Network Market Design Part I: Bandwidth Markets

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1 COMMUNICATIONS NETWORK ECONOMICS Network Market Design Part I: Bandwidth Markets Rahul Jain, University of Southern California ABSTRACT Markets for network resources have become increasingly important. Such resources include bandwidth in wireline networks to wireless spectrum. It is clear that if the markets are not properly designed, they can function rather poorly, even leading to market failure. This then leads to suboptimal use of network resources. In this article, we present the game theoretic framework behind the market design principles for network resources. We present some of the extant work on network market mechanisms in the context of wireline networks in part I. Part II [1] relates to network market design in the context of wireless systems. We also discuss some of the key challenges for the future. INTRODUCTION The basic problem in communication networking is how to effectively share resources, such as the bandwidth of communication links or buffer space in routers, or the use of a shared wireless medium. From an engineering perspective, resource sharing underlies the design of protocols used throughout the network stack. Resource sharing is also at the heart of economics. Indeed, the role of a market is to allocate resources among competing agents. Often, engineers view the economic considerations as being separable from the design of the underlying technology used for resource sharing, and focus mainly on the latter issue. For modern communication infrastructure, which is owned and operated by multiple, independent, profitmaking entities and handles a heterogeneous mix of traffic in diverse environments, it is becoming increasingly clear that such a separation is not tenable. This will become even more apparent in next-generation networks, which are envisioned to be even more complex hybrid wireless/wireline networks with greater capacity, reliability, and capability. Moreover, new architectures may evolve that tap user cooperation in social networks to enhance network capabilities and provide ubiquitous connectivity. There are several challenges to such a vision. Network capacity will remain constrained due to the scarcity of resources such as spectrum, and sometimes bandwidth as well; network utilization will remain suboptimal due to the static and often centralized allocation mechanisms used, particularly with smarter selfish users; and service quality will remain variable and without guarantees in the absence of adoption of network architectures for quality of service (QoS) provisioning. Network capacity can be enhanced by sophisticated information-theoretic and network coding methods, but those will not be enough without incentivizing cooperation between increasingly sophisticated and selfish users. Network utilization will improve with better decentralized algorithms, but what if these can be manipulated by strategic users? A dependable service quality on the Internet shall remain a distant dream until economic impediments to network service provider cooperation in QoS-provisioning are removed. The basic thesis of this article is that a systematic and foundational understanding which synthesizes the economic and technical approaches for resource allocation will aid in meeting these challenges by facilitating scalable, flexible protocols that account for user incentives. FROM NUM TO NMD Dynamic decentralized algorithms for network flow optimization have a long history. Since Kelly s seminal work [], network theorists have used ideas from mathematical economics and optimization to achieve distributed flow control and optimization in large-scale networks. Such a theory has both informed Internet congestion control protocol design (e.g., TCP-Vegas) as well as yielded power control algorithms for wireless networks. The rich mathematical theory underlying this work is often referred to as network utility maximization (NUM). Although NUM has its roots in ideas from economics, it uses these ideas mainly as a metaphor to guide engineering design as opposed to truly incorporating economic incentives. It is not robust to manipulation by increasingly sophisticated users, who act strategically and often have economic interests at stake. In fact, it was shown in [] that the Kelly mechanism can have arbitrarily bad efficiency when users behave selfishly and strategically. Thus, there is a need to go beyond the current distributed optimization-based NUM framework to a game-theoretic and market economics-based network market design (NMD) /1/$ IEEE IEEE Communications Magazine November 01

2 framework that takes incentive issues into account in design of network resource allocation algorithms and protocols, which are robust to strategic manipulation. The theory of mechanism design in game theory [4, 5] indeed deals precisely with these problems but does not always provide solutions for communication-network-specific problems. For example, network resources are usually allocated in bundles (e.g., bandwidth on links that constitute a route) and are often well approximated as being divisible, meaning that they can be arbitrarily divided among users. Mechanism design theory can in principle handle bundles and divisible resources, but the mechanisms it prescribes quickly become too complex to implement both in terms of their communication as well as their computational costs. This is in stark contrast to the scalability considerations that are essential for networking protocols. Another consideration is that most of this theory is for single-sided auctions (allocations by one auctioneer to multiple buyer) as opposed to double-sided (exchange between buyers and sellers facilitated by an auctioneer), which is more relevant in many network settings. Also, network resource allocation problems are inherently dynamic where user population, utilities, network capacities and even topologies change over time while current design solutions treat these as static, one-shot problems. Partly, the reason for this is that dynamic mechanism design is itself under-developed and a very hard problem. Finally, mechanism design focuses on allocating a given set of assets to the users. In many networking contexts, by making different engineering choices, one can redefine the underlying assets in different ways, creating a new dimension in market design. In this article, we explain some of the basic issues that need to be addressed in designing network markets, and discuss