THE Internet is a heterogeneous body of privately owned

Transcription

1 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER Economics of Network Pricing With Multiple ISPs Srinivas Shakkottai, Student Member, IEEE, R. Srikant, Fellow, IEEE Abstract In this paper, we examine how transit customer prices quality of service are set in a network consisting of multiple ISPs. Some ISPs may face an identical set of circumstances in terms of potential customer pool running costs. We examine the existence of equilibrium strategies in this situation show how positive profit can be achieved using threat strategies with multiple qualities of service. It is shown that if the number of ISPs competing for the same customers is large then it can lead to price wars. ISPs that are not co-located may not directly compete for users, but are nevertheless involved in a non-cooperative game of setting access transit prices for each other. They are linked economically through a sequence of providers forming a hierarchy, we study their interaction by considering a multi-stage game. We also consider the economics of private exchange points show that their viability depends on fundamental limits on the dem cost. Index Terms Internet economics, peering transit, quality of service, repeated games, Stackelberg games. Fig. 1. Structure of the Internet consisting of local transit ISPs. I. INTRODUCTION THE Internet is a heterogeneous body of privately owned infrastructure. Roughly speaking, it consists of two types of networks: 1) densely meshed networks in geographically localized regions which specialize in providing consumers with connection points to the network 2) networks traversing large geographical distances which provide connectivity between the local networks [15]. All the networks are connected by means of an inter-operable protocol stack agreed upon by the Internet Engineering Task Force (IETF). Fig. 1 illustrates the Internet as it looks like today. There are local Internet Service Providers (ISPs) providing services in small regions, which compete for the same group of customers transit ISPs which transfer data between such local groups. Local providers as well as transit providers establish a point of presence (shown as small clouds ) at a Network Access Point (NAP) where traffic may be exchanged. Local ISPs in different regions have two options for traffic exchange they can use the services provided by a transit ISP (shown as large transit clouds) at a NAP, or can build their own means of private exchange (either at a NAP or independently). Recent maps of the Internet [2] indicate that private traffic exchanges are becoming popular [20]. Manuscript received May 13, 2005; revised November 17, 2005; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor A. Orda. This work was supported in part by the Defense Advanced Research Projects Agency (DARPA) under Grant F the National Science Foundation under Grant ANI An earlier version of this paper was presented at the IEEE INFOCOM, Miami, FL, March The authors are with the Department of Electrical Computer Engineering Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL USA ( sshakkot@uiuc.edu; rsrikant@uiuc.edu). Digital Object Identifier /TNET A topic of current interest is the development of good pricing models for the Internet. Most ISPs currently employ flat rate pricing, i.e., end-users pay a fixed sum of money every month for unlimited access (most broadb providers dial-up services use this scheme). Others, following the telephony model, price the time spent connected to the Internet (some dial-up services are priced this way). ISPs may also use percentile-based charging. Still others charge based on actual bytes transferred (many Australia/New Zeal [3] Indian ISPs use this scheme). Pricing models for Internet economics have been discussed in [15], [23], [25]. The authors discuss how pricing may be used to provide different qualities of service argue that differentiated pricing for different types of data transfers might be a good approach. While some aspects of the traditional telephony model [18] may provide a starting point for economic analysis of the Internet, one must keep in mind that data transfers on the Internet are usually biased very heavily in one direction, whereas telephone calls impose equal loads in both directions. The transfer of data in the Internet may be assumed to be unidirectional in most cases, we call such a transfer as a flow. We assume that traffic originates at websites terminates at end-users. In [16] a similar model is used to find a Nash equilibrium solution for prices charged to websites end-users. Another difference is with regard to the quality of service (QoS). Since telephony is a constant bitrate service, quality of service is determined by whether or not a user s call can be set up. Providers design their networks so as to minimize the blocking probability. The Internet, on the other h, is a variable bitrate service. Currently, users obtain different bitrates based on their access methods for instance cable-based access gives a much higher bitrate than dial-up. The reason why QoS depends heavily on access methods is that the core of the /$ IEEE

2 1234 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006 Local ISPs which are co-located in a small geographical region (for example Region 1 in Fig. 2) compete for the same customers. They form a regional group of equals with identical interests. Transit ISPs (shown as Tier-I Tier-II in the economic hierarchy) which transfer traffic between local ISPs. There could be several transit ISPs between two regions competing for customers (local ISPs) forming a group of equals of their own. Our main results are summarized in Sections I-A D. Fig. 2. Illustrating the economic hierarchy in the current Internet for traffic from Region 1 to Region 2. The solid arrows indicate monetary transfers. The arrows pointing upwards indicate the fact that the higher tiers charge regardless of traffic direction. Bilateral settlement may be accomplished at private exchanges as shown. Internet, based on transit infrastructure, is highly over-provisioned [24] bottlenecks arise primarily at the edges. An interesting aspect of QoS on the Internet is that, unlike most industry models, a local ISP can provide several different QoS levels simultaneously [4], [5]. Thus, ISPs can choose to provide a subset of access methods that would be most lucrative. There is also another difference in the economic interaction between players in the Internet as compared to the telephony model. In the telephony model a single party, usually the origin, funds the transport of traffic on the complete path from sender to receiver. Thus, each receiving ISP in the path of the flow charges each transmitting ISP a termination charge. This is called bilateral settlement. On the other h, there is a hierarchy of providers in the Internet [15], which is illustrated in Fig. 2 for a particular direction of traffic. The higher tiers charge the lower tiers regardless of traffic direction. The Tier-I ISPs have bilateral settlements among themselves, in which they reciprocally set their termination charges to zero. Such an agreement is called a peering arrangement. The set of connectivity paths within the Internet can be seen as a collection of path pairs, where the sender funds the initial path component the receiver funds the terminating path component. Higher level ISPs are said to provide a transit service to the lower levels. However, the hierarchy is not rigid. Traffic could be exchanged within local groups within a region. ISPs in different regions could also exchange traffic atprivate exchanges. Companies such as Equinix [21] provide the infrastructure for the establishment of such exchanges. These are indicated in Fig. 2 by a circle containing a P. Thus, traffic need not traverse the whole of the hierarchy. It is for this reason that only the Tier-I ISPs (which do not pay transit charges to any other ISPs) the local ISPs (which potentially have to pay for all traffic to from their infrastructure compete for the same customers) are well defined. Several models have been proposed to study peering versus transit relations [8], [9], [12], [17], [20]. A problem with some of these models is that they do not take into account the fact that the Internet is separated into different groups of equals