1. technology and computing
2. internet technology
3. isps

Ann. Telecommun. (2010) 65:171–188
DOI 10.1007/s12243-009-0133-y
How to price Internet access for disloyal users
under uncertainty
Tuan Anh Trinh · László Gyarmati
Received: 18 November 2008 / Accepted: 7 October 2009 / Published online: 17 October 2009
© Institut TELECOM and Springer-Verlag 2009
Abstract It has recently (Trinh 2008; Biczók et al. 2008)
been demonstrated that customer loyalty can have a
significant impact on Internet service provider (ISP)
pricing. However, the results in those works are valid
only under the assumption of complete information,
i.e., both the ISPs and the customers fully know about
each others’ decisions; the question of how Internet access prices are set by the ISP for disloyal users in uncertain circumstances is still largely unsolved. In this paper,
we provide a game-theoretic framework to understand
the impacts of customer loyalty on ISP price setting
under uncertainty. The contribution of the paper is
threefold. Firstly, we provide an empirical analysis of
the customer loyalty issue by carrying out a survey for
the Hungarian ISP market and combine the results with
other European ISP markets. Secondly, we model ISPs’
uncertain decisions by using Bayesian games. Based on
our game theoretic model, we quantify the effects of
uncertainty on ISPs’ price setting and derive strategies
to optimize ISPs’ profits under these uncertain conditions. After that, we generalize the results to mixed
strategy scenarios. Finally, we develop a simulation tool
to validate the theoretical results and to demonstrate
our novel loyalty models. We argue that our findings
can motivate researchers to incorporate a finer-grained
T. A. Trinh · L. Gyarmati (B)
High Speed Networks Laboratory,
Department of Telecommunication and Media Informatics,
Budapest University of Technology and Economics,
2 Magyar tudósok körútja, Budapest, 1117, Hungary
e-mail: gyarmati@tmit.bme.hu
T. A. Trinh
e-mail: trinh@tmit.bme.hu
user behavior model involving customer loyalty in their
investigations of such interactions.
Keywords Socio-economic issues · Cost and pricing ·
User behavior · Internet access pricing · Game
theoretic analysis · Uncertainty management
1 Introduction
During recent years, we have witnessed a significant
change in networking paradigms: from network-centric
to user-centric networking [3]. In terms of “quality,”
while technical quality-of-service parameters are still
important, user perceptions and expectations, in other
words, quality of experience (QoE), now increasingly
attract the attention of manufacturers, operators, and
researchers alike. In addition, user behavior also has
a significant impact on the design of next-generation
network architectures, as well as creating profitable
services running them.
Furthermore, the economic interactions among service providers of different levels and end-users have
been in the focus of interest for several years. These
interactions will continue to get special attention, since
initiatives like the NSF FIND [4] and Euro-NF [5]
promote economic incentives as a first-order concern in
future network design. Also, decision-makers trying to
work out a plausible solution for the recently surfaced
net neutrality debate would greatly benefit from an indepth understanding of economic processes inside the
user–Internet service provider (ISP) hierarchy. There
is broad literature in the area of modeling interactions
between ISPs with game-theoretical means [6–8]. While
these papers introduce and analyze complex models for
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the interaction of ISPs at different levels of hierarchy,
they mostly assume a very simple user behavior model
when investigating the market for local ISPs: end-users
choose the cheapest provider assuming that the quality
of certain services is the same. This assumption may be
plausible in certain scenarios, but it could be misleading
if there are loyal customer segments present in the market. A vivid example of customer loyalty in practice is
the loyalty contract between service providers and customers. The customers are charged with different prices
if they sign a contract, and this difference depends on
the length of the contract! Economists are well aware of
the notion of consumer or brand loyalty, which is very
much existing in realistic markets. Practically speaking,
a customer is loyal to a brand when he/she purchases
the product of that brand, even if there are cheaper
substitutions on the market. Brand loyalty is rooted in
both satisfaction towards a given brand and customers
being reluctant to try substitute products.
There is existing work dealing with classification
of buyers into loyalty groups [9], and a recent study
develops and empirically tests a model of antecedents
of consumer loyalty towards ISPs [10]. In [11], authors
use a game-theoretic framework to prove that, if loyalty
is an additional product of market share and penetration, customer retention strategies seem to be consequently more efficient for market leaders. Another
study [12] analyzes a duopolistic price setting game in
which firms have loyal consumer segments but cannot
distinguish them from price-sensitive consumers. They
demonstrate that consumer loyalty plays an important
role in establishing the existence and identity of a price
leader. Georges [13] presents a duopolistic price setting
game, where loyal and also disloyal customers are on
the market. The companies set their prices based on the
number of their loyal customers; therefore, the Nash
equilibrium of the game changes, resulting in higher
utilities.
These works and a recent work [2] initiate the discussion on customer loyalty and its impact on pricing
strategies of ISPs. However, a number of issues are
still to be solved. First, these works are mainly theoretical without firm empirical analysis that supports
the customer loyalty argument. Second, to the best of
our knowledge, these works mainly deal with complete
information and simple loyalty models, i.e., the ISPs
fully know about other ISPs’ pricing decisions—we
believe this is not usually the case in practice. From the
arguments above, in this paper, we address these issues
and try to give answers to these open questions.
We carry out—by our own—a survey on customer
loyalty on the Hungarian ISP market; our results are
combined with other surveys on European markets to
Ann. Telecommun. (2010) 65:171–188
substantiate our case. It is suggested by our survey,
among others, that customer loyalty is strongly correlated with the price difference of ISPs. Price-differencedependent customer loyalty means that a subscriber will
stay with his/her current ISP as long as another ISP
whose price is significantly lower than his/her current
access price does not exist.
To model uncertainty in ISPs’ pricing decision, we
use the tools of game theory and Bayesian games in
particular. Based on our game theoretic model, we
quantify the effects of uncertainty on ISPs’ price setting
and derive strategies to optimize ISPs’ profits under
these uncertain conditions.
The paper is structured as follows. First, in Section 2,
after reviewing European ISP loyalty trends, we provide a comprehensive empirical analysis of ISP customer loyalty in Hungarian Internet access market
based on a survey, carried out by ourselves. We present
the basic notions of game-theory in Section 3. Section 4
provides game-theoretic models for price-differencedependent customer loyalty issues with incomplete information to deal uncertain ISP price setting decisions.
Furthermore, we show the impact of price-differencedependent customer loyalty in terms of Nash and
Bayesian equilibrium by detailed game-theoretic analysis. Section 5 provides simulation analysis of different,
price-difference-dependent customer loyalty models.
Finally, Section 6 concludes the paper.
2 Customer loyalty in ISP markets: an empirical
perspective
A number of empirical studies deal with user loyalty in
the ISP markets. In this section, by reviewing some of
their findings, we try to give a global ISP loyalty picture,
the European, and in particular, the Hungarian, situation is presented later on. The 2005 Walker Loyalty
Report for Information Technology shows that 38%
of US enterprise customers were truly loyal to their
ISPs [14], while 30% of the customers were high-risk
users, meaning they have low commitment and typically
do not intend to stay at their current providers. Based
on the report, quality, value, and price are the key
drivers of loyalty. Choice survey states that 90% of the
respondents had not changed their ISPs in the previous
12 months, including contract-users as well [15]; the
most important factor in choosing a service provider
was the price of the access. The PC Advisor Broadband
Survey 2008, carried out in the UK, revealed that the
vast majority of respondents have been with their ISP
for ages well beyond the minimum subscription period
[16]. A survey, made in Taiwan in 2002, investigated
Ann. Telecommun. (2010) 65:171–188
customer loyalty toward ISPs [17]; the key factors of
customer loyalty were perceived value, service satisfaction, and future ISP expectancy.