some of the main impossibility results as well as fundamental design guidelines. We then present some of the work that has been done in the last decade on network market design. Part I presents this in the context of network bandwidth, while Part II will present it in the context of wireless spectrum. We conclude with some of the open and challenging problems for future research. THE NUM FRAMEWORK FOR DECENTRALIZED NETWORK RESOURCE MANAGEMENT Consider a network described by a graph G = (V, E). A simple example is illustrated in Fig. 1. Let there be n users or enterprise customers. User i wants capacity on a bundle of links R i, which could constitute a route. Let the utility to user i of obtaining x i bandwidth on links R i be u i (x i ). Let there be M network service providers (NSPs) who own capacity on various links in the network. For example, NSP j might own capacity on a bundle of links L j which could constitute a route. Let the total capacity on link l be denoted by C l. Let the cost of provisioning y j capacity on links L j be c j (y j ). The system objective is to SYSTEM :max Sxy (, ) = vi( xi) cj( yj) i j subject to Flow Constraints: (1) () S(x, y) is called the social welfare function. Typically, the utility functions v i (x) are assumed to be non-negative increasing concave functions, while the cost functions are assumed to be non-negative increasing convex functions. This makes the social welfare maximization problem a convex optimization problem that can be solved easily using standard methods available in optimization theory. The difficulty here is that both the utility and cost functions are private information of the users and network operators, respectively. Thus, this problem cannot be solved in a centralized manner. Assume there is only one provider for each link, and assume the capacities y j are given. It was shown by Kelly [] that the system problem above can be decomposed into a user problem: USER i : max xi v i (x i ) S lœri l l x i, and a network problem: NETWORK j : max yj S lœlj l l x j c j (y j ), such that they both together solve the SYSTEM problem. Here, l Lj is shorthand for (l l : l Œ L j ), and these are the dual variables corresponding to the flow constraints in the SYSTEM problem, and interpreted as (shadow) prices for per-unit capacity on the links. Moreover, Kelly also proposed a distributed algorithm (henceforth called Kelly s algorithm) that iteratively solves the SYSTEM problem by having each user and each operator solve their own problems. Since the introduction of this idea to networking, it has been very well developed, and a whole framework, NUM, has been developed that allows decentralized resource allocation in large-scale communication, transportation, and power networks. However, with non-cooperative and strategic users and network service providers (NSPs), distributed NUM algorithms can fail to achieve predicted performance, and are often highly inefficient. For example, Kelly s mechanism for network allocation can have social welfare that is arbitrarily close to zero with selfish strategic users []. Thus, a different framework that is more robust, and takes cooperation incentives and economic interests of users and NSPs into account is needed for such settings. We call it the NMD framework based as it is on market economics, auctions, and game theory. xi yj, l E, xi, yj 0, i, j. il : R jl : L i Network operators 1,1 Service provider 1, v1(x) Figure 1. A network with network operators and service providers/users. j Network operators, A B C Service provider, v(x) v(x) c(y) Quantity IEEE Communications Magazine November 01 79

3 MECHANISM DESIGN: IN ANUTSHELL Mechanism design (MD) in game theory provides a framework in which design of markets can be considered in a systematic manner. It indeed offers the basis for the NMD theory, but there are challenges specific to communication networks. For example, auction theory deals with mostly indivisible goods, whereas network resources such as bandwidth and spectrum are often regarded as divisible goods. If standard mechanisms such as the Vickrey-Clarke-Groves (VCG) mechanism are used, this introduces informational problems since they require infinite-length bid messages to yield incentive-compatible outcomes. Furthermore, bundles of goods are mostly considered in combinatorial auction contexts, which usually have computational feasibility issues due to the integer programming problems involved. But above all, network resource allocation frequently involves exchange between agents (some being buyers, others sellers), and thus, mechanisms (e.g., VCG) that are not budget-balanced (i.e., payments made by the buyers equal payments made to the sellers) are untenable. And last but not least, network environments are inherently dynamic, so we need to go beyond the one-shot mechanisms available in state-of-the-art theory. Below, we first present the MD framework, and some of the most useful and celebrated results in the static setting. Consider n agents with the set of outcomes X. Agent i s utility is u i (x, q i ) with type q i ŒQ i. A social choice function (s.c.f.) f(q 1,, q n ) Œ X maps agent types to outcomes. This can be interpreted to be the outcome that is socially desirable given agent types or utility functions. An example is f(q 1,, q n ) Œ arg max x S i u i (x i, q i ), where x = (x 1,, x n ) Œ X. An s.c.f. f Q ÆX is called ex post efficient if for no q is there an x Œ X such that u i (x i, q i ) u i (f i (q), q i ) for all i, with a strict inequality for some i. The question now is what s.c.f.s can be implemented (in a distributed manner) when agents types are private information? And how? We define a mechanism G = (S 1,, S n, g(.)) as a collection of strategy sets and an outcome function g(s 1,..., s n ) Œ X. A mechanism G implements a s.c.f. f if there is an equilibrium strategy profile s* = (s 1 *(q 1 ),, s n *(q n )) of the game induced by G such that g(s 1 *(q 1 ),..., s n *(q n )) = f(q 1,..., q n ), "q i ŒQ i, "i. That is, the game corresponding to G is such that the equilibrium outcome is what would be regarded as the outcome for the social good (i.e., the s.c.f. f). Now, we say that a strategy profile s* is a Nash equilibrium if for each player i, u i (g(s i *, s* i ), q i ) u i (g(s i, s* i ), q i ) for every strategy s i ŒS i of player i. That is, given other players play strategies s* i = (s 1 *,, s* i 1, s* i+1,, s* n ), player i s best response is to pick s i *. Note that s i * does depend on s* i, in general. However, it is quite