with different economic interests. We divide ISPs into the following groups: A. Interaction Among Equals We study how members of a group who are in an identical position in the hierarchy interact with each other. As an example, we consider the case of the economic interactions of local ISPs of a region with each other, which we model as a repeated game. The idea behind the repeated game model is that ISPs would declare prices at constant intervals (say every month). We call such a time interval as a step. Potential customers would decide on which ISP to join (or not to join at all) based on the prices. Although we assume for exposition that potential customers could choose their preferred ISP as soon as prices are declared, it is not difficult to show that our results are valid even when we consider user inertia, i.e., users decide to switch to a lower priced ISP only after a fixed number of steps. The repeated game is associated with discount factor. The discount factor is the number by which a payoff at the next step must be multiplied in order to obtain its value at the current step. For instance, a discount factor of 1/2 would mean that the same payoff would be worth half as much in the next step, one-quarter as much two steps later so on. We show that for local ISP interaction that there exists an optimal scenario where all ISPs peer with each other jointly maximize their profits. In a non-cooperative game situation, we show that this optimal price can be enforced by the threat of lowering prices to a minimal value. We show that the discount factor at which the threat fails to work scales with the number of ISPs,as converges to unity as gets large. This means that ISPs could co-exist in two possible ways: 1) There could either be a small number of ISPs interested in a short-term profit or a large number which are all interested in long-term profits (discount factor close to unity). They would be content to peer with each other split the customer revenues amongst themselves. 2) Some or all of a large number of ISPs could be interested in short-term gains try to set prices below others to make quick profits by getting a larger customer share. We show that the natural outcome of the low discount factor scenario is that prices fall to the bare minimum all profits to go to zero, the equivalent of a price war. In the current Internet, the number of data carriers in a local region is small, usually limited to a couple of ISPs providing a few QoS levels (DSL, cable/t1 T3 seem to be most popular), which means that is not too large. This corresponds to the first case above is what is happening today [22] in this segment. However, the Voice over IP (VoIP) market is very different. Customers are very mobile as they can choose any long-distance provider available with access through 1-800

3 SHAKKOTTAI AND SRIKANT: ECONOMICS OF NETWORK PRICING WITH MULTIPLE ISPs 1235 numbers. This implies that there are a very large number of providers, most of whom are interested in quick gains (low discount factor). The model predicts a price war in such a scenario, indeed this is exactly what is happening [11]. The above is a specific case of our general result, which states that the scaling law applies to any system of equals in a single level of the hierarchy, for instance multiple transit providers between two geographical regions. It also holds when we extend the model to include the possibility of ISPs providing several different QoS levels. B. Interaction Along the Hierarchy Fig. 3. Example game illustrating Nash reversion. We next study the interaction of different groups of the hierarchy. We now consider interactions between local groups transit providers, which we study as a multi-stage game. The main result of this section is that in economic terms, Tier-II providers can be considered to be price transfer agents, transferring prices between two levels in the hierarchy. Any charges of the Tier-I ISPs would get transferred to the ISP originating/terminating the traffic along with an additional charge which is the Tier-II s profit. This means that Tier-II ISPs are middlemen between the local groups Tier-I providers. This is the reason why no ISP wants to be seen as Tier-II [22]. The objectives of the two groups (local transit ISPs) are entirely different the division of ISPs into local transit is thus the basis for peering. Any sort of peering or mutual-benefit traffic arrangement can be made only between entities of the same class, i.e., local ISP with local ISP or transit with transit. C. Private Exchanges As indicated in Fig. 2, ISPs have the option of exchanging traffic at private exchange points. This provides a new dimension to the problem. The marginal cost of traffic transfer for a Tier-II ISP is fairly low due to the large volumes of traffic generated by an aggregate of local ISPs. If local ISPs were to establish a private exchange, the marginal cost of transfer would depend on the volume of traffic exchanged. A study of the cost tradeoffs involved in such exchange is present in [21], where it is assumed that such ISPs at such exchange points would have similar amounts of traffic destined for each other have a peering agreement. We show that the above scenario is a special case of a more general multi-stage game where the players could have any bilateral settlement, rather than having to peer. Our results show that in the general case, the decision of whether or not to go in for private exchange is decided by fundamental limitations on the cost function can be taken unilaterally by any ISP. So asymmetric traffic cases (where there is a difference in amounts of traffic exchanged) are possible, as well as cases where one ISP uses a private exchange the other uses a transit provider, with both making higher profits than using the transit provider alone. This suggests that private exchanges would become more popular as time goes on, this is confirmed by the maps in [1]. We conclude that there could potentially be an incentive for local ISPs to buy out transit providers between certain regions, causing a decline of pure transit providers. D. Organization of the Paper In Section II, we discuss the game theoretic concepts used in the paper. We specify the Internet model in Section III. In Section IV, we discuss the case of interaction groups of equals within the hierarchy using the example of local ISPs competing for customers. In Section V, we consider the interaction between different groups of equals in the hierarchy, in particular that of local groups transit providers, show how there is a natural Stackelberg solution. We study the next logical extension of private exchange in Section VI, derive conditions for private exchange. Finally, in Section VII, we extend the model still further to include different qualities of service pricing schemes show that the principles derived in the study are fundamental to the dynamics of Internet economics. II. BASIC NOTIONS We first introduce the game theoretic concepts 1 that are used in this paper using a simple matrix game, which we call. Consider Fig. 3. There are two players, who have two possible actions or moves High Low. The players have identical strategy spaces, with elements Play Play called pure strategies. These can be thought of as the rules by which they select their moves. A strategy profile is an element of the product-space of strategy spaces of each player is denoted by. An example of a strategy profile would be for Player 1 to choose for Player 2 to choose, denoted by. The payoffs that each strategy profile yields to each player are the elements of the matrix. The objective of each player is to maximize its individual payoff. Definition: A pure strategy Nash equilibrium is a strategy profile from which no player has a unilateral incentive to change his strategy. The only pure strategy Nash equilibrium for this game (easily verified) is the strategy profile (i.e., the move chosen by both players is Low), which yields a payoff of zero to both players. Suppose now that the game above is repeated infinitely many times. We then obtain a new game which we denote by. We assume that each player is aware of all the moves made by all players up to the previous step, which we call the history up to that step, i.e.,, where each is the vector of moves made by players at step. Similarly, 1 A good reference for this is [14].