2.1 Customer loyalty in European ISP markets
A lot of national communication authorities of the European Union carry out market research dealing with
customer loyalty toward local ISPs. This section reviews
the findings of these surveys providing a summary of
European ISP loyalty.
UK’s Office of Communications (Ofcom) published
its communications market report in 2008 [18]. Around
60% of households (15 million) had a broadband connection in 2007 in the UK, often purchased as part of a
bundle. Almost 90% of consumers said they were either
“very” or “fairly” satisfied with both value for money
and connection speed. Twenty-seven percent of broadband users have already switched providers, while, as
another type of switch, 45% of narrowband consumers
claim they are likely to subscribe to broadband within
12 months. It is interesting that only 61% of Internet
users find it easy to switch ISPs.
In Ireland, the Commission for Communications
Regulation made a representative consumer information and communication technology survey that dealt
with broadband Internet access and ISPs in 2008 [19].
Sixty-three percent had a choice of broadband service
providers, 21% did not know, and 16% had no options
choosing a provider, which had an impact on users’
loyalty. The majority (84%) had not changed their
ISP in the last 12 months, and 12% had had a switch,
which is more frequent than the churn rate of mobile
providers and less than that of fixed-line providers.
Most respondents rated their home Internet service as
being fairly good value for money (56%), while 27%
rated their Internet service as being poor value for
money, and were more likely to switch their ISPs.
Anacom, the Portugal communication authority,
made its survey on the use of broadband in 2006 [20].
As for the consumers’ evaluation of broadband Internet access, satisfaction was high; only 6.6% were dissatisfied with their service. Regarding user loyalty, 81% of
broadband customers said they did not intend to change
operators in the next 12 months; this percentage was
slightly less in the previous year (71.1%).
The first interesting finding of the survey carried
out in Finland in 2007 [21] was that only 20% of the
customers said they might change their wired Internet
access to a mobile connection if the price levels were
equal. In 2006, this ratio was 30%; thus, the fixed subscribers are more loyal to, or more satisfied with, their
173
technology than they were in the previous year. The
survey highlights that 16% of the subscribers switched
their ISPs in 2007 mainly because of a better offer
from a competitor; the price of the Internet access was
reduced, with 5% on average in this 1-year-long period.
The Malta Communications Authority carried out its
survey in 2007 [22]. Eighty-four percent did not switch
their ISPs in the last 2 years; these customers were
loyal to their providers. Sixty percent thought it was not
difficult to change their ISPs; only 14% found it hard to
switch. It is surprising that 31% thought that they did
not have enough information about the services and
the prices of the service providers. Only 20% would
switch their current Internet subscription if the price of
their Internet access would be increased by 5–10%. The
main reasons of this loyalty intention are that the rise
of the price is minimal, it is an inconvenience to switch,
and e-mail addresses. When choosing an Internet subscription, 25% were not aware of the access, but 41%
considered the price as expensive.
Consumer preferences regarding telecommunications services were surveyed representatively in Poland
in 2005 by the Office of Electronic Communications
[23]. In that study, 13.4% of respondents with home
Internet access were considering changing their service
providers; however, only 76.6% of the respondents had
a choice of ISPs.
To conclude the revision of the loyalty intentions
on European ISP markets, we present the aggregated
results in Fig. 1. Europeans are satisfied with their
providers as they usually do not select new ISPs. Different countries have different loyal segments; thus,
the subscribers’ loyalty depends on the countries’
Percentage of loyal subscribers
86
84
84
84
81
73
UK
Ireland
Portugal* Finland
Malta
Poland*
Fig. 1 Percentages of loyal Internet subscribers in Europe
(marked values are only intentions)
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Ann. Telecommun. (2010) 65:171–188
culture. The users tolerate price differences for a certain amount of money; they only change their providers
if the offer of the other ISP is much cheaper.
2.2 An empirical analysis of subscriber loyalty
intentions towards ISPs in Hungary
In this section, we examine user loyalty from two perspectives: the service providers and the subscribers’
opinion. The providers have exact information about
their users’ behavior based on selling data while users
can judge their own preferences and loyal attitudes.
For the first aspect, we have contacted major ISPs
in Hungary to get real-world data about user loyalty
and their price setting strategies. Unfortunately, they
refused to give out this kind of information because the
number of customers and their loyalty are very sensitive
company secrets; companies can have disadvantages on
the market if they make these data public. Moreover,
the pricing strategy and the profits of the companies are
also private information.
The National Communication Authority published
some cumulative statistics about the numbers of subscribers of the service providers in 2007 along with a
churn number, the number of people who left the company in the last 6 months [24]. Based on the results we
conclude that, at most, 10% of the users have switched
their ISP; the detailed subscriber numbers and churn
ratios, presented in Table 1, predict that the users are
loyal to their providers.
We have dealt with the users’ point of view by asking
them a few questions about their personal ISPs and
their loyal intentions. We got in touch with people in
different ways: we sent emails to lists, we asked help on
Internet forums, and we also used social networks to
get more and more answers.
Based on the received answers, we state that the
survey was filled out by a wide community (778 people) where every age group was represented (less
than 18 years of age 3%, 19–24 years of age 53%,
Table 1 The number of subscribers and switching users at seven
main ISPs in Hungary at the end of 2007
Name of the ISP
Average
subscriber
number
Number of
switching
users
Switch
percentage
DIGI
FiberNet
GTS-DataNet
Invitel
Magyar Telekom
UPC
Enternet
22,334
50,461
42,156
14,568
228,786
240,558
34,653
725
585
135
196
21,497
16,041
3,520
3.25%
1.16%
0.32%
1.35%
9.40%
6.67%
10.16%
25–35 years of age 34%, 36–45 years of age 6%, more
than 46 years of age 4%). The gender and the educational background of the answerers were diversified.
The properties of the sample confirm that the empirical
analysis of the survey is a good illustration of the loyalty
of the whole community.
To help the interpretation of the outcomes, we describe the Hungarian ISP market and its prices. In
August 2008, there were 497 thousand asymmetric digital subscriber line (ADSL) subscribers and 690 thousand cable subscribers in Hungary [25]. The volume of
the prices of Internet access is hard to judge in the case
of a foreign country. Therefore, we compared the prices
to the average Hungarian net income in the first half of
2008 [26]; in the following, we present the ratio of the
price and the average net income.
Table 2 shows the statistics of the monthly price of
the Internet subscriptions. Most of the surveyed people
had Internet access with moderate prices (4–8% of the
average salary), but there were also a few who had
really expensive Internet access.
User loyalty can be described by different approaches, e.g., based on the number of switches or on
the number of years to be a customer of an ISP. In terms
of the loyalty history of the subscribers, around 60%
of the questioned people had not changed their ISPs in
the last 5 years (Fig. 2), which implies significant loyalty
towards ISPs in Hungary.
The type of connection always affects the loyalty
intentions as every communication method has its own
specialities: wired Internet access does not allow users
to switch easily between providers, e.g., technological
(change between cable and ADSL) or deployment issues, contrarily, a mobile ISP can be changed easily.
We present in Fig. 3 the frequencies of how long a user
has its current ISP based on the type of connection. A
lot of users have not changed their service providers
in the last two or more years, these subscribers can be
considered as loyal users. Note that not all the connection types have a lot of users for long times because
they were not available earlier (e.g., mobile Internet,
FTTX).