possible that a player has one best strategy to play no matter what strategies others play. Such a strategy is called a dominant strategy for player i. If a Nash equilibrium consists of a dominant strategy for every player, we call it a dominant-strategy equilibrium. If the equilibrium strategy profile of the mechanism G is a dominant strategy (Nash) equilibrium, it is called a dominant strategy (or Nash) implementation of the s.c.f. f. The question now is are all s.c.f.s implementable in some way? Do we need to consider all possible mechanisms, which is a rather huge space? Fortunately, the revelation principle provides a very helpful answer to the latter question. REVELATION PRINCIPLE Suppose there is a mechanism G that implements the s.c.f. f in dominant strategies. Then the direct mechanism (wherein S i = Q i, g = f) is incentive-compatible; that is, truthfully reporting types is a dominant strategy equilibrium. This has great implications: If an s.c.f. is implementable at all (in dominant strategies), it suffices to consider the direct mechanism wherein each player is just asked to report his/her true type. Thus, it suffices to consider direct mechanisms alone for dominant strategy implementation. We now discuss some desirable properties a mechanism should have: A mechanism is ex post individual-rational (IR) if the payoff at any equilibrium is nonnegative for all agents. A mechanism is said to be efficient if it maximizes social welfare. A mechanism is said to be budget-balanced if the sum of payments of agents is zero. A mechanism is said to be incentive compatible (IC) if at equilibrium it is a dominant strategy for all agents to bid truthfully. We now present two common auction mechanisms. FIRST-PRICE AUCTIONS Consider n bidders for a good. X is the set of allocations of good to each of the bidders. Each player has a utility function u i (x i, q i ) = x i q i. The types q i have U[0, 1] distribution, and this is common knowledge. The s.c.f. is f(q 1,, q n ) = max xœx S i u i (x i, q i ), where the outcome space X is the good being assigned to each of the bidders. The auction rule is that bidders submit sealed bids; the highest bidder wins and pays his/her bid. In this case, it is easy to show that there is a Bayesian-Nash equilibrium such that each bidder bids exactly half of what s/he is willing to pay for the good. Thus, the good will go to the bidder who has the highest value for it, although he does not reveal his/her true valuation for the good. SECOND-PRICE AUCTIONS Now, consider the following auction rule: The highest bidder wins but pays the second highest price [6]. It is easy to show that in this case a dominant-strategy equilibrium is for each bidder to bid exactly his/her true value for the good. Thus, in this case not only is the equilibrium outcome efficient (i.e., the bidder who values the good the most gets it), but also each bidder truthfully reveals their true valuation. Moreover, this is a dominant strategy for each player, independent of what the others know or do. Thus, this is a strong implementation of the s.c.f. f defined above. 80 IEEE Communications Magazine November 01

4 We note that the celebrated VCG mechanism, which is a generalization of the secondprice auction, is IC, ex post IR, and efficient but not budget-balanced. This raises the question of whether there are mechanisms that have all four properties. The answer turns out to be negative. Buyer i (b i, R i ) (true v i ) Auction system compute allocation and prices Buyer i (x i *, P i ) HURWICZ IMPOSSIBILITY THEOREM (1975) There is no mechanism (for simple exchange with quasi-linear preferences) that is IC, ex post IR, efficient, and budget-balanced [7]. We say that a mechanism is Bayesian IC (BIC) if truth revelation is a Bayesian-Nash equilibrium of the associated incomplete information game. The dagva (expected externality) mechanism (d Aspremont and Gerard-Varet, 1979, Arrow, 1979), a variation of the VCG mechanism for the incomplete information setting, is ex ante IR, BIC, efficient, and strongly budget-balanced when agents have quasi-linear preferences. The Myerson-Satterthwaite Impossibility Theorem (198) [8] states that it is impossible to achieve efficiency, budget balance, and interim individual rationality in a BIC mechanism even with quasi-linear utilities. We refer the reader to [4, 5] for a more thorough treatment of the theory of mechanism design. NETWORK AUCTION DESIGN: DIVISIBLE BANDWIDTH Consider a network with L links, each with divisible bandwidth; that is, bandwidth can be allocated in any fraction. Let link l have capacity C l. There are n buyers, and buyer i wants a bundle of links R i that constitute a route. If buyer i gets bandwidth x i on route R i, it derives a utility v i (x i ). If it makes a payment P i, its net payoff is u i (x i ) = v i (x i ) P i. The goal is to determine an allocation x** that maximizes the social welfare S(x) = S i v i (x i ) subject to the flow constraints: S i:lœri x i C l for all links l. The difficulty in determining the allocation x** is that the utility functions are private information. And buyers may not reveal them. If the utility functions are not known, how can we still determine the optimal allocation x**, if at all? Well, one way is to elicit some information from the buyers by conducting an auction. Suppose buyer i reports a bid signal b i for route R i as indicative of his utility function v i. The auction system would then take the bids bi of all the buyers, and determine an allocation x*(b) and payments P i (b i, b i ). Of course, the buyers are selfish and will act strategically. They will pick a bid b i to maximize their payoff u i (b) = v i (x i (b)) P i (b). See Fig. for a graphical illustration. So, unless the right allocation function x*(b) and payment functions P i are picked, and the SYSTEM and buyer objectives are aligned, there is no reason to believe that the auction allocation x* will be the same as the optimal allocation x**. A larger question is whether this is even possible. It turns out that this is indeed so! For simplicity, consider a single-link network with capacity C. One way to allocate divisible bandwidth via an auction is what is called Kelly s Figure. An auction system for network bandwidth. mechanism: Each player submits a bid, b i. The allocation for user i then is x i = Cb i /S i b i, that is, the capacity C is allocated proportionally (w.r.t. their bids) among the users. This can be seen as a solution of the SYSTEM problem with the surrogate utility functions u