How to Price Internet Access for Disloyal Users under Uncertainty I L´aszl´o Gyarmati∗, Tuan Anh Trinh Department of Telecommunications and Media Informatics Budapest University of Technology and Economics Magyar tud´ osok krt. 2, Budapest, Hungary H-1117

Abstract In [1] it is demonstrated that customer loyalty can have a real impact on Internet Service Provider (ISP) pricing. However the paper only dealt with simple loyalty models and complete information, i.e. ISPs fully know about others pricing decisions. In this paper, a comprehensive analysis of ISP pricing for disloyal users under uncertainty is presented, both from the 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 issue in ISP market is strongly dependent on price difference. We have created a game theoretic model dealing with price difference dependent loyalty. After that we investigated the stability of the model and we formulated the conditions of the pricing strategies. Our game theoretic analysis shows that if the price difference constant of the market is large enough the Internet Service Providers can cooperate, namely do not compete in the prices, thus they can sell Internet access on the highest possible prices. Next, we investigate the price setting strategies of ISP under uncertainty by using Bayesian games. Our analysis shows that on the one hand providers have to change their prices time after time because of the market conditions. On the other hand each provider knows only her own subscribers, she has to make price decisions based on her beliefs. These uncertainties can effect the profits of the ISPs. Finally, we use these observations to quantify the effects I

This paper has been partially supported by HSNLab, Budapest University of Technology and Economics, http://www.hsnlab.hu ∗ Corresponding author Preprint submitted to Elsevier

November 5, 2009

of uncertainty on ISPs’ profits. Key words: 1. Introduction Local Internet Service Providers (ISPs) who are at the edge of the ISP hierarchy provide Internet access for billions of end-users. Economical perspectives of this subscriber-local ISP relation have been examined in the recent years still it will be in the spotlight in the forthcoming years because economic incentives are important aims in project dealing with future network design (e.g. NSF FIND [2]). A number of papers deal with the problem of modeling ISPs interactions using game-theoretical methods like [3] [4] [5]. Game-theory provides a mechanism to examine optimization problems where multiple players, usually with opposite interests, want to maximize their utilities (e.g. minimize their costs or maximize their profit). Almost all of these publications model users very simplified as the users are looking always for the cheapest Internet access subscription. However economists and market researchers have recognized long ago that users tend to be loyal to a company or a band. They developed pricing models taking into account that loyal users purchase their favorite product even if there exists cheaper substitute product on the market. [6] 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 there loyal customers, therefore the Nash equilibrium of the game changes resulting higher utilities. There are a number of publications dealing with user loyalty in the telecommunication sector. These surveys show that user loyalty does exists in the sector. To give a global ISP loyalty picture, hereinafter we shortly review some of them emphasizing the most important findings. The 2005 Walker Loyalty Report for Information Technology shows that 38 percent of the enterprise customers were truly loyal to their Internet Service Providers [7]. 30 percent of the customers were high-risk users meaning they have low commitment and typically do not intend to stay at their current providers. These ratios are almost the same as the ratios at other IT sectors. Based on the report, quality, value and price are the key drivers of loyalty. Choice survey states that 90 percent of the respondents had not changed their ISP in the previous 12 months including contract-users as well [8]. 29 percent of 2

the customers were satisfied with their current ISP in 2007. Based on Choice survey the most important factor in choosing a service provider is the price of the access. The PC Advisor Broadband Survey 2008 revealed that the vast majority of respondents have been with their ISP for ages well beyond the minimum subscription period [9]. These reviews present loyal intentions in the United States and UK. In 2002 there was a survey in Taiwan investigating customer loyalty toward ISPs [10]. The survey shows that there exists customer loyalty in Taiwan too. It was found that perceived value, service satisfaction and even future ISP expectancy have impact on customer loyalty. These works and a recent work [1] clearly demonstrate that customer loyalty has a significant impact on pricing strategies of ISPs. For example almost every provider offers Internet access with contract too. From ISP point of view this contract ensures the loyalty of the users for a period, resulting fix income. From user point of view her subscription price is lower with a contract or the user receives gifts for her loyalty. However, [1] only dealt with simple customer loyalty models. More precisely, the paper did not deal with price difference dependent customer loyalty issue, which is supported in part by the surveys. Price difference dependent loyalty means that a subscriber will stay at her current Internet Service Provider until there does not exist an other ISP whose price is significantly lower than her current access price. Furthermore [1] dealt only with complete information, i.e. the ISPs fully know about other ISPs’ pricing decisions. We believe that this is not usually the case in practice. From the arguments above, this paper tries to provide the answers for the above mentioned questions by investigating the issue from both empirical and theoretical perspectives. We carry out - by our own - a survey on customer loyalty in Hungarian ISP market. It is suggested, by our own survey as well as other surveys that customer loyalty is strongly correlated with price difference of ISPs. To model uncertainty in ISPs’ pricing decision, we use the tools of game theory and Bayesian games in particular. We applied a few assumptions in order to model the ISP price setting problem. We suppose that ISPs offers flat-rate subscriptions, because deploying a usage-based sophisticated price scheme would be in general 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 subscription is still popular nowadays in Japan, 20 percent of DoCoMo’s mobile users have this kind of subscription [11]. Our 3

