Market Mechanisms for Network Resource Sharing - CiteSeerX

Market Mechanisms for Network Resource Sharing Nemo Semret

Submitted in partial ful llment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

Columbia University 1999

c 1999

Nemo Semret All Rights Reserved

ABSTRACT

Market Mechanisms for Network Resource Sharing Nemo Semret

The theme of this thesis is the design and analysis of decentralized and distributed market mechanisms for resource sharing in multiservice networks. The motivation for a market-based approach is twofold. First, in modern multiservice networks, resources such as bandwidth and buer space have dierent value to dierent users, and these valuations cannot, in general, be accurately known in advance as users compete against each other for the resources. Second, the network resources themselves are distributed, and often, not subject to any single authority. We present the Progressive Second Price auction (PSP), a new decentralized mechanism for allocating variable-size shares of a resource among multiple users. Under elastic demand, the PSP auction is incentive compatible and stable, in that it has a \truthful" -Nash equilibrium where all players bid at prices equal to their marginal valuation of the resource. PSP is ecient in that the equilibrium allocation maximizes total user value. In a dynamic setting, we derive a bound on the time to converge to equilibrium, when users are using an optimal normal form strategy. We then extend the PSP auction to be applied by independent resource sellers on each element of a network with arbitrary topology, with players having arbitrary but xed routing/provisioning constraints. We derive an optimal truthful strategy for coordinated bidding for a player participating in auctions on multiple resource elements, and show that the equilibrium and eciency results still hold. We also show how our networked auction model can apply to virtual networks, virtual paths, and edge capacity allocation networks. We then turn our attention to the problem of reservations and admission control

for connection oriented network services. We propose a new approach to pricing of capacity in service systems with blocking, using spot and derivative market mechanisms. A second-price auction among arrivals grouped in batches gives rise to the spot market of usage charges. A reservation guaranteeing access for an arbitrary duration with a usage price below the bid can be made at any time before or during service, thus eliminating the risk { inherent to the spot market { of being dropped before service completion. We de ne the reservation as a hold option, which is analogous to derivative nancial instruments (e.g. options, futures) integrated over time. Based on a heavy-trac diusion model for the corresponding two-stage queueing system, we compute the reservation fee as the fair market price of a hold option. We validate this approach with simulations driven by a real trac trace at a dial-up Internet access modem-pool. Finally, we present a decentralized, distributed, exible software architecture implementing the above pricing systems.

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Contents 1 Introduction

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1.1. Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2. Contributions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 1.3. Bibliographical Notes : : : : : : : : : : : : : : : : : : : : : : : : : : : 12

2 The PSP Auction

2.1. Message Process : : : : : : : : : : : : : : : : 2.2. Allocation Rule : : : : : : : : : : : : : : : : 2.3. Analysis of PSP Games : : : : : : : : : : : : 2.3.1 User Preferences : : : : : : : : : : : 2.3.2 Equilibrium of PSP : : : : : : : : : : 2.3.3 Eciency : : : : : : : : : : : : : : : 2.4. Bidding algorithm and Convergence : : : : : 2.4.1 Convergence in Dynamic Play : : : : 2.4.2 Convergence Time : : : : : : : : : : 2.5. Single Node Game Simulation : : : : : : : : 2.6. Conclusion : : : : : : : : : : : : : : : : : : : A. Information-theoretic basis for the valuation

3 Networked Auctions

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ii 3.1. Networked PSP Mechanisms : : : : : : : : : : : : : : : : : : : : : : 3.2. Network PSP Analysis : : : : : : : : : : : : : : : : : : : : : : : : : 3.2.1 Players : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.2.2 Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.2.3 Eciency : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.3. Peering, Provisioning and Market-pricing of Edge-allocated capacity 3.3.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.3.2 The Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.4. Dynamic Provisioning of Dierentiated Services : : : : : : : : : : : 3.5. Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

4 Spot and Derivative Markets in Admission Control

4.1. Queueing Model : : : : : : : : : : : : : : : : : : : : : 4.1.1 Preliminaries : : : : : : : : : : : : : : : : : : 4.1.2 Product-form solution : : : : : : : : : : : : : 4.1.3 Diusion approximation : : : : : : : : : : : : 4.2. Diusion Models of the Market Prices : : : : : : : : : 4.2.1 Spot Price : : : : : : : : : : : : : : : : : : : : 4.2.2 Diusion model of the spot price : : : : : : : 4.3. Computing Reservation Fees: the Derivative Market : 4.4. Simulations : : : : : : : : : : : : : : : : : : : : : : : 4.5. Approximations related to g : : : : : : : : : : : : : : 4.5.1 Case mu(1 ? u) > V 2 : : : : : : : : : : : : : : 4.5.2 Case mu(1 ? u)  V 2 : : : : : : : : : : : : : : 4.6. Approximating the Drift and Diusion Coecients : 4.6.1 Example: F uniform : : : : : : : : : : : : : : 4.7. Optimal Seller Strategies : : : : : : : : : : : : : : : :

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5 Networking Games Software 5.1. Architectural Overview : : : : : : : : : : 5.2. Object Model : : : : : : : : : : : : : : : 5.3. Experiments : : : : : : : : : : : : : : : : 5.3.1 2 and 3 buyer game : : : : : : : : 5.3.2 The importance of being truthful

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List of Figures 1-1 Bandwidth sharing : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1-2 Networking games view of sharing : : : : : : : : : : : : : : : : : : : : 2 1-3 Some dimensions of a Taxonomy of Pricing : : : : : : : : : : : : : : : 10 2-1 Exclusion-compensation principle: the intuition behind the PSP rule 2-2 Truthful -best reply : : : : : : : : : : : : : : : : : : : : : : : : : : : 2-3 Utility u4(s4) for s1 = (100; 1), s2 = (10; 2), s3 = (20; 4), s5 = (20; 7), s6 = (30; 12) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2-4 Parabolic valuation with  = 0:5 and q = 70 : : : : : : : : : : : : : : 2-5 Mean (+/- std. dev.) number of bids { solid line. The dashed line is I + I 2=10. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2-6 Mean (+/- std. dev.) convergence time in seconds (for a 1 second bid interval). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2-7 Number of bids to converge vs  : : : : : : : : : : : : : : : : : : : : : 2-8 Pi i(ai ) { solid line, and maxa Pi i(ai) ? 4()?1=2 { dashed line, vs . 2-9 Small user joining a stable market of 72 players : : : : : : : : : : : : 2-10 T1 user joining a stable market of 73 players : : : : : : : : : : : : : : 2-11 T3 user joining a stable market of 74 players : : : : : : : : : : : : : : 2-12 The volume discount eect: a5 < a4 ) c5=a5  c4=a4 : : : : : : : : : 2-13 Distortion-rate based valuation for a Bernoulli p = 1=2 source : : : :

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v 3-1 Demand curve for a broker j . : : : : : : : : : : : : : : : : : : : : : : 3-2 2-tier auction pricing framework for di-serv internet : : : : : : : : : 3-3 Inter-Network provisioning coecients for dierent services: Olympic Gold, Silver and Bronze services, and the Virtual Leased Line service 3-4 Valuation and Marginal Valuation Functions, ai = 3 and i = 6 : : : : 3-5 Simulation of three networks with two stable classes : : : : : : : : : : 3-6 Spectral radius as a function of inter-network provisioning coecients 3-7 Simulation of one unstable class : : : : : : : : : : : : : : : : : : : : : 3-8 Dis-peering eect : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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4-1 Queueing Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 94 4-2 Pricing Mechanism : : : : : : : : : : : : : : : : : : : : : : : : : : : : 95 4-3 Modem pool call statistics (solid) and equivalent { same mean { exponential distributions (dashed) : : : : : : : : : : : : : : : : : : : : : 108 4-4 Histograms of P , for P0 = 0 and  = 0; 1; 2 minutes; the bins represent price levels [0; 0:1); [0:1; 0:2); : : :; [0:9; 1). : : : : : : : : : : : : : : 109 4-5 Simulation trace : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 109 4-6 Eciency: 1.0 = rst-come rst-served system : : : : : : : : : : : : : 111 4-7 The reservation fee is at most 1 duration. : : : : : : : : : : : : : : : 112 4-8 Reservations are proportionally more expensive for low bid prices. : : 112 5-1 5-2 5-3 5-4 5-5

Architecture for a Distributed Networking Game. : : : : Distributed object model : : : : : : : : : : : : : : : : : : Two and three bidder game as seen by player 1 : : : : : Truthful and Aggressive strategy results : : : : : : : : : Allocations for 20 rounds in a 6 player, 6 strategy game :

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vi 5-6 Performance of truthful strategy (solid lines) compared to average of all strategies (dashed lines) : : : : : : : : : : : : : : : : : : : : : : : : 129

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Acknowledgements I would like to thank my advisor, Prof. Aurel A. Lazar, for giving me the opportunity to pursue this work, and for his teaching and research guidance throughout my time at Columbia. His vision and experience, going beyond traditional boundaries, were essential. Aurel has been a great inspiration and support both professionally and personally. I would also like to thank Prof. Eli Noam for his crucial role in sparking this research, for serving on my thesis proposal comittee, as well as on the defense comittee; Dr. Christian Huitema for giving me a very fruitful work experience during the summer of 1997 at Bellcore, for opening new research horizons, as well as for serving on my defense comittee; Prof. Predrag Jelenkovic for working closely with me in my rst two years, for many helpful discussions since, and for serving on my defense comittee; Prof. Andrew Campbell for our joint work, for his great support and advice, and for serving on my defense comittee. I am very grateful to Drs. Sid Ahuja and Yannis Korilis for providing me with a very rewarding work experience during the summer of 1996 at Bell Labs. I would like to thank Prof. Chris Sanchirico for serving on my thesis proposal comittee. I am indebted to Dr. Sanford Marble for many useful discussions during my summer at Bellcore. Many members of the COMET group have contributed to my research and have made my time at Columbia very enjoyable. In particular, I would like to thank Raymond Liao for our collaboration on the work of Section 3.3., and Wang Ke for his help with the simulations in Chapter 2. I would also like to thank Merate Kibriye, Michael Abdi and Gabriella Muratore for being my surrogate family in New York. Special thanks to Giovanna Giammarino for always being willing to argue, and her invaluable friendship. This thesis is dedicated my parents Guenet Guebre-Medhin and Semret Medhane.

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Chapter 1 Introduction Networking Games? Game eng. isn't work! [87]

1.1. Overview Communication networks are characterized by what economists call externalities. The value a user gets from the network depends on the other users. The positive externalities are that a communication network is more valuable if more people are connected. The negative externalities are that resources are shared by users who { because of distance, population size, or sel shness { cannot or will not coordinate their actions suciently to achieve the most desirable allocation of resources. Indeed, the basic problem in networking is how to share dispersed resources, such as bandwidth and buer space, among users who are also dispersed. Consider the simple bandwidth contention scenario shown in Figure 1-1 { a network with four switches and three links. Each user (or player) cares only about its own allocations of resources, and has only localized information. For example, user 3 may know the total demand at links 2 and 3, but not the situation at link 1. Since its allocation at link 1 aects the amount user 1 will request at links 2 and 3, in general the users are not able to fully cooperate to achieve the best

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Figure 1-1: Bandwidth sharing

Figure 1-2: Networking games view of sharing

3 allocations on the three links, even if they want to. There is no explicit central authority controlling the overall network. But the quality, eciency, and fairness of its services are shaped by the \rules" of interaction (protocols, and algorithms) put in place by the designers of the network. The recognition of this reality in many aspects of networks and distributed computations has lead in recent years to the emergence of game theoretic or noncooperative approaches in networking research, in areas such as ow-control [32, 33, 10, 35, 50], channel access [55], scheduling [89], routing [75, 51] and other areas [19, 2, 56]. Broadly, we call this approach networking games. For the scenario of Figure 1-1, the networking games view can be represented as in Figure 1-2. By taking a game theoretic approach in the engineering of the network, one can design mechanisms where the intelligence and decision making is distributed and the objective of a more ecient and fair utilization of shared resources results from the induced dynamics. This approach gives rise to decentralized algorithms, and can play a key part in scalable architectures which sustainably provide Quality of Service (QoS) guarantees, the holy grail of modern networking. The players in the network are likely to be software agents, rather than humans. Agents acquire resources, such as bandwidth and buer space, from the network on behalf of applications (video, voice, data transfer). Under appropriate rules of interaction, the collective actions of all the agents constitute a distributed intelligence which can achieve outcomes as ecient as those that would be obtained by the best possible central controller. Centralized control is not possible, however, because of the diverse ownership of the network resources. More fundamentally, unlike for example a computer where all resources are governed by the operating system, a large multimedia network with QoS would be too complex and far- ung to be under a single authority, or

4 even to de ne centralized control objectives. Thus the assumption of sel sh users is not due to pessimism, but rather a recognition that technology and nature make it impossible to be unsel sh, or even to know what it is to be unsel sh. Finally, in the networking games approaches, the key issue of pricing can be resolved from the ground up in the engineering of the network, rather than having, as in existing communication networks, a price structure imposed post-facto. Prices, whether they relate to \real money" in a commercial network or \funny money" (based on quotas) in a closed system, play a fundamental role as resource allocation control signals. In networks, as elsewhere, this role becomes apparent when demand exceeds supply, i.e., in the presence of congestion [16, 62]. Even when the network resources are public goods, pricing is necessary to maximize the social bene t by making ecient allocations of, e.g., spectrum for wireless communications [72]. In commercial networks, in addition to controlling congestion, prices have a dual role, which is singalling the need for, and nancing, the development of infrastructure [27]. The telephone system and the current Internet represent two extremes of the relationship between resource allocation and pricing. The resources allocated to a telephone call are xed, and usage prices are based on the predictability of the total demand at any given time. On the Internet, the current practice of pricing by the maximum capacity of the user's connection ( at-rate pricing) decouples the allocation (actual use) of resources from prices. In the emerging multiservice networks (ATM, Next-Generation Internet), neither of these approaches are viable. The former because the wide and rapidly evolving range of applications (including some which adapt to resource availability) makes demand more dicult to predict. And the latter because, once the at fee is paid, there are no incentives to limit usage since increasing consumption bene ts the user individually, whereas limiting it to

5 sustainable levels brings bene ts which are shared by all. This makes it vulnerable to the well-known \tragedy of the commons" [30]. Traditionally, on the Internet, the congestion control built-in to TCP has made most trac streams behave cooperatively, since they \back-o" in the face of congestion [38]. However, as multimedia trac { with less-friendly congestion control { accounts for an increasing share of the trac, such cooperative behaviour without incentives cannot be assumed. With at rates alone, there is a tendency toward one of two things: increasing congestion which chases away high-value users; or increasing rates which exclude low-value users [26], thus resulting in an over-provisioned network. Either scenario may be valid in some contexts [74], and in fact both can coexist in a world where networks are separated or partitioned into cheap-low quality and expensive-high quality networks. But, when even a single (large) user's needs are highly variable, the more static the partitioning, the more inecient the network compared to one where all types of users can dynamically share all the resources. Thus there is a need to develop new approaches to pricing of network resources. Among the requirements are: sensitivity to the range of resource requirements (either through a suciently rich range of trac classes which are priced dierently, or by allowing users to explicitly quantify resource requirements); prices must be dynamically responsive to unpredictable demand (market based system); perhaps most importantly the pricing architecture should constrain as little as possible the eciency trade-os of the policies. For example, if the scarcity of resources occurs only at a few speci c points in the network, a pricing architecture which must be engineered (for signalling allocations, monitoring usage, etc.) for the whole network is clearly not desirable. The fundamental issue in designing pricing policies is the trade-o between engineering eciency and economic eciency. This trade-o, which is more or less

6 constrained by the underlying network technology, has many dimensions, including:

 how much measurement (from usage to capacity pricing),  the granularity of dierently priced service oerings (e.g., number of trac classes),

 the level of resource aggregation { both in time and in space { at which pricing is done (per packet/cell or per connection, at the edge of the network or at each hop), and

 the information requirement (how much a priori knowledge of user behavior and preferences is required/assumed by the network in computing prices). These dimensions de ne a taxonomy for classifying previous work in pricing. In the most common approaches to pricing, the seller assumes perfect a priori knowledge of demand and explicitly computes the prices that achieve some systemwide objective. In the simplest version, this is an oine calculation of optimal prices (e.g., time-of-day pricing based on historical trac patterns, as in traditional telephony). In more sophisticated versions, the seller assumes the functional form of demand, and adjusts prices by on-line optimizations [29, 3, 52, 77]. These pricing schemes are \model-based", in that the relationship between demand and price (and possibly time) is assumed in an a-priori formula. Knowledge of this model and its parameters is precisely the \information requirement" described above. In market-based approaches, no precise model need be assumed, so a given demand may lead to dierent prices, which emerge directly from the users valuation of the resources and, possibly, their decision strategies. The seller does not require a priori demand information. In other words, no \market-research" is required, rather the activity of the market itself, i.e., the play of the game, is essentially the \real-time market-research".