i (x i ) = b i log x i. A perunit price is computed as m = S i b i /C. Thus, user i pays mx i = b i. If the users are non-strategic (or price-takers), it can be shown that this mechanism is indeed efficient, in the sense of achieving the optima of the SYSTEM problem above. Unfortunately, when the users are strategic, the mechanism has equilibria wherein the social welfare is arbitrarily small []. To address this problem, the VCG-Kelly mechanism was proposed []. Users submit onedimensional bids b i. The mechanism determines an allocation x* that solves the SYSTEM problem with surrogate utility functions u i (x i ) = b i log x i. Bidder i makes a VCG payment P i (b) = max S j i b j log x j S j i b j log x j *. It can be shown that if there are at least two special buyers (with u (0+) = + ) on each link who make positive bids, the Nash equilibrium point exists, and is unique and efficient. A similar idea has been developed in [9] in more generality. One of the shortcomings of this mechanism may be that it seems impractical. In most auctions and markets, the bidders are rather used to not only specifying how much they would pay, but how much they want as well. A mechanism that has a philosophy very similar to the VCG-Kelly mechanism, and yet addresses the above concerns, is the following. In the network second price (NSP) mechanism [10], buyers submit bids b i = (b i, d i ), which is interpreted as willingness to pay $b ii per unit for up to d i units of bandwidth on route R i. The mechanism then determines an allocation x* that maximizes social welfare S i b i x i subject to the flow constraints, and ensuring that x i Œ [0, d i ] for all buyers i. Buyer i receives x i * bandwidth on route R i and pays P i (b i, b i ) = S j i b j (x j i* x* j ), where x i* is the auction allocation when bidder i does not participate. We illustrate how this works through a simple example. EXAMPLE 1 Suppose C = 6 units of capacity on a single-link network. There are three players. Player R bids ($/unit, units), Player B ($/unit,.5 units), and Player G ($1/unit, 4 units). Then it can easily be checked that the allocation is x* R =, x* B =.5, and x* G = 0.5, and the payments would be P R = $, P B = $.5, and P G = 0. This is illustrated in Fig.. In fact, it can be shown that there is a Nash equilibrium b* at which the corresponding allocation x* is efficient: S(x*) = S(x**) [10]. We IEEE Communications Magazine November 01 81

5 AOL: v1(b,r) Comcast: v(b,r) Earthlink: v(b,r) Figure 4. A network with bandwidth owners and ISPs. Buyer i (b i, R i ) (true v i ) Seller j (a j, L j ) (true c j ) Bids Seattle SF Chicago Houston Auction system compute allocation and prices Figure 5. A market system for bandwidth exchange. Allocation Figure. Example 1: Allocation and payments with NSP. 1 NY DC Payments Boston AT&T: c1(b,l) MCI: c(b,l) Sprint: c(b,l) Buyer i (x i 0, P ic ) Seller j (y j 0, -T j ) note, however, that there are many Nash equilibria, not all efficient. Although some of the inefficient Nash equilibria can be eliminated through reserve prices (i.e., by having each winner pay a reserve price plus the designed payment). Thus, the mechanism proposed above may be considered a weak Nash implementation, is ex post individual rational, but does not have the budget-balance property. NETWORK MARKET DESIGN: INDIVISIBLE BANDWIDTH One difficulty with the mechanisms we considered in the previous section is that often bandwidth is only be available in discrete units (e.g., in increments of 1 Gb/s). That is, it is an indivisible good. Another difficulty is that, often we have a market setting, in which the bandwidth exchange involves both multiple buyers and multiple sellers, facilitated by an auctioneer or a market maker. In this case, it becomes crucial that there be budget balance; that is, the payments made by the winning buyers equal the payments made by the winning sellers, and the mechanisms proposed earlier become irrelevant including any VCG variants. As before, consider that buyer i wants bandwidth on route R i. If s/he gets x i units, and pays $0 $.5 $ P i, his/her payoff is u i b (x i, P i ) = v i x i P i, where v i is the valuation that buyer i has for each unit of bandwidth on route R i up to some d i units. A seller j has s j units of bandwidth to sell on link L j. If it sells y j units and receives a payment T j, its payoff is u j s (y j, T j ) = T j c j y j, where c j is what it costs the seller to sell a unit of bandwidth. See Fig. 4 for a simple illustration. Such costs may arise because while the seller may have sunk in some fixed infrastructure cost, s/he would still incur some maintenance costs, and this could depend on how much bandwidth in the pipe is actively used. The goal in designing a network (bandwidth) market then would be to determine an exchange (x**, y**) that maximizes the social welfare S(x, y) = S i v i x i S j c j y j subject to the flow constraints: S i : l ΠR i x i S j:l=lj y j for each link l. Note that while buyers want routes, so bundles of links, sellers sell bandwidth on individual links. And if they own bandwidth on multiple links, they could offer them separately but need not bundle them. This makes the problem tractable. In a market mechanism, the buyers can be asked to report (b i, R i, d i ), which is interpreted to mean that the buyer is willing to pay up to b i per unit for up to d i units on route R i. Meanwhile, the sellers can be asked to report (a j, L j, s j ) which is interpreted to mean that the seller wants at least a j per unit and can sell up to s j units on link L j. This is graphically illustrated in Fig. 5. Note that unless the market allocation (x*(b, a), y*(b, a)) and the payment functions P i and T j are properly designed, the SYSTEM and player objectives are not going to be aligned to result in the social welfare maximizing outcome (x**, y**). In the combinatorial Seller s Bid Double Auction (c-sebida) that was proposed in [11], the allocation (x*, y*) can be determined by maximizing the trading surplus S i b i x i S j a j y j subject to the flow constraints above and x i Π{0,, d i } and y j Π{0,, s j }. Once the allocation is determined, we can determine a price on each link as p l = sup {a j : y j * > 0, l = L j }, that is, the highest ask price a j among all matched sellers on the link l. Now, a matched buyer pays the sum of prices p l on the links in his/her route R i times the allocation x i *, and a matched seller receives a payment equal to price p Lj times quantity q j. We now illustrate through an example how the mechanism works. EXAMPLE Consider a single link network with three buyers with valuations $.1,.1, and 1.1, respectively, for one unit, and three sellers with costs $1,, and, respectively, for one unit. If each of them bids their true valuation or cost, the optimal trading surplus subject to flow constraints would be. when buyers 1 and are matched, and sellers 1 and are matched, but buyer and seller are not. In this case, the price will be p =, the highest ask price among the matched sellers. This is graphically represented in the first part of Fig. 6. Note that what the c-sebida mechanism does not do is try to match each buyer with a seller such that each such matching has a positive surplus. If this were the case, buyer 1 could be matched with seller, buyer with seller and buyer with seller 1, and surplus in each match- 8 IEEE Communications Magazine November 01