4 1236 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006 let denote the entire sequence of moves made by all players for all time. Let us denote the set of all possible histories of the game from step 0 to by. The payoff that the players receive in this game is the sum of (discounted) payoffs in each step. Formally, if the payoff in step to player is denoted by, then each player now has to maximize where. Note that the payoff in each step depends on the moves of all the players in that step. The concept of a pure strategy has to be extended a little in this setting. A pure strategy is now a contingent plan on how to play in each step for a possible history, making it a map from to the set of possible moves. For example, in a strategy could be Play Low, whereas in a strategy could be Play High until the other player plays Low, then play Low for ever. The concept of a Nash equilibrium remains unchanged. We next consider the concept of subgame-perfection. Since all players know, we can view the game from step onwards with history, as a game in its own right, which we denote as. This new game is called a subgame of the game. Suppose there were some Nash equilibrium strategy profile for the game. Then, for some history, would recommend playing a move suggested by at step. Here we have used the notation to denote the fact that the move is chosen given the history. If the same strategy profile were a Nash equilibrium for the game, the strategy profile is called subgame-perfect. Definition: A game is continuous at infinity if for each player the payoff satisfies The condition says that events in the distant future are relatively unimportant. It is obvious that the discounted payoff function in our example is continuous at infinity since the payoff in any step is uniformly bounded by 10. One-step Deviation Principle: In an infinite-horizon multistep game which is continuous at infinity, profile is subgameperfect iff there is no player no strategy that agrees with except at a single step, such that a unilateral deviation by player to gives him a better payoff conditional on history being reached. In our example game, using the one-step deviation principle, it is straightforward to show that if the discount factor, then the strategy profile Play High until the other player plays Low, then play Low for ever (for both players) is a subgame-perfect Nash equilibrium. We make the following observations about the strategy profile. The strategy profile being used when everything is going fine is, the best possible outcome in if both players were to cooperate. The strategy used when any player deviates is, the Nash equilibrium of the single-step game. Such a threat strategy is called Nash reversion [13]. (1) III. INTERNET MODEL In order to underst the basic dynamics of Internet economics, we start with a simple model, which is used to obtain fairly general results. Our model is a generalization of two earlier models that in [16], which considers only a local group of ISPs as a single-step game, that of [7], which considers only the hierarchical aspect. We assume initially that the quality of service is a fixed parameter plays no role in decision of consumers. ISPs can charge different prices to websites endusers for intra- inter-regional traffic. ISPs also charge a price for terminating each other s traffic. We extend the model in Section VII to allow different QoS levels other pricing schemes. The following players are involved in our model: 1) Local ISPs: These are the ISPs which compete for consumers in a geographical region. They have to set prices for intra- inter-regional traffic. For intra-regional traffic, ISP charges prices per unit traffic to websites end-users, respectively. Note the superscript indicating the region. Each ISP must provide a guarantee of connectivity to all other users. This means that ISPs have to exchange traffic. They do this by means of a bilateral settlement, i.e., ISP charges ISP an access charge for termination of traffic, where ISPs both belong to region. Also, let the prices charged by local ISP to end-users websites in region for inter-regional traffic be denoted by. An ISP incurs a marginal cost per unit of traffic originating in a website subscribing to its services. Similarly, a marginal cost is incurred by an ISP for terminating traffic at an end-user. The sum total of these costs is denoted by. The costs for all ISPs in the region is assumed to be identical. All ISPs have identical fixed costs, which (since it makes no difference to any of the results) we take to be equal to zero. 2) Transit ISPs: These are ISPs which invest in transcontinental fiber bridges to carry traffic from one local group to another, charge them for this service. We divide them into two types: Tier-II providers, which charge the local ISPs for transit. Let the transit charges for outgoing (forward) incoming (reverse) traffic in region be. The cost incurred by the Tier-II provider is denoted by. The Tier-I provider, which charges the Tier II provider (in region ) for traffic from to region. Its cost for the transfer is. 3) Consumers: We model consumer dem in terms of the number of consumers that a local ISP would have. Consumers are free to choose any local ISP in a region so the dem depends on the lowest of the prices charged by the local ISPs of that region. We denote the dem functions for local traffic of websites end-users by respectively, where. We assume that ISPs that charge more than (alternatively ) would have no customers if multiple ISPs all charge (alternatively ), they obtain an equal number of customers. Since we have assumed that consumers could separately subscribe for the long-distance

5 SHAKKOTTAI AND SRIKANT: ECONOMICS OF NETWORK PRICING WITH MULTIPLE ISPs 1237 (inter-regional) service, we take the dem for this service to be. We assume that each website each end-user have one unit of traffic between them. Thus, the total traffic within the region is. Note that a popular website could be simply thought of as multiple websites similarly high usage end-users can be thought of as multiple end-users. We assume that customers are charged based on usage, i.e., the number of bytes transferred. The profit function at step for any history, of the local ISPs in region connected by transit providers to region is then given by Here, the first term is for traffic which originates terminates in ISP. The second is for traffic which originates from another ISP terminates in ISP. The third term is for traffic which originates in ISP terminates in some other ISP. The last two are for inter-regional traffic. The objective is the maximization of the discounted sum of profits given by (1). Note that some of the terms could be zero if the ISP charges higher than some other ISP for the same segment. The model is very useful for exposition since intra-regional inter-regional traffic pricing can be solved separately, as they do not share any