7 One such approach, which achieves economic eciency is the smart-market approach of [61], wherein each packet contains a bid, and if it is served, pays a clearing price given by the highest bid among packets which are denied service (dropped). This approach is incentive compatible in that the optimal strategy for a (sel sh) user is to set the bid price in each packet equal to the true valuation. In other words, like all pure market systems, it has the lowest possible information requirement. But the engineering cost (sorting packets by bid price, per-packet and per-hop accounting) is prohibitive. Indeed, line speeds (the rates at which data packets arrive at and leave from nodes) are generally high relative to the processing power in the switch or router, and so the smart-market approach is not implementable with current technology. In [47], users are charged according to a combination of declared and measured characteristics of trac. By taking an equivalent bandwidth model of resource utilization, and assuming appropriate trac models, a menu of pricing plans indexed by the declared trac can be oered which encourages users to make truthful declarations (e.g., of the mean rate), and also encourages the users' characterization eorts to be directed where they are most relevant to the network resource allocation. As the pricing is relative, [47] does not aim to address the problem of determining the actual monetary values of the market price (that users would be willing to pay). Thus, the information requirement is only partially reduced, that is, with respect to quantity and not price. Another pricing scheme which incorporates multiplexing gain is formulated in [43]. These and a number of other schemes are summarized in [42], in a comprehensive view of the connection establishment process, which identi es the user-network negotiation as the key \missing link" in network engineering/economic research. In terms of our taxonomy, this is part of the information requirement trade-o. Indeed,

8 in the absence of formal mechanisms to deal with the information problem, complex and (at least intuitively) undesirable things happen. For example, some providers oer expensive \front of the book" rates to uninformed customers, and lower \back of the book" rates to informed customers who may be about to defect to another carrier (see [27] and also the recent wars between AT&T and MCI in consumer long-distance service in the United States). In [90], it is argued that architectural considerations such as where charges are assessed should take precedence over the pursuit of optimal eciency, and spatial aggregation, i.e., edge pricing, is proposed as a useful paradigm. We present the design and analysis of two new pricing mechanisms for communication network resources. Our mechanisms are the Progressive Second Price auction, and the Hold Option. We are concerned with pricing in relation to resource allocation (i.e. demand based pricing), and thus will not consider pricing in its role in the competition among dierent networks for users. In Chapters 2 and 3, we propose the Progressive Second Price auction, a new mechanism which accommodates various dimensions of the engineering-economics trade-o. Compared to traditional auctions and other mechanisms in the economics litterature, the main strengths of this mechanism are that:

 unlike traditional auctions, which are for indivisible objects, PSP applies to a generic arbitrarily divisible and additive resource model (which may be equivalent bandwidth, peak rate, contract regions, etc., at any level of aggregation);

 unlike the usual \direct revelation" mechanisms [15, 28] in the economics litterature, PSP a) is designed with a very small message space, so it does not impose an undue signalling burden; and b) has an auction rule which is computationally inexpensive, and expressible in a simple closed-form expression, while still achieving the objective of economic eciency.

9 Furthermore, unlike the model-based pricing schemes, PSP does not assume any speci c mapping of resource allocation to quality of service, neither does it assume the mapping from quality of service to willingness to pay. Rather, users are de ned as having an explicit monetary valuation of quantities of resource, which the network doesn't or can't know a priori. Thus, in terms of our trade-o taxonomy, the PSP mechanism provides for unlimited granularity, exibility in the level of aggregation and minimal information requirement. This last feature, that no a priori information on demand is required, is somewhat subtle. Indeed, the analysis of the auction as a game assumes some forms of demand, or user preferences, to derive properties. However, the mechanism itself whether at implementation or run-time, does not need to know anything about the users, their demands, or willingness to pay. Previous attempts to apply auction models to communications [1], and computational systems [95], focused mainly on the software and hardware architecture of the systems, rather than on designing the auction itself. They used traditional secondprice indivisible object auctions, and thus did not posses the advantage of light signalling and computational load combined with complete exibility in granularity of the resource allocations (i.e., arbitrarily divisible resources). These advantages are the bene ts of our design of a new mechanism, which we believe is the rst auction-based approach to pricing that is realistic for real-time use in high-speed communication networks. PSP can be used for market-based \capacity pricing", as opposed to usage pricing or at-rate pricing. In the most likely auction scenaria, the bidders would be automated agents requesting bulk bandwidth for aggregate ows like virtual paths, virtual private networks, or edge capacity [13]. Figure 1-3 shows positions of different combinations of allocation and pricing schemes in our taxonomy, with the measurement and granularity dimensions omitted. \BE" is the best-eort Internet

10 econ. efficiency SM DS/PSP ATM/PSP

POTS/TOD BE/FR BE/U

ATM/EB DS/FR

disaggregation

information required

Figure 1-3: Some dimensions of a Taxonomy of Pricing service, \DS" is the dierential service Internet [21], \ATM" generically denotes networks where virtual paths and virtual networks can be allocated, \EB" denotes equivalent bandwidth-based allocation as in [47], and \POTS" denotes the traditional (\plain old") telephone system. \FR" means at-rate pricing, \U" usage pricing, \TOD" denotes time-of-day pricing, \SM" denotes a per-packet bidding system (a la [61]), and \PSP" denotes our proposed auction mechanism. We begin by formally presenting the design of our Progressive Second Price auction mechanism for sharing a single arbitrarily divisible resource in Chapter 2. After describing our model of user preferences and the elastic demand assumption, we prove that PSP has the desired properties of incentive compatibility, stability, and eciency. We provide analytical and simulation results on the convergence properties, and the eciency trade-os. We conclude with detailed simulations of a realistic scenario of a single node bandwidth auction. Appendix A. describes an information theoretic basis for valuations of the type that are assumed in the analysis. Chapter 3 generalizes the results to distributed auctioning of resources on arbitrary network topologies.

11 The auction can be viewed as a real-time spot market for network resources, just as there is for commodities such as oil or pork bellies. For bandwidth, as for the other commodities, there is a natural derivative market for network bandwidth or buer space. In a derivative market, one can buy or sell risk, where the risk is the spot market price's inherent variability. This is done by trading contracts that secure a given price in the future [36]. These contracts (options, futures and other \derivatives") have themselves a fair cost for the party who is reducing risk, just as an insurance policy holder has to pay a premium. In the bandwidth market, a user may, for example, want to establish a 1 hour connection with a maximum of 10$ per hour of use, in a network where the capacity in each minute of that hour is allocated by spot market, resulting in variable prices. Thus, connection oriented services form a natural derivative market in communication networks. In Chapter 4, we de ne this kind of resource reservation as a new derivative, a Hold Option, and derive its fair price. Our approach relies on a heavy trac approximation on the queueig system represteing the mechanism, to derive a diusion model of the spot market price, to which we apply an extension of the Balck-Scholes option pricing approach to price reservations. To the best of our knowledge, there is no published previous work in applying derivative pricing to price communication network resources. At the time we rst published our work, the theory of real options was just beginning to be discussed in economic assessments related to regulation of the telecommunications industry [17]. Chapter 5 describes a software architecture for a distributed agent system which implements the games discussed in the preceding chapters. This platform provides a valuable tool for understanding the games. Indeed, many of the insights behind the analysis in Chapters 2 and 3 arose directly from experimentation with this platform, which also allows us to explore issues of beyond the scope of the analysis.

12

1.2. Contributions The Progressive Second Price auction mechanism is a new invention. While the theory of auctions for indivisible objects is well developed (see [93, 69, 100, 67] for a representative sample of the fundamental work), this appears to be the rst auction for arbitrary size shares of an in nitely divisible resource. The design of an incentive compatible mechanism with a small (under-dimensioned) message space, and an in nite dimensional user-type space, is a signi cant departure from the conventional approach in the mechanism design litterature [15, 28, 64]. The properties of incentive compatibility, eciency and the generalization to arbitrary network topologies are rsts for such a mechanism. These properties also make PSP the rst decentralized dynamic pricing scheme for communications networks that is feasible in terms of the signalling, billing, and a priori information load. The Hold Option formulation of a connection/reservation, and the combination of the heavy trac approximation with the stochastic dierential equation approach to derivative pricing is a novel idea and constitutes a contribution to both nancial mathematics and network engineering. Our software implementation was the rst auction implementation with distributed sellers and decentralized strategic computation for the bidders, as well as an innovative use of a platform independent programming language in the Internet.

1.3. Bibliographical Notes The analytical method in Chapters 2 and 3 is game theory [94, 24]. We view the problem as one of \mechanism design", where incentive compatibility is the key idea. Our approach extends the classic work of Vickrey [93], and is related to Clarke-Groves mechanisms [15, 28]. The design is then analyzed as a strategic game

13 of complete information, where the solution concept is that of the Nash equilibrium [70]. Most of our results of Chapters 2 and 3 appeared in [58, 59, 85, 86]. In Chapter 4, the key building blocks are a heavy-trac approximation to derive a diusion model of the queueing system inspired by [37, 9], and the classic stochastic dierential equation approach to option pricing pioneered by MertonBlack-Scholes [8, 65]. Results from Chapter 4 appear in [84, 60].

14

Chapter 2 The PSP Auction Strange gem in Wok. [87]

Following [99], it is useful to expose the design in terms of its two aspects: realization, where a message process that enables a certain allocation objective is de ned; and Nash implementation, where allocation rules are designed with incentives which drive the players to an equilibrium where the (designer's) desired allocation is achieved. We de ne the message process in Section 2.1., and the allocation rule in Section 2.2.. In Section 2.3., we analyze the outcome (Nash equlibirium) and performance (eciency) of the game.

2.1. Message Process Here we make the fundamental choice which will constrain the subsequent aspects of the design. Our rst concern here is with engineering. For the sake of scalability in a network setting, we shall aim for a process where a) the exchanged messages are as small as possible, while still conveying enough information to allow resource allocation and pricing to be performed without any a-priori knowledge of demand (market research, etc.); and b) the amount of computation at the center is minimized. Given a quantity Q of a resource, and a set of players I = f1; : : :; I g, an auction

15 is a mechanism consisting of: 1) players submitting bids, i.e. declaring their desired share of the total resource and a price they are willing to pay for it, and 2) the auctioneer allocating shares of the resource to the players based on their bids. Player i's bid is si = (qi; pi) 2 Si = [0; Q]  [0; 1), meaning he would like a quantity qi at a unit price pi. A bid pro le is s = (s1; : : :; sI ). Following standard game theoretic notation, let s?i  (s1; : : :; si?1; si+1; : : : ; sI ), i.e. the bid pro le of player i's opponents, obtained from s by deleting si. When we wish to emphasize a dependence on a particular player's bid si, we will write the pro le s as (si ; s?i). The allocation is done by an allocation rule A,

A:

S ?! S s = (q; p) 7?! A(s) = (a(s); c(s));

where S = Qi2I Si. The i-th row of A(s), Ai(s) = (ai(s); ci(s)), is the allocation to player i: she gets a quantity ai(s) for which she is charged ci(s). Note that p is a price per unit and c is a total cost. An allocation rule A is feasible if 8s,

X i2I

ai(s)  Q

and 8i 2 I ,

ai(s)  qi; ci(s)  piqi:

Remark: The above formulation is a generalization of what is usually meant by an auction. The latter is the special case where aw (s) = Q for some winner w 2 I and ai(s) = 0; 8i = 6 w, i.e. the sale of a single indivisible object to one buyer, for which

16 the theory is well developed [67, 69]. In our approach, allocations are for arbitrary shares of the total available quantity of resource. Equivalently, one could slice the resource into many small units, each of which is auctioned as an indivisible object. But in a practical implementation of auctions for sharing a resource, a process of bidding for each individual unit would result in a tremendous signaling overhead. More importantly, since the users would be bidding on a discrete grid of quantities, analytical predictions of outcomes could be misleading since they could be sensitive to the particular choice of grid1. Most of the mechanism design literature in Economics makes use of the following \Revelation Principle": Given any feasible2 auction mechanism, there exists an equivalent3 feasible direct revelation mechanism which gives to the seller and all bidders the same expected utilities as the given mechanism. ([69], Lemma 1) In this sharing context, a direct revelation mechanism would be one where each user message consists of the user's type, which is the valuation4 of the resource over the whole range of their possible demands, i.e. a function i : [0; 1) ! [0; 1), and the budget (see Section 2.3.1). A consequence of the revelation principle is that the mechanism designer can restrict her attention to direct revelation mechanisms, nd the best mechanism in terms of her (economic) eciency objectives, and then { if necessary { transform it into an equivalent mechanism in the desired message space. For a more detailed discussion of this point, see [24] p. 34, and references therein. Myerson's feasible auctions satisfy a number of requirements, notably an incentive compatibility constraint. 3 By equivalent, it is meant that, at some equilibrium, all players get the same utility. There may be other, possibly ill-behaved, equilibria. 4 The valuation of a given amount of resource is how much the user is willing to pay for that quantity. The inverse of the valuation is the user's demand function, giving a desired quantity for each price. 1

2

17 This is convenient because one can exclude the in nitely many mechanisms with larger message spaces, without fear of missing any better designs. The design process is usually the solution of an optimization (mathematical programming) problem. For this reason, in the litterature, mechanism design problems are mostly solved for cases where the space of users' types is one dimensional, or at most nite dimensional [64], using message spaces that are of the same dimension. In our sharing problem, the conventional approach is unsatisfactory in two ways:

 First, a user's type is in nite-dimensional, as we do not restrict the valuation functions beyond some very general assumptions (see Section 2.3.1), and so the conventional \programming" approach of deriving the mechanism from the revelation principle would lead to an intractable problem.

 Second, while the conventional approach, even if it was tractable, would yield a mechanism with an in nite dimensional message space, for engineering reasons5 we choose a message space that is 2-dimensional. Thus, a given message can come from many possible types, so there is no single way to do the transformation from the direct revelation mechanism to the desired one. Thus, unlike most of the mechanism design literature, we will take a direct approach, where we posit an allocation rule for our desired message space, and then show that it has an equilibrium, and that the design objective is met at equilibrium. This is equivalent to guessing the right direct-revelation-to-desired-mechanism transformation and building it into the allocation rule from the start. The size of the signalling messages that must be exchanged grows with the dimensionality of the message space. 5

18

p5

q

p4

q 4

pi

si

p 2 p1 q

5

q

2

q1

p 0 0 Q

q i ai = Q - q5 - q4 ci =

Figure 2-1: Exclusion-compensation principle: the intuition behind the PSP rule

2.2. Allocation Rule De ne, for y  0

2 3+ X Qi (y; s?i) = 4Q ? qk 5 ; pk y;k6=i

and

(2.1)

2 3+ X qk 5 : Qi(y; s?i) = lim Q (; s?i) = 4Q ? &y i pk >y;k6=i

The \progressive second price" (PSP) allocation rule is de ned as follows:

ai(s) = qi ^ Qi(pi ; s?i); X ci(s) = pj [aj (0; s?i ) ? aj (si; s?i)] ; j 6=i

(2.2) (2.3)

where ^ means taking the minimum. Remark a: For a xed opponent pro le s?i, Qi(pi ; s?i) represents the maximum available quantity at a bid price of pi . The intuition behind PSP is an

19 exclusion-compensation principle: player i pays for its allocation so as to exactly cover the \social opportunity cost" which is given by the declared willingness to pay (bids) of the users who are excluded by i's presence (see Figure 2-1), and thus also compensates the seller for the maximum lost potential revenue. Note that this amounts to implicitly assuming that the bid price accurately re ects the marginal valuation i0 on the range [ai; qi]. In other words, by this rule the auctioneer is saying to the player: \if you bid (qi; pi ), I take it to mean that in the vicinity of qi, i can be approximated by a line of slope pi ." This is the (built-in) transformation from the direct-revelation mechanism to the desired message process discussed in the second remark at the end of Section 2.1.. The charge ci increases with ai in a manner similar to the income tax in a progressive tax system. For a xed opponent pro le s?i, imagine player i is increasing qi, starting from 0. The rst few units that player i gets will be taken away from the lowest clearing opponent (i.e. m = arg minj fpj : aj > 0g), and player i will pay a price (marginal cost) pm per unit. When am reaches 0, the subsequent units that player i gets will cost him pm0 > pm , where m0 is the new lowest clearing player, the one just above m. The PSP rule is the natural generalization of second-price auctions (or Vickrey auctions). In a Vickrey auction of a single non-divisible object, each player submits a sealed bid, and the object is sold to the highest bidder at the bid price of the second highest bidder, which is what happens here if qi = Q; 8i. This is widely known to have many desirable properties [93, 69, 24], the most important of which is that it has an equilibrium pro le where all players bid their true valuation. As we will presently show, this property is preserved by the PSP rule in the more general case of sharing an arbitrarily divisible resource, and this leads to stability (Nash equilibrium). The PSP rule is analogous to Clarke-Groves mechanisms [15, 28, 63] in the direct-revelation case.

