poly: No expected value colonization equilibrium?

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.

Consider a probe at position (r,t) traveling outward with
velocity v in a spherically symmetric colonization process.
Imagine that in equilibrium this probe has a non-zero
probability of stopping to attempt to colonize an oasis at r,
growing there till (r,t+dt), and then sending out new probes
at velocity v which pass (r+dr,t+dt+vdr), where there is
another oasis.

An alternative route to starting with a probe passing (r,t)
and ending up with probes passing (r+dr,t+dt+vdr) is for the
probe to continue on from (r,t) to (r+dr,t+vdr), stop and
colonize there for dt, and then send out probes.
If the oases have equal probability of being suitable, and
equal abilities to support growth, these alternative strategies,
colonize-new and colonize-later, would produce the same
expected number of probes at (r+dr,t+dt+vdr).

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).

Thus a probe concerned only about its expected number of
post-colonization descendants would continue until it reached
an oasis where there was a zero chance that any previous probe
had already colonized there, or that another probe might
simultaneously try to colonize that oasis.

If there were several competing such probes at (r,t), they
would all continue on until one was sure it was the only one
to have survived. And if, as seems more plausible, they
could never be sure they were the only one that had survived,
they would all keep going on forever, until none were left.

Note this analysis still applies to a probe who places
different values on descendants at different space-time points.
The only requirement is that at each space-time point, they
only care about the expected value there.

For a probe who cares about the variance of the distribution
of their descendant probes, the two alternative strategies
of colonize-now and colonize-later are not equivalent.
The colonize-later strategy produces a distribution of later
probes that has mass at two points: a large mass at zero
and a small mass at a large number of probes. The
colonize-now strategy is roughly a poisson distribution
with a lower expected value.

The analysis also does not apply to a probe which cares
directly about which oases it colonizes when.

The final conclusion is that to analyze the colonization game
one needs to carefully consider probe preferences over the
distribution, not just expected value, of their descendant
probes. Or maybe preferences over which oases are colonized.

This suggests that evolutionary dynamics are rather chaotic;
rather than a smooth steadily growing frontier, trillions or
more probes die for every one that finally colonizes an oasis,
starting a new big bulge of colonization.

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 12 19:44:38 1997