next assumption was that the consumer demand for Internet access is constant 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 modeled that the users have single reservation price. In this case users will buy Internet access until its price is lower than the reservation price otherwise the users will not purchase it. 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. The paper is structured as follows. First, in Section 2, we provide a comprehensive empirical analysis of ISP customer loyalty in Hungarian Internet access market based on survey carried on by ourselves. Section 3 provides game-theoretic models for price difference dependent customer loyalty issues with complete information. In this Section, we also show the impact of price difference dependent customer loyalty in terms of Nash equilibrium by detailed game-theoretic analysis. Section 4 extends the results of Section 3 to deal with uncertainty of ISP price setting decisions by using Bayesian games. Section 5 provides simulation analysis for different customer loyalty models for the price difference dependent case. Section 6 concludes the paper. 2. An empirical analysis of subscriber loyalty intentions towards Internet Service Providers in Hungary We can examine user loyalty from two aspects: the first one is the aspect of service providers and the other one is subscribers’ opinion. The providers have exact information about their users behaviour based on selling data while users can judge their own preferences and loyal attitudes. For the first aspect we have contacted mayor Internet Service Providers in Hungary to get real world data about user loyalty and their price setting strategies. They refused to give out this kind of informations because of the following reason: the number of customers and their loyalty is a very sensitive company secret, companies can have disadvantages on the market if they would make these data public, the price strategy as well as the profits of the companies are also private information. We found some cumulative statistics at National Communication Authority about the number of subscribers of the service providers in 2007 along with a churn number that is how many people left the company in the last

4

Table 1: The number of subscribers and switching users at seven main Internet Service Providers in Hungary at the end of 2007

Name of the ISP

Average subscriber number

Switch percentage

22334 50461 42156

Number of switching users 725 585 135

DIGI FiberNet GTSDataNet Invitel Magyar Telekom UPC Enternet

14568 228786

196 21497

1.35% 9.40%

240558 34653

16041 3520

6.67% 10.16%

3.25% 1.16% 0.32%

six months [12]. Based on the result we can conclude that at most 10 percent of the users have switched their ISP in this six-month long period. The detailed subscriber numbers are shown in Table 1. These ratios 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 ISPs and their loyal intentions. We asked people about their personal ISPs not about their companies’ contractors because we are interested in personal decisions and loyalties. 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. As a result we received empirical data about user loyalty. Based on the received answers we can state that the survey was filled out by a wide community (778 people) where every age-group were represented (less than 18 years of age 3%, 19-24 years of age 53%, 25-35 years of age 34%, 36-45 years of age 6%, more than 46 years of age 4%). The questions were answered by both women (32 percent) and men (68 percent) and their educational backgrounds were also diversified. These describing statistics confirm that the empirical analysis of the survey is a good illustration of the loyalty of the whole community. Accordingly we can argue that not only in the USA, UK and Taiwan but also in Hungary significant user loyalty exists 5

Table 2: Monthly price of current Internet subscription

Monthly Price (relative to the average net income) 1.5% 4% 8% 12.5% >12.5%

Frequency

Percent

21 249 426 64 6

2.7% 32.5% 54.8% 8.4% 0.8%

Cumulative Percent

2.7% 35.2% 90.9% 99.2% 100%

on the Internet Service Provider market. Before presenting the empirical results we want to say a few words about the Hungarian ISP market and about the prices to help the interpretation of the outcomes. In Hungary there are three main type of Internet connections: xDSL (mainly ADSL), cable and mobile Internet access (3G). In August 2008 there were 497 thousands ADSL subscribers, 690 thousands cable subscribers. [13] There are four mayor Internet access providers in the cable market, they own 70 percent of the whole market. There are three companies on the mobile Internet market, they provide Internet access up to 7.2 Mbps. We will present Internet access prices based on the survey but the volume of the price can be hard to judge in case of a foreign country. We compared the prices to the average net income of a Hungarian in the first half of 2008 [14], we illustrate the ratio of the price and the average net income in the followings. In Table 2 we present the statistics of the monthly price of the Internet subscriptions based on our survey. Most of the surveyed people had an Internet access with moderate price (4-8% of the average salary) but there were also a few ones who had really expensive Internet access. This price distribution shows that our survey managed to cover subscribers of all price category. User loyalty can be measured by different approaches for example based on the number of switches, number of years to be a customer of an ISP. As a start we examined the loyalty history of the subscribers. We asked in the survey how many times has the person switched her service provider in 6

the last five years. As the result on Figure 1 shows around 60 percent of the questioned people do not change their Internet Service Providers in the last five years. This result forecasts that there is significant loyalty towards Internet access providers in Hungary.

Figure 1: Number of ISP switches in the last five years

Loyal users usually buy their Internet access from the same ISP years after years thus a loyal user has the same ISP since years. The type of connection has always an effect on the loyalty intentions. Every communication method has its own specialties: wired Internet access do not allow users to switch easily between providers (e.g. change between cable and ADSL or deployment issues), contrarily mobile Internet access can be used in almost every place and the operator can be changed easily. We present on Figure 2 the frequencies of how long a user has its current ISP based on the type of connection. Not all the connection types has a lot of users for long times because they have not been available earlier (e.g. mobile Internet, FTTX). A lot of users have not changed their service providers in the last two or more years resulting that these subscribers can be named as loyal users. As we mentioned above Internet providers offer services with contracts in order to have the users their own subscribers for a period. Only 17 percent of our answerers do not signed a loyalty contract when they bought their Internet access. This ratio is really interesting because this means that 80 percent of the customers have to be loyal for at least the duration of the 7