20

Remark b: When two players bid at exactly the same price, and they are asking for more than is available at that price, (2.2) punishes both of them. For example, if Q = 100 and s1 = (60; 4) and s2 = (70; 4), the allocations would be a1 = 60 ^ (100 ? 70) = 30, and a2 = 70 ^ (100 ? 60) = 40. Since the bid prices are equal, there is no \right" way to decide who to give the remaining capacity to. One could divide it equally, or proportionally to their requests, etc. For the subsequent analysis, it turns out it is simpler to not give it to either one (of course, it will be allocated to the lower bidders if there are any). This is just a technicality since by deciding this, we ensure that it will never happen (at equilibrium), since the users will always prefer to change their prices and/or reduce their quantity. Considering the computational complexity of PSP, a straightforward implementation would at worst, sort the bids in time I log I , perform (2.2) in linear time, and (2.3) can be done in time I 2. Thus, the complexity of computing the allocations is O(I 2).

2.3. Analysis of PSP Games 2.3.1 User Preferences Since the allocation rule A is given by design, the only analytical assumptions we make is on the form of the players' preferences. Player i's preferences are given by its von Neumann-Morgenstern utility function [94]: ui : S ?! (?1; 1) s 7?! ui(s): Utility is the key concept in the game theoretic approach, and it is worthwhile to consider its philosophical basis:

21 It dooes not seem to us that these notions [utility and preference] are qualitatively inferior to certain well established and indispensable notions in physics, like force, mass, charge, etc. That is, while they are in their immediate form merely de nitions, they become subject to empirical control through the theories which are built upon them { and in no other way. Thus the notion of utility is raised above the status of tautology by such economic theories as make use of it and the results of which can be compared with experience or at least with common sense. The individual who attemps to obtain [the utility's] maxima is also said to act rationally.[94] Player i has a valuation of the resource i(ai(s))  0, which is the total value of its allocation. Thus, for a bid pro le of s, under allocation rule A, player i getting an allocation Ai(s) has the quasi-linear utility

ui(s) = i(ai(s)) ? ci(s)

(2.4)

which is simply the value of what it gets minus the cost. In addition, the player can be constrained by a budget bi 2 [0; 1], so the bid si must lie in the set

Si(s?i ) = fsi 2 Si : ci(si; s?i)  big:

(2.5)

In the proofs of the following section, we will assume that users have elastic demand, that is:

Assumption 1 For any i 2 I ,  i(0) = 0;

22

 i is dierentiable,  i0  0, non-increasing and continuous  9 i > 0, 8z  0, i0 (z) > 0 ) 8 < z; i0(z)  i0 () ? i(z ? ). The last item says that as long as the valuation is strictly increasing, it must also be strictly concave (with minimum curvature i ). However, it is allowed to \ atten" beyond a certain amount of resource. Valuations of this form have wide applicability. In addition to the obvious economic justi cation of diminishing returns (the value of additional units of capacity diminishes), such valuations can be justi ed by information theoretic fundamentals as well. For examples, see Section 2.4.2 and Appendix A..

2.3.2 Equilibrium of PSP The auction game is given by (Q; u1; : : :; uI ; A), that is, by specifying the resource, the players, and a feasible allocation rule. We analyze it as a strategic game of complete information. De ne the set of best replies to a pro le s?i of opponents bids: Si(s?i ) = fsi 2 Si(s?i ) : ui(si; s?i)  ui(s0i; s?i ); 8s0i 2 Si(s?i)g. Let S (s) = Qi Si(s?i ). A Nash equilibrium is a xed point of the point-to-set mapping S , i.e. a pro le s 2 S (s). For static games, Nash equilibria are \consistent" predictions of how the game will be played, in the sense that if all players predict that a particular Nash equilibrium will occur then no player has an incentive to play dierently. Thus, a Nash equilibrium, and only a Nash equilibrium, can have the property that the players can predict it, predict that their opponents predict it, and so on. ([24], p. 13)

23 In a dynamic game, where players recompute the best response to the current strategy pro le of their opponents6 , this iteration can only converge to a Nash equilibrium (if it converges at all). In addition, an important trend in modern game theory is the development of learning models, and there too, it has been shown that Nash equilibria result also from rational learning through repeated play among the same players [44]. A more general (and hence weaker) notion of stability is the existence of an -Nash equilibrium. Let the -best replies be Si (s?i) = fsi 2 Si(s?i ) : ui(si; s?i )  ui(s0i; s?i ) ? ; 8s0i 2 Si(s?i)g. An -Nash equilibrium is a xed point of S . In a dynamic auction game,  > 0 can be interpreted as a bid fee paid by a bidder each time they submit a bid. Thus, the user will send a best reply bid as long as it improves her current utility by , and the game can only end at an -Nash equilibrium. De ne Pi (z; s?i) = inf fy  0 : Qi(y; s?i)  zg : (2.6) Thus, for xed s?i, 8y; z  0,

and7

z  Qi(y; s?i) ) y  Pi(z; s?i )

(2.7)

y > Pi (z; s?i) ) z  Qi(y; s?i):

(2.8)

The graph of Pi(:; s?i) is the \staircase" shown in Figure 2-1, and that of Qi(:; s?i) is obtained by ipping it 90 degrees. This is known as Cournot iteration. Actually, since Qi (:; s?i) is upper-semi-continuous (jumps up), we have z  Qi(y; s?i ) , y  Pi (z; s?i ). 6

7

24 It is readily apparent that

ci(s) =

Z ai(s) 0

Pi(z; s?i) dz:

(2.9)

The key property of PSP is that, for a given opponent pro le, a player cannot do much better than simply tell the truth, which in this setting means bidding at a price equal to the marginal valuation, i.e. set pi = i0 (qi). By doing so, she can always get within  > 0 of the best utility. Let Ti = fsi 2 Si : pi = i0 (qi)g, the (unconstrained) set of player i's truthful bids, and T = Qi Ti.

Proposition 1 (Incentive compatibility) Under Assumption 1, 8i 2 I , 8s?i 2 S?i , such that Qi (0; s?i ) = 0, for any  > 0, there exists a truthful -best reply ti(s?i ) 2 Ti \ Si(s?i ). In particular, let

Gi (s?i ) =



z 2 [0; Q] : z  Qi(i0 (z); s?i) and

Zz 0



Pi (; s?i) d  bi :

Then with vi = [sup Gi(s?i ) ? =i0 (0)]+ and wi = i0 (vi ), ti = (vi ; wi) 2 Ti \ Si(s?i ).

A truthful best reply can be found in a straightforward manner, as illustrated in Figure 2-2. Proof: Fix s?i 2 S?i. Let zi = sup Gi(s?i ) and yi = i0 (zi). R By de nition of zi , 9fz (n)g  Gi(s?i ) such that limn z (n) = zi . Hence bi  limn 0z(n) Pi (; s?i) d = R zi P (; s ) d  c (t ; s ), where the equality comes from the boundedness of P and the ?i i i ?i i 0 i

Lebesgue dominated convergence theorem, and the second inequality from (2.9) and (2.2). Thus ti 2 T \ Si (s?i ). Next we show that ti 2 Si (s?i ). First, zi = limn z (n)  limn Qi (i0 (z (n)); s?i)  Qi (limn i0 (z(n)); s?i), where the inequalities follow respectively from z(n) 2 Gi (s?i ), and

25

θ’i

ti

Q ui = ci =

Figure 2-2: Truthful -best reply the upper semi-continuity of Qi (:; s?i). Now by the continuity of i0 ,8 and Qi (limn i0 (z (n)); s?i) = Qi (i0 (zi); s?i ) = Qi (yi ; s?i ), hence

zi  Qi (yi; s?i ):

(2.10)

Now, we claim that ai (ti ; s?i ) = vi . Indeed, if zi = 0 then vi = 0 and ai (ti ; s?i ) = 0. If zi > 0, then by (2.10), Qi (yi ; s?i ) > 0 and since by hypothesis Qi (0; s?i) = 0, we have i0 (zi) = yi > 0. Also, zi > 0 implies vi < zi . Therefore, by Assumption 1, we have wi = i0 (vi ) > i0 (zi ) = yi . Hence, since Qi (:; s?i) is non-decreasing, Qi (wi ; s?i )  lim&y Qi (; s?i) = Qi (yi ; s?i )  zi > vi . Thus, by (2.2),

ai(ti ; s?i ) = vi

(2.11)

Actually, i0 need only be continuous from the left, since, as can be easily seen, z(n) can be chosen to be increasing 8

26 Now 8si 2 Si (s?i ),

ui(ti; s?i ) ? ui(s) = i (ai (ti ; s?i )) ? i (ai (s)) ? ci (ti ; s?i ) + ci (s)

Z ai(s)   Pi(z; s?i ) ? i0 (z) dz: = a (t ;s ) Z zi  Z ai i(is) ?i   Pi(z; s?i ) ? i0 (z) dz Pi (z; s?i ) ? i0 (z) dz + = vi Zziai(s)   Pi (z; s?i ) ? i0 (z) dz ?   zi

(2.12)

where the inequality follows from (zi ? vi )  =i0 (0) and the fact that i0 is non-increasing. Thus, it suces to show that the integral is  0. If zi < ai (s), take any z 2 (zi ; ai(s)]. By the de nition of zi , z 62 Gi (s?i ). Now R R si 2 Si(s?i ) implies bi  ci (s) = 0ai(s) Pi (; s?i) d  0z Pi (; s?i) d. Therefore, we must have z > Qi (i0 (z )), which by (2.8), implies i0 (z )  Pi (z ) and the integrand in (2.12) is  0 as desired. Suppose zi  ai (si ). Since i0 is non-increasing, Qi (:; s?i) is non-decreasing and Pi (:; s?i)  0, any point to the left of zi is in the set Gi(s?i ), 8z < zi; z 2 Gi(s?i ), hence

z  Qi(i0 (z); s?i) which by (2.7), implies i0 (z)  Pi (z; s?i ), so the integrand in (2.12) is  0 as desired. 2

Figure 2-3 shows the utility function of player 4, u4(s4), in a PSP auction with I = 6 players, with s?4 xed, and a valuation 4(q) = 10q. The plateaus correspond h i+ to the points where q4  Q4(p4; s) = Q ? Pfj:pj>p g aj (s) , and a4(s) can no longer be increased at that bid price { see (2.2). At bid prices p4 > p6, the utility decreases when a4 > Q ? q6, because after that point, each additional unit of resource is taken away from player 6, and thus costs p6 = 12, which is more than 40 = 10 its value to player i. Thus, each additional unit starts bringing negative utility. This is what 4

27

500

utility −− u4

400 300 200 100 0 15

100 80 10

60 40

5 20 price −− p4

0

0

quantity −− q4

Figure 2-3: Utility u4(s4) for s1 = (100; 1), s2 = (10; 2), s3 = (20; 4), s5 = (20; 7), s6 = (30; 12) discourages users from bidding above their valuation. Proposition 1 is illustrated by the fact that for any given quantity q4, the utility u4 is maximized on the plane p4 = 40 = 10. Remark: When the players have linear valuations and no budget constraint (bi = 1), PSP becomes identical to a second-price auction for a non-divisible object. Then the existence of a Nash equilibrium follows directly from incentive compatibility. Were the message process such that players declared a price and a budget (rather than desired quantity), it may have been possible to design an allocation rule A such that they are inclined to reveal their true budget, thus obtaining incentive compatibility in both dimensions, and hence equilibrium. But such a rule A would likely not have a simple form like (2.2) and (2.3). In essence, the computational load of translating budgets into shares would be centralized at the auctioneer, thus making the system less scalable to large numbers of users. On the other hand, decentralization has a cost too, which is the signaling overhead resulting from players possibly adjusting bids based on opponent bids in the iterated game. Our design

28 is based on the premise that the latter approach is the more scalable of the two (indeed that was the reason for choosing a small message space). The next property is that the truthful best reply is continuous in opponent pro les (this can be seen in Figure 2-2: as the \staircase" is varied smoothly, the point of intersection with i0 moves smoothly, provided i0 is not at { which is given by the last time item in Assumption 1). To prove that, we will need the following:

Lemma 1 8s; s0 2 S ; 8y; z  0; 8 > 0, if jjs?i ? s0?ijj <  then p

p

Qi(y + ; s?i) +  I  Qi(y; s0?i)  Qi(y ? ; s?i) ?  I; and

(2.13)

p

p

(2.14) Pi(z +  I; s?i) +   Pi(z; s0?i )  Pi (z ?  I; s?i) ? : Proof: First, jjs?i ? s0?i jj <  implies Pk jqk0 ? qk j < pI , and pk +  > p0k > pk ? . p P p P P +  I  q0 1 0  q 1 Thus, q 1 ?  I: Then, using (2.1) k k fpk +>yg

k k fpk ?>yg k k fpk >yg (a + b)+  (a)+ + (b)+, the rst result follows.

and the identity p For any y < Pi (z; s0?i ), by (2.7), we have z > Qi (y; s0?i )  Qi (y ? ; s?i ) ?  I , which p by (2.8), implies y ?   Pi (z +  I; s?i ). Letting y % Pi (z; s0?i ), we get Pi (z; s0?i )  p Pi (z +  I; s?i ) + .

p

For any y > Pi (z; s0?i ), by (2.8), we have z  Qi (y; s0?i)  Qi (y + ; s?i ) +  I ,

p

which by (2.7) implies y +   Pi (z ?  I ). Letting y & Pi (z; s0?i ), we get Pi (z; s0?i) 

p

Pi (z ?  I ) ? .

2

Lemma 2 (Continuity of best reply) Under Assumption 1, 8i 2 I , the -best reply ti given in Proposition 1 is continuous in s?i on any subset Vi(P; P ) = fs?i 2 Si : 8z > 0; P  Pi (z; s?i)  P g, with 1 > P  P > 0.

29

Proof:

Let zi = sup Gi (s?i ). We will show zi is continuous, and the continuity of vi = [zi ? =i0 (0)]+ and wi = i0 (vi) follow immediately (recall that by Assumption 1, i0 is continuous). Suppose there is a discontinuity at some s?i . Then, 91 > 0, such that 8 > 0, 9s0?i 2 Vi(P; P ) with jjs?i ? s0?i jj <  and jjzi ? zi0jj  1, where zi0 = sup Gi(s0?i). Suppose zi + 1  zi0 = sup Gi (s0?i ) (the case zi0 + 1  zi is handled identically, with s?i and s0?i interchanged. ). Consider the de nition of Gi ; since i0 is decreasing and Qi (:; s?i ) is non-decreasing and Pi (:; s?i)  0, any point to the left of zi0 is in the set Gi (s0?i ), therefore zi + 1 2 Gi(s0?i ): (2.15)

h

i+

P

Thus, zi + 1  Qi (i0 (zi + 1 ); s0?i ) = Q ? k qk0 1fp0k >i0 (zi + )g . Therefore, 1

p

zi + 1  Qi (i0 (zi + 1 ) + ; s?i ) +  I; using Lemma 1. Also, by (2.7) zi + 1  Qi (i0 (zi + 1 ); s0?i ) ) i0 (zi + 1 ) > P (zi + 1 ; s0?i ). Now since s0?i 2 Vi(P ; P ), this last expression is  P > 0, hence i0 (zi + 1 ) > 0. Then using Assumption 1, i0 (zi + 2 )  i0 (zi + 1 )+ i(1 ? 2 ) > i0 (zi + 1 )+  , for  < 1 = p I ^ 1 i , p and 0 < 2 < (1 ? 1 I ) ^ (1 ? 1 = i). Therefore, since Qi (:; s?i) is non-decreasing, 1

p

zi + 2  Qi(i0 (zi + 2 ); s?i) +  I ? 1 + 2 < Qi(i0 (zi + 2); s?i): Now (2.15) also implies that

bi 



Z zi + Z0 0

1

zi +3

Pi (; s0?i) d Pi (; s?i) d +

Z zi+   Pi(; s0?i) ? Pi(; s?i) d + (1 ? 3)P; 3

0

(2.16)

30 and this holds 83 < 1 . Now, using Lemma 1,

Z zi +   Pi(; s0?i) ? Pi (; s?i) d 0 Z zi+ Z zi + p Pi(; s?i) d Pi( ?  I; s?i ) d ?  ?Q + 0 0 Z zi +  ?Q ? p Pi (; s?i) d zi +p  ? I  ?(Q + P ) I: 3

3

3

3

3

p

Let 2 = (Q+ PP)pI , and 3 such that 0 < 3 < 1 ? (Q + P )2 I=P . Then 1

bi 

Z zi+ 0

3

Pi (; s?i) d;

(2.17)

for  < 2 . Now choosing  < 1 ^ 2 , (2.16) and (2.17) imply that Gi (s?i ) 3 (zi + 3 ) ^ (zi + 2 ) >

zi = sup Gi (s?i ), a contradiction.

