An Improved Method for Auctioning Wireless Spectrum

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 7 No. 2 2003, 175-206 AN IMPROVED METHOD FOR AUCTIONING WIRELESS SPECTRUM Bezalel Gavish Cox School of Business Southern Methodist University Dallas, TX 75205, USA e-mail: [email protected] Abstract: Radio spectrum for wireless communications is a finite resource whose allocation to different technologies and application areas is governed by regulatory bodies. Radio spectrum has enormous economic value, which attracts many potential wireless operators. One of the functions of the regulators is to assign the available radio spectrum to operators. Spectrum auctioning became a favorable tool for selecting operators and assigning bandwidth to bidders. Many auctioning methods have been proposed in order to achieve efficient and fair allocation. It is common to decide before the auction begins on the number of operators that will be permitted to operate in each geographic area. We develop a model of the net revenues of an operator as a function of the number of operators that are permitted to operate in a given area. It is demonstrated through numerical examples that a priori decisions can lead to significant reductions in the value of the allocated spectrum. An improved auctioning method is proposed which captures the impact of the number of winning bidders on the overall revenues collected by the auctioneer and provides the regulators with data on the value of different allocations. AMS Subject Classification: 90B12, 94A05, Key Words: radio spectrum, models of net revenues of an operator, numerical examples Received: 15 February, 2003 c 2003, Academic Publications Ltd. 176 B. Gavish 1. Introduction Auctions have been used throughout history as efficient mechanisms to trade goods and services1 . They are used for selling houses, plots of land, cars, books, paintings, coins, stamps, flowers, wines and any other items that do not have a ready made price in the local market. More recently, the wide spread use of the Internet, led to the development of hundreds of sites that promote on-line auctions of items that are up for bid. Some of the auction sites also support combinatorial auctions. Combinatorial auctions are auctions that handle the simultaneous trade of multiple goods; they have many forms and can arise in a variety of real-world situations. Some of the most famous uses of combinatorial auctions have been in the process of government deregulation of industries and the allocation of rights to goods, which were controlled by the government. Some of the well-known applications of restricted versions of combinatorial auctions have been in bandwidth allocation for cellular phone services (Jackson [11], McMillan [18], Milgrom[19]), rights for high definition TV broadcasting, allocation of take-off and landing slots in airports (Rassenti, Smith and Bulfin [22]), Grether, Isaac and Plott [10], and McCabe, Rassenti and Smith [17]). Bidding for railroad right of way segments (Brewer [3], Brewer and Plott [4]), bidding for shipping and Transportation contracts (Caplice, [5], Ledyard et al [15]), bidding for multi-attribute items/contracts in electronic procurement (Bichler et al [2]), and Scheduling of manufacturing operations in a factory (Wellman et al [25]). Radio spectrum is used in many application areas; it is a highly valuable resource and carries significant economic benefits. Examples of the many uses of radio spectrum include: Analog and digital radio (in multiple bands), TV broadcasting (regular, digital and HDTV), wireless LANs (WiFi, Bluetooth, 802.11), LMDS, radio amateur operators, satellite communications and satellite radio, aviation and marine uses, PC and direct TV, CB bands, military, personal security, fire, medical, police and other rescue operations, wildlife monitoring, cellular phone and data services, global positioning services, microwave tower communications, wireless industrial control systems. These are just a few of the many application areas of radio spectrum. The number and uses of wireless communications are expanding and growing over time. Useful Radio spectrum is a limited resource, whose effective capacity depends on developing new and efficient methods for squeezing bits into available bandwidth. This becomes harder and harder over time as scientists and engineers are approaching the theoretical and practical limits on spectrum use for 1 Early auctions of slaves are mentioned a number of times in the bible. AN IMPROVED METHOD FOR AUCTIONING... 177 digital communications. Due to the limited availability of useful spectrum and the possibility of interference between different applications and operators, the allocation and use of radio spectrum is regulated by regulatory bodies (international, national, regional and local). Being a scarce resource, radio spectrum has to be controlled and allocated to multiple sometimes-competing application areas and to specific functions. Spectrum was initially allocated through a straightforward licensing process. Potential operators applied for a license, had to satisfy some minimal technical and financial requirements. Licenses were allocated using a FIFO policy among the acceptable applicants. This was extended to other licensing methods, including a lottery system, which used a lottery to pick license awardees among the approved set of applicants, this encouraged the formation of consortiums of lottery members (increasing their chances to win). No restrictions have been put on exchanging spectrum or on reselling it by early license winners. Some of the early awardees bought spectrum from other winners at what by today standards were bargain prices, amassing enormous fortunes in the process (multi billion dollars). The enormous wealth collected by a few individuals through their radio spectrum acquisitions has attracted the attention of journalists, the general population, politicians, scholars and regulators, who were alarmed that public goods are given out below their market value, to a few individuals without the public benefiting from such allocations. This set in motion the process which led to the development of proposals on how the society can reap some of the financial gains stemming from spectrum allocations. In the mid 80s, the Federal Communications Commission (FCC) undertook the mission of allocating spectrum to digital cellular services. Many methods were considered for the allocation process, with the objective to collect some of the economic benefits associated with the allocation of radio spectrum. Kwerel and Felker [14] proposed that licenses for wireless services be allocated by the FCC using a spectrum auction process. This proposal led to the Congress authorizing the use of auctions for assigning radio spectrum for Personal Communications Services (PCS). The FCC was assigned by the Congress with the responsibility of developing spectrum auctioning rules and regulations. Much research has been done on auction schemes and bidding rules for spectrum allocation. Interested readers are referred to the work by Milgrom [19], Jackson [11], Kelly and Steinberg [12], Cramton [6], Fritts [7], Ausubel at. al [1], Rothkopf and Harstad [23], Mcafee and McMillan [16], McMillan [18]. The basic auction idea had to be streamlined into an implementable set of procedures for spectrum allocation. The real difficulty was discovered when Spectrum auction designers became aware that they are dealing with a difficult 178 B. Gavish combinatorial auction problem. There are many examples of auctions in which the values of the auctioned items to a bidder depend on him winning a collection of items. This synergistic effect may be specific to the bidder (depending on the amount of information, knowledge and financial and technical resources available to him). Auction designers have the incentive to design their auctions to encourage bidders to take advantage of