terms. This will help illustrate two key concepts that will be used Nash reversion among equals price transfer through a Stackelberg game along the hierarchy. IV. INTERACTION BETWEEN EQUALS In this section, we consider the dynamics of interaction among equals (belonging a single group of the hierarchy). An instance of this type of interaction is pricing among local ISPs, which is considered as a Bertr game. In a traditional Bertr Game, firms set prices produce goods to supply the dem at these prices. In our model, the number of customers (end-users websites) is the dem which depends on the prices set by ISPs. We first find the optimum price, which would maximize the profit of individually rational ISPs. We then show that this price can be enforced by threat in a repeated Bertr game for a suitably high discount factor. Consider the scenario illustrated in Fig. 4. This shows a small section of the Internet. ISPs 1, 2, 3 are localized in a small geographical location, while the Tier-II ISP is involved in transit services. They are interconnected at a NAP where routing of traffic between ISPs is accomplished. As mentioned in the previous section, we study only intra-regional traffic consider bilateral interactions of ISPs 1, 2, 3. ISPs may exchange traffic in one of two ways as shown in Fig. 4. ISP could charge ISP an access charge for termi- (2) Fig. 4. Local ISP structure which shows how policy decisions can be implemented at NAPs. nation of traffic, i.e., they could have a bilateral settlement. Note that the access charge may be different for different ISPs. The other option is for both ISPs to pay a transit charge to a higher tier ISP (such as the Tier-II ISP in Fig. 4), which would provide connectivity. Let the charge that the transit ISP asks for be. Then the ceiling on the access charge which a terminating ISP can ask for in a bilateral settlement is also as otherwise no ISP would want to directly interact with it. We study the game that such local ISPs play amongst themselves. At each step of the game ISPs must declare their prices access charges. Customers then choose their service providers. Each step models temporal changes of prices, for example on a monthly basis. The profit function of ISP in step (for any history ) is given by Note that these are exactly the first three terms in (2). Since all parameters refer to the same region, we have dropped the superscript. Using the same notation as in the example game in Section II, we will refer to the single-step game by the infinitely repeated game by. A. Properties of the Dem Functions We assume that the number of consumers is infinitesimally divisible. This a reasonable assumption since the number of customers is very large in practice. The dem functions, are assumed to belong to a class of functions, which satisfy the following conditions: 1) They are continuous, twice differentiable strictly decreasing. 2). The first of these implies that the population is finite normalized to be equal to one. The second implies that we have no customers at infinite price. 3) is unimodal with. This means that revenue obtained at infinite price is zero. Now, let us define the cooperative profit function as where over the set. We will (3)

6 1238 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006 see later that when ISPs cooperate set identical prices access charges the profit function (3) looks similar to. We note the following properties of the above function:. Follows immediately from properties This follows from property 3. The function is positive everywhere except for or. There exists (not necessarily unique) which maximizes, yielding a nonzero maximum. This is seen as follows. Since, we have that given any there exist finite, such that for all or. Thus, we can approximate the set over which we maximize, infinitesimally closely by the set. Now, since this is a compact set the function is continuous over this set, there must exist a (finite) maximum. B. Cooperative Maxima Between Local ISPs for the Single-Step Game Consider the single-step game. We would like to know if there exists any cooperative maxima, i.e., is there a set of prices access charges whereby the all the ISPs could maximize their joint profits? We will then show how threat strategies may be used to enforce cooperation. Theorem 1: Suppose there are ISPs in the region. Assuming that the ISPs cooperate set equal access charges prices, that the dem functions belong to the class, that equal prices implies equal number of customers, there exists a set of prices (consisting of pairs of form ) which maximize the profits. Proof: Recall that the profit function of ISP in step (for any ) is given by (3). We can simplify (3) in the manner of [16] as follows: Simplifying, we obtain By assumption, since the ISPs cooperate (i.e., all ISPs charge identical prices, so they have an equal number of customers), we have that (4) (5) Also, as the ISPs are individually rational. The individual profit functions are now The proof follows from the results on dem functions. We will use to denote the strategy profile in which all the ISPs play cooperative prices, all set the access charges to zero (peering). In general, however, ISPs have no incentive to cooperate. Therefore, a relevant question is the following: is the cooperative maximum found above enforceable by some means? We show the existence a threat strategy Nash equilibrium which would achieve the cooperative maximum. Our result is an example of Friedman [13] type punishment, wherein players revert to the single-step Bertr Nash equilibrium if a deviation occurs. C. The Threat: Nash Equilibrium for the Single-Step Game In the illustrative example considered in Section II, we saw that the single-step Nash equilibrium could be used to construct a subgame-perfect Nash equilibrium for the infinitely repeated game. We find the single-step Nash equilibrium below. Suppose that the ISPs set prices access charges as follows: We refer to the strategy profile in which all players play the above as. From (6), this means that all profits are zero. We will show that the above is actually the Nash equilibrium for the single-step game. First of all note that any unilateral increase in or would have the effect of losing customers leading to a loss of revenue (, of course, cannot be raised beyond, as it is the maximum possible). Also note that reducing the termination charge has absolutely no effect on the number of customers. Any reduction in only causes loss of revenue. So the only unilateral decision which has potential to increase profits is to decrease one or both of. We then have the following three cases. Case I: Set. An ISP doing this will get all the end-users as customers. Its profit for each unit of traffic can easily be shown to be by considering two cases: Traffic entirely within the ISP s infrastructure yields a profit of. Traffic originating in another ISP terminating within the ISP which charges the reduced price yields a profit of. Case II: Set. An ISP doing this will get all websites as customers. Again, we can show that the profit per unit traffic is. (6)

