, all of whom transmit continuously.
Considering the use case, we show that if there are only two Wi-Fi users, the users do not
have an incentive to join the Wi-5 mechanism. This is independent of the location of their
transmitter and receiver and the monthly contract fee they pay for their internet plan. In the
case that there are three Wi-Fi users, the incentive of the users to join the Wi-5 mechanism
depends on the location of the receiver and transmitter of the users and their monthly contract
fee. We demonstrate that at most two of the three Wi-Fi users have an incentive to join the
Wi-5 mechanism in the current scenario, in which none of the users join the Wi-5 mechanism. It
remains a point of discussion whether a Wi-Fi user has a different incentive if he considers other
conditional scenarios.
We have made significant first steps as to answering the question: ‘What is the ratio of Wi-
Fi users willing to join the Wi-5 mechanism?’. Given our findings from the three user example,
we are confident that there will be users that have the incentive to join the Wi-5 mechanism
under certain conditions. The Wi-5 model can be used to fully answer the research question, but
in order to do this, additional steps need to be made.
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