the nonlinearties involved in bidding on combinations of items, in such a way that will be both fair to the bidders and practical to implement. Spectrum auction designers were exposed to the negative experiences and the complexities involved in earlier combinatorial auctions of public goods. For example during the auctioning of take-off and landing slots in airports, bidders faced the following impediment. When a bidder bids on a take-off slot from JFK airport in New York City, he has to make sure that he has won a landing slot for that plane in another airport (Chicago for example) within the time frame of the arrival time of the flight to Chicago. At the same time, the bidder has to make sure that an earlier flight with the same plane type has landed in JFK, so that a plane will be available for the flight from JFK to Chicago. The above bidding problem is an example of a combinatorial auction. The value of a bid depends on the complete set of items (landing and take-off slots) won in this and other bids by the same bidder. Unfortunately, such full knowledge is available only when the auction is terminated and the winning bids and bidders are publicized. Bidding for bandwidth for cellular services has similar characteristics that can be handled in principle through combinatorial auctions. Two main difficulties are encountered when applying combinatorial auctioning procedures to the auctioning of radio spectrum for cellular services. The first difficulty stems from the computational complexities associated with non trivial size combinatorial auctions which lead quickly to a computational explosion, unless severe restrictions are placed on the types of combinations that are acceptable as part of the bidding process. The second type of difficulty follows from the inability of human beings to understand and process the vast amount of data generated during an open-ended combinatorial auctions. These two difficulties led to the formation and use of restricted versions of combinatorial auctions2 . The innovation of the Federal Communications Commission (FCC) spectrum auctions was in having an open auction for many licenses at the same time. The auction method selected was that of a simultaneous multiple-round auction. This auction form was proposed by auction experts Milgrom [20], 2 More details on combinatorial auctions and their implementation in auctioning of radio spectrum to cellular phone services is given in Section 2. AN IMPROVED METHOD FOR AUCTIONING... 179 Wilson and McAfee. A simultaneous multiple-round auction is similar to a traditional ascending-bid “English” auction, except that, instead of selling one license at a time, collections of licenses are auctioned simultaneously during distinct rounds, the auction winners are set when the bidding is concluded on all licenses. A bidder can bid on any of the collections of licenses put up for bid. The PCS auction process proceeds in rounds. At the end of each round, the highest bids become the leading bids. Before starting the next round, the winning bidders and their bids are made available to all potential bidders. The auction terminates when bidding has concluded on all licenses; that i s, until a round goes by in which there are no new bids on any of the licenses. Or until an auction termination criterion has been met (total revenues collected, elapsed time, a preset date, number of iterations). At the end of the last round, the leading bidder on each collection of licenses is designated as its winner. Rothkopf, Pekec and Harstad [24] provide additional details on the development and implementation of the PCS auction rules. Although, not optimal when compared to an open-ended combinatorial auction (i.e. all combinations are open for bidding), the simultaneous multiple-round auction represented a vast improvement over single license auctioning methods. Some of the important features of ascending-bid procedure include: its design allows the bidders to react to information revealed in prior rounds. This reduces the likelihood of the winners curse3 , enabling the bidders to bid more aggressively (Milgrom and Weber [21]). By auctioning on a large set of related licenses simultaneously, bidders are able to value collections of licenses. Since bidder valuations depend on the collection of licenses they win, the ability to value and bid for collections of related licenses permits a bidder to form efficient aggregations of licenses. This efficiency is much more difficult to achieve in sequential single item auctions, where one license at a time is put up for bid. A bidder who is interested in a certain collection of licenses, does not know what the prices will be in later auctions, when considering whether to change his bid at the present auction. Some licenses are complements, i.e. one set of licenses adds value to other licenses, whereas others are substitutes. The simultaneous sale of related licenses in an ascending bid auction, gives the bidders some of the flexibility they need to express these value interdependencies. By keeping the bidding on all licenses open until there are no new bids, bidders are provided with the flexibility of switching among license aggregations as prices and winners change. Another benefit of the FCC simultaneous ascending auction is its full transparency. Bidders and other interested parties can verify that 3 A situation in which a bidder wins and discovers after the auction that he would have been better off if he did not win. 180 B. Gavish the auction and biding rules are followed. If problems are discovered, they are highlighted and resolved before significant damage is done. Moreover, since secrecy is not an issue, costly procedures and protocols to preserve secret data are unnecessary. The paper is structured as follows; we present in the next section the difficulties involved in spectrum allocation and explain why combinatorial auctions have been developed as a possible solution to this difficulties. In Section 3, we outline and explain the steps used by the regulatory bodies to allocate bandwidth to wireless operators. In Section 4, we develop an economic model for deriving the net revenues of an operator when he is the only one permitted to operate in a service area, followed by a model for the case in which multiple operators are permitted to operate in a service area. Section 6 uses numerical examples to demonstrate the impact of the number of operators permitted to operate in a service area on the net revenues of an operator and on the number of cellular subscribers. Section 7 proposes a new parallel auctioning scheme, which eliminates many of the difficulties that are associated with existing bidding methods. The paper concludes with suggestions for possible extensions of the basic models developed in this paper. 2. Cellular Auctions as Combinatorial Auctions Spectrum auctioning for cellular phone services is one of the first important uses of combinatorial auctions. The combinatorial nature of the auction stems from the fact that bidders attach a much higher value to contiguous geographical areas, than to winning disconnected areas that have the same number of potential subscribers. A cellular user is more likely to value his ability to receive service over a larger contiguous geographic area, closer to the area in which he resides. Figure 1 contains the details of an example consisting of twenty service rectangles. Assuming that each rectangle has the same potential number of subscribers, we consider two possible configurations and compare their attractiveness to cellular bidders: • Configuration 1 consists of five areas {A, B, C, F, G} and all other areas won by other bidders. • Configuration 2 consists of five areas {A, D, O, R, K} and all other areas won by other bidders. We also assume that the licensing regulatory bodies do not permit secondary license trading markets (winning bidders trading areas between them). Under AN IMPROVED METHOD FOR AUCTIONING... 