7 SHAKKOTTAI AND SRIKANT: ECONOMICS OF NETWORK PRICING WITH MULTIPLE ISPs 1239 Case III: Set. An ISP doing this would get all consumers as customers. However, its profit would be. We see that a unilateral shift by any ISP from the above prices would not yield a higher profit. Thus, the above set of parameters, which gives zero profits, defines the Nash equilibrium solution for the game. Noting that any ISP can guarantee itself a profit of at least zero, we see that this set of prices also yields the guaranteed minimum payoff to any player. No ISP would play this threat unless its profits are hurt by some other ISP s choice of parameters. Thus, the threat of playing this set of parameters can be used to enforce that set of parameters which jointly imply a maximization of profit. D. Subgame-Perfect Nash Equilibrium for the Repeated Game We are now ready to address the question of whether there exists a subgame-perfect Nash equilibrium which ensures that the ISPs set prices according to. It is easy to see that is continuous at infinity, since the maximum payoff at any step is bounded. We can now use the one-step deviation principle to prove subgame-perfection. Theorem 2: The strategy profile Play until any player deviates then play for ever (for all players) is a subgame-perfect Nash equilibrium for the repeated game under the condition that the discount factor where denotes the optimal profit for every step under cooperation. Proof: Define the profit made in the first steps as Note that since the game is continuous at infinity, the above is consistent with the definition of. Consider the implications of a one-step deviation, i.e., suppose an ISP sets the prices lower than the others at some step. Since the ISP must maximize its profit it would do so by a reduction of by for end-users, thereby obtaining all end-users as customers. Also, depending on the value of it would decide on whether it would be more profitable decrease (to get all the websites as customers) or increasing (thus ensuring that it has no websites making a profit by terminating all the traffic). Thus, its profit for that one step is bounded in the following manner: The profit from step to if the ISP chooses not deviate from the cooperative values is given by, where by assumption. Thus, By assumption, since we obtain from (8) (9) that. Thus, by the one-step deviation principle the strategy profile is a subgameperfect Nash equilibrium. The fact that the discount factor varies with the number of ISPs as for the threat to be credible is worth noting. For large the required discount factor tends to unity. This has the implication that ISPs have to be interested in long-term profits rather than quick gains if cooperative prices are to be viable. In market terminology, the Bertr Nash equilibrium is the same as a price war. This means that if there are a large number of ISPs in the market, price wars are bound to follow since some of them would be interested only in short-term gains resulting in a downward spiral of prices. The only solution to this would be for larger players to buy up the smaller ones. Then the threat strategy would be credible prices would be stable. The framework we have developed above applies to any level in the hierarchy. For instance, if there were multiple transit providers (either Tier-II or Tier-I) spanning the same levels in Fig. 2 then they all must charge identical transit charges (we will see how exactly this charge would be determined in the next section). This can be enforced by appropriate Nash reversion threat strategy for scaling as. In the case of VoIP providers, the case is that of transit providers directly offering long-distance services to customers, who naturally pick the cheapest option. It follows that a price war would result if there are too many such providers in the market. If we model user inertia as a finite number of steps before changes can take place, the same kind of scaling law still applies. Finally, note that we have not gone into the details of the evolution of Nash equilibria, since we do not consider the dynamics of the system. Since the optimum is not unique, one would actually have to solve for the possible optimums given the dem functions then decide which one would actually work for instance an optimum in which the prices were higher than another optimum would not be chosen. (9) V. INTERACTION ALONG THE HIERARCHY The other ISPs would employ the threat strategy from the next step onwards, causing all profits to go to zero. So the profit from step to, given by is simply (7) (8) A. The Single-Step Stackelberg Game We model the interaction of ISPs as a Stackelberg game. In such a game players play in a definite sequence. Each player plays knowing that the next player would optimize his play based on what he does currently. The equilibrium in such a case is called a Stackelberg equilibrium. We assume for now that the local ISPs in a region cooperate all them charge the same prices (in the next subsection, we

8 1240 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006 will show how cooperation can be enforced among the local ISPs). This means that the local ISPs in a region can be grouped together as one player. Recall that in Section III, we noted that we could consider intra- inter-regional traffic as two separate problems. The only problem now is that of setting prices for inter-regional traffic (the last two terms in (2)). The Stackelberg solution, which we denote by, for the single-step game is found as follows. In Fig. 2, consider traffic that originates in Region, travels through the transit infrastructure terminates in Region (for traffic in the reverse direction, the roles of the two regions are reversed). We refer to the collection of local ISPs in a region as a group. Then the two economic segments are Group Tier-II Region Tier-I Group Tier-II Region Tier-I. Consider Group s problem: (10) where are the dem functions for inter-regional traffic in regions 1 2, respectively, we require that (individual rationality). Note that we have dropped the subscript identifying a particular local ISP since the whole group of local ISPs in a region is taken as a single player. Lemma 3: The solution to the maximization above is a strictly increasing function of. Proof: Differentiating (10) equating to zero we have that satisfies (11) where if lies in the interior of. Since is a maximum, we also have rearranging, we ob- Differentiating (11) with respect to tain (12) Proof: From the corollary of lemma 3 we have that is strictly decreasing in. Then proceeding as in the proof of Lemma 3, we obtain the result. Theorem 5: is a strictly increasing function of. Proof: The proof is a direct consequence of the Lemmas 3 4. The theorem defines the role of the Tier-II provider in the Internet