# poly: ESS for HPLD

From: Robin Hanson <hanson@econ.berkeley.edu>
Date: Fri Dec 05 1997 - 09:36:32 PST

Carl F. considers technological statis among competing entities expanding
out to colonize the universe, jumping from oasis (e.g. star sys) to oasis.
Carl F. assumes an evolutionary pressure for maximum speed (which Carl A.
questions) and shows a example calculation of optimal jumping distance.

I didn't follow his analysis - I think he needs to give more explicit math.
But I found the question interesting enough to do my own analysis.

Assume (with Carl F.) that spores traveling between oases at a constant
speed V have a 1/A chance of being destroyed per unit distance. Thus given
N1 initial spores the number of spores surviving after distance X is
N1*exp(X/A).

Assume that spores deliver unit wealth to an oasis, and at an oasis wealth W
grows a fractional rate of 1/G per unit time, up to a maximum Wmax. At any
point growth can be stopped and N2 = W/C spores can be sent out, where C is
the cost per spore. Thus after time T, N2 = exp(T/G)/C spores can be sent.

Finally, assume a delay D for spores to accelerate, decelerate, and set up
shop.
And assume only a fraction S of oases are discovered on arrival to be
suitable.

The maximum possible speed is where on average only one spore survives to
grow in a new oasis per each previous colonized oasis. So assume N1 = N2.

A little algebra shows X = A*ln(WS/C) and T = D + g*ln(W) + X/V .
So now we just vary things like W,A,V,D to minimize T/X, being sure to
consider their effects on cost C.

Varying W we find that we want W=Wmax, regardless of the other parameters.
So you always stay at an oasis until you completely exhaust it, then move on
all at once.

Varying A we find we want to maximize A*ln(N), so engineers should be willing
to spend huge sums to improve this parameter. Doubling A is just as good as
squaring the number of probes send out.

I haven't put in plausible functional forms for C(V,D) and D(V) to learn more
about optimal V. Anyone want to try this? Also, anyone want to flesh out
this model with plausible values for parameters?

I've been working out a differential equation to address the question of
whether
max speed is really the best idea, but that will have to wait for another
post.

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 Fri Dec 5 09:37:34 1997

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