Table 2 Monthly price of current Internet subscription
Monthly Price
(relative to the average
net income)
Frequency
Percent
Cumulative
percent
1.5%
4%
8%
12.5%
>12.5%
21
249
426
64
6
2.7%
32.5%
54.8%
8.4%
0.8%
2.7%
35.2%
90.9%
99.2%
100%
Ann. Telecommun. (2010) 65:171–188
175
Duration of the loyalty contract
0 month
24 months
18,34%
28,10%
6 months
18 months
1,06%
0,66%
12 months
51,85%
Fig. 4 Duration of the contract
Fig. 2 Number of ISP switches in the last 5 years
As we mentioned above, Internet providers offer
services with contracts in order to keep their subscribers for a long period. Only 17% of our answerers
did not sign a loyalty contract when they bought their
Internet access. This ratio is really interesting because
80% of the customers have to be loyal for the duration
of the contract. The causes of signing a contract verify
that service providers set their prices based on loyalty
intentions. Almost half of the persons (49.1%) signed
an optional contract due to a cheaper price. Contrarily,
in 29.4% of the cases, it was compulsory to sign a contract in order to have the specific subscription plan. The
duration of the contracts is also a significant parameter,
which we show in Fig. 4; people without a contract are
represented with 0 month. The most frequent lengths
are 1 and 2 years; with these contracts, ISPs are able to
keep customers for long periods.
One of the questions of the survey was a bit provocative; we wanted to know what the minimal price difference was between the users’ current subscription and
a subscription of another ISP when they would switch
their ISP, supposing that the two ISPs offer exactly the
same services, including connection speed, help desk,
etc. We received surprising answers, as shown in the
bar chart of Fig. 5 and in Table 3, where the exact
numbers and percentages are presented. Only around
5% of the answerers said a price difference where they
would leave their current ISPs did not exist (last row
in the table); these subscribers are really loyal to their
service providers. At the same time, the remaining 95%
would become disloyal and switch their providers at
a certain price difference. Based on the results, we
argue that modeling user loyalty based on the minimal
price difference to switch is a realistic description of the
ISP pricing problem. The first idea of what everybody
would say is to model loyalty based on the price ratios.
Minimal price difference to switch
4,71%
4,05%
4,97%
0.5%
1%
12,55%
1.5%
4%
8%
never
35,29%
38,43%
Fig. 3 User loyalty at different type of connection
Fig. 5 Minimal price difference to switch to an other ISP
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Ann. Telecommun. (2010) 65:171–188
Table 3 Minimal price difference to switch ISPs
Frequency
0.5%
1%
1.5%
4%
8%
Never
38
96
294
270
31
36
Percent
Cumulative
percent
5.0%
12.5%
38.4%
35.3%
4.1%
4.7%
5.0%
17.5%
55.9%
91.2%
95.3%
100.0%
1
125
162
5
0
0
23
202
40
4
1
1
14
14
1
5
17
11
2
0
2
3
4
More
years to be
a subscrib
er
at the ISP
1%
3
64
26
3
0
1
0.5%
11
15
11
0
1
0
Number of
4%
1.5%
4%
8%
12.5%
>12.5%
0
1.5%
Minimal price difference to switch (relative)
0.5%
1%
1.5%
4%
8%
never
20
ence
r
e diffe
al pric ch
Minim to swit
r
Relative
monthly
price
40
neve
Table 4 The connection between the monthly price and the
minimal price difference to switch (number of answers)
60
8%
On the one hand, it represents the relationship between
the two prices, but on the other hand, it does not give
any information about the socio-economic aspects. The
following short example illustrates the importance of
socio-economic aspects. Consider two countries, a rich
one and a poor one; in each country there exist two ISPs
providing Internet access for 5$ and 10$, respectively.
The price ratios are the same in both countries (0.5) but
there will probably be much more switchers in the poor
country, where 5$—the price difference—is worth a lot
more than in the richer country.
In our survey, both the price of the current subscription and the minimal price difference had to be
selected from a list of possible values; thus, the results
are discrete probability variables. Correlation analysis
is only useful in case of continuous variables; therefore,
we used crosstabs to investigate the possible connections between the monthly price of the current Internet subscription and the minimal price difference. In
Table 4, we present the crosstab where every cell stores
the number of occurrences of the specific pair.
In addition, we examine the connection between the
number of years to be a customer of the current ISP
and the minimal price differences (Fig. 6). Regardless
of the years, there are similar price differences where
the customers will switch their ISPs. We asked several
more questions about user loyalty in the survey, which
are published at [27].
80
Number of answers
Price difference
(relative to the average
net income)
Fig. 6 Relation of the user loyalty and minimal price
To conclude this section, we summarize its key
observations:
–
–
–
User loyalty has an impact on price-setting strategies of ISPs. On the Internet access market, the
majority of the users have loyal intentions towards
their service providers regardless of the countries.
Subscriber loyalty depends on the price difference of the current and the possible future service
providers; users would become disloyal if the price
difference is large enough.
Numerous factors have an impact on users’ loyalty;
some of them were presented above. However,
not all the factors can be examined or measured;
thus, ISPs do not have exact information about
the customers of their competitors; they have only
beliefs about them. The users select their access
providers based mostly on their impression, not on
exact parameters. Therefore, the price competition
between ISPs has uncertain parameters, resulting in
non-deterministic decisions.
3 Basic notions of game theory
Game theory provides efficient methods to handle
multi-person decisions, this section reviews basic notions of game theory that we use in this paper. For a
detailed introduction to game theory, we refer to [28].
Ann. Telecommun. (2010) 65:171–188
We will only deal with rational decisions; namely, every
person wants to select his/her best possible choice that
will maximize his/her utility.
A non-cooperative game, where players do not cooperate with each other, can be formalized as follows:
N = {1, 2, . . . , n} is the set of players, where 1, . . . , n are
the individuals who are playing and Si is the strategy
set of player i, who selects his/her strategy si ∈ Si from
the set. Every player has his/her own payoff function,
which gives the utility of the possible cases; if S =
S1 × S2 × · · · × Sn , then the payoff function of player
i is fi : S → R, which can be ordered; thus, a player can
select the best possible strategy from his/her strategy
set. s = (s1 , s2 , . . . , sn ) ∈ S is a strategy profile where si is
the strategy of player i while s−i denotes the strategies
of players except player i.
Nash equilibrium describes a strategy profile that has
good properties, namely, none of the players can have
more payoff if only one of them changes his/her strategy. Formally, the s∗ ∈ S strategy profile is a Nash equilibrium point, if fi (si∗ , s∗−i ) ≥ fi (si , s∗−i ) ∀si ∈ Si , ∀i =
1, . . . , n. Games can be partitioned based on several
aspects:
•
•
•
Strategy: A player plays with pure strategy if he/she
selects only a single strategy with one probability. Contrarily, if a player selects more strategies
with positive possibilities, he/she plays with a mixed
strategy. If every player plays pure strategy, then a
Nash equilibrium is pure equilibrium; otherwise, it
is a mixed-strategy Nash equilibrium.
Number of rounds: If the players play only once, we
call the game a single-shot game; otherwise, if they
play multiple rounds, it is a repeated game.
Information: An important partitioning of games
is based on the amount of information. If every
player knows all the information necessary for the
decision and this knowledge is common, the game
is a complete information game. In contrast, in an
incomplete information or Bayesian game, not all
the players have the same knowledge.
