From: carl feynman <carlf@atg.com>

Date: Mon Dec 15 1997 - 11:46:01 PST

Date: Mon Dec 15 1997 - 11:46:01 PST

At 11:50 AM 12/12/97 -0800, Robin Hanson wrote:

*>I've been trying to apply game theory to the colonization
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*>situation Carl F. raised, and I seem to have proven that
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*>conclusion that there is no colonization equilibrium when
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*>probes only care about the expected value of the number of
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*>descendant probes they produce at each point in space time.
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*>
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*>...
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*>If, however, v is the maximal speed at which probes are going,
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*>and if anyone ever colonizes, then it seems that the probability
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*>of the oasis at (r,t) being already occupied must be greater
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*>than the corresponding probability for the oasis at (r+dr,t+vdr).
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*>If so, the colonize-later strategy must produce a strictly
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*>larger expected number of probes at (r+dr,t+dt+vdr).
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If probes can continue indefinitely with no risk, then this is the correct

conclusion. However, consider the case where probes have a certain

probability per unit time of being lost. In this case, what we are

interested in maximizing is the probability of the proble surviving to time

t, multiplied by the probability of finding an unoccupied oasis at that

time. Let's call these probabilities A(t) and S(t) repsectively. We are

interested in maximizing AS. This is equivalent to maximizing ln(AS) =

ln(A)+ln(S). So we want to find a t for which d(ln(AS)/dt = 0. This is

equivalent to

- d(ln(A))/dt = d(ln(S))/dt

Under the assumption that the main mortatilty to probes is caused by random

uniformly distributed events such as collisions with dust grains, -

d(ln(A))/dt is a positive constant, inversely proportional to the expected

time between fatal events. S is an increasing function of t, so

d(ln(s))/dt is positive. It is not guaranteed that this equation has a

solution, but it is at least possible, for many plausible functions

relating S to t.

To put the rule in words: a probe should drop out of the race when the

percentage rate of increase in the suitability of oases starts to exceed

the (fixed) percentage rate of decrease in survival.

--CarlF

Received on Mon Dec 15 19:38:20 1997

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