Figure 2: User loyalty at different type of connection

contract. The causes of signing a contract verifies that service providers set their prices based on loyalty intentions. Almost the half of the persons (49.1 percent) signed a contract because a cheaper price, in this case the signing was optional. Contrarily in 29.4 percent 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 representative parameter which we show on Figure 3. We showed persons without a contract with zero month. The most frequent length is a one-year long contract but a two-year long one is also popular, with these contracts Internet Service Providers can keep customers for a really long period. One of the questions of the survey was a bit provocative in sense that we wanted to know which is the minimal price difference between their current subscription and a subscription of an other ISP when they would leave their current ISP for the other one. It was supposed that the two ISPs would offer exactly the same service including connection speed, help center, etc. We received surprising answers as we can see on the bar chart at Figure 4 and in Table 3 where the exact numbers and percentages of the cases are shown. Only around 5 percent of the answers sad that there do not exists a price difference where they would live their current ISPs. These subscribers are really loyal to their service providers. At the same time the remaining 95 percent sad that there is a price difference where they would become 8

Figure 3: Duration of the contract

disloyal and would switch their providers. Based on the results we think that modeling user loyalty based on the minimal price difference to switch can be a realistic description of the ISP pricing problem. The first idea what everybody would say is to model loyalty based on the price ratios. On the one hand it can represent the relationship between the two price but on the other hand it does not have any information about the social aspects. We illustrate this with the following short example: consider two countries, a rich one and a poor one. In each country there exist two IPSs, they provide Internet access for 5$ and for 10$. The price ratios are the same in both countries (0.5) but it is clear that there will be much more switchers in the poor country, where 5$ (the price difference) worths a lot more than in the richer country. The price of the current subscription and also the minimal price difference had to be selected from a list of possible values thus the results are discrete probability variables. The connection of two variables can be expressed with correlation but it 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 this crosstab where every cell stores the number of occurrences of the specific pair. There is a correlation between the two variables because the minimal price difference is usually smaller than the actual 9

Figure 4: Minimal price difference to switch to an other ISP

subscription price. We examined the relationship between the number of years to be a customer of the current ISP and the minimal price differences as well what we present on Figure 5. It can be noticed that regardless of the years there are similar price differences where the customers will switch their ISPs.

Figure 5: Relation of the user loyalty and minimal price

10

Table 3: Minimal price difference to switch Internet Service Provider

Price difference (relative to the average net income) 0.5% 1% 1.5% 4% 8% never

Frequency

Percent

38 96 294 270 31 36

5.0% 12.5% 38.4% 35.3% 4.1% 4.7%

Cumulative Percent

5.0% 17.5% 55.9% 91.2% 95.3% 100.0%

Table 4: The connection between the monthly price and the minimal price difference to switch (number of answers)

Relative Minimal price difference to switch (relative) Monthly Price 0.5% 1% 1.5% 4% 8% never 1.5% 11 3 1 0 1 5 4% 15 64 125 23 1 17 8% 11 26 162 202 14 11 12.5% 0 3 5 40 14 2 >12.5% 1 0 0 4 1 0

11

We have asked several more questions about user loyalty in the survey. These questions and the answers of them can be seen in Appendix C. We can summarize this section with the following conclusions: there do exists user loyalty toward Internet Service Providers regardless of the countries, service providers set their subscription prices taking into account loyal customers (e.g. offering discounted prices with contracts), furthermore price difference has an impact on user loyalty, loyal users can become disloyal if there is a cheap enough other offer on the market. To conclude this section we present its key observations: • User loyalty has an impact on price setting strategies of Internet Service Providers. On the Internet access market the majority of the users have loyal intentions towards their service providers. • 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. • Internet Service Providers do not have exact information about the customers of their competitors, they have only believes about it. The users select their access providers based mostly on their impression not on exact parameters. Because of this two reasons the price competition between ISPs has uncertain parameters, resulting non-deterministic decisions. 3. Price difference dependent loyalty In the previous section we have seen that user loyalty is an important factor in the ISP price competition. User loyalty can be formalized in a game theory model and its properties can be examined. For an introductory paper we refer to [1] 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 the old and the possible new service providers has a significant impact on user loyalty. Therefore we create a price setting game where this type of loyalty is taken into consideration. After reviewing the basic notions of game theory we will present our game theory model where the price difference has an important role and we will examine the stability criteria of the game. After that we will analyze how much is the price difference where the ISPs can play their equilibrium strategy months after months. Finally we will see what is the situation if there are disloyal users on the market. 12

3.1. Basic notions of Game Theory Game theory provides efficient methods to handle multi-person decision theory, this section reviews basic notions of game theory that we use in this paper. For a detailed game theory introduction we refer to [15]. We will only deal with rational decisions namely every person wants to select her best possible choice which maximize her utility. A non-cooperative game, where players do not cooperate with each others, can be formalized as follows: N = {1, 2, . . . , n} is the set of players, where 1, . . . , n are the individuals who are playing, Si is the strategy set of Player i, who can select her strategy si ∈ Si from the set. Every player has 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 her strategy set. s = (s1 , s2 , . . . , sn ) ∈ S is a strategy profile where si is the strategy of player i. s−i denotes the strategies of players except Player i. Nash equilibrium describes a strategy profile which has good properties, namely none of the players can have more payoff if only she changes her strategy. Formally, s∗ ∈ S strategy profile is a Nash equilibrium point, if fi (s∗i , 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 she selects only a single strategy with one probability. Contrary, if a player selects more strategies with positive possibility 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 as 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 a non-complete information or Bayesian game not all the players have the same knowledge.