6 ing would be positive. But this does not maximize the trading surplus, or the social welfare. For the c-sebida market mechanism, it has been shown in [11] that: The mechanism is budget-balanced and ex post IR. On a single link, every Nash equilibrium (NE) allocation (in weakly rationalizable strategies) is efficient. Moreover, there is an NE at which all players except the highest matched seller bid truthfully (almost IC). In the network case, it was established that if an NE exists with a trade for every link, every such NE allocation is efficient. The case of incomplete information has also been addressed. It was shown that the market mechanism is asymptotically Bayesian-IC and efficient, ex post individual rational, and budget balanced. Recall from the Myerson-Satterthwaite theorem [8] that we cannot have a mechanism which has all four non-asymptotic properties. So, in some sense, this is the best we can hope for. Network market design with indivisible bandwidth is an important problem to run bandwidth markets efficiently. However, there is a paucity of work on it. The only known related work is [1] which proposes a double auction mechanism that only has the weak budget balance property (i.e., the auctioneer s expected payoff is nonnegative). OPEN PROBLEMS AND FUTURE DIRECTIONS From the above discussion, it should be clear that auction and market design solutions for network resources exist for a wide variety of scenarios. Single-sided auction designs for divisible goods have been fairly well explored. Designing suitable double-sided auctions, or markets, has proved to be rather challenging though. Part of the reason is Hurwicz negative result: It is impossible to have all four properties. We must make a compromise on one, and it proves rather difficult to trade off incentive compatibility and efficiency while ensuring budget balance and individual rationality. Designing markets for indivisible goods proves to be even more challenging. Not only the computational issues become important, but the paucity of appropriate mathematical tools to deal with non-smooth problems makes the analysis rather difficult. There is thus scope to introduce new design methodologies and mathematical tools for such network market design problems. The other issue we have ignored is that markets usually operate dynamically. When markets are dynamic, participants not only glean valuable information from past plays, but they may also have the option of waiting until the next round. This belongs to the realm of dynamic mechanism design, a theory yet underdeveloped, but receiving increasing attention. It should be obvious that in dynamic mechanisms and markets, strategic learning and experimentation issues also become important, yielding new models such as multi-armed bandit games that have yet to be understood. b 1 =.1 a = b =.1 a = b =1.1 a 1 =1 Figure 6. Example : C-SeBiDA outcome and equilibrium. Designing markets for spectrum opens up a whole new host of issues: What is a good here? What do property rights mean? And, in the context of cognitive radio systems, what is the right framework for spectrum contract design? Some of these issues are addressed in the companion article [1]. REFERENCES [1] R. Berry, Network Market Design II: Spectrum Markets, IEEE Commun. Mag., this issue. [] F. P. Kelly, Charging and Rate Control for Elastic Traffic, Euro. Trans. Telecommun., vol. 8, 1997, pp. 7. [] S. Yang and B. Hajek, VCG-Kelly Mechanisms for Divisible Goods: Adapting VCG Mechanisms to One-Dimensional Signals, IEEE JSAC, vol. 5, no. 6, 007, pp [4] V. Krishna, Auction Theory, nd ed., Academic Press, 009. [5] A. Mas-Colell, M. D. Whinston, and J. R. Green, Microeconomic Theory, Oxford University Press, [6] W. Vickrey, Counterspeculation, Auctions And Competitive Sealed Tenders, J. Finance, vol. 16, no. 1, 1961, p. 87. [7] L. Hurwicz, On Informationally Decentralized Systems, C. McGuire and R. Radner, Eds., Decision and Organization: A Volume in Honor of Jacob Marschak, North- Holland, [8] R. B. Myerson and M. A. Satterthwaite, Efficient Mechanisms for Bilateral Trading, J. Economic Theory, vol. 8, 198, pp [9] R. Johari and J. N. Tsitsiklis, Efficiency of Scalar-Parameterized Mechanisms, Operations Research, 010. [10] R. Jain and J. Walrand, An Efficient Nash-Implementation Mechanism for Divisible Resource Allocation, Automatica, vol. 46, no. 8, Aug. 010, pp [11] R. Jain and P. P. Varaiya, An Asymptotically Efficient Mechanism for Combinatorial Network Markets, submitted to Operations Research, Nov [1] L. Y. Chu and Z.-J. Max Shen, Truthful Double Auction Mechanisms, Operations Research, vol. 56, no. 1, 008, pp BIOGRAPHY b 1 =.1 a = a 1 =1 RAHUL JAIN (rahul.jain@usc.edu) is an assistant professor and the K. C. Dahlberg Early Career Chair in the Electrical Engineering Department at the University of Southern California. He received his Ph.D. in EECS and an M.A. in statistics from the University of California, Berkeley, and his B.Tech from IIT Kanpur. He is a winner of numerous awards including the NSF CAREER award, an IBM Faculty award, and the ONR Young Investigator award. His research interests span wireless communications, network economics and game theory, queueing theory, power systems, and stochastic control theory. (a) b =.1 a =.1 c = b =1.1 (b) IEEE Communications Magazine November 01 8