2

We introduce one additional player, player 0, whose valuation is 0(z) = p0z, and whose bid can therefore be xed at s0 = (q0; p0) = (Q; p0). Player 0 can be viewed as the auctioneer, and p0 > 0 as a \reserve price" at which the seller is willing to \buy" all of the resource from himself. From (2.1), the presence of the bid s0 = (Q; p0) implies 8i 2 I ; Qi(y; s?i) = 0; 8y < p0. In particular, setting y = 0, the condition of Proposition 1 holds. Thus, we can restrict our attention to truthful strategies only, and still have feasible best replies. This forms a \truthful" game embedded within the larger auction game, where the strategy space is T  S , the feasible sets are Ti \ Si(s?i), and the best replies are Ri (s) = Ti \ Si(s). A xed point of R in T is a xed point of S  in S . Thus an equilibrium of the embedded game is an equilibrium of the whole game.

Proposition 2 (Nash equilibrium) In the auction game with the PSP rule given

31 by (2.2) and (2.3) and a reserve price p0 > 0, and players described by (2.4) and (2.5), if Assumption 1 holds, then for any  > 0, there exists a truthful -Nash equilibrium s 2 T .

Proof: 8s 2 T , 8i 2 I , 8z > 0, we have z > 0 = Qi(p0=2; s?i), which by (2.8) implies

Pi (z; s?i)  p0=2 = P . Let P = maxk2I[f0g k0 (0). Then, the conditions of Lemma 2 are satis ed and t = (v; w) is continuous in s on T . By Assumption 1, i0 is continuous therefore v (q; p) = v (q; 0(q )) (as de ned in Proposition 1), can be viewed as a continuous mapping of [0; Q]I onto itself. By Brouwer's xed-point theorem (see for example [68]), any continuous mapping of a convex compact set into itself has at least one xed point, i.e. 9q  = v (q ) 2 [0; Q]I . Now with s = (q ; 0 (q )), we have s = t(s ) 2 T .

2

2.3.3 Eciency The objective in designing the auction is that, at equilbrium, resources always go to those who value them most. Indeed, the PSP mechanism does have that property. This can be loosely argued as follows: for each player, the marginal valuation is never greater than the bid price of any opponent who is getting a non-zero allocation. Thus, whenever there is a player j whose marginal valuation is less than player i's and j is getting a non-zero allocation, i can take some away from j , paying a price less than i's marginal valuation, i.e. increasing ui, but also increasing the total value, since i's marginal value is greater. Thus at equilibrium, i.e. when no one can unilaterally increase their utility, the total value is maximized. Formally, the mechanism is ecient if at equilibrium, the allocation maximizes Pi i(ai). Since the valuations i are concave and the constraint set A =4 fa : Pi ai  Qg is convex, the well known Karush-Kuhn-Tucker [54] optimality conditions are that there exists a Lagrange multiplier  such that i0 (ai) = , if ai > 0, and i0 (ai)  , otherwise.

32 For the analysis of this section, we will make the additional assumption that the valuation function has a nite rst and second derivative (in economic terms, the marginal valuation and the elasticity of demand are bounded).

Assumption 2 9 > 0, 8i 2 I ,  8z; z0; z > z0  0, i0 (z) ? i0 (z0) > ?(z ? z0);  i0 < 1  bi = 1. If i0 is dierentiable, the rst condition is simply 0  i00 > ? . In the PSP game as de ned in the previous sections, players only care to maximize their utility to within  > 0, so one naturally expects that the degree of eciency attained should somehow depend on , and tend to optimality as  ! 0. That is indeed the case, as we will now show. Given any  > 0, for any -Nash-equilbrium s 2 T , let a  a(s), and let a  mini2I[f0g;ai>0 ai , the smallest non-zero allocation. The following is the \ version" of the Karush-Kuhn-Tucker conditions. q Lemma 3 Suppose Assumptions 1 and 2 hold. If for some j , aj > =, then 8i 2 I [ f0g, p i0 (ai ) < j0 (aj ) + 2 : q An immediate corollary is that if a > = then

p

p

 ? 2  < i0 (ai ) <  + 2 

q

if ai > = and

p

i0 (ai ) <  + 2 ;

33 if ai = 0, for some   0.

Proof:

p

Suppose i0 (ai )  j0 (aj ) + 2 . Since j0 (aj )  j0 (qj)  pj , we have i0 (ai ) 

pj + 2p: Since aj > 0, if player i bids at a price above pj , it can take all of player j 's allocation, without losing anything of its own, i.e. ai + aj  Qi (pj ; s?i ): By (2.7), this implies pj  Pi (ai + aj ; s?i ):

p

Let qi = (ai + =) and si = (qi ; i0 (qi )). Then

ui (si ; s?i ) ? ui (s ) =



Z ai +p= 



ai

i0 (z) ? Pi(z; s?i ) dz

q

i0 (ai + =) ? pj

q

q

> i0 (ai ) ?  = ? pj

=

q

p q q  2  ?  = =

=

=  which contradicts the fact that s is an -Nash equilibrium.

2

q

Proposition 3 (Eciency) Suppose Assumptions 1 and 2 hold. If a > =, then max A

X i

i(ai) ?

X i

p

i(ai ) = O( );

where A = fa 2 [0; Q]I +1 : Pi ai  Qg.

Proof: Take any a 2 A. Let I + = fk : ak > ak g and I ? = fk : ak < ak g. For i 2 I + , p

we have i0 (ai )   + 2 . For i 2 I ? , we have ai > ai  0, therefore by Lemma 3,

34 i0 (ai ) >  ? 2p. Therefore,

X I

P

i (ai) ? i (ai ) 



X

i0 (ai )(ai ? ai ) ?

X

i0 (ai )(ai ? ai )

I? p I+ p   ( + 2 )
? ( ? 2 )
;

P

where
= I (ai ? ai ) = I ? (ai ? ai ). Since
 Q the result follows, with the bound p 4Q . 2 +

Remark: The condition bi = 1 is sucient, but not necessary to achieve ecient outcomes. In fact with any budget pro le, eciency can be achieved if the users cooperate. For example, if they all choose a bid quantity close to what they can actually obtain (which they do if they use the strategy given by Proposition 1), then the price paid would be p0 per unit for all the allocations, and if p0 or the shares ai are not too large, then budget constraints are irrelevant and a is ecient. More generally, eciency is attained if the budgets are not too far out of line with the valuations, i.e. there are no players with very high demand and very low budget. Remark: (Welfare and Eciency) A more common measure of eciency is the social welfare, which is the sum of all the players' utility Pi ui, including the seller i = 0. The natural de nition of the seller's utility is the value of the leftover capacity a0 = Q ? Pi6=0 ai plus the revenue, i.e.

u0 = 0(a0) +

X i6=0

ci:

Then, Pi ui = Pi6=0(i ? ci)+ u0 = Pi6=0 i + 0(a0) = Pi i. Thus, Pi ui is equivalent to the eciency measure used above, which is Pi i. Another measure is the sellers revenue. Even though PSP is not, in general, revenue-maximizing, it tends to the revenue maximizing allocations and prices as demand increases [57].

35

Remark: Proposition 3 provides a key to understanding the basic trade-o between engineering and economic eciency. The smaller , the closer we get to the valueoptimal allocations. But in a dynamic game, where players iteratively adjust their bids to the opponent pro le, a player will bid as long as he can gain at least  utility (since that is the cost of the bid), thus a smaller  makes the iteration take longer to converge, i.e. entails more signaling.

36

2.4. Bidding algorithm and Convergence A prototype software agent based implementation of the auction game, has been developed and extensively used since December 1995 [83]. Much of the intuition behind the mechanism design and the analysis in this work came from experiments done on this inter-active distributed auction game on the World Wide Web, using the Java programming language. The game can be played in real-time by any number of players from anywhere on the Internet (see Chapter 5). Each user plays in the dynamic auction game using the following:

Algorithm 1

1 Let si = 0, and s^?i = ;. Start an independent thread

which receives updates of s^?i .

2 Compute the truthful -best-reply of Proposition 1, ti 2 Ti \ Si(^s?i). 3 If ui(ti; s^?i) > ui(si; s^?i) + , then send the bid si = ti. 4 Sleep for 1 second. 5 Go to 2. No assumption is made on the order of the turns. Players join the game at dierent times, and depending on the execution context of the client program, the sleep time of 1 second is more or less approximate. This, along with communication delays which make the times at which bids arrive at the server and updates at the clients essentially random times, make the distributed game completely asynchronous. Algorithm 1 can be described as sel sh and short-sighted. Sel sh because it will submit a new bid if and only if it can improve it's own utility (by more than the fee for the bid). Thus, the game can only converge to an -Nash equilibrium, if it converges at all. And short-sighted because it does not take the extensive form of

37 the game into account, i.e. does not use strategies which may result in a temporary loss but a better utility in the long run.

2.4.1 Convergence in Dynamic Play Proposition 2 established the existence of a Nash equilibrium, which is interpreted as a stable \operating point" of the decentralized resource allocation mechanism. We now seek to justify this interpretation by asking how and when can such a point be reached? Let in 2 I denote the identity of the player making the n-th bid, and sn 2 S the pro le after. The game ends at time n if 9m < n such that fin?m ; : : :; ing  I and sn = sn?1 = : : : = sn?m , because no player can unilaterally improve his utility by submitting a new bid, so no player will ever move thereafter. Thus, by de nition, the algorithm can only converge to an -Nash equilibrium, if it converges at all. But does it?

Proposition 4 Algorithm 1 converges in O(Pi i(Q)=) moves to a truthful -Nash equilibrium.

Proof:

P

Let V (a) = i i (ai ). First, we show that each move increases the value. Let the superscript n denote the state just after a move at time n. Suppose sn 6= sn?1 and let i = in. Then,

V (an) ? V (an?1 ) = i (ani) ? i (ani ?1 ) +

  = =

X

j (anj) ? j (anj ?1 )

j 6=i X n ? 1 n i (ai ) ? i (ai ) + j0 (anj)(anj ? anj ?1 ) j 6=i X n n ? 1 i (ai ) ? i (ai ) + pnj(anj ? anj ?1 ) j 6=i n ? 1 n i (ai ) ? i (ai ) + cni ?1 ? cni: uni ? uni ?1

38 > ; where the second line follows from the fact that j0 is decreasing, the third from anj  qjn = qjn?1 ) j0 (anj)  j0 (qjn?1 ) = pnj ?1 = pnj , the fourth and fth by the de nitions of ci and ui , and the last by sn 6= sn?1 and the de nition of Algorithm 1.

P

Thus the sequence fV (an ) : n  0g is increasing, bounded above (by i i (Q)) and therefore converges. Therefore, 9N such that n  N ) V (an ) ? V (an?1 ) <  ) sn =

sn?1 = s , i.e. the algorithm converges in nite time. By de nition of the algorithm, sn 2 T for all n, and since T is closed, the limit s 2 T . 2

2.4.2 Convergence Time An issue of obvious concern is how the convergence time scales with the number of bidders. We consider this experimentally using the software described in Section 2.4.. In all our simulations we let Q = 100. For each user, the valuation is strictly increasing and concave up to a maximum corresponding to a physical line capacity, and at beyond that. Since, as shown by Proposition 3, only the second derivative of the valuation is needed to measure the eciency of the PSP auction, a second order (parabolic) model is deemed sucient. Thus we use valuations of the form:

i(z) = ?i(z ^ qi)2=2 + iqi(z ^ qi); where qi is the line rate, and i > 0 (see Figure 2-4). We generate our user population with independent random variables fi0 (0)gI (corresponding to the maximumunit price the user would pay) uniformly distributed in [10; 20], and i = i0 (0)=qi, and qi uniformly distributed on [50; 100]. All players

39 1200

1000

valuation

800

600

400

200

0 0

10

20

30

40

50 quantity

60

70

80

90

100

Figure 2-4: Parabolic valuation with  = 0:5 and q = 70 have a budget bi = 100. The bid fee is xed at  = 5. Each user has a bidding agent which can submit at most one bid per second (see Algorithm 1). With this set-up, the results are shown in Figures 2-5-2-6. Simulations were run for 11 population sizes ranging from 2 to 96 player. Each point is simulated 10 times with new random valuations for all players. The overall mean is 11.9 bids per player. From Figure 2-5, the number of bids seems to grow as the square of the number of players. The actual time to converge, shown in Figure 2-6, grows more slowly, since the computation of bids is done in parallel by all the players. In fact, for small numbers of players, the time decreases. This can best be explained as follows. Suppose there are only two players, with similar valuations. They will both start by asking for their maximum quantity, at their marginal valuation (which at their maximum quantity is near zero). Then as each sees the other's bid, each will reduce the quantity and increase the price a little bit. And they go on taking turns, gradually raising the market price until they reach an equilibrium. However if there are 10 players, in between two bids by the same player, the 9 others will already have bid up the

40 3000

number of bids to converge

2500

2000

1500

1000

500

0 0

20

40

60

80 100 number of players

120

140

160

Figure 2-5: Mean (+/- std. dev.) number of bids { solid line. The dashed line is I + I 2=10. price, so he will jump to higher price than if there was only one opponent. Thus the equilibrium market price will be reached more quickly. For large populations, this eect becomes small compared to the sheer volume of bids, and the convergence time starts to grow. The trade-o between signaling and economic eciency discussed in light of Proposition 3 is illustrated by Figures 2-7-2-8. Increasing the bid fee speeds up convergence, at a cost of lost eciency. A resource manager should select a bid fee which optimally balances the two in a given context. Figure 2-8 also illustrates the validity of the lower bound given by Proposition 3.

2.5. Single Node Game Simulation Auctions, as formulated here, are applicable in a setting where the resources are arbitrarily divisible, and these divisions enforceable in some sense. Thus, this approach would t well in a system where arbitrary amounts of resources can be requested and obtained, rather than systems with a small xed menu of service

41

120

time to converge (seconds)

100

80

60

40

20

0 0

20

40

60

80 100 number of players

120

140

160

Figure 2-6: Mean (+/- std. dev.) convergence time in seconds (for a 1 second bid interval).