13

3.2. Game theoretic modeling of price difference dependent loyalty First we investigate a price setting game where only two service providers (players) exist. Customers are split into two partitions upon their loyalty: in the first group are l1 customers loyal to ISP1, in the second group are l2 loyal users of ISP2. 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 user’s ISP is more than the other ISP’s price plus d then the user will be a switcher thus the user leaves 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. The service providers set their prices simultaneously then the users select their access providers. The payoff function of the ISPs can be expressed as Πi =

   (li + lj )pi  

li pi 0

if pi < pj and |pi − pj | > d if |pi − pj | < d if pi > pj and |pi − pj | > d

(1)

Figure 6 illustrates the payoff function (Equation 1) in two different scenarios. We present the payoff of Player 1 at different prices (p1 ) while the price of Player 2 is fixed. Figure 6(a) presents the payoff function if the border price is not smaller than the price of Player 2 plus the price difference, while Figure 6(b) shows the payoff if the border price is lower.







   

  



(a) p2 + d ≤ α



(b) p2 + d > α

Figure 6: Illustration of the payoff function

The formal definition of the game is the following, we will refer it as G1 : • Players: the Internet Service Providers, N = 2, Player i has li loyal customers 14

• Strategies: the price of the Internet access, the decision of Player i is pi , pi ∈ [0, α], players can have only pure strategies, they play only once, it is a single-shot game • Payoff functions: the payoff of the Players i is described in Equation 1 • Information: complete, players know α, d, l1 , l2 Proposition 1. Consider G1 , the ISP price setting game with price difference dependent loyalty. (α, α) is the only pure strategy Nash equilibrium of 1 2 the game, if l1 l+l ≤ αd and l1 l+l ≤ αd . At the Nash equilibrium point the 2 2 payoffs of the players are Π1 = l1 α and Π2 = l2 α. Before the formal proof of the proposition we illustrate on Figure 7 what is the minimal price difference where a Nash equilibrium do exist. We can see that an equilibrium can exist on a market where one of the ISPs has a lot more subscribers than the other if the price difference is large enough.

100

Minimal price difference

95 90 85 80 75 70 65 60 55 0 50 0

20 20

Numbe

ers l us 40

60

40

r of loya

60

l users

80

80

of ISP2

100

100

ber

Num

of IS

P2

ya

of lo

Figure 7: 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

Proof 1. None of the players would set a price higher than α because then their payoff would be zero. Thus, the support of the equilibrium is [0, α]. In which cases is it worth to undercut the other ISP’s price more than d to get the whole market? If ISP2 sets a price p2 the first ISP can grab all 15

users if 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. First consider the [0, α − d] interval where the above mentioned prices are exact. ISP1 would compete when the following holds: (l1 + l2 )(p2 − d) > l1 (p2 + d) l1 p2 − l1 d + l2 p2 − l2 d > l1 p2 + l1 d l2 p2 > 2l1 d + l2 d 2l1 d p2 > d + l2 Let A1 := d + 2ll12 d . ISP1 would not compete if the price of ISP2 is below A1 , in this case its price will be p1 = p2 + d. In the [α − d, α] interval ISP1 can grab the users of the other ISP if the payoff in the first case is larger than in the second one. The maximal upper price is α because at p2 + d the payoff would be zero: (l1 + l2 )(p2 − d) > l1 α p2 > d +

l1 α l1 + l2

If ISP2 sets p2 = α ISP1 competes if the following stands: (l1 + l2 )(α − d) l1 α + l2 α − (l1 + l2 )d l2 α l2 l1 + l2

> l1 α > l1 α > (l1 + l2 )d d > α

If we look at Figure 8 we can see that this game can have a pure Nash equilibrium point at (α, α), if none of the ISPs have incentive to compete at α. At the other prices, where the graphs are very close to each other, do not exist any equilibrium because on the first hand the price difference have to be greater the d to grab, on the other hand to hold users price difference has to be less or equal to d thus the graphs do not have any intersection. 16











  







Figure 8: Best response function of the service providers

The requirement of the equilibrium is that all of the ISPs have best response at α: l2 d ≤ l1 + l2 α l1 d ≤ l1 + l2 α Because ISP1 has the larger user base, the requirement of the pure Nash 1 2 equilibrium is l1 l+l = 1 − l1 l+l ≤ αd . ¤ 2 2 The proposition means that if the market’s price difference value is large enough than the access providers do not have to compete with each others, they can sell Internet access on the highest possible price (α) resulting maximal profits. The game can be generalized to N players, which we denote as G2 . The formal definition of G2 game is the following: • Players: the Internet Service Providers, N = 1, . . . , n, Player i has li loyal customers 17

• Strategies: the price of the Internet access, the decision of Player i 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 Players i is based on the described function of Equation 1 • Information: complete Proposition 2. The N-player game G2 , where ISPi has li loyal users and the minimal price difference is d, has only one pure Nash equilibrium, (α, α, . . . , α) if the following holds: minj lj d 1− P ≤ (2) α j lj The payoff of Player i is Φi = li α at the Nash equilibrium point. Proof 2. See Appendix A.1. This pure Nash equilibrium is also Pareto efficient equilibrium, because none of the ISPs can have larger payoff without harming the others. We can summarize the findings of N-player ISP price setting game where price difference dependent loyal subscribers exist as follows: P j lj ≤ 1. 1 − min lj j

