Classroom Companion: Business

 
Chapter 9 · Network Effects

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141
9
9.6.4 
 Reed’s Law
In massive multiplayer online games (MMOGs), the players form groups. The 
number of possible groups among n players is N~2
n
(assuming that n is large). 
Therefore, it is reasonable to assume that the value of a digital service in which 
groups are formed is:
V
n
Reed
n
 
~
.
2
This is Reed’s law (Reed, 
2001
). Reed is an American computer scientist and one of 
the developers of the TCP and UDP protocols.
Reed’s law determines, in general, the value of a network in which interactions 
take place in groups. Again, we may use Odlyzko’s argument that the contribution 
from large groups is too big and the actual network effect of group formation is 
smaller than what is predicted by Reed’s law. One way to modify Reed’s law is to 
use Dunbar’s number, which is the average number of people an individual knows 
(see 
7
Box 
9.2
). A commonly used value of Dunbar’s number D is 150.
Let us set the maximum size of a group that can be formed by people—for 
example, in an online game—as equal to D (=150). The number of groups smaller 
than D that can be formed by n people is:
N
n
k
n
k
D
D
~
~
,








1
It can be shown that the ER graph 
becomes connected if p = c ln n/n for 
some constant c. This threshold is 
sharp. If the link probability is increas-
ing slightly slower with increasing n, for 
example, p = c ln n/n
a
, a > 1 + ε and ε is 
an arbitrarily small number, large parts 
of the graph will be unconnected. If the 
probability is increasing slightly faster 
with increasing n, for example, 
p = c ln n/n
a
, a < 1 − ε and ε is an arbi-
trarily small number, the graph will be 
tightly connected. It is reasonable to 
assume that the graph representing 
relationships between people is con-
nected—there exists a path from one 
person to another either directly or via 
other people. This path is rather short, 
as revealed by observations made by 
Milgram, leading to his law of six 
degrees of separation; that is, the dis-
tance between people is seldom more 
than six links, in which a link is from 
one person to another person person-
ally know. Since each of us has few 
direct links to other people, it is reason-
able to assume that the graph is lightly 
connected so that p~ ln n/n is a good 
approximation of the link probability 
of the relationship graph between peo-
ple. Since there are n
2
possible links, the 
total number of links N and the value 
of the network V(n)~N is 
V
O − T
(n)~N~n
2
ln n/n = n ln n, and, 
again, Odlyzko-Tilly’s law has been 
derived.