economy. It says that it acts as a middleman who transfers the access charge from Tier-I to Group makes a profit in the bargain. If there are multiple Tier-II providers in sequence then the entire sequence can be condensed into a block whose input is the access charge whose output is the transit charge. Thus, the role of the Tier-II provider in the network is that of a price transfer agent between levels of the hierarchy. Now, the theorems proved above hold for the other segment of the transfer as well. In this case price is transferred to Group. Thus, the problem faced by Tier-I is as follows: We define (14). It is easy to show that both of these functions belong to the class. Then we have that the Tier-I provider must solve (15) This looks identical in form to (6) there exist values of which maximize it. B. Example We illustrate the Stackelberg game (for traffic from Region to Region ) by taking the dem functions. We then have the optimization problems as follows. Group s problem is (16) The last inequality follows from the properties of the dem function (12). Suppose (boundary condition), then it is automatically increasing in. Hence, the proof. Corollary: is a strictly decreasing function of. The proof follows trivially from the lemma the properties of. Now consider the maximization problem faced by the Tier-II Region provider: (13) where. Lemma 4: The solution to the maximization above is a strictly increasing function of. The Tier-II Region ISP s problem is (17) where the second line follows from (16). We see that the transit provider has transferred the access charge to Group along with its cost a profit of one unit. Group Tier-II Region have similar problems to the above. Their Stackelberg solutions are (18) (19)

9 SHAKKOTTAI AND SRIKANT: ECONOMICS OF NETWORK PRICING WITH MULTIPLE ISPs 1241 Finally, the Tier-I provider s problem (after substituting the above parameters) is (20) The Stackelberg solution here is non-unique. One would expect that in this case, the Tier-I provider would charge both sides equally, but this is not required. C. A Local Threat to Enforce the Stackelberg Equilibrium The Stackelberg solution found above applies to each local ISP group as a whole assuming that they cooperate. We must construct a Nash reversion strategy for the ISPs in a local group, which would ensure such cooperation for a sufficiently high discount factor. While constructing such a Nash equilibrium, we must take into account the interactions that occur both among equals as well as along the hierarchy. When a player in a local group cheats, the others in the group immediately shift to the threat strategy. The higher tier ISPs have full knowledge of the lower tiers hence they too would shift their prices to the appropriate Stackelberg prices to maximize their profits. Suppose that the local ISPs in a region sets prices as follows: Fig. 5. Illustrating the cost tradeoffs in private exchanges. for ever (for all players) is a subgame-perfect Nash equilibrium for the infinitely repeated game if the discount factor where is charged by the Tier-II ISP to the originator of traffic for traffic originating terminating in the same region is determined using (21) which is the profit obtained by the Tier-II ISP with the knowledge that the local ISPs would play their threat strategy. The transit prices are determined using a Stackelberg formulation developed in (13), (14) by making the substitutions The transit providers have no interest in making the cheating ISP behave they just maximize their profits knowing that the cheating provider will be punished by the other local ISPs. The local ISPs know this fact set their threat prices accordingly. It is easy to verify that the above is a Nash equilibrium for the single-step game that no ISP in the group can make a nonzero profit in such a regime. We note that the threat strategy of Section IV is a special case of this Nash equilibrium. We denote this more general threat strategy using the same notation. The threat of playing this set of parameters can be used to enforce that set of parameters which jointly imply a maximization of profit. Theorem 6: (Proof is identical to that of Section IV) The strategy profile Play the for intra-regional traffic for inter-regional traffic until a group member deviates then play the single-step Nash equilibrium prices where denotes the optimal profit under cooperation, is the number of local ISPs in a group, is the cooperative strategy for intra-regional traffic (found in Section IV),. Thus, the scaling law under which local ISPs can make positive profits holds for the extended model as well. Finally, since the Tier-I ISPs peer with each other, we may consider a sequence of ISPs at the top of the hierarchy as a single hop. So the results hold for traffic that traverses several Tier-I ISPs to get to its destination. VI. THE CASE FOR PRIVATE TRAFFIC EXCHANGE Suppose that an ISP in each local group in Fig. 2 wishes to establish a private exchange point, bypassing the transit infrastructure. Norton [21] discusses the economic tradeoffs involved in this. According to [21], the cost per unit traffic passing through a private exchange point is inversely proportional to the amount of traffic carried on it, as shown in Fig. 5. Consider two ISPs, 1 2, in regions 1 2, respectively. s profit is given by (22) where the notion is similar to that of the previous section with the addition that are the termination fees charged by the two ISPs under consideration is a proportionality constant. s profit function is the dual of the above. The difference in termination fees that the ISPs pay each other is. Supposing that, where is small, i.e., the ISPs have a similar number of websites end users, then the difference in fees is small, which in practice makes a

10 1242 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006 case for them to peer. Also, assuming peering, comparing with (10), it is obvious that for (23) it increases profits if a private peering point were established. This is exactly what Norton requires in [21] the break-even condition is shown in Fig. 5. Now suppose that the two ISPs are not constrained to peer. We then have a new Stackelberg game with two players. Since every member of the group would establish a private exchange point if it were profitable to do so, we need consider only one ISP in each group. We first consider the asymmetric case where it is profitable for one group to use a private exchange, while the other uses the transit provider. As before, consider traffic from Region to Region.Now, s problem is where satisfies (30). The case where both providers use private exchange can be looked upon as two optimization problems similar to that considered above, only the providers maximize over the sum of revenues obtained from origination termination of traffic. The providers could also potentially bargain with each other, under the constraint that the charges would have