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: [email protected] T. A. Trinh e-mail: [email protected] 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 172 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) 174 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 176 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 178 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. 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