4 Price setting under uncertainty
In Section 2, we have seen that user loyalty is an important factor in the ISP pricing competition. User loyalty
can be formalized and its properties can be examined
in a game-theoretical model. For an introductory paper
we refer to [2], where a useful loyalty model is introduced. That model is a good starting point, but we have
seen in our survey that the difference of the prices of
177
the old and the possible new service provider has a
significant impact on user loyalty.
ISPs compete with each other under uncertainty,
which affects their pricing strategies. On the one hand,
ISPs might set access prices with certain probabilities, playing a mixed strategy because a pure strategy
does not exist, in order to have maximal profits. Thus,
the prices of the ISPs are variable, resulting in uncertain pricing strategies. On the other hand, an ISP
only knows exactly the number of its users, but other
important properties, like the number of other ISPs’
subscribers and the price difference, are unknown. The
ISPs set their prices based on their beliefs, which also
results in uncertain price setting strategies. Accordingly, throughout this section, we analyze the impact
of uncertainties on the ISPs’ pricing strategies on a
market, where the subscribers are loyal based on the
price differences.
We apply a few assumptions in order to model the
ISP price setting problem. We suppose that ISPs offer
flat-rate subscriptions because deploying a usage-based
sophisticated price scheme would, in general, be too
costly for ISPs. However, mobile operators do not like
flat-rate subscriptions because mobile access generates
high operational costs. The first successfully provided
flat-rate pricing was NTT DoCoMo’s i-Mode service
in Japan at the end of 1990s, which is still popular
nowadays in Japan; 20% of DoCoMo’s mobile users
have this kind of subscription [29].
The optimal case for ISPs would be if they would
be able to identify the personal reservation price of
every single user, which would be hard to carry out.
Therefore, we further assume that the consumer demand for Internet access is constant, until a maximum
price, meaning the demand function of the subscriptions is inelastic. This assumption is realistic in developed countries where Internet access is a must and
almost everyone can afford it. Finally, we suppose that
the users have a single reservation price, meaning the
ISPs cannot discriminate the subscribers with different
prices. The users buy Internet access until the price is
lower than the reservation price; otherwise, the users
do not purchase a subscription. Complex markets can
be modeled with these assumptions by dividing them
into several smaller markets, e.g., based on connection
speed requirements, where the assumption of single
reservation price and constant demand is realistic.
4.1 Pricing with incomplete information
ISPs do not know exactly the characteristics of the
market; they have to set their prices using incomplete
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Ann. Telecommun. (2010) 65:171–188
information. Because they only have beliefs about the
other ISPs’ properties, they set their prices under uncertainty. We examine a price-difference-dependent
loyalty game with incomplete information, where two
ISPs exist on the market; each of them knows exactly
the number of their own loyal subscribers but they
do not know the loyal base of their competitors. Customers are split into two partitions upon their loyalty:
l1 customers are loyal to ISP1 while ISP2 has l2 loyal
users. For simplicity reasons, we suppose that the first
service provider has more subscribers than the second
one (l1 > l2 ). Let d be the price difference, meaning
that if the price of a user’s ISP is more than the other
ISP’s price plus d, then the user will be a switcher,
i.e., the user leaves his/her ISP for the other one. The
demand function is modeled as a constant function
until a border price (α), if at least one of the ISPs set
a price less than α, the demand is l1 + l2 but above
α, none of the users buy Internet access. In terms of
the incomplete information game, the number of loyal
customers is the type—the private information—of an
ISP, while the ISPs’ beliefs about their opponents’ type
is common knowledge. In addition, the minimal price
difference (d) at which a loyal customer will switch is
also commonly known by the players. The expected
payoffs of the ISPs are:
⎧
⎨ (li + El j) pi pi < p j − d
pi − p j ≤ d
Ei ( p) = li pi
(1)
⎩
0
p j < pi − d
Figure 7 illustrates the payoff function (Eq. 1) in two
different scenarios. We present the payoff of ISP1 at
different prices ( p1 ) while the price of ISP2 is fixed.
Figure 7a presents the payoff function if the border
price is not smaller than the price of ISP2 plus the price
difference. Until p2 − d, ISP1 has every subscriber; its
profit is proportional to its price. If ISP1 ’s price is
higher than p2 − d but lower than p2 + d, ISP1 provides
1
1
α
Internet access only for its own customers. If ISP1
sets too high a price, larger than p2 + d, ISP1 does
not realize any profit because its subscribers switch to
ISP2 . Figure 7b shows the payoff if the border price is
lower than p2 + d. In this case, ISP1 keeps its customers
and realizes profit until it sets a price smaller than
α. In the figures, the highest point of the first linear
segment is lower than the second’s because the profit
is proportional to the price. However, this is not always
the case, e.g., if ISP2 has much more users, then ISP1
realizes higher payoff with low prices.
We illustrate the incomplete information ISP price
setting game with a short example. There are two ISPs
on the market, ISP1 has l1 = 100 loyal customers, which
is commonly known by both players. In contrast, the
number of ISP2 ’s customers is not known exactly by
ISP1 ; it only believes that ISP2 has l2 = 20 users with
0.3 and l2 = 80 users with 0.7 probability. Because
we are only interested in the relation of the prices,
L denotes that the player has a low-enough price,
i.e., he/she will have all customers on the market, T
denotes when each ISP has only its own subscribers,
and H denotes when the price is too high, i.e., the
ISP looses all of its customers. Table 5a shows the
payoffs of the ISPs if ISP2 is a small company while
Table 5b shows payoffs in the case of a larger ISP2 .
For calculating the expected payoffs of the cases, we
suppose that the price cases (L, T, H) are equally distributed. Thus, the expected payoff of ISP1 is l12 =20 =
1
0 + 13 100 p1 + 13 120 p1 = 220
p1 in the small competi3
3
tor case and l12 =80 = 13 0 + 13 100 p1 + 13 180 p1 = 280
p1
3
l2 =20
in the other case. Similarly, ISP2 can have 2
=
l2 =20
140
260
p
and
=
p
payoffs
in
the
cases.
Us2
2
2
3
3
ing these payoffs and the probability of ISP2 ’s size,
which is the belief of ISP1 , we calculate the overall
expected payoffs: E1 = 0.3 220
p1 + 0.7 280
p1 = 262
p1
3
3
3
140
260
224
and E2 = 0.3 3 p2 + 0.7 3 p1 = 3 p2 . The expected
payoffs of the ISPs depend on the beliefs of ISP1 —
different beliefs result in different payoffs.
Table 5 Payoff matrices of Bayesian example game and their
probabilities
L
p2-d p2 p2+d α
(a) p 2 + d <
− α
p1
p2-d p2 p2+d
(b) α < p 2 + d
p1
Fig. 7 Illustration of the payoff function. a p2 + d ≤ α. b α <
p2 + d
(a) [P(l2 = 20) = 0.3]
L
X
T
X
H
0,120 p2
(b) [P(l2 = 80) = 0.7]
L
X
T
X
H
0,180 p2
T
H
X
100 p1 ,20 p2
X
120 p1 ,0
X
X
X
100 p1 ,80 p2
X
180 p1 ,0
X
X
Ann. Telecommun. (2010) 65:171–188
179
After this short illustration, we present the formal definition of the Bayesian ISP price setting game
G1 ; afterwards, the conditions of the equilibrium are
computed.
–
–
–
–
Players: the ISPs, N = 2, ISPi has li loyal customers,
the type of the players is the number of their loyal
customers: i denotes the set of types of ISPi , i =
li1 , li2 , . . . , lim are the possible loyal bases.
