=Paper=
{{Paper
|id=None
|storemode=property
|title=Should I quit using my resource? Modeling Resource Usage through Game Theory
|pdfUrl=https://ceur-ws.org/Vol-862/REDp3.pdf
|volume=Vol-862
}}
==Should I quit using my resource? Modeling Resource Usage through Game Theory==
https://ceur-ws.org/Vol-862/REDp3.pdf
Should I quit using my resource?
Modeling Resource Usage through Game Theory
Paraskevas V. Lekeas
Department of Applied Mathematics,
University of Crete, Crete, Greece
plekeas@gmail.com
Abstract. Existing web infrastructures have supported the publication
of a tremendous amount of resources, and over the past few years Data
Resource Usage has become an everyday task for millions of users all
over the world. In this work we model Resource Usage as a Cooperative
Cournot Game in which a resource user and the various resource services
are engaged. We give quantified answers as to when it is of interest for
the user to stop using part of a resource and to switch to a different one.
Moreover, we do the same from the perspective of a resource’s provider.
Keywords: Resource Usage, Cournot Competition, Game, Core
1 Introduction
Data Resource1 Usage is an everyday task for millions of users all over the world.
Exchanging information, communicating, working and various other aspects of
our life have been inevitably affected by data repositories which can be accessed
through various channels, such as the Web and the Internet via different tech-
nologies, interfaces and infrastructures [1]. Usually once someone has identified
an appropriate resource, he interacts with it by exchanging information. This
sort of interaction is heavily commercialized, and a huge industry2 has been
established, which invests a great amount of money in marketing web services
and products that provide access to resources quite often freely. This is why
from now on we will use the word ”provider” to refer to the underlying struc-
ture responsible for a resource. These providers most of the time are extremely
interested in developing integrated resource services3 in order to attract users,
and, more importantly, to convince them to keep using these. This is because
users are valuable: They provide information to the resource by interacting with
1
When we refer to a resource we have in mind that in the background there exists a
set of electronic mechanisms or internet infrastructures that created this resource for
the purpose of value generation (either a profit when there is an underlying company
or some other social gain, such as [2]). See also [3] for a related taxonomy.
2
End-user spending for IT services worldwide estimated to be $763 billion in 2009 [4].
3
We prefer this term instead of the term ”web service” since many other alternative
channels exist like satellite and cellphone grids, ad hoc networks, etc.
81
it, they bring money through the adds or subscription fees, and of course they
bring new users to broaden the profit cycle.
A living example is Google which proposes a web resource experience through
the integration of different technologies in order for users to continue using its
services. Opening a Google account is a fairly easy one-minute process and in-
stantaneously the new user has access to different cutting-edge services like An-
droid OS apps, adaptive web search through the Google search engine, cloud
services, access to landline phone calls, teleconferencing services and much more.
A ”perfect” user of Google would be the one who uses all these services explicitly
through the Google APIs, sharing no data with any other competitive resource
(e.g. AWS [5] or Ubuntu One [6]) and thus enriching only Google’s resource
knowledge repositories. However, many times it is the case that not all services
or technologies of a resource are welcomed by users and sometimes users tend
to accept only specific services from a resource ignoring some others. Also a sit-
uation that is not so good for a provider is the case where users decide to quit
its resource and switch to a different one that provides similar or better services
[7].
In this work we investigate the following problem. When do users tend to
partially4 abandon a specific resource? What can resource providers do about
that? Is there a way to formulate the above trends in order to be evaluated and
measured? In order to approach the above questions we model the various user
- service interactions within a resource with different plays of the user, which
are engaged either in a cooperative or in a non-cooperative manner. Each of
these plays generates a value, which is to be conceived as a measure of the user’s
satisfaction for the appropriate service.
In the rest of the paper we proceed as follows: Section 2 gives a motivating
example and formulates Resource Usage as a cooperative Cournot game. Section
3 studies the cases of partial rejection of a resource and also the possible reactions
of the provider to prevent that. Section 4 concludes with a discussion and future
work.
2 Modeling Resource Usage
Before describing our model let us give a stimulating example.
2.1 The case of Zoogle+ resource
Imagine the following scenario5 : Zoogle Inc. decides to offer a new web integrated
data resource, named Zoogle+ that will provide its users with the following set of
4
Partially means that the user is unsatisfied only with some of the services and wants
to switch but likes the rest and wants to keep them.