200

180

number of bids

160

140

120

100

80 0

10

20

30

40

50 Bid fee

60

70

80

90

Figure 2-7: Number of bids to converge vs 

100

42 1965

total value of allocations

1960

1955

1950

1945

1940

1935 0

10

20

30

40

50 Bid fee

60

70

80

90

100

Figure 2-8: Pi i(ai ) { solid line, and maxa Pi i(ai) ? 4()?1=2 { dashed line, vs . classes to choose from. For example, a PSP auction is well suited for pricing the \expected capacity" pro les in an Internet with dierentiated services as proposed in [13]. Another application of PSP { one that motivated this work { is an auction of radio spectrum where frequency band allocations would be variable [72]. For network resource allocation, it is natural to assume that bidding would be on a per ow basis, or per unit time on an appropriate time scale. Not only is active bidding on the time scale of packets probably not feasible, but the relationship between the resource allocation of a single packet and the user-perceived quality of service is a daunting, if not impossible, thing to estimate [90]. On the ow level though, this is certainly possible, particularly if the player in the auction is a software bidding agent, possibly embedded into the applications, which develops (over time and with human feedback) an accurate estimate of , the relationship between resources and value or perceived QoS. We simulate, on the platform described in Chapter 5, the following scenario. The seller is an Internet service provider, whose network has a bottleneck bandwidth of 45Mb/s. Using the dierential service model, the ISP allocates capacity to all its

43 users (see Section 3.3. for a more detailed study of the provider's point of view). In that service model, the allocation is a trac pro le for each user, enforced by a pro le meter at the point of connection to the network. The network oers premium service to all in-pro le trac (e.g. lower loss and/or delay), and any out-of-pro le trac gets best-eort service. The provider must provision the network to carry all the in-pro le trac that it expects at a given time, or conversely, must not sell more capacity (\sum" pro les) than it has in the worst case scenario. Depending on how strictly the ISP wants to honor the contract and how well it can predict the routes it expects the trac to take, it will sell a total capacity which is equal or greater than the bottleneck capacity. Here we will assume the ISP is extremely conservative, so it will only sell 45Mb/s. The pro les can be any trac descriptor. Suppose they are such that the allocation will give the users a minimum guaranteed bandwidth (which will be the allocated quantity), with each user being allowed to burst up to their physical line rate9. Clearly the quality of service for most Internet trac will be better if the allocated premium bandwidth is increased. But, because the pro le allows for bursts, the improvement in quality will decrease as the premium bandwidth approaches the physical line rate, and of course, there is no use in having a premium bandwidth greater than the peak rate. Therefore, a reasonable model of valuation is strictly increasing and concave up to the peak rate, and at above the peak rate. Since, as shown by Proposition 3, only the second derivative of the valuation is needed to measure the eciency of the PSP auction, a second order model is sucient. Thus This can be done for example with a 'leaky-bucket' trac meter. Assume that bucket sizes are xed by the ISP, and users are allocated (and pay for) a token rate. The bucket size determines the maximum in-pro le burst size, and would, say, correspond to the worst case buer occupancy { or delay { that the ISP can tolerate for premium trac 9

44 we use valuations of the form:

i(z) = ?i(z ^ qi)2=2 + iqi(z ^ qi); where qi is the line rate, and i > 0. To represent a realistic range of the connection speeds of users who would request premium bandwidth, from home or small business users to a peer network connecting with a T3, our user population has line rates qi 2 f256kb/s; 1:5Mb/s; 45Mb/sg. What is a realistic range of values for i? The highest unit price any user would pay is i0 (0) = iqi. As of this writing, ISPs charge T1 connections at at rates of US$500-$3000 per month [79, 27]. This works out to $0.008-$0.045 per Mb/s per minute. Similarly, dial-up users are charged at-rates of $10-$30 per month for 30 Kbps connections, which works out to $0.007-$0.021 per Mb/s per minute. The closeness between the two cases indicates that one can take these at rates as a rough indication of average valuations. Thus we generate our user population with independent random variables fi0 (0)gI uniformly distributed in [0:005 ? 0:05],10 and i = i0 (0)=qi. In this simulation, the total capacity is Q = 45 Mb/s. The bid fee is  = $0.00125, and the reserve price is p0 = $1:125  10?5 . All times are in seconds. We begin at t0 < 0 with a population consisting of:

 20 users (players 1-20) with a peak rate of 1.5 Mb/s  2 users (players 21 and 22) with a peak rate of 45 Mb/s  50 users (players 23-72) with a peak rate of 256 kp/s These numbers are given for illustrative purposes, and are not an attempt to accurately model today's reality { indeed, the diculty in doing the latter is one of the reasons for using auctions. 10

45 The valuations are random, as described above. We let this set of users play until an equilibrium is reached (i.e. no new bids have been received for a few minutes), at some time t < 0.

 At t = 0, player 73 (a new 256 kb/s user) joins. Initially, the new user assumes the market is empty, so he asks for a large pro le. As his knowledge of the opponent pro le is updated, he realizes this is too expensive, decreases his quantity, and raises the bid-price. Figure 2-9 shows that after three bids by the new player (and reactions by others), the market has stabilized at a new equilibrium by t = 170, where player 1 has an allocation of a premium bandwidth a1 = 128 kb/s.

 At t = 550, player 74 (a new 1.5 Mb/s user) joins. Fgure 2-10 shows that after two bids by player 2, by t = 600, a new equilbirium is reached with a2 = 728 kb/s. At this new equilbrium, the unit price paid by player 1 is increased slightly from 0.00625 to 0.00629 (barely visible on Figure 2-9).

 At t = 890, player 75 (a new 45 Mb/s user) joins. Unlike the previous two, this arrival has a signi cant impact on the market. Figure 211 shows that by t = 1200, after 28 bids by the new player, a new equilibrium is reached, where a3 = 19 Mb/s. At the new equilibrium, player 2's share of the premium bandwidth drops from 728 kb/s to 540 kb/s, and the price paid per unit to rises from 0.0063 to 0.008. Between time t = 1600 and 1700, 50 new users with peak rates of 256 kb/s join the game. This causes the market to uctuate for a couple of minutes, and a new equilibrium is reached where the prices are slightly higher, and allocations slightly lower for the existing players (see Figures 2-9 to Figures 2-11).

46 256 kbps line rate user −− requested qty

Mb/s

0.2 0.1 0 0

500

1000

1500

500

1000

500

1000

2000 allocation

2500

3000

3500

1500 2000 bid and paid price

2500

3000

3500

1500

2500

3000

3500

Mb/s

0.3 0.2 0.1 0 0 $/(min x Mb/s)

0.01

0.005

0 0

2000 time (s)

Figure 2-9: Small user joining a stable market of 72 players

1.5 Mbps line rate user −− requested qty

Mb/s

1.5 1 0.5 0 0

500

1000

1500

500

1000

500

1000

2000 allocation

2500

3000

3500

1500 2000 bid and paid price

2500

3000

3500

1500

2500

3000

3500

Mb/s

1.5 1 0.5 0 0 $/(min x Mb/s)

0.01

0.005

0 0

2000 time (s)

Figure 2-10: T1 user joining a stable market of 73 players

47 45 Mbps line rate user −− requested qty

Mb/s

60 40 20 0 0

500

1000

1500

500

1000

500

1000

2000 allocation

2500

3000

3500

1500 2000 bid and paid price

2500

3000

3500

1500

2500

3000

3500

Mb/s

60 40 20 0 0 $/(min x Mb/s)

0.015 0.01 0.005 0 0

2000 time (s)

Figure 2-11: T3 user joining a stable market of 74 players Just before t = 2500, players 1-20 (all 1.5 Mb/s line rate) suddenly leave. This causes a sharp drop in the market, which takes about 200 seconds to stabilize. Note that the subsequent equilbrium is not much dierent from the previous one, in terms of both allocations and prices, despite the fact that potential demand has been reduced by about 30 Mb/s. This is because many users with lower valuations who were not fully satis ed before now use up the freed capacity, and there is still enough unsatis ed demand just below the market clearing price to keep prices at almost the same level. At t = 3140, players 21-22 (both 45 Mb/s rate) suddenly depart. The freed capacity is enough to satisfy a lot of pent-up demand, so the prices drop by about half, to around $0.004, and allocations increase for the remaining players. Note that throughout, of the three players we follow, the T3 user (player 75) pays a lower price per unit than the other two. This \volume discount" follows from (2.3). Figure 2-12 illustrates this eect. Player 4 is getting a larger allocation

48

p5

q5

p4 p3

q

4

q 3

p 2

q2 q1

p

1

p 0

q0

Q a4 = q4 c4 =

+

a5 = q5 c5 =

Figure 2-12: The volume discount eect: a5 < a4 ) c5=a5  c4=a4 than player 5, and this results in a lower price per unit.

2.6. Conclusion Auctions are one of oldest surviving classes of economic institutions [...] As impressive as the historical longevity is the remarkable range of situations in which they are currently used. [67] We proposed the progressive second price auction, a new auction which generalizes key properties of traditional single non-divisible object auctions to the case where an arbitrarily divisible resource is to be shared. We have shown that our auction rule, assuming an elastic-demand model of user preferences, constitutes a stable and ecient allocation and pricing mechanism. Even though we are motivated by problems of bandwidth and buer space reservation in a communication network, the auction was formulated in a manner which is generic enough for use in a wide range of situations.

49

A. Information-theoretic basis for the valuation In general, valuations are simply assumed to be given as external factors. Indeed, the fundamental assumption in any market theory is that buyers know what the goods are worth to them. The \elastic demand" or \diminishing returns" nature of Assumption 1 is fully justi ed from a purely economic standpoint for virtually all resources in everyday life. In the case of variable bandwidth, we can go even further by better quantifying what the goods are. For a user sending video, say, how much value is lost when the channel capacity goes from 1.5 to 1.2 Mbps? Ultimately, the value lies not in the amount of raw bandwidth but in the information that is successfully sent. Our goal in this section is to give a brief description of how Information Theory can be used for a bottom-up construction of bandwidth valuations { based on the fundamental thing the user cares about which is communication of information { and that such valuations will generally be of the type in Assumption 1. Any information source has a function D(:), such that when compressed to a rate R, the signal has a distortion of at least D(R)[7]. The distortion is the least possible expected \distance" between the original and compressed signals, where the minimization is over all possible coding/decoding schemes. In this context, we make the distance measure the monetary cost of the error. This cost can be chosen, for example, to be proportional to some common measures like the mean squared error, the Hamming distance (probability of error), the maximum error, etc., or heuristic measures based on experiments with human perception. Given that modern sourcecoding techniques can, given a distortion measure, achieve distortions close to the theoretical lower bound[25], it is not unreasonable to use the rate-distortion curve as an indication of the value of the bandwidth share. Let Di (:) be the distortion-rate function of fXi (t)g, a stochastic process modeling

50 the source of information associated with user i. In the most likely auction scenaria, sources would be aggregates of many application streams for which bulk capacity is being purchased, e.g. Virtual Paths, Virtual Private Networks, or edge capacity[13]. Xi is encoded as Yi which has a rate of R bits per second. Shannon's channel coding theorem[88] states that Yi can be received without errors if and only if the channel has a capacity C > R. In our auction context, user i has capacity (bandwidth allocation) C = ai, and thus has to suer a distortion of at least Di (ai). The value of the bandwidth is then

i(ai) = i ? Di (ai);

(2.18)

where i is the value of the full information. The relevant properties of the distortion-rate functions are:

 when the rate is greater than the entropy of the source, the distortion is zero, and

 for many common source models and cost functions, the distortion-rate function is convex, and has a continuous derivative. It is easy to see that, with these properties, (2.18) satis es Assumption 1. Example 1: Let fX g be a Bernoulli source, taking two values with probabilities p and 1 ? p. Without loss of generality, let p 2 [0; 1=2]. In this case, since the source is i.i.d, one can de ne the distortion on a per symbol basis. Using a Hamming cost function d(X; Y ) = 1fX 6=Y g; i.e. assuming it costs one unit of money every time one bit is wrong, we have the

51 0.5

Valuation

0.4

0.3

0.2

0.1

0 0

0.2

0.4

0.6 Rate

0.8

1

Figure 2-13: Distortion-rate based valuation for a Bernoulli p = 1=2 source distortion D = IEd(X; Y) = IP(X 6= Y). the rate-distortion function is

R(D) = [H (p) ? H (D)]+ ; where H (x) = ?x log(x) ? (1 ? x) log(1 ? x), and the distortion-rate function is the inverse function. It can be easily seen that D(R) is strictly convex and decreasing for 0  R < H (p), and D(R) = 0 for R  H (p). The continuity of D0 on 0  R < H (p) and R > H (p) is obvious. At the critical point (R = H (p); D = 0), we have lim D0(R) = Dlim 1=R0 (D) &0

R%H (p)

= Dlim 1= log(D=1 ? D) &0 = 0

= R&lim D0(R): H (p) Thus continuity of D0 holds throughout, and Assumption 1 is valid for the valuation of the form (2.18) for this source { see Figure 2-13.

52

Example 2: Let fX g be a Gaussian source with Markovian time-dependency,   i.e a covariance matrix  = 2rji?jj i;j ; r 2 [0; 1). Suppose we use the squared

error cost, i.e. it costs one unit of money for one unit of energy in the error signal. Then, we have for low distortions D  (1 ? r)=(1 + r), R(D) = 21 log 1?Dr , or 2

D(R) = (1 ? r2)22?2R; and Assumption 1 clearly holds for (2.18). In the i.i.d. case (r = 0), the formula holds for all R. As the source models get more complex, it rapidly becomes impossible to give closed-form expressions for either R(D) or D(R). Often parametric forms are available, and the functions can be evaluated numerically. Fortunately, the convexity property extends to a wide class of models, including for example auto-regressive sources, even when the generating sequence is non-Gaussian (see [7] for a full treatment of R(D), including the above cases). It can happen, e.g. for some video source models, that the R-D curve, which gives the best (R,D) pairs achievable by any coder/decoder, is not convex. But, for tractability, practical codecs are usually optimized on a convex hull of the space of possible (R,D) pairs[76, 82]. Thus, even when the theoretical D(R) curve is not convex, the actual distortion achieved in real-life systems almost always varies in a convex manner with the available bandwidth. Further investigation needs to be done on relating more complex recent trac models [40, 41] to valuation models based on information theoretic considerations.

53

Chapter 3 Networked Auctions A segment grow ink. [87]

In this chapter, we extend the auction game of Chapter 2 to a network setting. Instead of a single divisible resource, we now have a network of divisible resources. In other words, the resource space is a set of network resource elements, and each element has a certain quantity of a shareable resource. An element may be a physical network element, such as a bandwidth on a communication line or buer space on the output port to which a line is attached. This is the case in connection oriented networks, such as those based on ATM protocols [22]. An element may also be abstract, e.g. the capacity of a subnetwork. In the latter case the \network of elements" on which the game takes place is a set of interconnected subnetworks, and the granularity of capacity allocation is the subnetwork rather than individual switches, routers or links. This is the case for \edge-allocated" capacity frameworks, such as the Dierentiated Services Internet [21]. Recall that the single element PSP mechanism of Chapter 2 is decentralized in the information sense. Like all auctions, by de nition, it recognizes the fact that the demand information is not known centrally, rather it is distributed in the bidders'

54 valuations. Incentive compatibility is the key to achieving eciency in spite of the inherent limitation that the seller cannot accurately know the demand in advance. In the network (multiple shareable element) context, a basic goal is that the mechanism also be distributed in that the allocations at any element depend only on local state: the quantity oered by the seller at that element, and the bids for that element only. In Section 3.1., we formulate a distributed PSP network mechanism. In that design, each element is sold by an independent seller. This is useful not only because in reality the resource elements may be owned by dierent entities, but also because, in communication networks, maintaining common state information across the large distances separating the elements is very dicult, and when done accurately, incurs a large signalling overhead. Both aspects (decentralized and distributed) can be considered design goals (aiming to distribute computation among buyers and multiple sellers) or simply as a recognition of the prevailing reality in the intended contexts. Since the mechanisms at each element are independent, each player is responsible for coordinating its bids at the dierent elements in such a way that its utility is maximized. Any player can bid for resources at any combination of elements. By de ning players appropriately, we can synthesize auctions for dierent kinds of network \services". In Section 3.2.1, we introduce one kind of player, the bottleneck buyer, which we then use to model the players for virtual networks (e.g VPs and VPNs in ATM networks), and for edge-capacity (e.g. dierentiated Internet services [13, 21]). These players may be viewed as wholesale buyers. While it is of course possible for end users to participate directly at the wholesale level, in reality most markets are often separated into wholesale and retail levels. In Sections 3.2.2 and 3.2.3, we consider network-wide stability and eciency, and how a single seller can sell resources to end users in a coordinated manner at

55 multiple elements in an ecient manner.

3.1. Networked PSP Mechanisms Let the set of all players be denoted by I = f1; : : : ; I g. A player's identity i 2 I as a subscript indicates that the player is a buyer, and as a superscript indicates the seller. Suppose player i is buying from player j . Then he places a bid sji = (qij ; pji ), meaning he would like to buy from j a quantity qij and is willing to pay a unit price pji . Without loss of generality, we assume that all players bid in all auctions, with the understanding that if a player i does not need to buy from j , we simply set sji = (0; 0). A seller j places an ask sjj = (qjj ; pjj ), meaning he is oering a quantity qjj , with a reserve unit price of pjj . In other words, when the subscript and superscript are the same, the bid is understood as an ask. In this network-wide mechanism, there is only one seller per resource. Thus we can identify the resource with the identity of the player, i.e. \element j " refers to the network element for which player j is the seller. (qjj ; pjj ) plays the role played by (Q; p0) in Chapter 2. When sub/superscripts are omitted, the notation refers to the vector obtained by letting it range over all values, e.g. qi  (qi1; : : :; qiI ), and q is the I  I matrix. A subscript with a minus sign indicates a vector with that component deleted s?i  (s1; : : :; si?1; si+1; : : : ; sI ), and (xi; s?i ) denotes the pro le obtained by replacing si with xi. Based on the pro le of bids sj = (sj1; : : :; sjI ), seller j computes an allocation (aj ; cj ) = Aj (sj ), where aji is the quantity given to player i and cji is the total cost charged to the player i. Aj is the allocation rule of seller j . It is feasible if aji  qij , and cji  pji qij . We will assume that all the sellers use the PSP allocation rule of

56 Section 2.2..

3.2. Network PSP Analysis 3.2.1 Players We de ne a bottleneck buyer as a player i 2 I with utility of the form

ui = i  ei(a) ?