181 Figure 1: A service area consisting of 20 areas open for bidding the above assumptions, the potential number of subscribers in both winning bids is identical. However, it is obvious that configuration 1 is much more valuable to a cellular operator than a winning bid consisting of the areas in configuration 2. Figure 2a: Configuration 1 of service areas There are two main reasons for the desirability of configuration 1 over configuration 2. The first one has to do with the attractiveness of configuration 1 to customers who reside in areas A, B, C, F, G, they are assured that they will have uninterrupted cellular service in areas close to their home base, a feature that is desirable to potential subscribers. The second reason has to do with the cost to the service provider of providing cellular services. The system setup and marginal costs of providing cellular services in configuration 1 are lower than the ones in configuration 2. A higher portion of the costs is sharable in configuration 1 when compared to configuration 2. For example advertisement and sales costs can be shared, the possibility of sharing base stations and channels between adjacent areas, shorter base station interconnection costs, and more efficient service and technical support operations. In a sequential bidding process a potential bidder faces an exposure problem, he has to decide on how he should bid for sub-areas and not end up in a 182 B. Gavish Figure 2b: Configuration 2 of service areas situation in which he overpays for the sub-areas he wins. Consider the following valuations by the bidder for different configurations. He values winning one subarea at 10, winning two adjacent sub-areas at 27, winning three adjacent subareas at 45, winning four adjacent sub-areas at 65, and winning five adjacent sub-areas at 100. Using the above valuations, he values configuration 1 at 100, configuration 2 at 50. If he bids more than 10 for each of the single sub areas, he might win five sub-areas that are isolated from each other. Since he values such a win at 50, he overpaid for his winning bids. On the other hand if he offers only 10 for each sub-area, another bidder who makes an offer of 51 for five adjacent sub-areas will win them paying only 51, while our bidder who values them at 100, did not bid an amount in th e range of 51 to 100. Both cases are bad, both for the auctioneer who would have collected a higher sum, and for the bidder who will not extract his full expected value from the bidding process. Combinatorial auctions have been developed in an attempt to resolve the bidder exposure problem. In combinatorial auctions, a bidder can specify a combination of sub-areas he wishes to bid for and specifies how much he is willing to pay for that combination. A bidder can submit many bids at the same iteration of the auction each with a different combination and price. Combinatorial auctions bring up a different question, how does the auctioneer determine which combinations of bids are the winning bids? And how are the winners determined? The FCC and wireless communications regulatory bodies in other countries have devised bidding schemes, in which the spectrum allocated to an application domain, is put up for bid by existing and potential service providers who can bid for parts of that spectrum. The regulatory bodies are trying to balance a few conflicting objectives. Under directive from the legislators, they would like to collect as much as possible for the right to use the allocated bandwidth, i.e. maximize the total amount paid by the bidders for the spectrum allocated AN IMPROVED METHOD FOR AUCTIONING... 183 to them. Another objective is to ensure that several cellular phone operators will operate in each service area, so that potential users will be able to select among several competing service providers, hoping that this will encourage multidimensional competition (in price, quality of services, coverage, length of contracts, etc.) and will result in better service and lower prices to consumers. As shown in Gavish and Sridhar [8, 9], the two objectives can conflict. By increasing the number of cellular operators in a service area, the level of competition between service providers is increased. However, at the same time it reduces the effective bandwidth available to each operator. When a cellular operator is allocated less bandwidth to operate the cellular system, he is forced to use (out of his own optimal system configuration) inferior technologies and inferior system configurations (number of cells, location of base stations, transmission power levels) when compared to the ones in which more spectrum is available to him (less operators in each service area). This may result in reduced overall economic benefits from operating a cellular service, and increasing the costs to customers and the economy. This observation is analyzed in Section 4 and Section 5 using appropriate economic models. 3. Current Spectrum Allocation Steps The regulatory bodies have to play a careful-balancing act between allowing for the profitable operations of cellular service providers, impact on the economy of the new technologies and services, and to take into consideration the social and political implications of their allocation decisions. To accomplish these, sometimes-conflicting objectives, the regulatory body decision-making process consists of the following main steps: Step 1. Allocating bandwidth and frequency spectrum to application domains. There are many existing and potential applications of wireless bandwidth. Some of the potential application areas include uses for: broadcast radio (analog and digital), amateur radio operators, wireless local area networks, broadcasting analog TV, broadcasting digital TV (high definition), aviation based communication, radar detectors, police, fire and emergency medical services, cellular phone services, local broadband wireless services, security and control, military communications. These are just a few of the many application areas of wireless communications. The regulatory bodies have to decide how much bandwidth out of the finite useful bandwidth should be allocated to each potential use. In addition, they also set limits on transmission power on fixed and mobile units, and specify the frequency ranges that are allocated to each 184 B. Gavish service class. This is done in order to reduce the likelihood of interference between different application areas and technologies, and to increase the likelihood of standardized operations. The regulators have also to take into consideration when they decide on bandwidth allocations, past spectrum allocations and existing commitments, current technologies and emerging application areas and technologies. Simultaneously, they have to make sure that spare spectrum is available to support future possible technologies and application areas which will emerge or be discovered at a later point in time. Step 2. Dividing the spectrum into sub-bands: During this step, the regulatory bodies decide how to divide the overall geographic area into biddable service areas (the size of a biddable area depends on the application or service), they also decide for an application or service, how many cellular operators will be allowed to provide services within a geographic service area. In addition, they also decide on the allocation of the assigned spectrum between the different operators. If the objective is of economic parity, it implies equal division of the applicable spectrum between the different service providers, however in some cases, due to social and political considerations the regulatory bodies decided on unequal spectrum allocation. In recent years more and more of the bandwidth allocation process was done through an auctioning process which initially used ascending sealed bid mechanism with the attempt to maximize the amount collected by the