Re: poly: ESS for HPLD

From: Robin Hanson <hanson@econ.berkeley.edu>
Date: Mon Dec 08 1997 - 16:51:25 PST

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