Re: poly: No expected value colonization equilibrium?

At 11:50 AM 12/12/97 -0800, Robin Hanson wrote:
>I've been trying to apply game theory to the colonization
>situation Carl F. raised, and I seem to have proven that
>conclusion that there is no colonization equilibrium when
>probes only care about the expected value of the number of
>descendant probes they produce at each point in space time.
>
>...
>If, however, v is the maximal speed at which probes are going,
>and if anyone ever colonizes, then it seems that the probability
>of the oasis at (r,t) being already occupied must be greater
>than the corresponding probability for the oasis at (r+dr,t+vdr).
>If so, the colonize-later strategy must produce a strictly
>larger expected number of probes at (r+dr,t+dt+vdr).

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