A refutation of Metcalfe's Law and a better estimate for the value of networks and network interconnections Andrew Odlyzko1 and Benjamin Tilly2 1 Digital Technology Center, University of Minnesota 499 Walter Library, 117 Pleasant St. SE Minneapolis, MN 55455, USA [email protected] http://www.dtc.umn.edu/»odlyzko 2 ben [email protected] Preliminary version, March 2, 2005 Abstract. Metcalfe's Law states that the value of a communications network is proportional to the square of the size of the network. It is widely accepted and frequently cited. However, there are several arguments that this rule is a signi¯cant overestimate. (Therefore Reed's Law is even more of an overestimate, since it says that the value of a network grows exponentially, in the mathemat- ical sense, in network size.) This note presents several quantitative arguments that suggest the value of a general communication network of size n grows like n log(n). This growth rate is faster than the linear growth, of order n, that, according to Sarno®'s Law, governs the value of a broadcast network. On the other hand, it is much slower than the quadratic growth of Metcalfe's Law, and helps explain the failure of the dot-com and telecom booms, as well as why net- work interconnection (such as peering on the Internet) remains a controversial issue. 1 Introduction The value of a broadcast network is almost universally agreed to be proportional to the number of users. This rule is called Sarno®'s Law. For general communication networks, in which users can freely interact with each other, it has become widely accepted that Metcalfe's Law applies, and value is proportional to the square of the number of users. This \law" (which is just a general rule-of-thumb, not a physical law) has attained a su±ciently exalted status that in a recent article it was classed with Moore's Law as one of the ¯ve basic \rule-of-thumb 'laws' [that] have stood out" and passed the test of time [26]. Even as far back as 1996, Reed Hundt, the then-chairman of the Federal Communications Commission, claimed that Metcalfe's Law and Moore's Law \give us the best foundation for understanding the Internet" [9]. Metcalfe's Law was frequently cited during the dot- com and telecom booms to justify business plans that had the infamous \hockey-stick" revenue and pro¯t projections. The hopes and promises were that once a service or network attained su±cient size, the non-linear growth of Metcalfe's Law would kick in, and network 2 Andrew Odlyzko and Ben Tilly and bandwagon e®ects would start to operate to bring great riches to the venture's backers. This led to the argument that there was little need to worry about current disappointing ¯nancial results, and all e®orts should be devoted to growth. The fundamental insight, that the value of a general communication network grows faster than linearly, appears sound. Although it had not been enunciated explicitly until a few decades ago, it had motivated decision makers in the past. For example, it a®ected the development of the phone system a century ago, as shown by the following quote [19]: The president of AT&T, Frederick Fish, believed that customers valued access and that charging a low fee for network membership would maximize the number of subscribers. According to Fish, the number of users was an important determinant of the value of telephony to individual subscribers. His desire to maximize network connections led the ¯rm to adopt a pricing structure in which prices to residential customers were actually set below the marginal cost of service in order to encourage subscriptions. These losses were made up through increased charges to business customers. Serious quantitative modeling of network e®ects dates back to the 1974 paper of Je®rey Rohlfs [24] (see also [25], and some of the references in [24] to earlier work in the area), which was motivated by the lack of success of the AT&T PicturePhoneTM videotelephony service. (For a survey of economics literature in this area, see [7].) Metcalfe's Law itself dates back to a slide that Bob Metcalfe created around 1980, when he was running 3Com, to sell the Ethernet standard. It was dubbed \Metcalfe's Law" by George Gilder [8] in the 1990s, and citations to it started to proliferate [9, 15, 16]. The foundation of Metcalfe's Law is the observation that in a general communication network with n members, there are n(n ¡ 1)=2 connections that can be made between pairs of participants. If all those connections are equally valuable (and this is the big \if" that we will discuss in more detail below), the total value of the network is proportional to n(n ¡ 1)=2, which, since we are dealing with rough approximations, grows like n2. Metcalfe's Law is intuitively appealing, since our personal estimate of the size of a network is based on the uptake of that network among friends and family. Our derived value also varies directly with that metric. We therefore see a linear relationship between the perceived size and value of that network. Reed's Law [23] is based on the further insight that in a communication network as flexible as the Internet, in addition to linking pairs of members, one can form groups. With n participants, there are 2n possible groups, and if they are all equally valuable, the value of the network grows like 2n. The fundamental fallacy underlying Metcalfe's and Reed's laws is in the assumption that all connections or all groups are equally valuable. The defect in this assumption was pointed out a century and a half ago by Henry David Thoreau. In Walden, he wrote [28]: We are in great haste to construct a magnetic telegraph from Maine to Texas; but Maine and Texas, it may be, have nothing important to communicate. Now Thoreau was wrong. Maine and Texas did and do have a lot to communicate. Some was (and is) important, and some not, but all of su±cient value for people to pay for. Still, Metcalfe's Law 3 Thoreau's insight is valid, and Maine does not have as much to communicate with Texas as it does with Massachusetts or New York, say. In general, connections are not used with the same intensity (and most are not used at all in large networks, such as the Internet), so assigning equal value to them is not justi¯ed. This is the basic objection to Metcalfe's Law, and it has been stated explicitly in many places, for example [12, 14, 20, 25, 26]. This observation was apparently ¯rst made by Metcalfe himself, in his ¯rst formal publication devoted to Metcalfe's Law [15]. Some users (those who generate spam, worms, and viruses, for example) actually subtract from the value of a network. Even if we disregard the clearly objectionable aspects of communication, such as spam, many users do complain about the problem of dealing with too many options. There are additional scaling arguments that suggest Metcalfe's and Reed's laws are incorrect. For example, Reed's Law is implausible because of its exponential (in the precise mathematical sense of the term) nature [10]. If a network's value were proportional to 2n, then there would be a threshold value m such that for n below m ¡ 50, the value of the network would be no more than 0.0001% of the value of the whole economy, but once n exceeded m, the value of the network would be more than 99.9999% of the value of all assets. Beyond that stage, the addition of a single member to the network would have the e®ect of almost doubling the total economic value of the world. This does not ¯t general expectations of network values and thus also suggests that Reed's Law is not correct. Metcalfe's Law is slightly more plausible than Reed's Law, as its quadratic growth does not lead to the extreme threshold e®ect noted above, but it is still improbable. The problem is that Metcalfe's Law provides irresistible incentives for all networks relying on the same technology to merge or at least interconnect. To see this, consider two networks, each with n members. By Metcalfe's Law, each one is (disregarding the constant of proportionality) worth n2, so the total value of both is 2n2. But suppose these two networks merge, or one acquires the other, or they come to an agreement to interconnect. Then we will e®ectively have a single network with 2n members, which, by Metcalfe's Law, will be worth 4n2, or twice as much as the two separate networks. Surely it would require a combination of singularly obtuse management and singularly ine±cient ¯nancial markets not to seize this obvious opportunity to double total network value by a simple combination. Yet historically there have been many cases of networks that resisted interconnection for a long time. For example, a century ago in the U.S., the Bell System and the independent phone companies often competed in the same neighborhood, with subscribers to one being unable to call subscribers to the other (see [20], or [17] for much more detail). Eventually interconnection was achieved (through a combination of ¯nancial maneuvers and political pressure), but it took two decades. In the late 1980s and early 1990s, the commercial online companies such as CompuServe, Prodigy, AOL, and MCIMail, provided email to subscribers, but only within their own systems, and it was only in the mid-1990s that full interconnection was achieved.