d α

is the condition of Nash equilibrium

2. The equilibrium strategies of the players are (α, α, . . . , α) 3. The payoffs of the players at the equilibrium are l1 α, l2 α, . . . , ln α 3.3. Repeated game The condition of the pure strategy Nash equilibrium is important if we play only one game. Internet Service Providers are playing usually their price setting game repeatedly, e.g. they set a new price every months. We can investigate our price difference dependent loyalty model as a repeated game. There is two ISPs in the market, ISP1 has l1 loyal customers while ISP2 has l2 . We model the repeated game using discounted payoffs where the price is discounted at each step with discount factor Θ ≤ 1. As usually, d is the minimal price difference where a loyal user will switch its service provider. We will calculate the discount factors for the following strategy: an ISP will cooperate until the other ISP will set lower price than her price with at least 18

d then will not cooperate any longer. Cooperation means, that both ISP set (α, α) as their price and they milk their own loyal customers. They will not compete with each other in order to hold the prices high. If an ISP does not cooperate she always want to preserve her loyal customers, thus she will set d as her price. We suppose that an ISP can loose her loyal customer only for that specific round of the game, in the next round she will have again her loyal users, e.g. ISP1 will have l1 loyal user at the beginning of every round. An ISP’s profit is Πcoorp = li α if she cooperate and Πnot = li d if she does not i i cooperate. We refer this game as G3 , its formal definition is the following: • Players: the Internet Service Providers, N = 2, Player i has li loyal customers • Strategies: the price of the Internet access, the decision of Player i is pi , pi ∈ [0, α], players can have only pure strategies, the players play multiple round, the discount factor of the repeated game is Θ • Payoff functions: the payoff of the Players i is described in Equation 1 • Information: complete, players know α, d, l1 , l2 and the strategy Proposition 3. The strategy profile ”Set α as a price until the other player deviates than play d as a price” is a sub-game perfect equilibrium for the price difference dependent loyalty repeated game G3 , if Θi >

(l1 + l2 )(α − d) − li α (l1 + l2 )(α − d) − 2li d

(3)

holds, where Θi is the discount factor of ISPi. Proof 3. The prices and the payoff functions are continuous that means we can proof the sub-game perfect equilibrium using the one-step deviation property. Namely, the game is sub-game perfect if none of the ISPs has incentive to change their strategy because if she deviates her discounted payoff will be less than if she does not deviate. The strategy is sub-game perfect if the following holds: not Πcoorp (k, ∞) > Πdev i i (k) + Πi (k + 1, ∞)

19

Without loss of generality let us suppose that ISP2 deviates at step k then the discounted payoffs are: ∞ X

Θi Πcoorp > Θk π2dev + 2

i=k

∞ X

Θi Πnot 2

i=k+1

We know the payoffs of the different cases: ∞ X

Θi l2 α > Θk (l1 + l2 )(α − d) +

i=k

∞ X

Θi 2dl2

i=k+1

The price of ISP2 can be at most α−d at the deviation step in order to get all the users but after that she can only set d + d = 2d as her maximal price to keep her own users. Let’s calculate the minimal discount factor where this game is sub-game perfect! ∞ X

> π2dev + Θi Πcoorp 2

∞ X

Θi+1 Πnot 2

i=0

i=0

Πcoorp 2

ΘΠnot 2 1−Θ 1−Θ > (1 − Θ)π2dev + ΘΠnot Πcoorp 2 2 dev not dev − π > Θ(Π − π ) Πcoorp 2 2 2 2 π2dev − Πcoorp 2 Θ > π2dev − Πnot 2 > π2dev +

Using this finding we can create lower bound constraints on ISPs’ discount factors: (l1 + l2 )(α − d) − l1 α (l1 + l2 )(α − d) − 2l1 d (l1 + l2 )(α − d) − l2 α Θ2 > (l1 + l2 )(α − d) − 2l2 d

Θ1 >

If these constrains are satisfied and the ISPs use the described strategy the repeated game will be sub-game perfect. ¤ The meaning of this proposition is that if the price difference is large enough the ISPs can play the Nash equilibrium strategy repeatedly resulting the maximal payoff. 20

3.4. Dealing with disloyal users We have seen that the price difference dependent loyalty ISP price setting game has one pure strategy equilibrium. In this game each ISP had the same price difference (d) but this assumption is not always true in the real world. One ISP can have more hard-core loyal subscribers than the others. We can model this if we introduce separate minimal values (di ) for every ISP. Using this we can model disloyal users if we add a new virtual service provider to the market with users who have zero minimal price difference. First we deal with the two players game afterwards we generalize our model for N players. In this game we have ISP1 with l1 loyal users and with d1 minimal price difference and ISP2 with l2 subscribers and d2 price difference. The demand is constant between 0 and α. The payoff functions of the ISPs are:    (l1 + l2 )p1

Π1 =   Π2 =

l1 p1 0

if p1 < p2 and |p1 − p2 | > d2 if |p1 − p2 | ≤ d1 if p1 > p2 and |p1 − p2 | > d1

(4)

l2 p2 0

if p2 < p1 and |p2 − p1 | > d1 if |p2 − p1 | ≤ d2 if p2 > p1 and |p2 − p1 | > d2

(5)

   (l1 + l2 )p2  

The formal definition of G4 game is: • Players: the Internet Service Providers, N = 1, . . . , n, Player i has li loyal customers with di price difference • Strategies: the price of the Internet access, the decision of Player i is pi , pi ∈ [0, α], players can have only pure strategies, they play singleshot game • Payoff functions: the payoff of the Players i is based on the described function of Equation 4 and 5 • Information: complete Proposition 4. If the ISPs’ subscribers have different price differences there exists a pure strategy Nash equilibrium of game G4 if the following conditions

