SITUATION: OPERATING SYSTEMS: MAC OR WINDOWS?
Well, the idea that the more users you get, the more valuable something is, is even stronger today when it’s so easy for people to connect up and build communities. I do think it goes back [to Windows]. As more people used Windows it became sort of a standard thing that people would learn; as people used Office they would do add-ons. Everyone who had an Office-for- matted document who wanted to exchange it as an attachment with some- body else was kind of encouraging that person. And so these network effects, that was really [the] business model that created Microsoft. —BILL GATES6
Since the early 1980s, most consumers in developed countries have had to make a choice between doing their computing on a Windows-based PC or on a product of Apple. The PC was originally developed by IBM, with micro- processors built by Intel and an operating system designed by Microsoft (first DOS and then Windows). In contrast, Apple supplied both hardware and soft- ware components, initially with Apple II and then with the Mac. Apple hit the market before IBM, but eventually the PC would dominate. Today, about 95% of personal computers run the Windows operating system.
Although most product markets are not dominated by a single product— unlike the market for operating systems—dominance is not uncommon when there are network effects. A product has network effects if its value to a con- sumer is greater when more consumers use it. Classic examples are commu- nication networks such as telephone and e-mail. For example, e-mail is not of much value if no one else has it, and furthermore, its value increases as more and more people can be accessed with it. The case of operating systems is a more indirect version of network effects, but no less important. As more peo- ple use an operating system, more applications are written for it—word pro- cessing, spreadsheets, games, etc.—that increase the value a consumer attaches to using the operating system.
The decision as to which operating system to buy is a game played not among two, or ten, or even a hundred people, but millions of people. Nevertheless, the choice has all of the essential ingredients of a strategic setting, in that the best operating system for someone depends on what she thinks other consumers will do. No consumer wants to use an operating system that is “stranded” with few applications.
Suppose, then, that there are people contemplating which operating system to purchase. Each has a strategy set composed of Mac and Windows. The payoff to buying a Mac is assumed to be where m is the number of people who choose Mac. The payoff for choosing Windows is
100 � 10 � m,
n � 2
*Before you cancel that airline reservation, keep in mind that an important departure from reality in these experiments is that the participants were not allowed to communicate with one another. Permitting such communication prior to play could make it more likely that players would coordinate on the best equi- librium, in which all choose strategy 7.
Assume that the effective security level is now determined by the highest (not the lowest) security measures chosen by airlines. Letting max{s1, . . . , sn} denote the highest of the airlines’ strategies, we find that airline i’s payoff is now
50 � 20 � max{s1, . . . , sn} � 10 � si.
Assuming the same strategy sets, find all Nash equilibria.
5.2 CHECK YOUR UNDERSTANDING
126 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
where w is the number of consumers who buy Windows. Assume for simplicity that everyone buys one or the other, so that
Note that the value of an operating system is higher when more people buy it. The part of a consumer’s payoff that depends on how many people buy the good—which is for a Mac and for Windows—measures the network effect. A second property is that the Mac is presumed to have superior features, and they are worth 100 in terms of payoff.
One Nash equilibrium has every consumer buying the superior Mac sys- tem, and that system then becomes the standard. To prove this claim, consider the strategy profile in which everyone buys a Mac. Given that the other consumers buy a Mac, a consumer receives a payoff of from doing so, but a payoff of only from buying Windows. Since
it is optimal to buy a Mac. Because both the game and the strategy profile are symmetric, this conclusion applies to each and every con- sumer. It is then a Nash equilibrium for every player to choose Mac.
It can also be a Nash equilibrium, however, for the inferior Windows sys- tem to prevail. If all other consumers buy Windows, then the payoff to a con- sumer from doing likewise is while the payoff from buying a Mac is 110 (since, in that case, If
10 � n � 110, or equivalently, n � 11,
then it is better to do as everyone else does and buy Windows. When there are at least 11 consumers, the advantage that Windows has from a bigger network effect exceeds the additional value coming from the superior Mac technology. Hence, every consumer then chooses Windows.