Strategies: the price of the Internet access, the
decision of ISPi is pi , pi ∈ [0, α], players can have
only pure strategies, they play once as a single-shot
game.
Payoff functions: the payoff of the ISPi is presented
in Eq. 1, where the expected values are calculated
based on the players’ own beliefs.
Information: incomplete, it is a Bayesian game,
where ISPi knows exactly α, d, li and has a belief
probability distribution on the values of l j, which is
common knowledge
Proposition 1 The G1 incomplete information twoplayer price setting game has a pure strategy Bayesian
equilibrium at (α, α) with payoffs l1 α, l2 α if the following
hold:
where an equilibrium does exist if the ISPs would have
common knowledge. Note that an equilibrium exists
even on a market where one of the ISPs has a lot more
subscribers than the other if the price difference is large
enough.
Proof None of the ISPs would set a price higher than
α; otherwise, their payoff would be zero. Thus, the
support of the equilibrium is [0, α]. In which case, is it
worth to undercut the other ISP’s price more than d
to get the whole market? If ISP2 sets a price p2 , ISP1
grabs all users if he/she sets a price lower than p2 − d.
If ISP1 would not compete, its maximal price can be
p2 + d without losing its loyal customer base.
We divide the [0, α] interval into three parts where
we will look for an optimal price for pure strategy
Bayesian equilibriums. First, we look at the [0, α − d)
interval; ISP1 will not compete if the following holds:
E (l1 + l2 )( p2 − d) ≤ l1 ( p2 + d)
(l1 + El2 )( p2 − d) ≤ l1 ( p2 + d)
p2 ≤ d +
The same can be said for ISP2 ; thus, ISP2 will not
compete on the [0, α − d) interval if p1 ≤ d + 2lEl21d . Secondly, what is the condition of ISP cooperation at the
[α − d, α) interval? For ISP1 we get:
El2
d
≤
l1 + El2
α
(2)
d
El1
≤
El1 + l2
α
(3)
E (l1 + l2 )( p2 − d) ≤ l1 α
Before the formal proof of the proposition, we illustrate in Fig. 8 what the minimal price difference is
(l1 + El2 )( p2 − d) ≤ l1 α
p2 ≤ d +
l1 α
l1 + El2
α
Similarly, ISP2 will not compete if p1 ≤ d + Ell12+l
. Fi2
nally, we investigate the (α, α) case. ISP1 will not
compete if:
100
95
Minimal price difference
2l1 d
El2
90
E (l1 + l2 )(α − d) ≤ l1 α
85
80
(l1 + El2 )(α − d) ≤ l1 α
75
70
El2 α ≤ (l1 + El2 )d
65
60
55
0
50
0
20
40
20
60
Numbe
40
60
r of loya
l users
80
80
of ISP2
100
100
al u
f loy
er o
b
Num
P2
f IS
o
sers
Fig. 8 Existence of Nash equilibrium in game G1 at different
loyal customer numbers (l1 , l2 ) and minimal price difference (d)
at a fixed α = 100 border price
El2
d
≤
l1 + El2
α
1
If this condition is true and, for ISP2 , ElEl
≤ αd holds,
1 +l2
we argue that (α, α) is a pure strategy Bayesian equilibrium of the game.
If we look at Fig. 9, we see this game does not have
any other pure Bayesian equilibrium because there is
180
Ann. Telecommun. (2010) 65:171–188
p2
Proof If none of the ISPs have an incentive to compete
in α, there exists a Nash equilibrium in the N-player
game when:
NEP
α
B1
⎡
E⎣
j
ISP1
ISP2
A1
⎤
l j(α − d)⎦ ≤ li α
j
=i
El j
j El j
≤
∀i
d
α
d
d
A2
B2 α
p1
The condition of the equilibrium follows from this
expression.
Fig. 9 Best response function of the service providers
4.2 Mixed strategy Nash equilibrium and expected
payoffs
no more intersection between the two best response
functions. ISP1 does not compete until p2 is less than
A1 = d + 2lEl12d ; it offers Internet access for p2 + d as a
price. Then it is worth to set a lower price than ISP2
l1 α
( p2 − d) until p2 > d + l1 +El
= B1 . Afterwards, ISP1
2
selects α as its price to maximize its profit. ISP2 has
a similar best response function with its own border
prices (A2 , B2 ). At prices other than (α, α), where the
graphs are very close to each other, any equilibrium
does not exist because, on the one hand, the price
difference has to be greater than d to grab, on the other
hand, in order to hold users, the price difference has to
be less than or equal to d. Accordingly, the graphs do
not intersect each other.
The proposition means that, if the market’s price
difference value (d) is large enough, and the users are
not too sensible for the prices, the ISPs do not have to
compete; they can sell Internet access at the highest
possible price (α). We emphasize that the conditions
of the equilibrium include the uncertainty of the loyal
customer bases as the expected values are based on the
ISPs’ beliefs.
This incomplete information game and its Bayesian
equilibrium can be easily generalized to N players.
Each player knows only his/her loyal customers (type)
and they have commonly known beliefs about the types
of their opponents. The N-player price-differencedependent incomplete information ISP price setting
game has only one pure strategy Bayesian equilibrium
(α, . . . , α) if the following equations hold:
∀i : 1 −
li +
li
j
=i
El j
≤
d
α
We have seen the conditions of the equilibrium in
case of incomplete information. In the following, we
suppose that every ISP has common knowledge about
the number of customers. This complete information
scenario has the same conditions like before (Eq. 3)
without the expected values. However, a pure strategy Nash equilibrium where providers have to play
mixed strategies does not exist in every possible market
scenario.
We present the mixed strategy equilibrium for the
following game. There are two ISPs (ISP1 , ISP2 ) with l1
and l2 price-difference-dependent loyal subscribers on
the market. The subscribers leave their ISP if its price is
larger than the other’s price plus an additional d value.
The demand function is constant on [0, α]. The payoff
function of the ISPs is formulated as:
⎧
⎨ (li + l j) pi pi < p j − d
pi − p j ≤ d
i ( p) = li pi
(4)
⎩
0
p j < pi − d
The formal definition of game G2 is as follows:
–
–
–
–
Players: the ISPs, N = 2, ISPi has li loyal customers.
Strategies: the price of the Internet access, the decision of ISPi is pi , pi ∈ [0, α], players can have only
li
mixed strategies; thus, li +l
> αd holds, they play a
j
single-shot game.
Payoff functions: the payoff of the ISPi is described
in Eq. 4.
Information: complete, players know α, d, l1 , l2 .
Proposition 2 In the two-player price setting game G2 ,
where the ISPs have price difference dependent loyal
Ann. Telecommun. (2010) 65:171–188
181
users, the mixed equilibrium strategies have the following cumulative distribution functions:
F1 ( p) =
F2 ( p) =
⎧
0
⎪
⎪
⎨ p−
l1
l1 +l2
l1
α− l +l
1 2
⎪
⎪
⎩
1
⎧
0
⎪
⎪
⎨
⎪
⎪
⎩
l2
p− l +l
1 2
l2
α− l +l
1 2
1
d
d
p1 <
l1
d
l1 +l2
l1
d
l1 +l2
≤ p1 ≤ α
α < p1
l2
d
p2 < l1 +l
2
d
d
l2
d
l1 +l2
f
ISP1
ISP2
(5)
≤ p2 ≤ α
α < p2
*
p-d
Proof What is the minimal profit that an ISP can have
regardless of the other player, and what is the minimal
price that results in this payoff? If an ISP sets d as its
price, it can never loose its loyal customers because
the other ISP cannot set a lower price than d − d = 0.