X j

cji ;

(3.1)

where ei : [0; 1)I ! [0; 1) is the expected bottleneck (or simply bottleneck) function of player i, and i is the buyer's valuation function, which is private information. As the name indicates, a bottleneck buyer's valuation depends only on a scalar bottleneck ei(a) which is a function of the allocated quantities at all the resources. As before, the player is possibly subject to a budget constraint, i.e. the set of feasible bids for player i is

X Si(s?i) = f cji (si; s?i )  big: j

(3.2)

The speci c de nition of ei determines what services can be represented by bottleneck buyers. Here we specify a general form which, as we will show in the following sections, captures virtual path, virtual network and edge-capacity services. Let player i's type now include, in addition to the valuation function and the budget of Section 2.3.1, a generic route-provisioning vector ri . De ne, for any allocation a, 4 aji j ei (a) = j + aij ; ri

57 and let

ei(a) =4 min ej (a): j 6=i i

(3.3)

3.2.1.1 Virtual Network Buyer Consider a player seeking to construct a virtual network by buying resources from a subset of the network elements. Such a player i can be modeled as a bottleneck buyer. rij denotes the fraction of i's trac that is routed through element j . For the elements j that are not part of i's desired virtual network, and obviously a rational (utility-maximizing) player will set sji = 0. Since i is not a seller, qii = 0, and ai = 0. The form (3.3) is justi ed as follows. Suppose the total trac on the virtual network constructed by player i is xi. For the service to function as desired, the allocated capacity at each element in the virtual network must be no less than the expected ow of trac through that element: aji  xirij : This will hold simultaneously for all j if and only if xi  minj aji =rij = ei(a): Thus ei(a) upper-bounds the total trac that the virtual network can carry, which is precisely the notion of a bottleneck. Arguing that the value of a network is the value of the potential trac that it can carry, we get a valuation i  ei(a), as postulated in (3.1). For a virtual path, we set rij = 1 for all the elements j along the path, and rij = 0 for all other j . This models the fact that the virtual path buyers value only the end-to-end \thickness" of their \pipe" { given by the thinnest allocation along the route, which is ei(a). When the path consists of a single element, (3.1) reduces to the form (2.4), i.e. a simple buyer at a single resource element as de ned in Chapter 2.

58

3.2.1.2 Aggregate Capacity Broker We now consider brokers, i.e. players whose aim is to buy capacity from many dierent sellers in order to sell capacity to end users. This aggregated capacity oered by a broker is one resource element. Thus, a broker is both a buyer and a seller. The justi cation for the form (3.3) is the same as in the virtual network case, but with an added twist. We assume a broker i 2 I can explicitly recognize which of the sellers it is buying from are themselves brokers which are buying from i. We call these the peers of broker i. This is useful because in many contexts, there is no \loop-back" trac, i.e. trac going from i through j and back to i, either because it serves no purpose (in the case of physical network elements), or because it is considered parasitic behaviour that is intentionally blocked. Suppose i 2 I is a broker. It oers a capacity qii for sale to its users. It must get enough capacity from the other sellers to provide the service implied by the allocations ai on its sell-side. Let l denote one such peer broker, and ril be the \fraction of trac" generated by j 's customers that is routed to the network where player l is the peer (see Section 3.3.2.2 below for interpretations of ri). Then, i must satisfy X (3.4) ril aij  ali; j 6=l

for all l. If that i does not block \loop-back" trac from l, then the summation would be over all i.

Proposition 5 (Broker's sell-side constraints) Let i 2 I be a broker, and x its buy-side allocation (ai ; ci). Then, on the sell-side, the quantity oered must satisfy

qii  ei(a)

59 For a broker who does not sell at a loss, the reserve price must satisfy

X pii  q1i cli: i l

Proof: Suppose 9l 6= i such that qii > eli. Then when all the oered quantity is bought,

l l P P we have j aij = qii > eli = arili + ail , j 6=l aij > arili , and condition (3.4) is violated. This proves the rst assertion.

P

Since l cli is the total cost of the capacity that i is buying, the second assertion follows immediately from the our assumption that the broker will not sell at a loss. 2

Remark: The obvious way for a broker to satisfy Proposition 5 is simply setting

qii = minj6=i eji (a). Alternately, the seller can leave qii equal to the maximum physical capacity, and place in its own market an arti cial \buy-back" bid equal to q0i = (qii ? e)+ , pi0 = i0 (e); where e = minj6=i eji (a). Note that this arti cial player 0 62 I . This buy-back bid eectively limits i's users to precisely the capacity that i can honor in forward to its peers. In other words, the buy-back bid ensures that the quantity constraint of Proposition 5 is automatically satis ed. However, unlike reducing qii, it leaves open the possibility of increasing it back again, if there is demand at prices greater than i0 (e). As we will become apparent through Proposition 7 below, i0 (e) is precisely the price at which i could obtain more capacity at its bottleneck to a peer network. Mathematically, the broker is an extension of the virtual network buyer, the only dierence being that the valuation i, instead of being externally given, is a part of the game (depends on the users on the buy-side). The natural utility is the potential pro t ui = i  ei(a) ? Pj cji ; where i, the broker's buy-side valuation, is

60 j y = f (z)

j q i j p i j z = d (y)

Figure 3-1: Demand curve for a broker j . the potential revenue from the sale (on the sell-side) of the capacity obtained on the buy-side aggregated into a scalar bottleneck value ei(a). The potential revenue is derived from the demand on the sell-side: let 8y  0,

di (y) =4

X pij y

qji ;

the demand at unit price y. Its \inverse" function is de ned by,

n o f i (z) =4 sup y  0 : di (y)  z : See Figure 3-1. Remark: Note that we chose f i to be continuous from the left. For a given demand function di (:), 8z  0, f i(z) represents the highest unit price at which j could sell the z-th unit of capacity (the actual prices charged to users depend on the speci c allocation mechanism Ai used).

Proposition 6 (Broker's buy-side valuation) Let i 2 I be a broker with inverse demand f i (z). Its buy-side valuation is

i(x) =

Zx 0

f i (z) dz:

61 Thus i  ei (a) =

R ei (a) f i (z) dz: 0

Proof: Since the broker seeks to maximize pro t, for a given allocation a, it will sell as

much as possible; thus by Proposition 5, qii = ei . If ei decreases by  , then qii must be reduced by  . The value to i of the lost quantity is the revenue i could have gotten from it. By de nition, this lost potential revenue is f i (emi ) . Thus, by abuse notation, writing i as a function of eli , i (ei) ? i (ei ? ) = f i (ei )

2

and the result follows.

3.2.2 Equilibrium We now extend the analysis of Section 2.3.2 to the network case. Consider a bottleneck buyer i 2 I , participating in many auctions simultaneously. By the nature of its utility (3.1), capacity allocations are valuable to the broker only insofar as they increase its expected bottleneck capacity ei. Thus, i must coordinate its buy-side bids to maximize its overall utility. The following result states the intuitively obvious fact that, without loss of utility, a buyer i can always reduce the bid quantities qil to the level where the bottleneck value eli is the same for all l.

Lemma 4 (Bottleneck buyer coordination) Let i 2 I be a bottleneck buyer. For any pro le s, si = (qi ; pi ), let a  a(s) be the allocations that would result. For a xed s?i , a better reply for player i is xi = (zi; pi ), where 8l = 6 i i h zil = ei(a) ? ail ril :

62 That is, ui(xi ; s?i )  ui (s). Moreover,

ali(zi; pi) = zil:

(3.5)

Proof:

To avoid cluttered notation, since s?i is xed, we will omit it, writing, e.g., ui (:; :)  ui ((:; :); s?i). Also, the argument of the function will be omitted when it is simply s, so that ui  ui (si )  ui (si ; s?i ). Note that, since we are holding all the other players xed, and varying only the buy-side of player i, only the quantities with subscript i will change. In particular, ail remains the same throughout. We will show that ui  ui (qi ; pi)  ui (zi; pi) (3.6) Now, 8l 2 I ,

h i h i  eli (a) ? ail ril = ali 2 3+ X qkl 75 ;  64qll ?

zil = ei (a) ? ail ril

plk pli ;k6=i

(3.7)

where the last line follows from (2.2). Now using (2.2) again, we get

3+ 2 h i X qkl 75 ^ zil = zil = ei (a) ? ail ril; ali (zi ; pi) = 64qll ? plk pli ;k6=i

where the second equality follows from (3.7), and the last is by de nition. This proves (3.5). Thus, we have eli (a(zi; pi)) = ali (zi ; pi)=ril + ail = ei (a), and this holds 8l 6= i. Therefore, i  ei (a(zi; pi)) = i  ei (a), i.e., changing the bids from (qi ; pi) to (zi ; pi) does not change i's bottleneck value. Therefore,

ui(zi ; pi) ? ui =

X l6=i

cli ? cli(zi; pi)

63 =

X Z ali

l l6=i ai (zi ;pi )

f l (qll ? z) dz:

Now 8l, ei (a)  eli (a) ) zil =ril + ail  ali =ril + ail ) ali  zil  ali (zi ; pi); where the last in-

equality follows from (2.2). That along with the fact that f l  0 implies ui (zi ; pi) ? ui  0.

2

Remark: If si is feasible, i.e. Pl cli(s)  bi, then clearly xi is also feasible, since

coordinating the bids by reducing the bid quantities from qi to zi cannot cause the total cost to increase. Note that for Proposition 1, we do not require that i0 be continuous. Nonincreasingness and continuity from the left suce. Whenever these same conditions are satis ed for the bottleneck buyer (which is the case even for a broker { see Proposition 6), we can expect that the same principle (optimality of truth-telling) should hold. Indeed, as we will now show, it turns out that the optimal strategy is very similar to that of a single resource user. But instead of searching directly for the optimal quantity, the broker nds the optimal expected bottleneck e, which is the largest one such that the marginal value is just greater than the market price. The role of the market price is played by the sum of the market prices at the dierent auctions, weighted by the route provisioning factors. The actual bids are obtained by transforming the desired optimal expected bottleneck e back into the corresponding quantities to bid at each buy-side market. As with a single resource user, truthtelling is optimal for the broker, i.e. at each buy-side market, the broker sets the bid price to the marginal value. For each element l, let Qli and Pil be as de ned in Chapter 2.

Proposition 7 (Network incentive compatibility) Let i 2 I be a bottleneck buyer,

64 and x all the other players' bids s?i , as well as the sell-side sii (thus ai is xed). Let

8 9 < =   X zi = sup :h  0 : i0 (h) > Pil (h ? ail)ril ril ; ; l6=i 8 9 Z < =  hX l i = sup :h  0 : 0 Pi (h ? ail)ril ril dh  bi; ; l6=i

(3.8) (3.9)

e = (zi ^ i ? =i0 (0))+ , and for each l 6= i, vil = (e ? ail)ril; and

wil = r1l i0(e): i

Then a (coordinated) -best reply for the bottleneck buyer is ti = (vi; wi), i.e., 8si, ui(ti; s?i) +   ui(si; s?i ).

Proof: First suppose e = zi. Since i0 is non-increasing and 8l; Pil is non-decreasing, (3.8) P

implies i0 (e) > l6=i Pil (vil)ril, and therefore 8l 6= i,

wil > Pil (vil) ) vil  Qli(wil) = qll ? dl(wil); using (2.8). Then

) ali(ti; s?i) = vil; ) eli  a(ti; s?i) = e:

Therefore

ui (ti; s?i ) =

X Z vil l 0 Pi (z) dz i ( ) d ? 0 l= 6 i 0

Ze

65 =

Ze 0

i0 ( ) d ?

XZ e





rlP l ( ? ail )ril d: i i i

l6=i al

Now suppose 9si = (qi ; pi) such that ui (si ; s?i ) > ui (ti ; s?i ) + :   Let  = mink6=i eki  aki (s), and 8l 6= i, il =  ? ail ril and i = (i; pi). From (3.5) in Lemma 4, aji (i ; s?i ) = ij , therefore

ui(i; s?i ) =

Z 0

i0 ( ) d ?

XZ 





rlP l ( ? ail)ril d: i i i

l6=i al

By Lemma 4, ui (i ; s?i )  ui (si ; s?i ): Therefore, ui (i ; s?i ) > ui (ti ; s?i ) + , which is equivalent to Z  XZ  l j  ri Pi ( ? ail )ril d > : (3.10) i0 () d ? e

j 6=i e

R

Let e = e + =i0 (0). Since i0 is non-increasing ee i0 ( )  i0 (0)(e ? e) = . That, along with the fact that Pij is non-negative, and (3.10), implies

Z

 XZ  l j  0 i () d ? riPi ( ? ail )ril d > 0: e j= 6 i e P





If  > e, then for some  > 0, i0 (e +  ) > j 6=i ril Pij (e +  ? ail )ril ; which contradicts (3.8).   P If   e, then i0 (e) < j 6=i rilPij (e ? ail )ril d: But, since both i0 and Pij are contin  P uous from the left,(3.8) implies that i0 (e)  j 6=i ril Pij (e ? ail )ril d , which is a contradiction. Now consider the case e = i , i.e. the optimal bottleneck is now the largest such that the cost is no greater than the budget. As before, for any si , let i be the corresponding coordinated bid. If ui (i ) > ui (ti ) + , then necessarily  > e, which implies ci (i ) > bi . Thus, any bid si which has higher utility than ti is not feasible, therefore ti is a best reply.

2

66 Recall that in Chapter 2, to prove the existence of -Nash equlibria in PSP auction games, we assume that demand is elastic for all players. However, the broker does not satisfy the smoothness (continuous derivative) condition. From Proposition 6, the broker's valuation, as a function of the (scalar) expected bottleneck capacity ei, is piecewise linear and concave (the derivative is the \staircase" function shown in Figure 3-1). Thus, we assume that brokers apply some smoothing in deriving the buy-side valuation from the sell-side demand, e.g. by tting a smooth concave curve to the piecewise linear one. The following result establishes the stability of a game with any combination of bottleneck buyers (brokers, virtual network and single element buyers), where the sellers all apply the PSP allocation rule independently, and the oered quantities are not dependent on other players' actions.

Proposition 8 (Network Nash equilibrium) In the network auction game with the

PSP rule applied independently by each seller, with xed sii and reserve prices pii > 0, for all sellers i 2 I , and players described by (3.1) and (3.2), if Assumption 1 holds, then for any  > 0, there exists a truthful -Nash equilibrium.

Proof: De ne P~i (z; s?i ) =

X l2L

Pil(z; sl?i )ril;

n

o

Q~ i (y; s?i) = sup z  0 : P~i (z; s?i) < y : By inspection, we see that the proof of Lemma 2 applies with Pi and Qi replaced by P~i and Q~ i respectively. Thus the best-reply correspondence is continuous provided each seller has a positive reserve price. Thus, the result is established by the same reasoning as in the proof of Proposition 2.

2

67

Remark: Note that for the above result to hold, we assume that sellers are static players, in that sii does not change in reaction to other players actions. In other words, brokers, for example, only actively play the game on the buy-side, where the valuation is derived from the bids they receive on their sell-side. But Proposition 5 indicates that the reverse in uence should occur as well, i.e., the resources obtained on the buy-side contrain the quantity oered and the reserve price on the sell-side. This has implications on the stability (existence of a Nash equilibrium) of the overall game which are investigated in Section 3.4.