to be less than that charged by the transit provider. We do not consider the evolution of the system in this paper so do not consider such bargaining. In any case, the decision as whether or not to establish a private exchange point depends on fundamental limits on dem functions costs, can be taken unilaterally by any of the ISPs in a group. We conclude that when the requirement that the two groups must peer is relaxed, it increases the space where private exchange is possible. This suggests that local ISPs would buy out transit ISPs between some regions the hierarchy is likely to become flatter as time progresses. s problem is (24) (25) It is easy to show that is monotone decreasing in we have a very similar set of optimization problems as before. Then, if it turns out that there exists such that (26) it would make sense for to ask for private exchange of traffic with. For since from (13) we know that, it is simple to see that the profit when the transit provider is eliminated is always higher. For the reverse problem of traffic from Region to Region the required condition is similar to (26). A. Example Similar to the example considered before, let the dem functions be exponentially decreasing i.e.,. Then from (24) we have that maximizes From (25) we have that s problem is (27) (28) (29) (30) Then we have from (26), (28), (30), (20) (assuming that the Tier-I provider charges both halves equally) that the condition for which private exchange is better for is (31) VII. QUALITY OF SERVICE AND FLAT RATE PRICING In the traditional industry model [10], it is assumed that each industry can choose only one quality of service, usually modeled by a real number. However, in the Internet the quality of service seen by a customer depends on a finite number of access methods provided by a local ISP. 2 The fundamental difference between the industry model the Internet is that providers can choose to provide any number of service levels. Many local ISPs today provide different QoS levels to satisfy different sections of the market [4], [5]. Transit ISPs on the other h, invest in much larger bwidth solutions, measurement has shown that their bwidth is not a bottleneck in the QoS received by an end-user [24]. We model the market using an attraction model [6], [19], in which the total population of customers depends on all the prices QoS levels. Each service provided by an industry has an attractiveness index that depends on the price it charges for that service. The number of customers that the industry gets for each service is proportional to the attractiveness index. A. Market Model Let the vector of QoS levels be. Each ISP chooses whether or not to provide each QoS level sets prices associated with it, i.e., a vector of the form, where if the ISP provides QoS level otherwise. Note that the condition that ISPs have the option of charging different prices for on-site off-site traffic is not essential merely aids exposition. A customer interested in a particular QoS level (supposing it is offered by some ISP) would naturally pick the ISP that offers it at the lowest price. Let the vector of such lowest prices corresponding to the vector be. Then the populations of consumers for on-site off-site traffic are, respectively,, 2 In North America, they are limited by technology to T3/DS3 (45 Mb/s), FracT3 (3 Mb/s), T1/Cable (1.5 Mb/s), DSL (750 kb/s), ISDN (128 kb/s), dial-up (33.6, 28.8, 56 kb/s).

11 SHAKKOTTAI AND SRIKANT: ECONOMICS OF NETWORK PRICING WITH MULTIPLE ISPs 1243,. We assume that are decreasing in price increasing in, i.e., the population of customers increases if prices are low or the number of available choices is high. Note that the superscript refers to the region being considered all ISPs in a region see the same customer behavior, but it could vary between regions. Each QoS level that ISP chooses to provide has an associated attractiveness index (equivalently,, ). If an ISP charges more than the lowest price being charged for a particular QoS level, its attractiveness index for that QoS level is zero, i.e., if otherwise where is strictly decreasing in strictly increasing in. A similar condition holds for the attractiveness indexes for the other market segments. Finally, if the total number of ISPs is, the number of end-users that the ISP receives for QoS level for intra-regional traffic is given by (32) Thus, the population of customers is divided up amongst the local ISPs proportional to their respective attractiveness indexes for each QoS level that they provide. We can similarly define,. We assume that users in each QoS level would access a fraction of the Internet, for instance end-users with low QoS might access relatively fewer websites. The profit function for the local ISPs, which is similar to (2), is given by provide the same set of QoS levels. We are interested in how to enforce such a cooperative maximum hence identify the single-step Nash equilibrium. Lemma 7: (Proof is straightforward) The Nash equilibrium, which we call, in region for the single-step game consists of all ISPs offering all QoS levels at prices given by the parameter set As before, the transit charges are determined using the Stackelberg formulation. It should be clear that under this scenario ISPs make no profits, while consumers have the choice of all the QoS levels at rock bottom prices. Theorem 8: (Proof is identical to that of Section IV) The strategy profile Play the for intra-regional traffic for inter-regional traffic until a group member deviates then play the single-step Nash equilibrium prices for ever (for all players) is a subgame-perfect Nash equilibrium for the infinitely repeated game if where denotes the optimal profit under cooperation, is the number of local ISPs in a group,. Thus, the scaling law for cooperation is robust to modifications in the model. The system can operate in two phases many ISPs providing a large number of QoS levels having price wars, or consolidation of ISPs into a small number of blocks providing fewer services making profits. C. Other Forms of Charging In today s Internet customers are often charged a flat rate. They might also be charged based on percentile usage, i.e., the usage is sampled over a time period the th percentile value is considered to be the usage for the entire duration. We show here that the idea of cooperation enforced by the single-step Nash equilibrium is unchanged in either scenario. Using the same notation as before, the profit function assuming flat rate pricing is given by In the above equation the first two terms refer to intra-regional traffic the other two to inter-regional traffic. We define the single-step game the repeated game as before. B. Cooperation Its Enforcement Although the profit function looks complicated, we may bound the prices in a compact set hence a cooperative maximum exists. The exact parameters of the maximum could be calculated given the dem functions costs. We call the parameters found assuming cooperation as, wherein all the local ISPs would have to peer with each other