5
Any explicit or implicit references to facts or persons is accidental and imaginary.
Beware also that in Greek, Zoogle is pronounced almost the same as the Greek
word ”” which means Jungle referring to the chaotic and controversial informational
nature of the WWW.
82
services: N = {email, cloud, voip}. A user u decides to try Zoogle+ for a certain
period of time, and for this reason he signs up creating an account. Since u wants
to be accurate in his calculations he uses a worth function v(·) to rate how good
his experience is. Since it is Zoogle+’s policy to prohibit the exclusive use of
only one service, ignoring the rest, the single use of a service for u is worth 0,
i.e. v({i}) = 0, ∀ i ∈ N . Moreover, when u uses any two of the services he rates
his experience with a value of 2, i.e. v({i, j}) = 2, ∀ i, j ∈ N with i 6= j. Finally,
when using all three of the services he calculates a value of 2.5, i.e. v(N ) = 2.5.
What should the user do? Should he be totally loyal to Zoogle+ or not?
On the one hand, when u uses all the services he gets on average 0.83 for
each one, but, on the other hand, when he selects only two of the services he
gets 0 for the one not selected and 1 for each of the rest. Therefore, u decides to
maximize his satisfaction and thus selects to use only two services of Zoogle+
and to seek the third in an external resource. So the answer in this case is that u
will partially leave. What would happen if v(N ) is worth 3 to u? Clearly then he
should stay loyal to Zoogle+ because he would maximize his satisfaction in this
case, since there is no combination of services that would give him more than 1
(on average).
Underlying the intuition of the above example is the idea of the core in
cooperative game theory [8]. Briefly speaking, a cooperative game is a situation
in which a group of N players, by making decisions that take into account each
other’s actions and responses, decides to act together to generate some value.
This value (or what is left if we put aside the costs of playing the game) must
later be split according to certain rules agreed upon by these players. The various
ways this split can be made are defined as solution concepts. One of the most
used solution concepts is that of the core. According to this, the split is in the
core if no subset of players can benefit (earn more) by breaking away from the
whole set N . Under this light, user u decided to partially leave Zoogle+ when
v(N ) = 2.5 because the core of the game was empty, while when v(N ) = 3 the
core became non-empty and thus u used all the services. In what follows we
formulate Resource Usage as a cooperative Cournot game.
2.2 The model
Suppose that a user u decides to sign up for a resource R in order to use its
i, i ∈ {1, · · · , n}, different services. The phrase ”use a service” refers to the
interaction of u with the service in order for some desired tasks to be completed.
For example, using the GUI of a service, entering data by typing, downloading
files, writing scripts and compiling them online can be perceived as parts of such
an interaction. Let pi denote the interaction of u with service i. Call each pi a
play that u does with the service. Every play6 generates some value for u. Since
u needs to make an effort to generate this value we assume that the per unit cost
to u for playing pi is c. For example, if a play generates 2 units of value for u,
then the cost of the play would be 2c. In general the cost function for qi units of
6
from now on the terms ”play” and ”interaction” will be used interchangeably.
83
value produced in a play would be cqi . We assume for clarity of the exposition
that c is the same for every pi , i = 1, · · · , n. If we now take the value generated
from a play and subtract the cost spent to produce it, then we will find how
much the play is worth to the user, or in other words we will find the profit of
the user from using the specific service, which intuitively represents a measure
of how satisfied the user is with the service. So each play contributes some worth
to the user and if we add all these contributions from all the plays we will have
the total worth to the user u from using all the services of R.