Accordingly, the ISPs can have li d profit in each case.
What is the condition of a mixed-strategy equilibrium? ISPs have a pure-strategy equilibrium (α, α) if
l1
l2
l1
the following hold: l1 +l
≤ αd , l1 +l
≤ αd . Thus, if l1 +l
>
2
2
2
l2
d
d
,
> α , the ISPs will play mixed strategies during
α l1 +l2
the price competition and compete with each other.
We compute the minimal price where the ISPs still
compete. Firstly, ISP1 competes if it can have a larger
payoff than its minmax profit: (l1 + l2 ) p1 > l1 d, from
l1
which we get that p1 > l1 +l
d. We also have a similar
2
l2
lower bound price for ISP2 : p2 > l1 +l
d.
2
To be able to compute the expected payoffs of the
ISPs, we have to create their probability distribution
function. We have the lower ( pi ) and the upper bounds
(α) of the distributions but we do not know the proper
function yet. With the following indirect proof, we
argue that each ISP has uniform distribution between
the bounds. Suppose that ISP1 plays with a distribution
where there exists a specific price ( p∗ ) that it plays
more often than the other prices. In this case, ISP2 can
set deterministic a lower price ( p∗ − d) to get ISP1 ’s
loyal customers. The illustration of the indirect idea
is shown in Fig. 10. Therefore, the expected payoff of
ISP1 will always be less than it would be if it plays
with non-uniform distribution. The proof is the same
for ISP2 ; both ISPs have to play their mixed strategies
with uniformly distributed prices.
The boundaries of the mixed strategies’ supports
are known, in addition, the ISPs have uniform price
distribution; thus, the cumulative distribution functions
of the ISPs can be created, which is identical to Eq. 5.
The calculation of ISPs’ expected utilities based on
the payoff function of the game and their probability
p*
p
Fig. 10 Illustration of prices in case of non-uniform distribution
function
density functions is hard. The expected profit of an ISP
can be threefold: firstly, if its price is small enough,
it can keep its loyal customers and also grab the customers of the other ISP. Secondly, if the price of ISP1
is close to the other’s price, e.g., the price difference
is smaller than the critical d difference, it can keep its
loyal customers but the subscribers of the other player
will not switch to ISP1 . Thirdly, if ISP1 has too large
a price, none of its customers will stay at its resulting
zero expected profit. Unfortunately, the expected profit
of the ISPs cannot be formulated in closed form; for
details, see [27].
In the proof, we have mentioned that the expected
payoffs of the ISPs are hard to express in a closed form.
In order to investigate the payoffs, we calculated them
numerically in different market scenarios with different
minimal price difference dependencies; the findings are
presented in Section 4.4.
4.3 Mixed strategy under incomplete information
As we have seen in the complete information game, the
ISPs will use mixed strategies to compete for customers
if several conditions are valid. If we model the ISP price
competition game with incomplete information, where
an ISP knows only its own loyal user base, it can only
predict the loyal user base of other ISPs based on its
beliefs. Under these uncertainties, the service providers
play a mixed-strategy incomplete information game.
We deal with the two-ISP case, but it can be generalized for N players. The providers will use mixed
strategies if the following hold as we have already seen,
182
Ann. Telecommun. (2010) 65:171–188
where each ISP knows the number of its loyal users li
and can guess the expected value of the other’s user
base.
d
El1
> ,
El1 + l2
α
price difference is used as another parameter, with a
value between 0 and 100.
Figure 11 shows the effect of the loyal user base
size on the expected payoffs. We plotted the profits
of the ISPs and the total profit as well. The minimal
price difference was 30 and 60 in Figs. 11a and 11b,
respectively. It can be seen that the price difference
has an effect on the expected payoffs. The plots of the
smaller value are continuous, meaning that the ISPs are
always playing their mixed strategies. The profit of ISP2
increases better as it has more and more loyal users.
The jump in Fig. 11b presents the change between the
mixed and the pure strategies—where the conditions
of the pure Nash strategy are true. Until the jump, the
ISPs play mixed strategies, where their prices are varying. They realize more profit after the jumps because
they do not have to compete; the Internet access is sold
at the highest possible price (α). The expected payoff of
ISP1 is constant or slightly decreasing as ISP2 has more
loyal users.
In Fig. 11c, we show the expected profits of ISPs
and the total profit when ISP2 has 30 loyal users. The
horizontal axis shows the value of the minimal price
difference. The profit of ISP1 is increasing as the minimal price difference grows because ISP1 has more loyal
users and it can keep them easier and easier as ISP2
has to cut its price more and more. By contrast, the
expected profit of ISP2 is constant or slightly decreasing
because it can grab users from ISP1 harder as the
minimal price difference rises. The change between the
mixed and the pure strategy is around 80; that is why
there is a huge jump in the payoff functions.
If an ISP does not have exact information about the
user base of its competitors, its expected profit depends
on its beliefs. Figure 12a and b plot the expected profits
of two ISPs when they play on a market. The significant jumps on the surface are the consequence of the
d
El2
>
l1 + El2
α
We calculate the cumulative distribution functions
the same way as we did in the complete information
case; we only have to add the expected value of the user
bases to the formulas of Eq. 5.
It is also hard to formulate the expected payoff of the
mixed strategy in a closed form in case of incomplete
information similarly as it was at complete information.
Therefore, we compute the expected payoffs of the
service providers numerically; we present the results in
the following section.
4.4 Expected payoff based on analytical calculations
In this section, we present the expected payoffs of
ISPs under uncertainty. The expected payoffs of the
ISPs depend on various factors: the number of own
loyal customers, the maximal price, and the minimal
price difference. In addition, the volume of the other
ISP’s loyal user base has a significant impact on the
expected payoff. An ISP usually does not have accurate
information about the other ISP’s customer base; it
has only beliefs about it. The payoff functions and
the probability distribution functions are formulated in
Matlab to calculate the ISPs’ expected payoffs numerically. We examine a game where only two ISPs exist
on the market. The parameters of the simulations are
as follows: ISP1 has 100 loyal users, and the maximal
possible price (α) is also 100. We scale up the number
of loyal customers of ISP2 from 0 to 100; the minimal
Expected Profits
Expected profit
Expected profits
10000
ISP1
ISP2
Total
6000
Mixed
strategy
12000
Pure
strategy
10000
4000
20
40
60
80
Number of loyal users of ISP2
(a) d = 30
8000
Pure
strategy
6000
2000
Mixed strategy
0
0
Mixed strategy
4000
5000
2000
ISP1
ISP2
Total
10000
Profit
15000
Profit
Profit
8000
14000
ISP1
ISP2
Total
100
0
0
20
40
60
80
100
0
0
20
Number of loyal users of ISP2
(b) d = 60
(c)
40
60
Price difference
ISP 2 has 30 users
Fig. 11 Expected payoffs if ISP2 has different number of loyal users (a, b) and at different minimal price differences c
80
100
Ann. Telecommun. (2010) 65:171–188
183
details, including the impact of the loyalty model and
the impact of the minimal price difference on market
shares and profits.
5.1 Discussion on loyalty models
The price-difference-dependent loyalty game, which
we proposed in Section 3, is only a simple model of the
problem; it may not describe all the characteristics of
the price setting game exactly. First, we present a loyalty model, where the loyalty of the subscribers is based
on the ratio of the prices; then we propose three novel
extensions to handle the price difference dependency.
Each model is constructed to be used in a repeated
price setting game, where the Internet providers simultaneously set their prices in every round.