controlling agency. The regulatory bodies decide before hand how to divide to available spectrum to chunks of bandwidth that are put up for bid. Recent auctions have used in some form or another multi-round simultaneous ascending auction methods. Step 3. The auctioning process: In the past, bandwidth was allocated using a licensing process, in which spectrum was given away for a licensing fee to petitioners who fulfilled some technical and financial requirements, in some cases a simple FIFO policy was used to assign the available spectrum to petitioners. Licensing made sense when the general believe was that spectrum was a plentiful resource. This view changed a decade ago, when policy makers and the public observed how many of the early licensees made vast returns on their investments. They also began to realize that the useful portion of the spectrum is a finite resource, with an enormous economic value. This led them to look on the spectrum, as a finite public good. Pressure mounted on the regulatory bodies, to capture part of the economic benefits associated with the allocation process. Auctioning of wireless spectrum became the main vehicle for collecting some of its significant monetary benefits. The auction process is typically an iterative process, with multiple rounds. At the beginning of a round, the bid- AN IMPROVED METHOD FOR AUCTIONING... 185 ders submit to the auctioneer the bundles of service areas and spectrums they wish to bid on and how much they are willing to pay if that bundle is assigned to them. In many cases, they are allowed to submit multiple bundles and bids per round of the auction. At the end of each round, the auctioneer uses the submitted bids to compute the optimal spectrum allocation and determines the winning bids. The winning bids and their prices are publicized, and the bidders can modify their bids. This auctioning process continues until there is no change in bids between two successive rounds, or some upper limit on elapsed time or number of rounds has been reached. After the auction is complete and the final bids have been submitted, the auctioneer selects the winners, collects the amounts that they agreed to pay, and assigns the geographic areas and spectrum to the winners. Each one of the above steps has enormous economic implications. This has been recognized by many investigators who looked into methods for extracting the economic value of bandwidth allocation. Allocating too much bandwidth to an application domain reduces the bandwidth that is available to all other existing and to future application areas. Allocating too little to an application area might create a situation in which a service or viable technology does not make it in the marketplace due to limited bandwidth. How much bandwidth to allocate to each service provider is a complex balancing act between conflicting economic, political and social objectives. In the next section, we develop an economic model of the expected net revenues to the cellular operator as a function of the number of operators that are permitted to operate in a service area. Given the finiteness of available bandwidth, the number of assigned operators determines the bandwidth allocated to each operator. 4. The Economic Model This section develops an economic model of a cellular operator valuation of spectrum, which affects how much he is willing to bid for the bandwidth auctioned in a service area. The decision on how much to bid depends on his ability to attract subscribers who will use the cellular services that he offers, the revenues they generate, and the costs of providing them with cellular services. The value of the wireless bandwidth in a given service area, depends on how many subscribers will be attracted to the cellular service, the overall revenues that will be generated by the users of the service, the cost of providing the service to that number of subscribers and the net revenues that are generated. We develop first an economic model for a single operator (a monopolist) and use 186 B. Gavish it in the second part of this section to develop a model to calculate the expected revenues that a single operator can expect when K operators have been assigned to provide cellular services in the same geographic service area. The decision of a potential subscriber to actually subscribe and use the cellular service depends on many factors. The main input to the decision making process of a potential subscriber in deciding to join a specific service provider. Seems to be: the price of the service as charged by the service provider, the structure and length of the service contract which determines the magnitude and length of the subscribers financial commitment, the quality of cellular services (quality of voice, support services, blocking and drop-off probabilities, and customer service) offered by the service provider. In many cases the regulatory bodies set the minimal levels of service quality that have to be satisfied by a service provider. In order to simplify the analysis, we assume that the market is competitive and all competing cellular service providers offer the same level of quality of service and support services. Under the above assumptions, the demand D for cellular services depends on the price p charged for the cellular service as set and charged by the service provider. In what follows, we assume that p is the present value of all the revenues generated by a subscriber who decides to join the service. The demand D(p) for cellular services as a function of p, is given by the following relation: D(p) = f (p) . (1) The analysis concentrates on an average customer and assumes that the price charged for cellular services is identical for all customers4 . For each service area, an upper bound DM ax exists on the overall demand for cellular services that can be generated by all potential customers in this service area. The maximal demand level DM ax is equal to the long-term level of demand generated when the price charged for the cellular service is equal to zero, i.e. DM ax is equal to f (0). The analysis assumes that the demand is nonincreasing with the increase in price, and that the demand function is defined over the range of zero demand to f (0) = DM ax , the demand function does not have to be differentiable in its full range of existence5 . Given the relationship in (1), the present value of the expected revenues R(p) collected by the service provider when he sets the price to p, is calculated 4 Cellular service providers use differential pricing to attract different segments of the consumers population. We believe that the results of the analysis apply also to differential pricing and will deal with this extension in a future paper. 5 As we show later, the demand function is unlikely to be differentiable everywhere. AN IMPROVED METHOD FOR AUCTIONING... 187 as a function of price as: R(p) = D(p) · p = p · f (p) . (2) Assuming that for every fixed price level a unique demand level exists, price can be expressed as a function of demand. Let f − (D) be the reverse function of demand, i.e. p(D) = f − (D) 0 ≤ D ≤ f (0) = DM ax . (3) The revenues can therefore be expressed as a function of demand, R(D) = D · f − (D) 0 ≤ D ≤ f (0) = DM ax . (4) When f − (D) can be derived analytically it is possible to obtain closed form expressions for the revenues as a function of D. Below are a few examples of such closed form solutions: 1. Exponential demand function, the demand is expressed as a function of price as: D = f (p) = Ae−Bp . Taking natural logarithms, of both sides leads to ln D = −Bp + ln A and ln(A/D) ln A − ln D = p = B B R(p) = p · A · e−Bp thus,   D A R(D) = D · p(D) = · ln . B D 2. Exponential power function, the demand function is given as: µ D = f (p) = Ae−Bp . Taking natural logarithms, leads to ln D = −Bpµ + ln A and ln(A/D) ln A − ln D = pµ = B B  1/µ   ln A − ln D ln(A/D) 1/µ p = = and B B