21

hold: l2 d2 ≤ l1 + l2 α l1 d1 ≤ l1 + l2 α

(6) (7)

Proof 4. See Appendix A.2 Why is it so important to have different price differences? The answer is simple: disloyal users. With the above introduced model we can handle disloyal users as well. We can create a virtual ISP which has the disloyal users. The price sensitivity of the subscribers of the virtual ISP is small (around zero). If we look at the constraints we can see that if we have disloyal users the game will no longer have pure strategy Nash equilibrium: the value of dαv will be zero but the left side of the inequations will be positive. As a closing word we can state that if there are disloyal users on the market the Internet Service Providers can not play their pure equilibrium strategies, they have to set their prices under uncertainty. 4. Price setting under uncertainty Uncertainty can have at least two different meaning in our price setting games. On the one hand a service provider might play mixed strategy because there do not exist a pure strategy equilibrium. In this case the ISP has to set access prices with certain probabilities in order to have maximal profit. The price of the ISP is not the same every time thus the price is uncertain. On the other hand an Internet Service Provider only knows exactly the number of her users. The number of other ISPs’ subscribers as well as the price difference are unknown. The ISP can set her price based on her beliefs which results an uncertain price setting strategy. In this section we will come around this two kind of uncertainty. 4.1. Mixed strategy Nash equilibrium and expected payoffs In Section 3 we presented a game theoretical model of ISP price setting game where loyalty was price dependent. We have seen that there exist pure strategy Nash equilibriums but not in every possible market scenarios. In these cases providers have to play mixed strategies instead of the pure ones. 22

We will find 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 can be formulated as: Πi (p) =

   (li + lj )pi

li pi   0

pi < pj − d |pi − pj | ≤ d pj < pi − d

(8)

The formal definition of game G5 is as follows: • Players: the Internet Service Providers, N = 2, Player i has li loyal customers • Strategies: the price of the Internet access, the decision of Player i li is pi , pi ∈ [0, α], players can have only mixed strategies thus li +l > αd j holds, they play a single-shot game • Payoff functions: the payoff of the Players i is described in Equation 8 • Information: complete, players know α, d, l1 , l2 Proposition 5. In the two-player price setting game G5 , where the ISPs have price difference dependent loyal users, the mixed equilibrium strategies have the following cumulative distribution functions and supports: F1 (p) =

 0     p−

p1
, > El1 + El2 α El1 + El2 α We can 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:  0    

F1 (p) = 

F2 (p) =

p1
d | min pj − pi | ≤ d

5.1.3. Uniformly distributed price difference dependent loyalty The threshold based loyalty model is a good reference but it may not describe accurately the user behaviour. Our next model is the uniformly distributed price difference dependent loyalty where all of the 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 pj −pi | d

| min pj − pi | > d | min pj − pi | ≤ d

5.1.4. Normally distributed price difference dependency The Internet Service Providers usually have large number 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 will be 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: Ã

Ã

1 | min pj − pi | − d √ G= 1+Φ 2 σ 2

!!

We note that any kind of cumulative distribution function can be used to model the loyalty of the population. 32

5.2. Simulation results We have used the above introduced loyalty models to study the ISP price setting game in different market scenarios. We used a discrete event simulator [16] where each ISP has some users as their actual market share. We suppose that the local ISP market is saturated meaning that no user leaves or enters the market. This market situation is valid for those places where everybody can afford an Internet access and this connectivity is essential. We normalized the market size to 100%, if a ISP has the half of the market, his market share is 50%. For simplicity reasons we suppose that the market is infinitely dividable between the ISPs. Each ISP can set her 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 users still buy connections. In every simulation the ISPs have an initial market share. This market share 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 her ISP she will choose the cheapest ISP as 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 assumption the ISP searches for a price where her 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 on Figure 12 where the thresholdbased price difference dependent loyalty model was used. In this simulation 40 percent of the users can change their ISPs in a single round. Three service providers compete on the market, they have 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 this 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 on Figure 12(a). As we have mentioned, each player uses the same strategy, thus sets 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 want to loose their actual users. Then the players 33

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want to maximize their profit so they set a high price where they still can 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 to whole market to have maximal payoffs resulting a price which is smaller then the previous one minus the minimal price difference. This pattern continues until a price where it is better dealing with own customers than setting a too small price. On the next figure (12(b)) we plotted the market shares (number of subscribers) of each ISPs. Because of the deterministic model everyone will retain her initial market share, 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 it can be seen on Figure 12(c). The profit graph inherited the shape of the price’s graph but the amplitude of the variation is third of the minimal price difference. This is the consequence 34

of the market share ratios. Finally we can examine the total profits of the ISPs (Figure 12(d)). The total profit is calculated based on the previous profits which are discounted with the discount factor. 5.3. The impact of the loyalty model 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 percent of the users can change their service provider. The discount factor is 0.99 which we used to calculated the total profits. The minimal price difference is 40. On Figure 13 we plotted the prices of the ISPs. The effect of the models is significant because if a price difference based loyalty model is used the prices are larger than at 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 14 shows the market shares of the ISPs. Before the first round ISP1 has 100 percent of the market, the other two providers have 0 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 on Figure 15.) is 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. Figure 16 presents the total profits of the ISPs which is the sum of the discounted profits. The graphs illustrate that the loyalty model has a great impact on the total profits. The total profits at threshold-based case are at least double of the original model’s total profits. 5.4. Impact of minimal price difference We have seen in the game theoretical analysis that the minimal price difference has an effect on ISPs’ strategies therefore it has an effect on the payoffs as well. We investigate the impact of the price difference in a simulation case where two players exist on the market with unequal initial market shares (100 and 0 percents), 10 percent of the users can change ISP in a single round, the discount factor is 0.95 and the border price is 100. On Figure 17 35

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Figure 13: Impact of loyalty model on prices

we plotted the market shares of three different price difference (0,10,30) while Figure 18 shows the profit of these cases. In case of zero price difference, i.e. every user chooses always the cheapest subscription the market shares are oscillating around the fair distribution. If there exist price difference dependent loyalty on the market than the market shares converge to the fair distribution without oscillation. 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.

