Subject: Re: near/medium future digital media economics
From: "Stephen J. Turnbull" <turnbull@sk.tsukuba.ac.jp>
Date: Fri, 19 May 2006 18:27:38 +0900

 Fri, 19 May 2006 18:27:38 +0900
>>>>> "Ben" == Ben Tilly <btilly@gmail.com> writes:

    Ben> Metcalfe's Law (BTW, note the spelling) is going to apply
    Ben> best in networks where people actively wish to connect with
    Ben> fairly random strangers.  Very few networks look like this.

That's because historically commodity markets were the only practical
place where that would work.  It is not an accident that the huge
complex of networks we call the stock market are concentrated in maybe
a dozen institutions worldwide, and they coalesced into an Intermarket
so long ago that we no longer think of them when we talk about
networks.

What search technology *may* make possible is scaling up search in a
heterogeneous network to a billion participants or so.

    Ben> I suggest that you should look up Benford's Law and explain
    Ben> why it doesn't apply to this type of content when it does to
    Ben> other kinds.

What in the world do you think Benford's Law has to do with anything
here?  Benford's Law is about decimal representations of a
monotonically increasing sequence, not about values.  Eg, if I were
going to bet on leading "digits" in web content, I'd go for 60 base
256, wouldn't you?

    Ben> This argument, like Metcalfe's original argument, has the
    Ben> implicit assumption that everyone is equally interested in
    Ben> all potential connections.  I'm not.  Nor is anyone else.

Oh, come on, Ben, give us a little!  You're capable of presenting
the math explicitly, and it's worth doing so here.

The question is "given an increasing size of network, how does the
value distribution change?"  If we assume that to some agent (note the
abstraction!) the probability distribution over (gross) value of
connection to a random other agent is constant, then the expected
(gross) value of the network *is* proportional to network size for
each individual.

So what you need is marginal cost of connecting bounded below that
average value, and vol! net value per agent is linear in network
size, and you got Metcalfe's Law.

At one extreme of marginal cost, we have the marriage market.  For me
it was zero for q=1 and infinite for q=2.  So then the value of the
network to me was an order statistic (max Vi over i = 1, ..., N).
Anything that increases network size increases personal value, but
definitely not linearly (bounded above by "10", of course!)  So
effectively we've increased the value of a rival good, but not changed
it into a network good  l Metcalfe.

Can we get to the other extreme, constant marginal cost below average
value?  If we're talking about communication networks among humans,
no.  Social contact of positive value requires time bounded below on
the order of a second or so (however long it takes to perceive your
baby's smile), and there are only a finite number of seconds available
to each of us.

However, anonymous networks (eg, markets) or machine-mediated
communication (sorry, no examples---that's Tom's job! ;-) might come
pretty close, at least up to population levels feasible for humans on
this planet.

I think the jury's out on just how far Metcalfe's Law can extend its
domain---but the financial markets are a pretty compelling example for
"hey, this really does work sometimes!"

Note that a market is a pretty trivial network topology (star).  But
then, the basic property of the Internet that these laws depend on is
that it turns Life, the Universe, and Everything into a star topology.

    Ben> You've just counted connections and assumed that they all
    Ben> have equal value.  But that assumption flatly contradicts
    Ben> observations of how people really network in social
    Ben> situations.

First, it's not an assumption of equal value, really.  It's an
assumption of equal expected value per connection, independent of the
number of connections.  Apply Law of Large Numbers, live Metcalvian
ever after.

Second, consider Mr. Lynch's problem.  Have you ever watched a
headhunter network a LUG meeting? ;-)  If such people (excuse the
exaggeration) are a constant fraction of the population, whammer
jammer, Metcalfe's Law!  (In the long run, nobody else will matter if
they obey Metcalfe's Law---their Metcalfe value is a lower bound on
social value.)

Third, remember that many participants in networks are not people,
they are firms.  To the extent that a firm can have a bounded labor
force with the additional costs of networking borne by capital, a
network of firms (or capital-enhanced humans, for that matter) could
exhibit rather different networking behavior.  (Then again, maybe they
wouldn't.  That was Jamie Zawinski's point about groupware.)

    Ben>  And Google's power derives from the fact that good searching
    Ben> allows people to come closer to perfectly extracting the
    Ben> value of being on the network.

Uh, what are you trying to say here?  I think it's arguable that
Google increases the size of the network in some sense; I'd like to
see a careful argument why that isn't so.

    Ben> As for your defence of Reed's Law, well what you're saying is
    Ben> that you think that it is reasonable that adding one person
    Ben> to the internet could double everyone's perceived quality of
    Ben> life.  If you think that that's reasonable, then I don't
    Ben> really know what to tell you.

"The King!"  ;-)

    >> This is not to say that at some point, every new member has
    >> that effect.  The constant factor on these approximations
    >> matter and we are lucky/happy when a few great minds per
    >> generation or 10 have such effect.

    Ben> Oh no.  Reed's Law *explicitly* says that every new member
    Ben> has that effect.  Which is why it doesn't pass the "smell"
    Ben> test.

Strawman!  Strawman!  Out to the cornfields with you!

Reed's Law would work if the fraction of great minds per generation
were constant regardless of size of generation, and everybody else was
worth precisely zero in this sense.  I think that's unlikely but
arguable.

The problem with Reed's law is elsewhere.  It is that it assumes that
all needed groups actually form.  If there's one great mind, then she
has to participate in all (N-1)(N-2) possible groups for her to get
Reed's law to work.  Again, you could adjust this for some sort of
average or probabilistic formation, but I don't think that makes much
more sense than assuming all groups form---you still need the number
of groups forming to be proportional to the number of possible groups,
and eventually Ms. Great Mind will get tired of all that networking.

And it needs to be the case that having formed the
group-of-one-billion, *you* would get an equal increment of value from
also forming the group-of-one-billion-less-*me*.  Hardly seems likely!
:-)

Please note, I agree with you, not Tom.  My intuition is that in the
long run Metcalfe's law will fail, and network values will be
sublinear.  But I strongly believe that the reason why they fail
matters, both to the degree to which Metcalfe's Law approximates value
for "small" networks, and to long-run industrial organization of
networks that start out small enough for that approximation to be
valid.

-- 
Graduate School of Systems and Information Engineering   University of Tsukuba
http://turnbull.sk.tsukuba.ac.jp/        Tennodai 1-1-1 Tsukuba 305-8573 JAPAN
        Economics of Information Communication and Computation Systems
          Experimental Economics, Microeconomic Theory, Game Theory