As is generally accepted, maximizing user satisfaction constitutes the key
issue to every service. This forces every play pi to seek to maximize its contri-
bution to the total worth earned by u. But this happens under the following
restriction. The user u has a limited time to spend interacting with the services
and thus he must split this into his needs wisely in order to acomplish his differ-
ent tasks through R. This means that no play can monopolize all the available
time of u. Moreover, u’s multitasking abilities are limited by nature. So spend-
ing more time on service i might, on the one hand raise satisfaction from i but
on the other might lower satisfaction from service j (j 6= i). A mathematical
model that describes such interactions among pi ’s is the Cournot competition
[9]. Under this model it is assumed that the plays do not cooperate with each
other but instead decide independently and at the same time about how much
value they should produce for u. Thus if with πi we declare the profit of each pi
we will have that:
πi = max(value generated from i-th play − cost spent) (1)
i
Assume that u in play i creates qi , i = 1, · · · , n units of value. In economics
usually the function that describes the total value generated from the i-th play
Pn
in its simplest form is given by (a − qj )qi . Here a is a positive constant
j=1
that represents the size of the environment into which interactions happen. For
our model, a could e.g. represent the total size of data held by the resource
available for use or the total time that u wants to dedicate to R. If, for example,
a represents the time available, then the demand function intuitively says that
the more value is generated by all the plays the less time remains to be used and
vice versa. Since now the cost spent for i-th play is cqi using the above in (1) we
will have:
n
X
πi = max(a − qj − c)qi , i = 1, · · · , n (2)
i
j=1
The solutions of (2) are the Nash equilibria of the plays of u in the case that
these plays do not cooperate. These equilibria will help us find the worth of u
and reason about his loyalty to R. It is easy to prove (the proof can be found in
the Appendix, or for a more general case, in [9]) that the solutions of (2) give
84
the following worth for each play pi in the case that these act independently
(non-cooperatively):
2
a−c
πi = (3)
n+1
Assume now that all the pi ’s decide to act in a cooperative manner. In this
case due to symmetry we can imagine all the plays combined together into a
unique play, so (2) collapses into a single equation the maximization of which
gives:
(a − c)2
v(N ) ≡ (4)
4
where with v(N ) we denote the total worth produced when all plays cooper-
ate. Before applying our model to quantify u’s loyalty to R let us discuss what
cooperation and non-cooperation means for the plays pi . When we say that a
set of plays cooperate, we mean that they do not harm each other but instead
act as a unity to produce a common value. On the other hand, when there is
non-cooperation each play does not care about the other plays but wants only
to maximize its own value. Since this idea might sound a bit subtle we give the
following example: In the case of Zoogle+ if the 3 services acted in a cooperative
2
manner then their total worth according to (4) would be (a−c) 4 . If, on the other
hand, they acted in a non-cooperative manner, from (3) each would be worth
2 2 2
a−c
3+1 and their total worth would be 3 a−c 4 < (a−c)
4 . So it is clear that
when under cooperation the plays produce more.
3 User Loyalty
Let us now turn our attention to the loyalty of u to R. As said earlier pi ’s can act
in a cooperative or non-cooperative manner in order to achieve their goals. Apart
from total cooperation and non-cooperation there exist intermediate situations
in which some of the plays might decide to cooperate and some others will
decide to deviate (non-cooperate). It is exactly these cases that will shed some
light on u’s loyalty to R. Consider the following scenario: User u is thinking
of abandoning a non-empty set S ⊂ N of services because he discovered that
on a different resource R0 he earns v(S) from these. In order to make his final
decisions he first reasons as follows: ”If I stay loyal to R, my total worth from
(4) is v(N ) and on average I earn v(N )
n . On the other hand, if I partially switch
v(S)
to R0 I would earn on average s , where |S| = s. So I partially switch to R0
if v(S)
s >
v(N )
n .” The idea just described describes the notion of the core in the
cooperative game played by pi , i = 1, · · · , n. We say that the core is non-empty
when for any non-empty set of services chosen, u on average earns less than if
all the services were chosen. In other words, the core is non-empty when:
85
v(S) v(N )
≤ , ∀S ⊂ N (5)
s n
So finally the user partially deviates from R if he discovers at least one set
∗
of services S ∗ that violates (5). For this set we will have that v(S
s∗
)
> v(N
n
)
⇒
∗ 2
v(S ∗ ) > s (a−c)
4n . In this case u would have an incentive to partially leave R.
3.1 Resource provider’s view
Our approach is not only valuable to users but also to a resource provider. This
is because under our model the provider can adopt some strategies to fight a
potential rejection of a user. Using equation (5) we see that the provider must
try to keep the core of the game non-empty. So first of all the provider must
decide to provide its services in a cooperative way. We can observe this trend in
many resource providers nowadays. For example, Google recently (March 2012)
unified its services in order to act cooperatively. In this way there is the trend
that any two services will cooperate in order to produce a common value. Thus
through Gmail you can view or process your attachments through GoogleDocs
or use the Dashboard to synchronize all your e-mail contacts with your Android
device. In this manner Google tries to keep its core non-empty and thus give
incentives to users for more satisfaction. Of course there is always the case of
a user using the services in a non-cooperative manner, but then from (3) he
2
would earn less. Moreover, the provider should try to increase the ratio (a−c) 4n .