3.2.3 Eciency In Section 2.3.3, we showed that the PSP auction reaches, at equilibrium, allocations which are arbitrarily close to maximum eciency. Thus, in the distributed PSP mechanism of this chapter, each element in isolation reaches ecient allocations. However, because the players bids are dependent across the elements, but the state information is not shared, it is not obvious that the allocations maximize the network-wide eciency. The following result shows that the \local" eciency result of Section 2.3.3 extends to the whole network, provided some technical conditions are satis ed. For clarity, in this section, we will exclude brokers, so that we have a set of players I , and a set of sellers L  I , with 8l; k 2 L, k 6= l, rll = 1 and rlk = 0. Thus, we will have ei(a) = ali=ril. However the analysis follows through with only minor changes when brokers are included, i.e., ei(a) = ali=ril + ail. We are concerned with the eciency of the buyers' allocations, i.e. whether at equilibrium the total user value

X i2I

i  ei(a)

68 is maximized over the set of feasible allocations A = fa 2 (Ql[0; qll])I : Pi ali  qllg. As in Section 2.3.3, this is a convex optimization problem, and to test for the eciency of the equilbirum, we seek an  approximation to the Karush-Kuhn-Tucker conditions [54]. One would expect that Lagrange multiplier at each element l is the market clearing price P0l (0; s), which is the unit price of any player would have to pay to increase capacity beyond the equilibrium allocation (see Figure 2-1). Indeed, that is the case, as shown in Lemma 5 below, where some technicalities arise from the fact that a buyer's change of bid can itself change the market clearing price. We will make use of the following notation. Let

+l =4 alj(l)(s);

o n where j (l) =4 arg minj2I plj : alj (s) > 0 is the \lowest clearing player" at l. For example, player 1 is the clearing player at the single element auction of Figure 2-1. Let ?l =4 qk(ll); o n 4 where k(l) = arg maxk2I plj : alj (s) = 0 is the highest bidder that is not fully satis ed. In Figure 2-1, it is player 1 again.

Lemma 5 Suppose Assumptions 1 and 2 hold. For any  > 0, let s be a truthful1 -Nash equilibrium of the distributed network PSP game, and e = e  a(s), the vector of bottlenecks at equilibrium. Let 8l 2 L,

1

l+ = j0 (l)(ej(l))=rjl (l)

(3.11)

l? = pkl(l):

(3.12)

That is, at all players are bidding according to Proposition 7.

69 Then 8i 2 I , 8 < mini;l +l =ril ,

i0 (ei ) ?

X

l+ ril < = + :

(3.13)

l? ril > ?= ? :

(3.14)

l

And 8 < mini;l ?l =ril , if ei > , then

i0(ei ) ?

Proof: Fix  < mini;l +l =ril, and let

X l

= = +  . Suppose the contrary of (3.13), i.e.

i0 (ei ) ?

X l

l+ ril  :

(3.15)

We will show that player i can increase its bottleneck to ei +  . By (3.11) and (3.15), j (l) 6= i. Also, from (3.11) and Proposition 7, it follows that plj(l)  l+. Now from Figure 2-1, we see that

Pil (ali(s ) + ril; s?i )  Pil(ali(s ) + +l ) = plj(l)  l+ :

(3.16)

Suppose player i bids si = (qi ; pi), with qil = ali (s) + ril , pli = i0 (qil =ril)=ril, instead of si . Then

h   i pli = r1l i0 (qil=ril) > r1l i0 ali(s)=ril ?  i i   1 = l i0 (ei ) ?  ri  l+ + r l i l l  > Pi (ai (s ) + ril ; s?i ); where the rst inequality follows from Assumption 2, and the last two from (3.15) and (3.16) respectively. Thus, ali (si ; s?i ) = qil . The resulting bottleneck value of player i is, ei =

70 minl ali (si ; s?i )=ril = minl ali (s )=ril +  = ei +  , and player i has increased it's utility by

ui (si; s?i ) ? ui (s ) =

 >

Z ei +

X Z ali(s)+ril 

Pil (z; s?i ) dz l (s ) a ei l i Z ei + X i0 ( ) d ? Pil (ali(s ) + ril; s?i )ril ei l i0 ( ) d ?

"

#

X i0 (ei ) ?  ? rill+ 

 ( ? ) = ;

l

which contradicts the fact that s is an  Nash equilibrium. This proves (3.13). Now suppose ei >  and the contrary of (3.14) holds, i.e.,

i0 (ei ) ?

X l

l+ril  ?= ? :

(3.17)

Then player i can reduce its bottleneck by  , by bidding si = (qi ; pi ), with qil = ali (s) ? ril , and pli = i0 (qil=ril )=ril. Then,

Z ei

Z l  0 ( ) d + X ai (s ) P l (z; s ) dz ?  ?i ali (s )?ril  i ei ? i l " X l l# 0  > ?i (ei ) ?  + ri ?  l  ;

ui (si; s?i ) ? ui (s) =

where the second-last line follows from the fact that on the interval of integration z 

ali(s ) ? ril  ali(s ) ? ?l = ali(s ) ? qk(ll) ) Pil(z; s?i ) = pkl(l) = l? :

2

As these conditions are satis ed at equilibrium, the allocations are indeed ecient.

71

Proposition 9 (Eciency) Suppose Assumptions 1 and 2 hold. Then max A

X i

i  ei(a) ?

X i

i  ei(a) = O(= + );

where A = fa 2 Qj [0; qjj ]I : Pi aji  qjj g, for any   mini fei (a) : ei(a) > 0g:

Proof: The proof parallels that of Proposition 3. Let

= = + : Take any a 2 A. By way of shorthand notation, let 8i; ei  ei (a), and ei  ei (a ). Let I + = fi : ei > ei g and I ? = fi : ei < ei g. For i 2 I +, by (3.13), we have i0 (ei ) < Pl l+ril + : For i 2 I ? , we P have ei > ei   , therefore by (3.14), i0 (ei ) > l l? ril ? . Thus,

X I

i (ei ) ? i(ei ) 

X I

< 2

= 2

P

+

i0 (ei )(ei ? ei ) ?

X

l X

P

l


l +

X I?

X lX +

i0 (ei )(ei ? ei)

ril(ei ? ei ) ?

l I+ l l
(1 + + ? l? );

X lX l

?

I?

ril(ei ? ei )

where
l = I ril (ei ? ei ) = I ? ril (ei ? ei ). The equality of the two sums follows form the fact that, as in Chapter 2, the seller taking all of the leftover capacity ensures that the +

P

P

total capacity I ril ei = I ali = qll remains constant. Since
l  qll and i0 is bounded the result follows. 2

q The bound = +  is minimized when  = =. Thus, the strongest statement q that can be made here is that as long as minifei(a) : ei(a) > 0g > = we get q an ineciency which is O( =).

72

3.3. Peering, Provisioning and Market-pricing of Edgeallocated capacity 3.3.1 Introduction The recent development of the dierentiated service (di-serv) Internet model is aimed at supporting service dierentiation for aggregated trac in a scalable manner. The tenet of di-serv is to relax the traditional \hard" Quality of Service (QoS) models (e.g. end-to-end per- ow guarantee of Int-serv, and ATM) in two dimensions: slower time-scale network mechanisms and coarser-grained trac provisioning. The key idea in achieving this is that capacity is allocated at the edges of the network [14]. Incoming packets pass through a pro le which ensures that there are enough \expected" resources in the network to support the type of service needed by the packet. The focus of the proposed dierentiated services framework has been mainly on packet level behavior, with the purpose of de ning building blocks for scalable dierentiated services. Substantial progress has been made in the development and standardization of packet forwarding behaviors [39, 31]. A crucial aspect that has not been addressed systematically is the feasibility of maintaining consistent service level agreements (SLAs) { or di-serv pro les { across inter-connected networks with dynamic, market-driven, edge capacity allocation. Inconsistent SLAs would result in frequent recon guration of trac conditioners on the edges, and/or signi cant violations of service quality in the core of the networks. While the role of prices as essential resource allocation control signals has been established from the outset of di-serv [13, 71], the precise development of pricing mechanisms is still at its early stages. In [49], the service charge for a user is proportional to the nominal subscribed bit rate, and the price dierentiation between

73 dierent service classes is xed. Similarly, in the User-Share Dierentiation proposal [96], the pricing is based on the user share that is allocated over a long time scale. These schemes fall within the category of capacity-based pricing. Just as di-serv aims to provide a range of \better than best-eort" services without the complexity and per- ow state of hard-QOS, capacity-based pricing schemes can be thought of as \better than at-rates" (more rational and sustainable from the economic point of view), without the per- ow measurement and accounting required by usage-based pricing. Flat-rate pricing is the extreme of capacity pricing where the capacity equals the access line speed, while usage pricing can be thought of as the extreme where capacities are continuously adapted to t the actual transmission rate of each ow at each moment in time. A pricing scheme which explicitly covers the range between these two, as well as the service-type dimension, is that of [48]. The space of network resource pricing schemes has many dimensions, as discussed more fully in Chapter 1. One is \where" the capacity abstraction takes place: at each hop inside the network or at the edges [90]. Another is how much a priori information on demand is required. At one extreme, the seller assumes perfect a priori knowledge of demand and does an oine calculation of optimal prices (e.g. time-of-day pricing based on historical trac patterns). A less extreme approach is to assume only the functional form of demand, and adjust prices by on-line \tatonnement" algorithms [29]. Auctioning is the pricing approach with minimal information requirement. The more dicult it is for the seller to obtain demand information (or valuations), the stronger the case is for using auctions. In the Internet, because of the diverse and rapidly evolving nature of the applications, services, and population, the case is compelling. With suitably designed rules, auctions can achieve ecient (value maximizing) allocations with minimal a priori information.

74 User

User route provision coeff.

Class 1 SBB

i

r ij

Class 1 SBB

r RBS

j

j i

Sub-network

Sub-network RBS

Class 2 SBB

Class 2 SBB

User

User

Class 1 SBB

User

k

RBS

Sub-network Class 2 SBB User

Figure 3-2: 2-tier auction pricing framework for di-serv internet In the previous sections, we showed that when the Progressive Second Price (PSP) auction is used as the di-serv bandwidth market mechanism, it achieves economic objectives (incentive compatibility, and eciency), while being realistic in the engineering sense (small signalling and computation load). As such, it provides a useful baseline for understanding the conditions for the economic feasibility of wide-area dierentiated services. Here, we consider the capacity provisioning problem. Section 3.3.2 applies our game model to the di-serv context, giving the wide-area pricing architecture, and the model for provisioning and dierentiation of the services. Following this, in Section 3.4., we present our analytical and simulation results.

3.3.2 The Model 3.3.2.1 Distributed Market Framework

75 Our network model assumes that each network can be abstracted into a single bottleneck capacity (e.g. as a \Norton-equivalent" [34]). The capacity may be represented by an absolute amount of bandwidth, or some relative metrics like user share in the User-Share Dierentiation proposal [96] or resource token in Location Independent Resource Accounting [92]. Large networks can be modeled by subdivision into a set of interconnected networks, each of which can be abstracted into a bottleneck capacity. The degree of subdivision that is necessary depends on trac, topology and size constraints as well as the desired level of accuracy. Within each network, the routing of aggregated trac to each peer2 is stable over the resource allocation time scale (e.g. in the order of hours). Figure 3-2 presents the architecture of our proposed auction pricing framework for a set of interconnected networks as described above. A two-tier whole-seller/retailer market model is used to accommodate a network of goods (i.e. bandwidth) with multiple dierentiated service classes. We de ne three kinds of players: users, service bandwidth brokers (SBBs) and raw bandwidth sellers (RBSs), to play the roles of end-users, retailers and whole-sellers respectively. Each network has a single RBS and a separate SBB for each class of service being oered. The RBS can be thought of as the bearer, and the SBBs as service providers [73]. If the RBS and multiple SBBs on the same network are not owned by the same entity, a non-cooperative game formulation is the best way to model the problem. Even if they are owned by the same entity, a competitive framework is valuable, the idea being that competition among SBBs results in a dynamic and ecient partition of the physical network resources among the services being oered, based on the demands from users. The users, or retail buyers, are subscribers to We use the term \peer" in the most general sense, i.e. any network which inter-connects with a given network, and not just those that choose to exchange all trac free of charge. 2

76 a particular service oered by a particular provider. In the di-serv context, these will likely be large subscribers (web sites, intra/extranets, virtual private networks), rather than individual end-users.

3.3.2.2 Dierentiating Services We do not explicitly consider the per-hop behaviors per se, which of course are essential in assuring the service quality on the packet time-scale. On our level of abstraction, only the vector of provisioning coecients ri dierentiates broker i and the service it oers. A broker is characterized by the type of service level agreement (SLA) that it oers, e.g.:

 expected capacity SLA; on average, users will get the capacity they pay for, even when the trac enters peer networks. This could include for example services built on the di-serv assured forwarding (AF) per-hop behaviors [31]. In this case, rij is the expected3 fraction of the total trac entering i that is routed to j . rii is the fraction of trac that terminates with one of i's own customers, and Pj6=l rij = 1, where l is the RBS in i's network.4

 worst-case capacity SLA; another type of SBB may oer service agreements for worst-case bandwidth, i.e. each user always gets the amount of bandwidth they pay for, even if all of the trac is routed to the same peer. This could include for example services built on the di-serv expedited forwarding (EF) per-hop behavior [39]. In this case rij = 1 for all peers j . Thus r represents aggregate ow patterns. r is measured over a time-scale slow enough to make quasi-static estimates which average out instantaneous micro- ows. 4 Note that for expected capacity, a user m whose trac is entirely within the allocated pro le i am when it enters its broker i's network could temporarily be out of pro le in the peer network j, if i miscalculated rij , or if there is a sudden surge of trac from many of i's customers to j. 3

77 r

k i 1

1/3

V

S

G

B

2/3

1

r

j i

Figure 3-3: Inter-Network provisioning coecients for dierent services: Olympic Gold, Silver and Bronze services, and the Virtual Leased Line service

 local SLA; for an SBB which oers SLAs valid only within its own network, 6 i. rii = 1 and rij = 0; 8j = Figure 3-3 illustrates several service scenaria for an SBB i with two peers j and k. In all the cases, the steady-state aggregate trac pattern is such that 2/3 of i's trac ows to j 's network, and 1/3 ows to k's network (to visualize in only two dimensions, we assume no trac terminates within i's own network). Thus, if i is oering an expected capacity service, ri will lie along the line with slope 1/2. Here we show how the SBB would have to provision the three classes in the \Olympic service" based on AF [31]. Degrees of over-provisioning must be used to dierentiate among AF classes. A Bronze service class SBB would provision just enough capacity to carry the trac on average (circle marked \B" in the gure). If the SBB is providing Silver class service, then it must provision more generously to ensure that they are less loaded, and thus experience better service, and even more generously if the service is Gold class. For the Virtual Leased Line service based on the EF PHB [39], the more conservative provisioning that is required can be achieved by providing for the worst case ows, i.e. all the trac can ow to any

78 one peer and still be satis ed. Depending on the scheduling and buer management algorithms used to provide the PHBs, some amount of over-provisioning may be required [39]. These engineering needs can be represented in this model by simply factoring the over-provisioning into each coecient of r.

3.4. Dynamic Provisioning of Dierentiated Services It is useful to conceptually decouple the game into two. On one hand is a \demand game" wherein users and brokers compete for the available bottleneck capacities. On the other hand, we have what may be called the \supply game" among brokers which results in the setting of the bottleneck capacities. Since the brokers are driven by the users' demands, and the users are competing for the oerings of the brokers, the two games are inter-dependent, and may be played on the same or vastly dierent timescales. On the demand side, the strategic game analysis of Sections 3.2. and 2.3. gives the optimal bidding strategies for users and brokers, and establishes the existence of an ecient (value maximizing) equilibrium point among buyers, when sellers are static. We now consider the supply game among brokers by itself. We will then use simulations to verify that the insights are valid when the two games are coupled. De ne the vector e = (e1; : : : ; ei; : : : ; eN ) for any pro le of allocations a, where ei is the bottleneck capacity of seller i as given by (3.3), and f1; : : : ; N g is the subset of I consisting of all the sellers (RBS' and SBBs). Pure buyers (users) are assumed to be players numbered m = N + 1; N + 2; : : :. At the equilibrium point, from Section 3.2.2, the following conditions will hold 1  i  N :

ei =

X j 2I ;j 6=i

aij

79

aji = (ei ? aij )rij These conditions can be re-written in matrix form as

e = e + u;

(3.18)

where for 1  i; j  N , j 6= i,

1?1 0 N i rk X r k i A ; ui = aim @1 + i k m>N k=1;k6=i 1 ? rk ri X

i;i = 0; 0 1?1 N i rk X rji @ r k i A : 1+ i;j = i rk 1 ? r (1 ? rji rij ) i k k=1;k6=i The matrix  = (i;j )1i;jN is the key to determining the stability of the game. The spectral radius of a matrix , denoted (), is the largest of the moduli of the eigenvalues. Let jj = (ji;j j)1i;jN . Consider now the brokers dynamically playing against each other. Speci cally, on the buy side, each broker uses a best-reply strategy (Proposition 7), and on the sell side, limits the oered capacity to the bottleneck capacity that it can obtain. Mathematically, the brokers' game is equivalent to a distributed computation to solve (3.18).

Proposition 10 The provisioning game, where brokers play asynchronously (i.e. each broker can act at any time, with no assumed order of turns, and variable but nite delays between turns), will converge to an equilibrium if an only if (jj) < 1.