!1/µ A ln D 0 ≤ D ≤ A. R(D) = D · B 188 B. Gavish 3. Power function, the demand is given as: D = f (p) = A · p−B p > 1, B > 1 leading to p = and the revenues as a function of demand are given by  1/B A . R(D) = D · D  A D 1/B The present value of the net revenues collected by the service provider are the difference between the overall revenues collected by the cellular service provider and his costs of providing that service. The cost of providing cellular services is composed of many cost components, they include: the cost of acquiring the bandwidth, taxes paid to different governmental bodies (local, regional, country), sales, marketing and advertisement costs, real estate, labor, equipment, cellular phones, roaming and charges by other telecommunication providers, financing, accounting and billing costs. Among other effects, costs are also dependent on the specific technology deployed to provide the cellular services. A service provider can select between different technology choices, and for each one of them, he can decide on different system configurations (number of base stations, their locations, transmission power, base station interconnection network, antenna types, their direction and configuration, voice coding techniques to name a few of his parameter choices). Let the index set of the technology choices that a cellular service provider has available to him, to select from be T . t, t ∈ T is a particular technology choice. For a given technology type t, Ct (D) is the lowest cost of technology t when it is configured to support demand level D. Different demand levels result in different system configurations for the same technology, thus Ct (D) is dependent on the demand level. Without loss in generality we can assume that Ct (D) is not decreasing with the demand level D, Ct (D) can be expressed as: Ct (D) = gt (D) (5) To simplify the analysis we assume that if there is a demand range for which a given technology type cannot support that level of demand within the acceptable tolerances of quality of service; its associated cost function is set to a very high cost M , M >> CM where CM = Max t∈T 0≤D≤DM ax {gt (D); 0} . The cellular service operator has the goal of deploying a technology configuration that can support the level of demand generated by the subscribers to AN IMPROVED METHOD FOR AUCTIONING... 189 the cellular service, and wishes to do it at minimal cost. This is expressed by: C(D) = Min {gt (D)} 0 ≤ D ≤ DM ax t∈T (6) C(D) is the cost function of the cost-effective technologies as a function of demand level. When there exists a range of demand for which C(D) is equal to M , it implies that no feasible technology exists for that demand range. The net revenues N R(D) for a demand level equal to D are given as the difference between the present value of the overall revenues and the present value of the costs associated with that demand level. N R(D) = R(D) − C(D) = D · f − (D) − Min {gt (D)} , t∈T 0 ≤ D ≤ DM ax . (7) A cellular operator, who wishes to maximize the present value of his net revenues, wants to operate at a demand level a technology and a customers pricing level that maximizes his net revenues. He will attempt to operate at b with an associated price f − (D) b that satisfies the demand level D b = N R(D) Max 0≤D≤DM ax {N R(D)} . (8) b and N R(D) is a nondifferentiable function, thus the only way to compute D its associated net revenues is by using numerical procedures. This is done in Section 6, in which we develop and use a numerical example to illustrate the formulas and their implications. 5. Extending the Model to Multiple Cellular Operators As mentioned in Section 4, the demand function that a particular operator faces depends on how many operators operate in his service area. The number of operators determines the amount of bandwidth out of the overall bandwidth that will be allocated to an individual operator. It is unlikely that changing the number of operators (within a reasonable range) in a service area will have a significant influence on the maximal potential of the overall number of subscribers to cellular services. Thus, the overall demand function given in (1) applies to this case. However, changing the number of cellular operators will affect how many subscribers will be attracted to a particular operator. The overall actual demand for cellular services will be divided between the different 190 B. Gavish cellular operators. Given that cellular operators operate in a service area, we assume that in order to be fair to each of the winning operators, the spectrum will be divided equally between them. Assuming a perfect competitive market. Under the Nash equilibrium conditions, the cellular operators will charge similar prices for providing the cellular service, and if identical sets of technology choices are available to them6 , they will choose to use an identical technology and system configuration. Given that all cellular operators will use the same price, p the overall demand D(p) associated with p is independent of the number of operators and is given in (1). Given the number of operators K, the mean number of customers attracted to a single operator i is di , while the overall load D(p) on all K operators satisfies: K X di , (9) D(p) = i=1 i.e. the overall demand D(p) is divided between the K operators. As noted above, all operators use the same price p. Thus, the overall demand will be divided equally between them, or, di = D(p) K = d(p) i = 1, 2, .., K. For a given overall demand level, D we can associate a price p(D) that is equal to the maximal price that will generate an overall demand equal to D. Or, p(D) = Max {p|D(p) = D} . 0≤D≤DM ax Given the above definition it is possible to define the sum of revenues R(D) collected by all operators as a function of the demand level D, R(D) = D · p(D) . (10) Given that the system is in equilibrium, without loss in generality we can assume that d1 = d2 = .. = dK = d. Assuming that the demand imposed on each operator is equal to d, and using the definition in (10), the revenues associated with a single operator (out of the K operators) are given as: r(d) = d · p(K − d) . 6 In reality, due to existing ongoing relationships/commitments between cellular operators and equipment manufacturers, and prior system configuration decisions and handset choices made by the operators, the set of technology choices might be different for different operators. In order to simplify the analysis, we assume in this paper identical sets of technology choices. AN IMPROVED METHOD FOR AUCTIONING... 191 The cost function that each operator faces as a function of technology choice t and demand level (capacity) d is given by Ct (d) and is assumed equal for all operators. An operator will choose a technology which for a given demand level minimizes his overall costs, or Ci (d) = Mint∈T {Ct (d)}. Operator i who expects a demand level di will choose to maximize his net revenues and will operate at a capacity level de that maximizes his net revenues, which are given in the following expression: e − Ci (d) e = N Ri = r(d) Max 0≤d≤DM ax {r(di ) − C(di )} . Once de is determined the system overall net revenues as a function of K are given as:   e − Ci (d) e . N R(K) = K − N Ri = K · r(d) 6. Numerical Examples In this section, we present a numerical example that illustrates the concepts and formulas developed in Section 5. To simplify the exposition the numerical example is restricted to one geographic service area that is up for bid, and that the demand D for cellular services is a function of the price charged for such services. The numerical example uses a demand function D that is dependent on the price p charged by the cellular operator for his services. The consumer demand function is given by D(p) = 1000 · e−0.2p . The graph of demand as a function of price is depicted in Figure 3. The example assumes that the service area has a maximal cellular subscribers potential of 