In sum, there is always a Nash equilibrium in which all consumers use a Mac, but if then there is also a Nash equilibrium in which all con- sumers use Windows. That equilibrium outcomes are extreme—either all go with the Mac or all go with Windows—is because network effects are a form of tipping. To see this more clearly, assume that there are 20 people FIGURE 5.5(n � 20).
n � 11,
m � 1). 10 � n,
100 � 10 � n 7 10, 10(� 10 � 1)
100 � 10 � n n � 1
10 � (n � m)10 � m
w � n � m. 10 � w,
FIGURE 5.5 Mac vs. Windows
20 4 6 8 10 12 14 1615 2018
Pa yo
ff
w
100
140 160
200
300
150 160
Payoff from Mac
Payoff from Windows
5.2 Symmetric Games 127
shows the relationship between payoff and the number of users of a system. On the horizontal axis is the number of adopters of Windows; a value for w corresponds to w consumers choosing Windows and buying a Mac. The vertical axis measures the associated payoff for a Windows user and a Mac user. For example, if then the payoff for using Windows is
and for using a Mac is Reflecting net- work effects, as the number of Windows users rises, then the payoff for using Windows increases and for using a Mac declines.
Suppose now that, of the other 19 consumers, 15 have chosen Windows. A consumer then gets a payoff of 160 from also choosing Windows (because, with her choice of Windows, there are now 16 Windows users) and a payoff of 150 from choosing a Mac. The consumer will then prefer Windows. The logic for Windows is even more compelling if, instead, 16 of the other 19 consumers use Windows. Now Windows delivers a payoff of 170, while the payoff from choos- ing Mac is only 140. In Figure 5.5, the gap between the Windows payoff and the Mac payoff rises as the number of Windows users goes from 16 to 17 and so on.
As long as a consumer thinks that at least 15 other consumers will choose Windows, she’ll choose Windows as well. If all consumers reason this way, then, in fact, all 20 consumers will buy Windows, and it will become the dom- inant operating system. However, if a consumer believes that only 14 other consumers will buy Windows, then he will prefer a Mac. Figure 5.5 shows that, with 14 users of Windows, the Mac payoff is 160, while the Windows pay- off is 150 (because, with this consumer, there are now 15 buying Windows). Since fewer consumers are expected to buy Windows, the relative attractive- ness of a Mac rises: as indicated in Figure 5.5, the gap between the Mac pay- off and the Windows payoff gets larger as w shrinks.
The tipping point is 15. If, on the one hand, each consumer believes that 15 or more other consumers will buy Windows, then each and every consumer will buy Windows. If, on the other hand, each consumer believes that 14 or fewer other consumers will buy Windows, then each and every consumer will buy a Mac. Depending on those beliefs, the market can tip to one standard or the other.
Consumers’ expectations are, then, critical when a product has network ef- fects. If most consumers come to believe that most consumers will buy a par- ticular product, then those beliefs will prove self-fulfilling, as all will, indeed, buy that product. So how does a company convince consumers that its prod- uct will be the popular one? Although advertising can help, it isn’t sufficient: a consumer who is swayed by an advertisement must also know that many other consumers are being thus swayed. That is, one wants to make it com- mon knowledge that the product is compelling. Junk mail—of either the paper or electronic variety—won’t do the trick. It may reach a lot of people, but those people who are reached don’t know how many other people have been simi- larly contacted.
Perhaps the best generator of common knowledge in the United States is a commercial during the Super Bowl.7 It is not simply that the Super Bowl is the most widely watched program, but that almost everyone knows that al- most everyone is watching it. During the 1986 Super Bowl, the Discover card was advertised extensively, and it is a classic case of a product with network effects. Retailers will accept the Discover card only if many consumers use it, and many consumers will use it only if many retailers accept it. Thus, the more consumers who use the card, the more retailers will accept it, which will
160(� 100 � 6 � 10).140(� 14 � 10) w � 14,
20 � w
128 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
induce yet more consumers to use it, which will induce more retailers to ac- cept it, and so on. Better yet, the Mac was introduced with a 60-second com- mercial during the 1984 Super Bowl.