5.1.1 Original model
Fig. 12 Expected payoff of ISPs in the price-differencedependent loyalty game. a Expected payoff of ISP1 . b Expected
payoff of ISP2
strategy change—from mixed to pure strategies. As a
conclusion, we argue that the expected payoff of an ISP
is really dependent on its beliefs about the other ISPs’
loyal users, as well as on the value of minimal price
difference. An ISP can realize higher profit if its beliefs
are closer to the real market situation.
5 The effect of price-difference-dependent loyalty in
different market scenarios
We have carried out extensive simulations to investigate the ISP price setting game when the user loyalty is
price-difference-dependent. During the simulations, we
use four loyalty models. First, we present our loyalty
models, then simulation results are shown. In particular, we examine a typical simulation scenario with
profits and market shares; after that, we provide more
This model effectively describes the price competition
between ISPs both in deterministic and in stochastic
ways. We only review this model shortly; for a detailed
description, we refer to [2]. The model calculates the
number of switchers who are changing their service
provider at the end of a round. The number of the
switchers is calculated based on the relative price difference between their current and the other ISPs. The
formula provides a ratio that describes the rate of the
switchers and the total users of the ISPs. Li is the user
migration threshold, which constrains the number of
users who switch their ISP at a single step. In practice,
this constraint is due to the fact that termination and
creation of contracts are time-consuming. Based on the
model, for a given ISPi , the number of users it loses to
or gains from other providers in round k is defined as
(k)
min j p(k)
j − pi
(k)
(k−1)
U i = U i
max min
, Li , −Li
(k)
min j p(k)
j + pi
The population change is described by the fraction;
the minmax conditions constrain the number of switchers to the [−Li , Li ] interval.
Next, we will present our three novel pricedifference-dependent loyalty models, which expand
this original model. We will use F =
(k)
min j p(k)
j − pi
(k)
min j p(k)
j + pi
as a
short form of the original model’s function. The change
in the number of users of a given ISP i will be the following in all cases, where G is the distribution function
of the loyalty model:
U i(k) = U i(k−1) max (min (G · F, Li ), −Li )
184
Ann. Telecommun. (2010) 65:171–188
Our first model is the simplest one that can express
the price-difference-dependent loyalty behavior, where
d represents the minimal price difference. Like a step
function, if the price difference is larger than d, all
potential switchers can change their ISPs; if the difference is smaller than or equal to d, none of the users
will change. The distribution function of the thresholdbased model is the following:
1 | min p j − pi | > d
G=
0 | min p j − pi | ≤ d
5.1.3 Uniformly distributed price-difference-dependent
loyalty
The threshold-based loyalty model is a good reference
but it may not accurately describe the user behavior.
Our next model is the uniformly distributed pricedifference-dependent loyalty where all users switch if
the price difference is larger than the d threshold value,
while a fraction of the users change if the price difference is smaller than the threshold. This function involves uncertain user decisions better. The distribution
function of this loyalty model is:
G=
1
| min p j − pi |
d
| min p j − pi | > d
| min p j − pi | ≤ d
scenarios. We use a discrete event simulator [27] where
each ISP has some users as their actual market share.
We suppose the local ISP market is saturated, i.e., no
user leaves or enters the market. This market situation
is valid for those places where everybody can afford
Internet access, and this connectivity is essential. We
normalized the market size to 100%; if an ISP has half
of the market, its market share is 50%. For simplicity
reasons, we suppose that the market is infinitely dividable between the ISPs.
Each ISP can set its own price at the beginning of the
rounds. The price of the Internet access is always between 0 and 100; the upper bound of the price is actually
the maximal price at which users still buy connections.
The ISPs market shares can change at the end of the
rounds because of the user migration. The volume of
the switchers is a function of the applied loyalty model.
If a user leaves his/her ISP, he/she will choose the
cheapest ISP as his/her new service provider; if there
are more than one cheapest ISPs, then they split the
users evenly.
The ISPs want to maximize their current profit in
each round. In our simulations, every ISP uses the
same greedy price setting strategy: they suppose that
the others will not change their current prices. With this
Prices
5.1.2 Threshold-based price-difference-dependent
loyalty
5.1.4 Normally distributed price difference dependency
100
75
50
25
0
ISP1
ISP2
ISP3
5.2 Simulation results
We apply the above introduced loyalty models to
study the ISP price setting game in different market
20
Profits
40
60
Round
80
100
100
75
50
25
0
120
ISP1
ISP2
ISP3
0
20
40
60
Round
80
100
100
75
50
25
0
120
ISP1
ISP2
ISP3
0
Total profits
We note that any kind of cumulative distribution function can be used to model the loyalty of the population.
Market shares
0
The ISPs usually have large numbers of customers;
thus, the users’ price-difference-dependent loyalty distribution function can be modeled as a normal distribution based on the central limit theorem. The mean
of the function is d, which was the threshold price
difference. The distribution function of the normally
distributed price-difference-dependent loyalty model
is the cumulative distribution function of the normal
distribution:
| min p j − pi | − d
1
G=
1+
√
2
σ 2
20
40
60
Round
80
100
2000
1500
1000
500
0
120
ISP1
ISP2
ISP3
0
20
40
60
Round
80
100
120
Fig. 13 Simulation results of price dependent loyalty (threshold = 40)
Ann. Telecommun. (2010) 65:171–188
185
100
75
50
25
0
ISP1
ISP2
ISP3
0
20
40
60
80
100
120
want to loose their actual users. Then, the players want
to maximize their profit so they set a high price where
they can still keep their subscribers. This price is higher
than the previous price with 40, which is the loyalty
threshold. In the next round, the ISPs want to grab
the whole market to have maximal payoffs resulting a
price that is smaller than the previous one minus the
minimal price difference. This pattern continues until a
price where it is better dealing with own customers than
setting too small a price.
In the next figure (Fig. 13b), we present the market shares (number of subscribers) of each ISP. Because of the deterministic model, everyone will retain
their initial market shares, which was a third of the
market.
As the market shares and the prices are equal, the
profits of the ISPs are also the same in every round,
as can be seen in Fig. 13c. The profit graph inherited
the shape of the price’s graph, but the amplitude of
the variation is a third of the minimal price difference,
which is a consequence of the market share ratios.
Finally, we examined the total profits of the ISPs
(Fig. 13d). The total profit is calculated based on the
Market shares
Prices
assumption, the ISP searches for a price where its profit
is maximal and then plays it.
We present the profits of the ISPs, the total profits,
the market shares, and the prices of a price setting game
in Fig. 13, where the threshold-based price-differencedependent loyalty model was used. In this simulation,
40% of the users can change their ISPs in a single round
with three service providers competing on the market,
with equal initial market shares. The threshold value of
the price-difference-dependent loyalty model is 40. The
ISPs play repeated game, where the discount factor is
0.99, the total profit of an ISP is calculated using the
discount factor. Every part of the presented scenario is
deterministic (users choose their providers based only
on the prices; the companies choose their next price
based on common knowledge); thus, the charts of the
three ISPs are overlapped.
We can see the prices of the ISPs in every round
in Fig. 13a. As we have mentioned, each player uses
the same strategy, thus setting exactly the same prices.