1000, this is the demand level corresponding to a price level of zero. The revenues that a service provider is expected to collect as a function of price are given by the multiplication of price and demand, i.e. R = p · D. Given that the demand for cellular services is a function of the price charged for such services, the revenues are calculated as a function of price, R(p) = p · D(p). As we will see later, the revenues are compared to the costs of providing cellular services; these costs are a function of demand. Thus, to be able to compare the two costs using the same graphs, the revenues have also to be depicted as a function of the demand D. This relationship is depicted in Figure 4. The revenues begin at a value of zero, when the price charged for cellular services is so high that it discourages customers from subscribing to the cellular service, the demand level associated with that high price level is equal to zero. The 192 B. Gavish Figure 3: Demand for cellular services as a function of price revenues peak at some level and go down again to zero as the price charged for cellular services decreases to zero, its associated demand level is high, but the multiplication of price and demand is close to zero (due to a negligible price charged to the large number of subscribing customers). So far we have neglected the costs to the service provider for providing the cellular service, these costs are significant, they include paying for the bandwidth allocated to them, licensing costs, equipment costs, site costs, labor costs, payments to other providers, taxes, marketing, billing, accounting, and system maintenance. The cellular system that they put together have also to satisfy quality of service requirements imposed by the regulatory bodies on cellular operators. In this example, we assume that the operator selects between two technologies, each with its own cost function. The first technology has a cost as a function of demand given by C1 (D) = 300 + 3 · D, a relatively low setup cost but a high marginal cost. The second technology has a higher setup cost, but has a markedly lower marginal cost that is given by C2 (D) = 987.5 + 0.25 · D. The two cost functions intersect at a demand level of 250. We assume that each one of the technologies was selected and configured so that it satisfies the service requirements within the range of feasible demand levels. For a given demand level, the overall cost function is given by the minimal cost of the two technologies, i.e. C(D) = Min {C1 (D); C2 (D)}. The overall cost function is depicted in Figure 5. The expected net revenues for different demand levels are calculated as AN IMPROVED METHOD FOR AUCTIONING... Figure 4: Revenues as a function of demand Figure 5: Revenues and costs as a function of demand 193 194 B. Gavish the difference between revenues and costs N R(D) = R(D) − C(D). The net revenues for different demand levels are depicted in Figure 6. In this example when the demand levels are low, the expected revenues are low and the expected net revenues are negative, the cellular operator is going to loose money due to the low demand levels. The sum of his setup and marginal costs are so high that the demand level has to exceed a given bound, in order for the operator to be able to cover his costs. At the other extreme, when the demand levels are very high, such high levels of demand are induced by very low prices charged by the service provider, his overall revenues will be low and his costs will be high, resulting in significant loses. Figure 6: Net revenues as a function of demand, and the optimal operating point for a single service provider in the service area When only one operator is allowed to operate in the service area, (he is a monopolist) he will operate at the level of demand (and price), which maximizes his net revenues. In this example the highest net revenues are derived when the demand equals to 346.5. The net revenues associated with his optimal operating point are equal to 762.1. It is interesting to note that when a single service provider (a monopolist) is assigned to the service area and is given the full bandwidth, he will offer service to every customer, which is willing to subscribe, but will set the price to the customer at a price level, which will induce only 34.6 percent of the potential number of customers to actually subscribe and use the service. When the overall bandwidth is allocated to two operators, the example AN IMPROVED METHOD FOR AUCTIONING... 195 assumes that the bandwidth is divided equally between the two operators (this is done in order not to provide undue advantage to one operator over the other one) that are permitted to operate in the service area. Each operator receives half of the overall bandwidth that the FCC is willing to allocate to cellular services. Assuming a perfect competitive market, each operator will attract at most half of the overall demand level. Both operators will operate at their optimal operating system configurations and will charge similar prices for the service7 . Therefore, we can expect that for a given price level the overall demand generated by consumers will be divided equally between the two providers. This will result in a lower load on a service provider for the same price level. Going through the same steps as for the single service provider example, and using the same technological options used in that case, the best operating point for each service provider assuming that they use the best available technologies is at a demand level of 100.94. It is a different operating point than the one used for the single service provider case that was 346.5 (see Figure 7). It is interesting to note that in this example, each service provider has a lower net revenue when compared to the monopolist case, and that the sum of net revenues 409.48 of both service providers is less than the net revenue obtained by the single (monopolist service provider). The difference in net revenues between the two cases, is the loss suffered by the economy. In this example the economy is loosing an amount of 762.1 − 409.48 = 352.62 when the regulatory bodies decide to allocate the bandwidth and permit two operators to provide cellular services in the service area (keeping the overall bandwidth allocated to cellular services equal to the one provided to the single monopolist). Is such an economic loss worth the economic value generated by competition is up to the political system to decide. When the overall bandwidth is allocated to three service providers, each one of them receives one third of the overall bandwidth. The demand per service provider for the same price charged to customers will be one third of what it was in the single service provider case. The cost functions to the service provider will remain the same (see Figure 8), leading to much lower net revenues. For the example, their optimal operating point is 42.3 with a net revenue per operator of 36.49. The additional economic loss when compared to the two service providers case is 409.48 − 109.48 = 300 an additional economic loss of 300. The cellular service will attract only 20.189 percent of the overall potential customers to the three operators. When the overall bandwidth is allocated to four operators, each cellular 7 Empirical evidence exists showing that cellular operators use similar pricing in order to remain competitive within their areas of operation. 196 B. Gavish Figure 7: Net revenues as a function of demand and the optimal operating point for a two service providers case Figure 8: Net revenues and optimal operating point for three service providers AN IMPROVED METHOD FOR AUCTIONING... 