12 1244 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 6, DECEMBER 2006 Again, it is straightforward to show that a cooperative equilibrium exists. It can be enforced if a single-step Nash equilibrium exists (with scaling as ). Such a Nash equilibrium is defined for each QoS level by the equations the values of the transit prices are determined by the Stackelberg formulation as before. The above equations essentially say that the threat is to charge the cost price for data transfer, i.e., a price war. Thus, if the above equations have a solution, it is a Nash equilibrium. Finally, the problem of finding Nash equilibria with percentile charging has been dealt with in [27]. The paper suggests that ISPs cannot make a profit in a price competition. However, a ruinous Nash equilibrium is actually a threat in a repeated game can be used to enforce cooperation under the same conditions of discount factor as before. We thus see that the basic ideas on Internet economics hold for a variety of pricing schemes QoS levels provided. VIII. CONCLUSION In this paper, we have studied the interactions of ISPs at different levels when they compete directly for customers, when they belong to different levels of the hierarchy when they try to bypass a section of the hierarchy. The conclusions we derive on how the Internet is likely to evolve are: Local ISPs would coalesce until only a few providers in each region remain. The ISPs in each region would peer, set similar prices provide similar QoS levels. The hierarchy of providers would flatten out as private exchange becomes more popular. Local ISPs would buy out the transit providers between some regions as they realize that private exchange with a non-peering bilateral settlement is possible. ACKNOWLEDGMENT The first author would like to acknowledge Prof. N. Brownlee (University of Auckl) for interesting discussions on Internet architecture pricing. REFERENCES [1] Internet Mapping Project [Online]. Available: [2] AS Internet Graph [Online]. Available: [3] The Australian ISP Directory [Online]. Available: australianispdirectory.com [4] Broadb ISP List [Online]. Available: [5] U.S. Nationwide ISPs [Online]. Available: com/misc/usa [6] R. A. El, E. Altman, L. Wynter, Telecommunications network equilibrium with price quality-of-service characteristics, in Proc. ITC, Berlin, Germany, Sep [7] B. Hajek G. Gopal, Do greedy autonomous systems make for a sensible Internet?, in Conf. Stochastic Networks, [8] P. Baake T. Wichmann, On the economics of Internet peering Berlecon Research, Berlin, Germany, Tech. Rep., [9] N. Badasyan S. Chakrabarti, Private peering among Internet backbone providers, Economics Working Paper Archive, Series on Industrial Organization, Washington University in St. Louis (WUSTL), Jan [10] F. Bernstein A. Federgruen, A general equilibrium model for industries with price service competition, Ann. Oper. Res., vol. 52, no. 6, Nov [11] B. Charny, VoIP providers face price war, Nov. 3, 2003 [Online]. Available: [12] R. Dewan, M. Freimer, P. Gundepudi, Interconnection agreements between competing Internet service providers, in Proc. 33rd Hawaii Int. Conf. System Sciences, Maui, HI, Jan. 2000, p [13] J. W. Friedman, A non-cooperative equilibrium for supergames, Rev. Economic Studies, vol. 38, no. 113, pp. 1 12, Jan [14] D. Fudenberg J. Tirole, Game Theory. Cambridge, MA: MIT Press, 1991, 7th printing, [15] G. Huston, ISP Survival Guide: Stratagies for Running a Competitive ISP. New York: Wiley, [16] J.-J. Laffont, S. Marcus, P. Rey, J. Tirole, Internet interconnection the off-net cost pricing principle, The RAND J. Economics, vol. 34, no. 2, pp , 2003, No. 130, IDEI Working Papers from Institut d Economie Industrielle (IEDI), Toulouse, France, [17], American economic review, Internet Peering, vol. 91, pp , [18] J.-J. Laffont J. Tirole, Competition in Telecommunication. Cambridge, MA: MIT Press, [19] P. S. H. Leeflang, D. R. Wittink, M. Wedel, P. A. Naert, Building Models for Marketing Decisions, 1st ed. Dordrecht, The Netherls: Kluwer Academic, [20] W. B. Norton, Internet service providers peering, 2001 [Online]. Available: [21] W. B. Norton, A business case for ISP peering, 2002 [Online]. Available: [22], The Evolution of the U.S. Internet Peering Ecosystem. White paper from Equinix [Online]. Available: equinix.com [23] A. M. Odlyzko, Paris Metro pricing for the Internet, in Proc. ACM Conf. Electronic Commerce (EC 99), Mar. 1999, pp [24], Internet growth: Myth reality, use abuse, J. Comput. Resource Manag., pp , Nov [25], Pricing architecture of the Internet: Historical perspectives from telecommunications transportation, 2004 [Online]. Available: [26] S. Shakkottai R. Srikant, Economics of network pricing with multiple ISPs, in Proc. IEEE INFOCOM 2005, Miami, FL, Mar. 2005, pp [27] H. Wang, H. Xie, L. Qiu, A. Silberschatz, Y. R. Yang, Optimal ISP subscription for Internet multihoming: Algorithm design implication analysis, in Proc. IEEE INFOCOM 2005, Miami, FL, Mar. 2005, pp [28] C. Wolter, Consumer VoIP price war heating up, Dec. 1, 2004 [Online]. Available: Srinivas Shakkottai (S 00) received the B.Eng. degree in electronics communication engineering from Bangalore University, India, in 2001, the M.S. degree in electrical engineering from the University of Illinois at Urbana-Champaign in He is currently pursuing the Ph.D. degree in the Department of Electrical Computer Engineering at the University of Illinois at Urbana-Champaign. His research interests include the design analysis of peer-to-peer systems, pricing approaches to resource allocation in fixed wireless networks, game theory, congestion control in the Internet measurement analysis of Internet data.

13 SHAKKOTTAI AND SRIKANT: ECONOMICS OF NETWORK PRICING WITH MULTIPLE ISPs 1245 R. Srikant (S 90 M 91 SM 01 F 06) received the B.Tech. degree from the Indian Institute of Technology, Madras, in 1985, the M.S. Ph.D. degrees from the University of Illinois at Urbana-Champaign in , respectively, all in electrical engineering. He was a Member of Technical Staff at AT&T Bell Laboratories from 1991 to He is currently with the University of Illinois, where he is a Professor in the Department of Electrical Computer Engineering a Research Professor in the Coordinated Science Laboratory. His research interests include communication networks, stochastic processes, queueing theory, information theory, game theory. Dr. Srikant was an associate editor of Automatica, is currently an associate editor of the IEEE/ACM TRANSACTIONS ON NETWORKING IEEE TRANSACTIONS ON AUTOMATIC CONTROL. He is also on the editorial boards of special issues of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS IEEE TRANSACTIONS ON INFORMATION THEORY. He was the chair of the 2002 IEEE Computer Communications Workshop in Santa Fe, NM, will be a program co-chair of IEEE INFOCOM 2007.