The pattern of the graph is a good illustration of the
price-difference-dependent loyalty. The first price is 50
because they want to have some profit but they do not
100
75
50
25
0
0
100
75
50
25
0
ISP1
ISP2
ISP3
20
40
60
80
100
120
60
80
100
120
60
Round
80
100
20
40
60
80
100
120
20
40
60
80
100
120
(c) Uniform distribution loyalty model
ISP1
ISP2
ISP3
40
120
ISP1
ISP2
ISP3
0
120
(d) Normally distributed loyalty model
Fig. 14 Impact of loyalty model on prices. a Original model.
b Threshold-based loyalty model. c Unform distribution loyalty
model. d Normally distributed loyalty model
Market shares
Prices
100
75
50
25
0
20
100
100
75
50
25
0
(c) Uniform distribution loyalty model
0
80
(b) Threshold-based loyalty model
ISP1
ISP2
ISP3
40
60
ISP1
ISP2
ISP3
0
Market shares
Prices
100
75
50
25
0
20
40
100
75
50
25
0
(b) Threshold-based loyalty model
0
20
(a) Original model
Market shares
Prices
(a) Original model
0
ISP1
ISP2
ISP3
100
75
50
25
0
ISP1
ISP2
ISP3
0
20
40
60
Round
80
100
120
(d) Normally distributed loyalty model
Fig. 15 Impact of loyalty model on market shares. a Original
model. b Threshold-based loyalty model. c Unform distribution
loyalty model. d Normally distributed loyalty model
186
Ann. Telecommun. (2010) 65:171–188
previous profits, which are discounted with the discount
factor.
5.3 The impact of the loyalty model
100
75
50
25
0
ISP1
ISP2
ISP3
0
20
40
60
80
100
120
Profits
(a) Original model
100
75
50
25
0
ISP1
ISP2
ISP3
Market shares
Profits
The following plots show the price, market shares, and
profits when three ISPs are on the market; their market
shares are not equal, ISP1 has all the users at the first
round. In a single round, 40% of the users can change
their service provider; the discount factor is 0.99, which
we used to calculate the total profits while the minimal
price difference is 40.
In Fig. 14, we present the prices of the ISPs. The
effect of the models is significant because, if a pricedifference-based loyalty model is used, the prices are
larger than in the original model. We can read the
minimal price difference from the graphs because the
changes of the prices are based on this difference.
Figure 15 shows the market shares of the ISPs. Before the first round, ISP1 has 100% of the market; the
other two providers have zero percentage. The variance of the market shares is lower at price-difference-
dependent loyalty models than it was at the original
model. The threshold-based model can create almost
constant market share distribution after the initial inequalities are equalized.
The profits of the service providers, shown in Fig. 16,
are derived from the above two properties; the profit in
a specific round is the product of the actual price and
the market share. These graphs have the same oscillation as the prices had. The total profits are calculated as
the sum of the discounted profits; therefore, the loyalty
model has a great impact on the total profits. The total
profits at the threshold-based case are at least double
the original model’s total profits.
The original model introduces the loyalty-based pricing; however, it is not realistic that the prices of the
ISPs are so cheap and fluctuating. The threshold-based
loyalty model captures the socio-economic aspect of the
users’ loyalty; a too-small price difference is not enough
for the users to change their ISPs. The combination
of these two models is presented under the uniformly
distributed model, where the uniform loyalty results in
a more realistic market description. The last model, the
normally distributed, incorporates the uncertainty of
user decisions due to its stochastic behavior. A pattern
of real-world scenario would have similar subscriber
loyalty trends as this model. To close this short discussion, we note that any kind of user loyalty profile can
100
75
50
25
0
ISP1
ISP2
0
0
20
40
60
80
100
20
20
40
60
80
100
120
Market shares
Profits
ISP1
ISP2
ISP3
0
20
40
60
Round
80
100
100
120
ISP1
ISP2
0
20
40
120
(d) Normally distributed loyalty model
60
Round
80
100
120
(b) Threshold: 10
Market shares
Profits
ISP1
ISP2
ISP3
0
80
100
75
50
25
0
(c) Uniform distribution loyalty model
100
75
50
25
0
60
Round
(a) Threshold: 0
(b) Threshold-based loyalty model
100
75
50
25
0
40
120
100
75
50
25
0
ISP1
ISP2
0
20
40
60
Round
80
100
120
(c) Threshold: 30
Fig. 16 Impact of loyalty model on profits. a Original model.
b Threshold-based loyalty model. c Unform distribution loyalty
model. d Normally distributed loyalty model
Fig. 17 Impact of minimal price difference on market shares. a
Threshold: 0. b Threshold: 10. c Threshold: 30
Ann. Telecommun. (2010) 65:171–188
187
be included in our price-difference-dependent loyalty
framework; therefore, real-world loyalty samples, e.g.,
created based on a survey, can also be used.
5.4 Impact of minimal price difference
Profits
We have seen in the game theoretical analysis that the
minimal price difference has an effect on ISPs’ strategies; therefore, it has an impact on the payoffs as well.
We investigate the impact of the price difference in a
simulation where two players exist on the market with
unequal initial market shares (100% and 0%), 10% of
the users can change their ISPs in a single round, the
discount factor is 0.95, and the border price is 100. In
Fig. 17, we present the market shares of three different
price differences (0, 10, 30), while Fig. 18 shows the
profit of these cases.
In case of zero price difference, i.e., every user
always chooses the cheapest subscription, the market
shares are oscillating around the fair distribution. If
there exist price-difference-dependent loyalty on the
market, then the market shares converge to the fair
distribution without oscillation. In addition, the profits
of the ISPs are also constant after the end of convergence at price-difference-dependent loyalty models.
The cause of different profits is that the equilibrium
price is higher if the minimal price difference is larger.
100
75
50
25
0
ISP1
ISP2
0
20
40
60
Round
80
100
120
Profits
(a) Threshold: 0
100
75
50
25
0
ISP1
ISP2
0
20
40
60
Round
80
100
120
Profits
(b) Threshold: 10
100
75
50
25
0
6 Conclusion
In this paper, we have demonstrated how ISPs price
Internet access for disloyal users under uncertainty,
taking into account their impact on prices, profits, and
market shares, from both theoretical and empirical
points of view. In empirical terms, we carried out a survey on the customer loyalty issue for the Hungarian ISP
market. The analysis of our own survey, as well as the
results from other empirical researches, showed that
customer loyalty in ISP market exists, and it depends
strongly on price difference.
We have created a game-theoretic model that contains this kind of customer loyalty and we have formulated price-setting strategies for ISPs. It turned out that
if the customers of the ISP market are loyal enough, the
ISPs can cooperate; namely, they do not compete in the
prices; thus, they can sell Internet access at the highest
possible prices.
ISPs have to deal with uncertainties during their
price decisions; we have seen that these uncertainties
have an effect on the profits and market shares of the
ISPs. On the one hand, providers have to change their
prices time after time because of the market conditions
as an uncertain behavior. On the other hand, each
provider knows only its own subscribers; thus, it has
to make price decisions based on its beliefs. Using
Bayesian games, we showed how much the profit has
changed under different beliefs.
In addition, we have presented three novel loyalty
models that describe the properties of price-differencedependent loyalty. Based on simulation results, we
have presented the impact of the different loyalty models and the impact of the minimal price difference
on ISPs’ prices, market shares, and profits. Service
providers have to change their prices paying attention
to the price difference; a too small price reduction
might not result in enough new customers. An ISP can
expect more profit if it knows better the price difference
of the population, as it has an important impact on the
prices. As a future research, we plan to investigate the
impact of operational costs on ISP pricing for disloyal
customers, with or without uncertainty.
ISP1
ISP2
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0
20
40
60
Round
80
100
120
(c) Threshold: 30
Fig. 18 Impact of minimal price difference on profits. a Threshold: 0. b Threshold: 10. c Threshold: 30
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