36

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Figure 14: Impact of loyalty model on market shares

6. Conclusion In this paper, a comprehensive analysis of ISP pricing for disloyal users under uncertainty is presented, both from the 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 issue in ISP market is strongly dependent on price difference. We have created a game theoretic model which contains this kind of customer loyalty. After that we investigated the stability of the model and we formulated the conditions of the pricing strategies. It turned out that if the price difference of the market is large enough the Internet Service Providers can cooperate, namely do not compete in the prices, thus they can sell Internet access on the 37

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Figure 15: Impact of loyalty model on profits

highest possible prices. In some cases the providers can apply this strategy month after month without a price competition. Disloyal users who are always looking for the cheapest access can force companies to alter their prices in order to maximize their profits. Next, we investigated the price setting strategies of ISPs under uncertainty by using Bayesian games. We found that on the one hand providers have to change their prices time after time because of the market conditions which is some kind of uncertain behaviour. On the other hand each provider knows only her own subscribers thus she has to make price decisions based on her beliefs. These uncertainties can effect the profits of the ISPs. We also showed that how much the profit can change under different believes. In addition, we presented three novel loyalty model which can describe the properties of price difference dependent loyalty. Similarly any kind of price difference density function can be used which can 38

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Figure 16: Impact of loyalty model on total profits

model the subscriber population the best. We have seen that the service providers have to change their prices paying attention to the price difference, a too small price reduction might not results enough new customers. An ISP can expect more profit if she knows better the price difference of the population as it has an important impact on the prices. As future work, we plan investigate the impact of operational costs on ISP pricing for disloyal customers, with or without uncertainty. References [1] Bicz´ok, G., Kardos, S., Trinh, T.A.: Pricing internet access for disloyal users: a game-theoretic analysis. In: NetEcon ’08: Proceedings of the 3rd international workshop on Economics of networked systems, pp. 55– 39

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60. ACM, New York, NY, USA (2008). DOI http://doi.acm.org/10. 1145/1403027.1403040 [2] NSF: Future Internet Network Design Initiative. http://find.isi.edu [3] He, L., Walrand., J.: Pricing and Revenue Sharing Strategies for Internet Service Providers pp. 205–216 vol. 1 (2005) [4] X.-R. Cao H.-X. Shen, R.M., Wirth, P.: Internet Pricing with a Game Theoretical Approach: Concepts and Examples. In: IEEE/ACM ToN, pp. 208–216 (2002) [5] Shakkottai, S., Srikant, R.: Economics of Network Pricing with Multiple ISPs. IEEE Infocom (2005) [6] Georges, C.: Econ 460: Game Theory and Economic Behavior. Course handouts and exercises, Department of Economics, Hamilton College (2008) 40

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Figure 18: Impact of minimal price difference on profits

[7] Walker: The 2004 walker loyalty report for information technology shows. http://www.walkerinfo.com/knowledge-center/walkerlibrary/article.asp?id=761 (2004) [8] Choice: Isp satisfaction survey. http://www.choice.com.au/viewArticle.aspx?id=105998 (2007) [9] Advisor, P.: Pc advisor broadband survey 2008. http://www.pcadvisor.co.uk/news/index.cfm?newsid=105129 (2008) [10] Chiou, J.S.: The antecedents of consumers’ loyalty toward internet service providers. Inf. Manage. 41(6), 685–695 (2004). DOI http: //dx.doi.org/10.1016/j.im.2003.08.006 [11] DoCoMo, N.: http://www.nttdocomo.com [12] NHH:

National

communication 41

authority.

http://webold.nhh.hu/hirszolg/szolg/szolgaltatokLekBeforeAction.do (2007) [13] NHH: National communication authority’s internet access report of august 2008. http://www.nhh.hu/dokumentum.php?cid=16830 (2008) [14] KSH: Average net income. http://portal.ksh.hu/pls/ksh/docs/hun/xftp/gyor/let/let20808.pdf (2008) [15] Fudenberg, D., Tirole, J.: Game Theory. MIT Press (1991) [16] Game Theory Group, D.o.T., Media Informatics, B.U.o.T., Economics: Website. http://malna.tmit.bme.hu/cgibin/twiki/view/GameTheory/WebHome (2008)

A. Proofs A.1. Proof of Proposition 2 We will present the proof for three players then we will generalize the steps for N players. A.1.1. Three-player game In this game we have three IPSs setting their prices for Internet access. The different ISPs have loyal user bases with different sizes, let these be l1 > l2 > l3 . An ISP can lose its users if there exists an other ISP with smaller price, where the difference of the prices is larger than d. The demand function is the same in this case as in the two-player game and can be formulated as: Πi =

 P   (li + z:pz −pi >d lz )pi  

li pi 0

pi < pj , ¬∃j : pj < pi − d ∀j|pi − pj | ≤ d ∃j : pj + d < pi

(12)

For the same reason as it was in the two-player case the pure Nash equilibrium of the game can be at (α, α, α) if there none of the ISPs have incentive to compete.