7 COMMUNICATIONS NETWORK ECONOMICS Network Market Design Part II: Spectrum Markets Randall A. Berry, Northwestern University ABSTRACT Market-based approaches have promising potential for allocating network resources. Part I of this article introduced the game theoretic underpinnings of market design and argued for the need to jointly consider market design with the underlying engineering issues in communication networks. Example research questions in this area were reviewed for wireline networks. In this part, we turn to network market design for wireless systems and in particular for the flexible sharing of wireless spectrum. We use this as a vehicle for discussing challenges that arise in the design of markets in the presence of externalities. INTRODUCTION Resource sharing is the basic issue underlying both the design of communication protocols as well as the design of economic markets. The first part of this article made the case that these two approaches to resource sharing are not separable and espoused the need for a theory of network market design, which combines both economic and technical considerations. Examples of such an approach were discussed in the context of designing markets for bandwidth in a wireline network. In this part, we turn to wireless networks and discuss several other issues that such a theory would need to address. In particular, we focus on market-based approaches for spectrum sharing. Currently, most licensed spectrum for wireless services is allocated on very coarse scales in both time and space. For example in the United States, many licenses are allocated for 10-year timeframes and typically cover a large portion of the country. It has been widely argued that this approach has to led to underutilization of spectrum; this in turn has led to an array of techniques being proposed to enable more efficient use of limited spectrum resources (e.g., [1] provides an overview of this area). These techniques include various market structures for trading and/or leasing spectrum on finer temporal and spatial scales (e.g., see []). Here, we use such markets as a way to discuss a number of issues related to network market design, which may also be relevant in other settings. We highlight a few key differences between wireless spectrum markets and the bandwidth markets considered in Part I. First, in a bandwidth market, to the first order it is fairly clear how to define the set of assets being allocated. For example, on a single link, the assets are units of bandwidth whose sum is no greater than the link s capacity. Of course, there are secondary considerations. For example, as discussed in Part I, there is still an issue that in practice bandwidth is not infinitely divisible, so there is a choice to be made on how this asset is divided up into indivisible bundles. If temporal dynamics are taken into account, there is also an issue of deciding on the timescale at which allocations occur. However, in the context of wireless spectrum, even such a first order approximation is not clear. For example, one approach to allocating a band of spectrum to a group of users is to require that all users transmit using spread spectrum over the entire band, treating interference from others as noise. With such an approach, the allocation of spectrum might correspond to determining the allowed transmission power of each user or each set of users. Another approach is to use frequency-division multiplexing (FDM) and allocate exclusive use of frequency bands to different users. These two approaches lead to very different definitions of the set of assets being traded. These are engineering choices, but the resulting choice also affects any market that emerges. Jointly considering such effects is a key dimension of network market design, which clearly requires both engineering and economic insights. A second basic property of wireless spectrum is that two users utilizing the same frequency band at nearby locations mutually interfere with each other. Hence, an agent s value for a given spectrum asset may depend in part on the allocation of other assets to other agents. Such effects are known in economics as externalities. When spectrum is allocated on coarse geographic scales, such interference externalities are a relatively minor problem, in part because the boundaries can be drawn through sparsely populated areas and in part because the boundaries comprise a much smaller portion of the total area. However, if spectrum was allocated on a finer spatial scale, these issues become more relevant. It is well known in economics that the presence of such externalities can greatly complicate market design. We begin in the next section, with a more complete discussion of such externalities /1/$ IEEE IEEE Communications Magazine November 01

8 We follow this with a discussion of several approaches for dealing with interference externalities in spectrum markets. Transmitters p 1 n 0 Receivers EXTERNALITIES AND TRAGEDIES In the model for bandwidth allocation considered in Part I, a user s utility is simply a function of the amount of bandwidth she is allocated, and in particular does not depend on the bandwidth allocated to other users. In the case of wireless spectrum, this may not be the case. Due to interference, the utility an agent derives from an allocation may decrease if a nearby agent increases his allocation. Economists refer to such effects as externalities. This interference effect is a negative externality, since increasing the allocation to one agent has a negative effect on the performance of all other agents. Networking problems may also exhibit positive externalities when the actions of one agent lead to higher utility of another. An example of this is a peer-to-peer file sharing system in which users bring resources that help other users as well as themselves. Here we focus on negative externalities. More formally, as in Part I, consider a general resource allocation problem of selecting an outcome x = (x 1,, x N ) for N agents from a set of feasible outcomes X, where x i represents agent i s share of the resource. Loosely, without externalities, each agent i s utility for a given allocation will depend only on x i, that is, it will be a function u i (x i ) and not depend on the values of x j, for j i. On the other hand, when externalities are present, agents will have utilities that depend on both x i and x j for j i, that is, these will be functions of the form u i (x 1,, x N ). This is not a precise definition. In particular, note that the choice of labels for each outcome is arbitrary, and by simply changing the labels, one can change the dependence of the utilities on these labels. As an extreme example, for any problem, we can simply label each outcome with the corresponding utility received by each agent, so we would have u i (x i ) = x i and thus would never