From: Robin Hanson <hanson@econ.berkeley.edu>

Date: Mon Dec 08 1997 - 16:51:25 PST

Date: Mon Dec 08 1997 - 16:51:25 PST

Hal writes:

*>It is an interesting thought that an earlier generation of replicators
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*>burned through some currently unavailable energy source and produced the
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*>dark matter. They neglected stars because either they didn't exist yet
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*>or their output was too paltry to be concerned with. ...
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*>Carl's reasoning emphasizes the behavior of replicators at the edges
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*>of the sphere. If their whole point is speed, they might well ignore
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*>resources which take more time to develop. I think Robin's equations
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*>will show the same thing. You have to look behind the frontier though and
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*>try to understand why there were no low-budget colonies left behind to
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*>use the remaining resources. It's possible that there was too much of a
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*>parameter-space gap to evolve from the fast-moving replicators to the
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*>kind which could use stars and other low energy sources. In a sense,
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*>we may be the first replicators able to utilize this source of energy.
*

Here's a relevant calculation, that generalizes my previous calculation.

Let R(T) be the resources available at an oasis after growing there

for time T. Assume ln(R(T)) is increasing and concave in T with R(0) = 1.

This models Carl's observation that some resources at an oasis allow

fast growth, and others require more patience to exploit.

As before, assume a cost C > 1 to make a probe which may eventually

deliver a resource of 1 to a new oasis, a delay D to speed up and slow

down, a constant velocity V while traveling, a chance of A per unit

distance of being destroyed, and a chance Q that an oasis is suitable.

The maximum sustainable speed S is then given by

1 1 (D + T)

- = - + -------------- ,

S V A*ln(Q*R(T)/C)

and no probes leave an oasis until some optimal T*, at which point

all of R(T*) is used to make probes. The optimal T* is the tangency

point where a line from the point (-D,-ln(Q/C)) touches the ln(R(T))

curve.

Note that if a probe falls behind the front of the colonization wave,

it faces a smaller chance of finding a suitable oasis not already

occupied by a previous probe. Thus Q declines behind the frontier.

Thus the max sustainable speed is less behind the front, so the

"depth" of the front region should spread out with time. Furthermore,

since T* increases as Q declines, selection seems to favor probes who

use up more local resources R(T*) before moving on. As Q approaches

zero, T* grows until sustained colonization becomes impossible, at

Qmin, where C(V) >= Qmin*R(infinity) for any V.

Thus it seems replicators won't leave much resources untouched.

It seems that any earlier replicators must have been stupid

in this sense: they couldn't adjust their strategy to crowding,

or they just couldn't make replicators out of the resources we use.

Robin Hanson

hanson@econ.berkeley.edu http://hanson.berkeley.edu/

RWJF Health Policy Scholar, Sch. of Public Health 510-643-1884

140 Warren Hall, UC Berkeley, CA 94720-7360 FAX: 510-643-8614

Received on Tue Dec 9 00:45:08 1997

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