42

(l1 + l2 + l3 )(α − d) ≤ li α (l1 + l2 + l3 )(α − d) ≤ l1 α (l1 + l2 + l3 )(α − d) ≤ l2 α (l1 + l2 + l3 )(α − d) ≤ l3 α Resulting the following inequations: l1 + l2 d ≤ (l1 + l2 + l3 ) α l2 + l3 d ≤ (l1 + l2 + l3 ) α l3 + l1 d ≤ (l1 + l2 + l3 ) α ISP3 has the smallest loyal user base, thus the maximal term is: l1 + l2 l3 d =1− ≤ (13) l1 + l2 + l3 l1 + l2 + l3 α If this equation holds the game has a pure strategy Nash equilibrium. ¤ A.1.2. Price competition with N players We can generalize the three-player game constructing the criteria of the pure Nash equilibrium. If none of the ISPs has an incentive to compete in α there exist a Nash equilibrium in the N-player game when: X j

lj (α − d) ≤ li α P

j6=i lj α

P P

j lj

j6=i lj

P

j lj

≤ d ≤

d α

The requirement of the equilibrium is that the equation holds for the smallest ISP as well: 43

minj lj d 1− P ≤ α j lj

(14)

A.2. Proof of Proposition 4 We calculate the minimal prices where an ISP will compete in the [0, α − di ] and [α − di , α] intervals. First we deal with ISP1: p2 ∈ [0, α − d1 ] (l1 + l2 )(p2 − d2 ) > l1 (p2 + d1 ) l2 p2 > l1 d2 + l1 d1 + l2 d2 l1 (d1 + d2 ) p2 > d2 + l2 p2 ∈ [α − d1 , α] (l1 + l2 )(p2 − d2 ) > l1 α l1 α p2 > d2 + l1 + l2 We can have similar conditions for ISP2: p1 ∈ [0, α − d2 ] (l1 + l2 )(p1 − d1 ) > l2 (p1 + d2 ) l1 p1 > l2 d1 + l2 d2 + l1 d1 l2 (d1 + d2 ) p1 > d1 + l1 p1 ∈ [α − d2 , α] (l1 + l2 )(p1 − d1 ) > l2 α l2 α p1 > d1 + l1 + l2 This game has a similar best response figure as it was in the original game that is there do not exist an intersection of the graphs except at (α, α). The 44

original game had a pure strategy Nash equilibrium at (α, α), we will now see this for this game as well. ISP1 will compete if her payoff can be larger if she grabs the users of ISP2. With numbers: (l1 + l2 )(α − d2 ) > l1 α l2 α > d2 (l1 + l2 ) d2 l2 > l1 + l2 α The conclusion is the same for ISP2. This means that if the ISPs’ subscribers have different price sensitivity there exists a pure strategy Nash equilibrium at (α, α) if the following conditions are satisfied: l2 d2 l1 d1 ≤ ≤ (15) l1 + l2 α l1 + l2 α We can generalize this game for N players. We give a sketch proof of the three-player game, the N-player game can be derived similarly. If the market has three players there can be a pure strategy equilibrium if none of the service providers have an incentive to compete for others’ subscribers. We provide the proof of conditions only for ISP1, the others go all the same way. There will be equilibrium if: (l1 + l2 + l3 )(α − d2 ) ≤ l1 α (l1 + l2 + l3 )(α − d3 ) ≤ l1 α (l2 + l3 )α ≤ d2 (l2 + l3 ) (l2 + l3 )α ≤ d3 (l2 + l3 ) l2 + l3 d2 ≤ l1 + l2 + l3 α d3 l2 + l3 ≤ l1 + l2 + l3 α We can see that the game has similar conditions as it had in the two player game. ¤ 45

A.3. Proof of Proposition 5 What is the minimal profit that an ISP can have regardless of the other player, and what is the minimal price resulting this payoff? If an ISP sets d as price she can never loose her loyal customers because the other ISP can not set lower price than d − d = 0. This means that the ISPs can have li d profit in each cases. We want to have a mixed strategy equilibrium but what is the condition of it? ISPs play mixed strategies if they have to compete with each other. 1 ISPs have a pure strategy equilibrium (α, α) if the followings hold: l1 l+l ≤ 2 d l2 d l1 d l2 d , ≤ α . Thus, if l1 +l2 > α , l1 +l2 > α the ISPs will play mixed strategies α l1 +l2 during the price competition. We have to calculate the minimal price where the ISPs will still compete. First, ISP1 will compete if she can have larger payoff than her minmax profit: 1 (l1 + l2 )p1 > l1 d from which we can get that p1 > l1 l+l d. We also have a 2 l2 similar lower bound price for ISP2: p2 > l1 +l2 d. To be able to calculate 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. With the following indirect proof we can argue that each ISP has uniform distribution between the bounds. Suppose that ISP1 plays with a distribution where there is a specific price (p∗ ) that she 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 (Figure 19). The expected payoff of ISP1 will be always less if she plays with non-uniform distribution than it would be. The proof is the same for ISP2 which means that both ISPs have to play their mixed strategies with uniformly distributed prices. Now we have the boundaries of the mixed strategies’ supports and we know that the ISPs have uniform price distribution, thus we can create the cumulative distribution functions of the ISPs:

46



 


  







Figure 19: Illustration of prices in case of non-uniform distribution function

    

F1 (p) = 

0

p1