see any dependence on x j in user i s utility. A more precise definition of externalities is somewhat subtle. To see this, note that even in the case of allocating the bandwidth of a single link, one agent s bandwidth allocation reduces the amount of bandwidth available and thus will have an effect on the utility that can be received by other agents. Is this an externality? One approach to answer this is to define an externality not simply in terms of the underlying resource, but in terms of a market for that resource. Given a market, an externality is defined as an effect that one agent causes others that is not accounted for in the market []. For example, there are no externalities in a market based on the Kelly mechanism discussed in Part I for allocating the bandwidth of a single link, since the dependency of the users via the common capacity constraint is represented via the per-unit price, m. Although this definition is more precise, the term externality is more often used in the former sense to simply reflect the dependence of one agent s utility function on the resources obtained by another, under some natural parameterization of the resources. p. p M Figure 1. Channel model for M transmitter/receiver pairs sharing spectrum via power allocation. To further illustrate the role of a market in determining an externality, consider an open market for sharing a single link with capacity C among N agents. In this market, each agent simply requests some amount of bandwidth and is allocated their request, without paying any charge, provided the sum of their requests is less than C; otherwise, they receive nothing. As in Part I, each agent i receives a utility u i (x i ) that depends on the amount of bandwidth x i she is allocated. Now additionally assume that the agents receive a disutility per unit flow that is proportional to the total traffic on the link, divided by C, which models some form of congestion on the link. Agent i s utility is now given by ( j x ) u ( x ) 1 i i C x i j, where the second term represents a negative externality not accounted for in this market. To see the effect of this, further assume that u i (x i ) = x i for each agent i. It can be shown that the game defined by this market structure has a unique Nash equilibrium in which each agent requests C xi = 1 + N units of bandwidth. For comparison, the efficient allocation in this setting is to give each agent C xi =. N The difference between these two allocations is because the market does not provide a way to account for the negative costs an agent s allocation has on other agents utilities, leading agents to over-consume. To see the effect of this, note that the total welfare obtained by the equilibrium allocation is NC We =, (1 + N) h 11 h 1 h 1 h n 0 n 0. IEEE Communications Magazine November 01 85

9 CDF Efficiency Figure. CDF of the efficiency of the second-price sequential auction from [8] for randomly placed nodes; in all cases the efficiency exceeded the worst case bound in [8], and there was no efficiency loss in 80 percent of the realizations. which goes to zero as N increases. With the efficient allocation, the total welfare is C/4, independent of the number of agents. One common measure of the degradation of welfare in such a market compared to the optimal allocation is given by the market s efficiency, which is the ratio of the equilibrium allocation to the optimal allocation. For this example, the efficiency is 4N, (1 + N) which approaches 0 percent as the size of the market grows, a situation sometimes referred to as a tragedy of the commons. PRICES One classical solution to the previous tragedy is to use a price to internalize the externality. In this model, the externality each user contributes to the overall welfare is given by 1 C x i x j. j i Note that when every other user has an efficient allocation, the marginal change in this term for each user i is given by 1 1 p =. N It follows that if we instead charge each user a price of 1 1 p = N per unit bandwidth, the resulting market would have a unique equilibrium in which the users receive the efficient allocation. This price is also known as a Pigovian tax after the economist Alfred Pigou, who proposed such a solution to externality problems [4]. Note that in this problem all agents were charged a common Pigovian tax of p. This sufficed due to the symmetry of the problem. Without this symmetry, the marginal change in the welfare of the other users due to an change in user i s allocation may depend on i, so N different Pigovian taxes may be required. Actually determining the optimal Pigovian tax required knowledge of the utilities of each user. Of course, the main motivation for using a market to allocate resources is that this information is not known a priori. To illustrate this difficulty in the context of spectrum, consider a model for sharing a band of spectrum among M agents in a given area. For simplicity assume that each agent corresponds to a single transmitter/receiver pair and that spectrum is shared by specifying the power, P i, that each agent uses to transmit over the common band. Interference from other agents is then treated as additional noise. Each agent i receives a utility u i (g i ) that is an increasing function of their signal-to-interference-plus-noise ratio (SINR) g i, which is given by hp ii i i = n + h P, 0 j i ji j where h ij denotes the channel gain from transmitter i to receiver j and n 0 is the noise power (Fig. 1). Here, each user generates a negative externality due to the interference. If users ignore this externality and simply choose their own power P i to maximize their own utility u i (g i ), they would all choose to use the maximum power possible, since their own utility is increasing in their own power. This can result in a total welfare that is much smaller than the optimal. What would Pigovian taxes look like here? They can be expressed in terms of interference prices as introduced in [5]. The interference price p j of user j is the marginal decrease in that user s utility due to an increase in the total interference at that user (i.e., an increase in S j i h ji P j ). Given the set of interference prices, the Pigovian tax charged to user i (per unit transmit power) is given by the sum of the interference prices from each user j i weighted by the cross channel gains h ij. In general, each user will have a different interference price, which depends in part on that user s utility and SINR, and these prices will be weighted differently for each user, resulting in a different tax for each user i. Determining these taxes appears to require knowledge of all channel gains as well as the utilities of each agent. However, in [5] it is observed that each agent can compute their own interference price given only local knowledge. Based on this observation, a distributed algorithm is developed in which the agents iteratively exchange interference prices and update powers. With certain restrictions on the class of utilities this is shown to converge to the socially optimal power allocation. Like the Network Utility Maximization (NUM) framework described in Part I, the interference pricing algorithm from [5] uses these prices mainly as a metaphor to develop a distributed algorithm and is not truly modeling economic incentives. In particular, the agents have an incentive to announce inflated interference 86 IEEE Communications Magazine November 01