This can be done in the following ways: First, the provider can help u to reduce
his cost c. This can be achieved in many ways, e.g. by upgrading its hardware,
by hiring a qualified service [10], by adopting process completeness strategies
[11] or by improving the service tutorials and introducing online help desks [12,
13]. Second, it might consider increasing the factor a, which as we said might
represent the size of data or the time available. For example, a social network
might strive to attract more users thus increasing a so that the current users
will belong to a bigger society and become more satisfied, resulting in more
options for interaction. The same idea also applies to the critical mass of Service
Overlay Networks [14]. Finally, it might consider reducing the number of services
it provides (reducing the denominator of (5) increases the fraction), for example
by obsoleting outdated services or not so popular ones.
Another important factor for a provider is to collect user rating information
data for its services so as to compute its own estimations of how satisfied the
users are. According to (5) the closer the provider’s estimation vprovider (S) is
to the user’s one vuser (S), the more an effective strategy can be adopted to
maintain its customers, and this is because in this way the provider has a clear
image of what its users like. Also providers should follow user trends to estimate
the potential S ∗ ’s that make its resource vulnerable either by asking for feedback
from the users or by outsourcing this task to experts [15, 16]. Moreover, more
complex scenarios can be adopted from the provider’s point of view in order to
86
further refine its strategies. For this the provider can design more complicated
games into which users would engage themselves. For example, the provider will
not only consider how to satisfy the user but also how to earn more money, so
instead of having the players compete in order to find the equilibria between value
produced and cost spent we could have players competing between satisfaction
offered and money earned and cost spent.
4 Discussion and future work
As stated previously, the value generated by the players and the cost spent are
related to the user’s loyalty to the resource. But can the above be calculated by
the user? After all, there are many controversial metrics that can be used for
rating. For example, user u in the Zoogle+ example might have various concerns
when using the resource: how user friendly are Zoogle+’s interfaces? How much
storage space am I allowed? What are the privacy policies of Zoogle+? Do most
of my friends belong to Zoogle+ too? And there are even more. An approach to
these concerns is the following: On the one hand, the value generated through a
user - service interaction must be perceived as a combination A) of the amount
of information in the form of data created, exchanged, stored, or retrieved by
the user, and B) of the user’s personal metrics. For example one such metric is
the one we adopted as a cost factor, i.e. the amount of the user’s time invested
to produce the value through interaction (programming, typing, asking queries,
etc.). And this is something natural, but other functions can be used too, such
as money spent by the user, bandwidth or CPU resources used, or a combination
of the above. One can consult [17] and most of the references therein for a recent
treatment of QoS properties and measurement metrics of services.
In our cooperative game we used as a notion of fairness one in which the
value generated by the players must split equally among them. This is called
a game with transferable utilities since user u makes the decision based on the
average value calculated by all the plays. This means that the transfering of profit
between any two services x, y is allowed, so as to maintain the same average. But
since there exist many different notions of fairness such as the Shapley Value, it
is of particular interest to extend the analysis to these notions as well.
Finally, since under our model we assumed that each service is somewhat of
the same nature as every other service, in a more realistic scenario in which the
services differ, we could have used a differentiation parameter γ and the demand
P
n
from service i would change to (a − qi − γ qj ), thus resulting in a more
j=1,j6=i
complex worth function. This case is a subject of ongoing work.
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Appendix
We prove below the Nash equilibria for the game described in section 2.2. The
first order derivative conditions of (2) ∀i, i = 1, · · · , n give:
P
n
n n a−c− qj
∂ X X j=1,j6=i
[(a − qj − c)qi ] = a − qj − c − 2qi = 0 ⇒ qi =
∂qi j=1
2
j=1,j6=i
The system gives: q̃ ≡ q1 = · · · = qn , thus q̃ = a−c−(n−1)q̃
2
a−c
or q̃ = n+1 . Now
plugging q̃ in (2) gives the result.
88