197 operator will operate at a loss −47.63. Thus, it is unlikely that they will bid for such an option (unless they make a mistake). If by mistake they do bid and four operators win their bids, it is likely that at some point in time, one or more of them will declare bankruptcy in which case the remaining operators will move into the black. Figure 9: Net revenues and optimal operating point for four service providers Table 1 tabulates for the numerical example the results of different performance measures when the bandwidth is allocated to one up to four cellular operators. If the bandwidth is allocated to more than three operators all of them will loose money and can not offer a viable cellular service. The overall net revenues decrease with the number of cellular operators in the service area, when a single operator operates in the service area his net revenues are equal to 762.10, this decreases to 36.49 per operator when three operators are permitted to operate in the service area, or 109.48 total revenues for the three operators, a decrease of more than 85.6 percent in overall net revenues. This reduces how much a potential bidder will be willing to bid for the right to provide cellular services in the service area. An increase from one to two operators creates a reduction in overall net revenues of 352.62, each additional operator decreases the overall net revenues by an additional 300 units. Table 1 also displays the total number of subscribers being served by the cellular operators as a function of the number of operators permitted in the service area. The number of subscribers to the cellular service when one operator operates the service is 346.46 (out of a maximum of 1000) or 34.646 percent, it 198 B. Gavish goes down to 201.89 (20.189 percent) for two or three cellular operators. This shows a decline of the number of subscribers, this decline is a negative consequence of increasing the number of operators in a service area, and contrasts the regulatory body stated objective which is to make the service more affordable and available to a high percentage of the population. The actual numbers of subscribers will depend on the exact structure and parameters of the demand function and the operators cost functions. At present, it is up to the regulatory body to decide on the number of operators that will be permitted to operate in the service area. AN IMPROVED METHOD FOR AUCTIONING... 199 Table 1: Performance measures as a function of the number of operators 200 B. Gavish Table 2: Impact of demand sensitivity to price changes and to number of operators on the number of subscribers and net revenues A second set of experiments examined the impact of the demand function AN IMPROVED METHOD FOR AUCTIONING... 201 on the net revenues and the number of subscribers as a function of the number of operators in the service area. The demand as a function of price was set equal to 1000 · e−αD , α was set equal to 0.1, 0.2, and 0.25. A higher value of α implies a higher sensitivity of demand to price changes. Table 2 displays the number of subscribers and net revenues, for an individual operator and for the overall system, obtained for the different values of α and number of operators K. In the above analysis we made an implicit assumption that the functional relationships between prices and demand are known, and that the operator cost functions for different technologies and service levels are readily available. In reality, estimating the demand as a function of prices for upcoming technologies and services, some of which the cellular operator and the consumers are not familiar with is a nontrivial task. The technological cost functions are even more difficult to estimate. They depend on existing and future technologies, some of which do not exist yet, or, are in the design and prototyping phase, or might exist in research laboratories (similar uncertainties existed in the bidding for second and third generation cellular systems bandwidth). Technological costs also depend on many details, which affects the system costs, examples for such cost factors include: the actual distribution of customers and their daily and weekly mobility patterns, the geographic topology of the service area which affects reception of transmitted signals which in turns influaces how many cells will be allocated to the service area and the exact locations of base stations, local zoning restrictions, 7. An Improved Auction Process In the previous section, we have developed and demonstrated the difficulties associated with the existing auctioning process for allocating spectrum to cellular services. In this section, we propose a change in the auctioning process that will eliminate some of the difficulties. The regulatory bodies decide how to divide the available spectrum into sub-bands that are auctioned to the highest bidders. Our analysis shows that the decision as to how many sub-bands to divide the available spectrum has a significant impact on the economic value of the auctioned spectrum. To illustrate this claim, consider an auction of a bandwidth of ten megahertz, in a desirable frequency range. Taking the bandwidth of ten megahertz and dividing it into auctioned sub-bands of one Hertz each (ten million of them), while enforcing a policy, that does not permit secondary markets for spectrum 202 B. Gavish purchases or exchanges, than the overall value of the ten megahertz bandwidth is almost zero. In order for the regulatory bodies to judiciously decide to how many sub-bands to divide the available bandwidth, they must have access to data that specifies how consumer demand is affected by price and to the efficient cost functions of optimal technological choices and configurations for the service area. It is highly unlikely that the regulators will have at their disposal reliable data on demand as a function of prices, and to the operators cost data. Under the current spectrum auction rules, the bidders do not have the incentive to provide the regulators with information that is uncertain to them. The bidders cost structure is also part of their competitive advantage and they will make every effort to guard this information. In many cases of highly advanced technologies, service providers face their own uncertainties regarding the cost structure associated with future technologies. In what follows, we propose a change in the auction process that shifts the burden of estimating the demand curve and the costs associated with providing the cellular services, from the regulatory bodies to the potential service operators in a geographic service area. The proposed change also provides the regulatory bodies with accurate data regarding the value of dividing the available spectrum into sub-bands. Thus, when a decision is made to permit K operators in a geographic service area, i.e. divide the available spectrum into K sub-bands. The regulatory bodies know how much they give up in monetary proceeds for making that decision. Once a decision has been made as to how much spectrum will be assigned to cellular services in a geographic area. The only remaining decision the regulatory body has to make before the auction begins is an upper bound N on the maximal number of service providers that it will permit to operate in the service area. The actual number of service providers selected for the service area, through the auction process can be less than the upper bound N . We assume that the available spectrum will be divided equally between the selected service providers. The auction consists of N parallel auctions. Auction n, n = 1, 2, ..N assumes that there will be n winners in the auction process and that the spectrum will be equally divided between the n winners. A bidder can participate at the same time in multiple auctions and he chooses to which ones he wishes to submit bids. When he selects to participate in auction n he is aware that this is an auction, which assumes that the overall bandwidth will be divided between n different winners. Let Vnb be the amount bid by bidder b, b ∈ Bn in auction n. Venb is the ordered AN IMPROVED METHOD FOR AUCTIONING... 