10 1 Figure. Example of an interference graph G for 5 spectrum assets. One independent set is shown in orange. prices, since this would reduce their interference and not cost them anything. The theory of mechanism design discussed in Part 1 of this article provides a framework for dealing with such incentive issues. Indeed, the Vickery-Clarke-Groves (VCG) mechanism can be used in settings with externalities to again provide an incentive compatible and efficient allocation. Recall, for a setting without externalities, i.e., one in which each user s utility only depends on his allocation x i, this mechanism requires agents to essentially submit their utility function u i (x i ) and the mechanism is then required to solve an optimization problem corresponding to maximizing the total welfare to determine the allocation and N related problems to determine the payments. In a setting with externalities, the required bids are again the utility functions, only now these must be given as a function of the joint allocation to all agents, i.e., agents must specify u i (x 1,, x n ). For the power allocation problem described above, this corresponds to reporting not only one s utility u i (g i ) as a function of the SINR, but also the relevant channel gains involved in determining the SINR as a function of the powers allocated to each agent. In practice, this could be an excessive amount of information to report and may not even be known accurately (e.g., cross channel gains could be difficult to measure). Once again the mechanism is required to solve N + 1 optimization problems: one for the allocation and N for the payments. These tend to be more difficult to solve in the presence of externalities. For example in this power allocation problem, these optimization problems may be non-convex due to the interference. We also note that the VCG payment can be viewed as the total externality that an agent imposes on the welfare of all other agents. This can be contrasted with the Pigovian tax, which gives the marginal change in the externality. Given the aforementioned difficulties with the VCG approach, it is natural to seek a simpler procedure for pricing such interference externalities, such as the Kelly mechanism discussed in Part 1 for bandwidth allocation. What would be a Kelly-like mechanism for this power allocation problem? In the case of bandwidth, the Kelly mechanism is a market-clearing mechanism, i.e., it sets a per unit price so that supply (the links capacity) meets demand (represented by the bids). For this wireless model, the supply is less clear. One way to think about this is that each receiver has a supply of interference it is willing to tolerate. This suggests having nodes bid for the supply of interference at each node, and then again set prices as in the Kelly mechanism so that supply meets demand. Such an approach was studied in [6, 7] for pricing the received interference at only a single measurement point with a fixed supply of interference. In such a setting even if user s are price taking, they are still strategically coupled due to the interference. It is shown in [6, 7] that the resulting games for both price anticipating and price-taking users have Nash equilibria. However, in general these equilibria are not efficient, even for price taking users, and the efficiency may go to zero. In this case, not only is this market clearing mechanism not accounting for price anticipating users, but the resulting single price is not accurately reflecting the externalities among the agents. As we have discussed, truly capturing the externalities requires a different interference price at each receiver. Why not simply run a Kelly-like mechanism for the interference at each of these? There are several difficulties with such an approach. First, how much interference a node is willing to supply may depend on how much power it can use (due to the available interference at other nodes). Second, how much interference a node is willing to buy at one node would depend on how much it may buy at other nodes, since its total power is determined by the smallest interference allocation it receives. Finally, implementing such an approach would require the seller to have knowledge of the received interference at each receiver, which may not be feasible in practice. Instead of pricing the received interference at each receiver, an alternative market for this scenario is to view the supply as the potential transmission power available at each node i. Each unit of this power can be allocated to either node i, allowing her to increase her transmission power, or to another agent j, which prevents i from using this power and thus reduces the interference at node j. A market for such a scenario is considered in [8], in which discrete units of power at one user were allocated to two users sequentially using a second-price auction for each unit. It was shown that such a mechanism can have an efficiency as small as 1/n, where n is the number of discrete units of power. As n increases, this shows that the efficiency can go to zero. However, this is a worst-case result over the set of possible utilities for two users. Assuming each user has a utility that is proportional to its rate, and users are randomly placed in a given area, numerical results [8] indicate that in average cases, there may be little if any efficiency loss in many cases (e.g., Fig. ). Extending the analysis in [8] to more than two users appears to be difficult, in part because whenever one agent j i is allocated a unit of agent i s transmission power, it reduces the interference for all agents 4 5 IEEE Communications Magazine November 01 87