Proof: This follows from the chaotic relaxation method [12, 10, 101].

2

80

Remark: Users can be viewed as external inputs driving a dynamic system, where the system dynamics are governed by the brokers: the system equation is then

e(t + 1) = e(t) + u(t):

(3.19)

The convergence of the game is the exactly the notion of stability of the dynamic system (3.19). Remark: When all the rij are equal, i.e., rij = r; 8i; j; i 6= j , we have:

i;j =  = 1 + (Nr? 2)r2 : In this case jj has a single eigenvalue equal to (N ? 1) and N ? 1 eigenvalues equal to , and (jj) = (N ? 1) = 1 +(N(N??1)2)r r2 : Speci cally, when N = 2; (jj) = r, so the convergence condition becomes r < 1. When N  3, the convergence condition (jj) < 1 is equivalent to: (1 ? N 2? 1 r)2 > ( N 2? 3 r)2 , r < N 1? 2 or r > 1: Therefore, the equal provisioning game over more than two fully connected networks does not converge if r 2 [ N1?2 ; 1]. Remark: Brokers of dierent service classes do not buy from each other. But dierent service brokers in the same network do compete with each other to buy capacity from the RBS, and the RBS does not buy from any other player (see Figure 3-2). Thus, we have the following matrix structures in, for example, a two

81 class network:

1 0 BB class1 0 0 CC BB 0 class2 0 CCC ; =B A @ Id Id 0

where Id is the identity matrix, which is in the rows corresponding to the RBSs. Since the eigenvalues of  comprise all the eigenvalues of the diagonal blocks (i.e. class1, class2 and 0), the dierent service classes are independent with regard to stability.   Therefore, for any class, we need only take rij i;j2I the matrix of the brokers' inter-network provisioning coecients, derive the corresponding jj, and compute its eigenvalues to test whether or not the game among brokers is stable. In what follows, we will use simulation to con rm the above observations under realistic service provisioning scenaria. We consider two classes of services, and hence, two SBBs in each sub-network:

 class 2 is for reliable and high quality service (e.g. the virtual leased line service considered by the EF PHB), and;

 class 1 is for adaptive multimedia applications with less stringent quality requirements (like the Olympic Bronze in Figure 3-3). In this scenario, best-eort service does not need any explicit capacity allocation. It is charged on at rate and does not participate in the bandwidth auction market. The simulation network has a mesh topology of three networks as shown in Figure 3-2. Two access networks, argo and bongo, connect to each other and to a backbone network maraca. Inter-network links are assumed to have a capacity equal to the capacity of the destination network. The dierent degrees of provisioning for the two service classes are re ected in the routing factors rji that are set according to Table 3.1.

82

buyer, class 1 SBBs

buyer, class 2 SBBs

buyer, RBS'

argo bongo maraca argo bongo maraca argo bongo maraca seller argo 0.3 0.2 0.1 class 1 bongo 0.2 0.3 0.1 SBBs maraca 0.5 0.5 0.8 seller argo 1.0 0.4 0.1 class 2 bongo 0.4 1.0 0.2 SBBs maraca 1.0 1.0 1.0 argo 1.0 1.0 seller bongo 1.0 1.0 RBS' maraca 1.0 1.0

Table 3.1: Inter-Network provisioning coecients: rji (empty entries are zero)

argo

40

available bandwidth (Mbps) bongo maraca

40

user distribution:

150

uniform across classes and networks

20 T1 users

10 T3 users

mean ON interval

mean OFF interval

max capacity: 1.5 Mbps max capacity: 40 Mbps 720 time units

72 time units

Table 3.2: Simulation parameters

83 10 Parabolic Valuation Marginal Valuation, Parabolic Logarithmic Valuation Marginal Valuation, Logarithmic

9 8

Valuation

7 6 5 4 3 2 1 0 0

0.5

1

1.5 Quantity

2

2.5

3

Figure 3-4: Valuation and Marginal Valuation Functions, ai = 3 and i = 6 The simulation parameters are given in Table 3.2. To simulate the dynamics of subscribers switching among service providers, each user is modulated by an ONOFF Markov process. At the beginning of an ON period, the user is connected randomly to one of the three networks (a uniform load distribution). During OFF periods, the user unsubscribes from the service. ON and OFF intervals are exponentially distributed with mean of 720 and 72 time units. The users are given randomly generated valuation curves, which model them as having elastic demand (as in Chapter 2). Thus, a class 1 user with a maximum capacity of 1.5Mbps will request a quantity ranging from 0 to 1.5 Mbps of class 1 service capacity. Both the quantity and price of a bid depend not only on the player's valuation, but also on the market conditions (the requested quantities and bid prices of the other players). In the Section 2.3., we assumed a very general form (i.e. elastic demand) for a user's valuation. Further speci cation of users' valuations requires a market study on actual Internet users (see for example [80]).5 In the simulations, we give our 5 Recall that the diculty in developing realistic models is one of the reasons why auctions are advantageous in the rst place, since the mechanism itself does not need to know the valuations.

84 users a parabolic valuation

i(a) = ? ai2 (a ^ ai)2 + 2ai (a ^ ai ); i

i

where ai is the line rate, i is the maximum valuation. The marginal valuation i0 (a) has the linear form with maximum at i0 (0):

i0 (a) = 2a2i (ai ? (a ^ ai)): i

In simulation, i is generated randomly from a uniform distribution on [0, 10]*ai so that i0 (0) is uniformly distributed in [0, 20]. As mentioned before Proposition 8, the broker's buy-side valuation must be smoothed. We select a logarithmic form:

i(a) = ai (a ^ ai)(1 + ln a ^ai a ) i

i

To t the curve to the demand, the broker dynamically sets ai = Pj qji and i = P qi pi . The marginal price function 0(a) has the form: j j j

0(a) = ai ln a ^ai a i

i

where as a approaches zero, the marginal valuation approaches in nity (see Figure 34.) In some circumstances, this can be useful. A nite marginal valuation would make it possible for the broker to be completely shut-out (i.e. alj = 0 at some peer l where enough users have very higher valuations). Both of the above classes are stable as shown in the simulation traces of Figure 35. In each plot, the solid line represents class 1 and the dotted line represents class 2. The long-term average demand is the same for all three networks, and for the

85 1.4

40 bottleneck capacity

1.2

price

1 0.8 0.6 0.4

30

20

10

0.2 0

0

100

200

300

400

0

500

0

100

200

300

400

500

100

200

300

400

500

100

200

300

400

500

(a) argo trace 40 bottleneck capacity

1

price

0.8 0.6 0.4 0.2 0

0

100

200

300

400

30

20

10

0

500

0

(b) bongo trace 0.7

150 bottlneck capacity

0.6

price

0.5 0.4 0.3 0.2

100

50

0.1 0

0

100

200

300

400

500

0

0

(c) maraca trace

Figure 3-5: Simulation of three networks with two stable classes two classes. In the shown interval, the demand is such that capacity of argo is partitioned equally between the two classes. As expected, the higher quality class 2 service is more expensive, since it has higher provisioning coecients { see (a). A similar situation occurs in maraca - see (c). In bongo, class 1 is more expensive because the demand is greater, which is also re ected in the larger share of capacity allocated to class 1 { see (b). Consider now three inter-connected networks, with just one class, i.e. three brokers f1; 2; 3g. Let r12 = r21 = x, r13 = r23 = y, and rij = 0:99 for all other pairs

86 1.6

1.8

0.95

1.2

y

1.6 1.4

1.1

1.2

1.05 1

1 0.8

0.95 0.6 0.9 0.4 0.85 0.2

0.2

0.4

0.6

0.8

1 x

1.2

1.4

1.6

1.8

Figure 3-6: Spectral radius as a function of inter-network provisioning coecients

i; j . Figure 3-6 shows (jj) as a function of x and y. The gure shows that when x > 1 and y < 1 or vice-versa, the provisioning of this class becomes unstable. It is interesting to note that simply overprovisioning x > 1 and y > 1 does not give rise to instability. Thus, instability can be due more to asymmetry in the ows rather than to the actual degree of over-provisioning. Neither can instability be simply attributed to the existence of \cycles" in the graph of the network. Figure 3-7 shows a scenario where a single class network { with a simple topology of two access networks connected to a backbone network { can be unstable even if the graph of the network has no cycles. In Figure 3-7 (b), the right-hand side shows the allocations for trac going from argo to maraca (lower curve), and the bottleneck capacity in argo itself (upper curve). The instability is re ected in the volatility of the allocated capacities. In a stable scenario, one must still worry about what kind of equilibrium is reached. Indeed, it can happen that the only equilibrium for a stable class is one where all the bottlenecks are zero. Figure 3-8 illustrates this possibility, which we

87

argo 40 Mbps

bongo 40 Mbps

1.1

1.1 1

1

maraca 150 Mbps

(a) Simulation topology (() = 1:02) 40

35

1.2

30

buy−side alloc and bottleneck

1

price

0.8

0.6

0.4

25

20

15

10 0.2

0 0

5

100

200

0 0

100

(b) Trace at argo

Figure 3-7: Simulation of one unstable class

200

88 bottleneck capacity

maraca

60 40 20 0 0

x=0.4

x=0.45

x=0.49

x=0.5

50

100

150

200

250

300

350

400

50

100

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250

300

350

400

50

100

150

200

250

300

350

400

argo

40

20

0 0

bongo

40

20

0 0

Figure 3-8: Dis-peering eect refer to as dis-peering. Here, we simulate the three network element with a single class that is provisioned identically in all directions, i.e., 8i; j; i 6= j; rij = x. As x approaches 0:5, the bottleneck becomes smaller, until nally, none of the brokers has any capacity to sell. Here, there is only one class, and the physical capacity as well as the average demand from the users remains constant (even though users do come and go { see Table 3.2). Thus, the reduction in bottlenecks is purely a result of the provisioning dynamics, and not of other trac \squeezing out" this class. Indeed, since capacity is edge-allocated, a broker must provision for all possible routes (here there are two, one to each peer network), with a degree of assurance x. When this required assurance x reaches a critical level (which depends on the topology), it becomes impossible for the broker to satisfy any demand. This is one of the \penalties" to be incurred in exchange for the simplicity and scalability of edge-capacity allocation with stateless service dierentiation. Indeed, if the broker could oer allocations tied to speci c routes, dis-peering would not occur. This eect may also be the converse of what has been observed in the current

89 (best-eort only) Internet. In recent years, some large ISPs have decided it is not in their interest to peer free of charge with some smaller ones because they would do better by selling the bandwidth directly to their own customers [6]. Here, with dierentiated services, a broker in a large network may decide to set rij = 0 in the direction of the smaller networks (i.e. not to buy any dierentiated service from the smaller network), when it is not worthwhile to get the allocations required for a high level of assurance in a congested network. Other related phenomena have been studied in the literature [5, 27, 91].

3.5. Conclusion The results of Section 3.2. apply to distributed PSP games with any combination of brokers, virtual network buyers and simple users. Which kind of players are most appropriate depends largely on the information structure of the game. Indeed, Proposition 5 gives the constraints on how much total capacity a seller can oer, given that the seller is itself a buyer and aggregator of resources at multiple elements, i.e. a broker. Consider the broker selling to -users. If the nature of the service is such that

 capacity allocations are independent of the destination (i.e. their trac goes on dierent routes at dierent times, but the users always expect to get the allocated capacity on average), and

 seller i's a-priori knowledge of the ows is only in the aggregate ri, then seller cannot do better than sell the aggregate capacity as a single element. In that case, Proposition 9 shows that the PSP rule results in an optimally ecient sharing among that sellers customers. What if each user speci es a single destination to the broker? This restriction on

90 the users is, from the point of view of the seller, a relaxation of the rst of the above conditions. An allocation that was ecient for aggregate expected ows may not be the most ecient when the routes taken by the ows are known deterministically. For example, say a given user has a relatively low bid price and gets a small allocation in an auction of the aggregated capacity. If the seller knows that that user's ow is going only to parts of the network where there is unused capacity, then that user's allocation can be increased any reduction on the other users. Thus, when routes are known accurately in advance, the virtual network buyer is the appropriate player model, while in an edge capacity framework, the users buying from brokers are more appropriate. In investigating the stability of provisioning dierentiated internet services using a distributed game theoretic model, our results indicate that, in an internet with multiple dierentiated classes competing for the same resources, even though the demand for one service aects the amount of capacity available for another, the stability of each class is independent of the others'. Thus, the good news is that dynamic market-driven partitioning of network capacity among services appears sustainable. The bad news is that very conservatively provisioned services can be unstable on this macro level, even in the simplest network topologies. Even in stable cases, the only sustainable outcome may be to not peer for dierentiated service trac. These results are not merely artifacts of PSP or of any particular pricing mechanism. They appear to be fundamental issues of market-driven peering under edge capacity allocation. The dynamic system formulation of (3.19) suggests an interesting direction for future work. It may be possible to achieve certain wide-area network objectives, (e.g. stability or avoiding dispeering) by exercising feedback control. If such controls can be derived and are not too large in magnitude, they could be applied by injecting

91 some service requests at multiple strategic edge points to drive the brokers of that speci c class to a bene cial equilibrium.

92

Chapter 4 Spot and Derivative Markets in Admission Control Net game now risk, G. [87]

In Chapters 2 and 3, we proposed a pure market approach for bandwidth pricing, where allocated capacities may vary during the lifetime of a ow, as players compete for resources through an auction game. In this chapter, we propose a mechanism for circuit switched calls, wherein calls are admitted or rejected at (or soon after) their arrival time, and if admitted, get a xed allocation of capacity, and have the option of securing the resource at a guaranteed maximum price for a guaranteed minimum duration. Thus the charge has two components:

 a market-based usage charge, where the user continuously pays the \instantaneous" market price (determined by second price auctions among recent arrivals); if the market price exceeds a user's bid price, that user is dropped unless

 upon arrival, the user pays a reservation fee (buys an option contract), which gives him the right to buy the capacity at any time in the future up to a

93 speci ed duration at his bid price. The user pays the market price as long as it is below his bid price. If at some point during the call, the market price exceeds the bid price, then the user automatically exercises the option, i.e., remains connected while paying no more than the bid price. In other words, we introduce a \derivative" instrument to reduce the uncertainty inherent in the \spot" market mechanism. Naturally, this contract (or reservation) must itself be sold for a \fair" price. In the context of nancial markets, the fair price is calculated with the Black-Scholes approach [8, 65], which is based on the idea that the option must be priced such that a perfectly hedged (i.e., riskless) combination of the derivative and the underlying equity will provide (locally in time) the same expected return as a risk free security [36] (otherwise the derivative would present an arbitrage opportunity, i.e., unfair advantage, which would be exploited until its price rises). In our context, the contract diers in that, rather than the right to buy once at a given strike price, it gives the right to buy repeatedly over a given duration, i.e., it is a series of \call options". Here the \fairness" we seek is that the reservation be priced in a way that re ects the probability that the system will become busier during the lifetime of the reservation, in order to avoid individually rational but socially sub-optimal behaviour. Speci cally, we seek to avoid customers connecting at low periods and making a reservation limiting their usage price to an excessively low price for a very long time, to avoid rejoining when the prices are higher1. A further dierence from usual options is that, rather than the standard geometric Brownian motion model of e.g., a stock price, we have an underlying spot market process that is derived directly from our queueing system, via a heavy-trac A simpli ed form of this arbitrage is commonly observed with at-rate priced dial-up Internet access, where some users remain logged on for very long periods of time even if they are not using the network, in order to avoid the chance of getting busy signals when they need to be on. 1

94 arrivals Poisson ~ λ bid price ~ F B lines

batch ouput every τ ~exp/µ of size m ~βm 1 0.

4.5.1 Case mu(1 ? u) > V 2 By the Central Limit Theorem, the binomial density tends to the normal density, i.e., for 0 < u < 1, we have ( 2) m! um?k (1 ? u)k  q 1 ? [ k ? (1 ? u ) m ] : exp k!(m ? k)! 2u(1 ? u)m 2mu(1 ? u) Thus, ! m?C +n?1 C ?n g(u; n; m) = (C ? n)!(m m ? 1 ? C + n)! u ( (1 ? u) 2) ? [ C ? n ? (1 ? u ) m ] 1 m ? C + n q exp :  u 2u(1 ? u)m 2mu(1 ? u) Dierentiation yields

# " @g (u; n; m)  g(u; n; m) C ? n ? m (1 ? u ) 1 @n m ? C + n + u(1 ? u)m and

@ 2g (u; n; m) @n2

8 #2