203 list (within auction n) of bids ordered from the highest bid to the lowest bid, i.e. Ven1 ≥ Ven2 ≥ .. ≥ VenBn . To simplify the exposition it is assumed that all the bids in Bn were submitted by different bidders. Under this assumption, it is clear that for the n-th auction, the winning bids are the n highest bids, and the value collected by auction n if awarded is: Vn = n X b=1 Venb . (11) Once the Vn values have been computed for all values of n = 1, 2, ..N . It is possible to compute the loss (or gain) for assigning h service providers in a service area instead of assigning h − 1 cellular operators; the gain or loss is given as the following difference: ∆Vh = Vh − Vh−1 h = 2, 3, ..., N . (12) Based on the above differences, the regulatory body can assess the monetary implications of a decision to allocate the available spectrum to different numbers of operators in the auctioned service area. In most cases increasing the number of cellular operators, decreases the overall expected net revenues collected from the service area. This in-turn reduces the overall amounts that potential cellular operators will be willing to bid for the right to operate in the service area. 8. Conclusions and Further Research This paper highlighted the difficulties caused by the FCC predetermining the number of winners in a spectrum auction process. It is argued and demonstrated through numerical examples that by slight changes to the auction process, i.e. allowing bidders to participate in simultaneous auctions, each with a different number of winners. The decision makers in the government can collect data on the loss generated by allowing more operators to operate in a given area. Based on the additional data, decisions that are more judicious can be made on how to allocate the available spectrum. It is interesting to note that the proposed bidding method reduces and perhaps eliminates the difficulty faced by regulators when they put K subbands up for bid, and by coincidence N bidders participate in the auction, where N ≤ K. Once the N bidders discover (through the iterative open auction process) that this is the situation, they do not have the incentive to bid beyond the minimal level set by the auctioneer. This situation happened in the 3G 204 B. Gavish auctions in Italy, Holland, Swiss, Belgium and Greece (Klemperer [13]). In the proposed parallel bidding process, the bidders do not know in advance which of the parallel auctions will be the one selected by the regulators, it makes it difficult for the bidders to enter into such implicit or explicit coalitions. Throughout the paper, it was assumed that spectrum is divided equally between the winning bidders. It will be interesting to extend the results to cases in which spectrum is not divided equally between the bidders. When spectrum in a parallel auction follows a nonequal division, bidders in each of the parallel auctions are given a division (possibly not equal) of spectrum to bands of spectrum and can choose on which band they are bidding and how much they are willing to pay for it. It is interesting to note that the governing bodies had accepted without much difficulty the notion of dividing a service area into subareas, and allowing as part of the bidding process to compose clusters of subareas that they bid on. What we propose in this paper is to extend the same concept to the spectrum domain. Another possible extension to the model is to incorporate into the analysis, queuing effects and quality of service requirements. Queuing, blocking and drop-off probabilities change the cost functions and prices that users and operators are facing. Incorporating them into the expected net revenue models will lead to bidding methods that are based on the idea of allowing for parallel bids which take into consideration the number of operators the overall bandwidth and its impact on QOS and the queuing (delay) that customers experience. Acknowledgements I wish to express my deep appreciation to the many comments made by participants in seminars and conferences in which preliminary versions and ideas, which led to this paper, have been presented. References [1] M. Ausubel, P. Cramton, R.P. McAfee, J. McMillan, Synergies in wireless telephony: evidence from the broadband PCS auction, Journal of Economics and Management Strategy, 6, No. 3 (1997), 497-527. [2] M. Bichler, M. Kaukal, A. Segev, A Multi-attribute auctions for electronic procurement, In: Proceedings of 1-st IBM IAC Workshop on Internet Based Negotiation Technologies, Yorktown Heights, NY (1999). AN IMPROVED METHOD FOR AUCTIONING... 205 [3] P.J. Brewer, Decentralized computation procurement and computational robustness in a smart market, Economic Theory, 13 (1999), 41-92. [4] P.J. Brewer, C.R. Plott, A binary conflict ascending price (BICAP) mechanism for the decentralized allocation of the right to use railroads tracks, International Journal of Industrial Organization, 14 (1996), 857-856. [5] C.G. Caplice, An Optimization Based Bidding Process: A New Framework for Shipper-Carrier Relationships, Ph.D. Thesis, MIT Department of Civil and Environmental Engineering (1996). [6] P. Cramton, The FCC spectrum auctions: an early assessment, Journal of Economics and Management Strategy, 6, No. 3 (1997), 431-495. [7] B.C. Fritts, Private property, economic efficiency, and spectrum policy in the wake of the C block auction, Federal Communications Law Journal, 51, No. 3 (1999), 849-885. [8] B. Gavish, S. Sridhar, The impact of mobility on cellular networks configuration, Wireless Networks, 7 (2001), 173-185. [9] B. Gavish, S. Sridhar, Economic aspects of configuring cellular networks, Wireless Networks, 1, No. 1 (1995), 115-128. [10] D. Grether, R. Isaac, C. Plott, The Allocation of Scarce Resources: Experimental Economics and the Problem of Allocating Airport Slots, Westview Press, Boulder (1989). [11] C. Jackson, Technology for Spectrum Markets, Ph.D. Thesis, School of Engineering, MIT, June (1976). [12] F. Kelly, R. Steinberg, A combinatorial auction with multiple winners for universal service, Management Science, 46, No. 4 (2000), 586-596. [13] P. Klemperer, How (not) to run auctions: the european 3G telecom auctions, European Economic Review, 46, No. 4-5 (2002), 829-845. [14] E. Kwerel, A.D. Felker, Using auctions to select fcc licensees, Office of Plans and Policy Working Paper No. 16, Federal Communications Commission, Washington, D.C. (May 1985). [15] J.O. Ledyard, M. Olson, D. Porter, J.A. Swanson, D.P. Torma, The first use of a combined value auction for transportation services, Social Science Working Paper, Caltech (March 2000). 206 B. Gavish [16] R.P. McAfee, J. McMillan, Analyzing the airwaves auction, Journal of Economic Perspectives, 10, No. 1 (1996), 159-175. [17] K.A. McCabe, S.J. Rassenti, V.L. Smith, Smart computer assisted markets, Science, 254 (1991), 534-538. [18] J. McMillan, Selling Spectrum Rights, Journal of Economic Perspectives, 8 (1994), 145-162. [19] P.R. Milgrom, Auctioning the Radio Spectrum, Auction Theory for Privatization, Cambridge University Press (1995). [20] P.R. Milgrom. Putting auction theory to work: the simultaneous ascending auction, Technical Report 98-0002, Department of Economics, Stanford University (December 1997). [21] P.R. Milgrom, R.J. Weber, A theory of auctions and competitive bidding, Econometrica, 50 (1982), 1089-1122. [22] S.J. Rassenti, V.L. Smith, R.L. Bulfin, A combinatorial auction mechanism for airport time slot allocation, Bell Journal Of Economics, 13, No. 2 (1982), 402-417. [23] M.H. Rothkopf, R.M. Harstad, Modeling competitive bidding: A critical essay, Management Science, 40 (1994), 364-384. [24] M.H. Rothkopf, A. Pekec, R.M. Harstad, Computationally manageable combinatorial auctions, Management Science, 44, No. 8 (1998), 1131-1147. [25] M.P. Wellman, W.E. Walsh, P.R. Wurman, J.K. MacKie-Mason, Some economics of market-based distributed scheduling, In; Proceedings of 18-th International Conference on Distributed Computing Systems (1998).