From: Robin Hanson <hanson@econ.berkeley.edu>

Date: Fri Dec 12 1997 - 11:50:28 PST

Date: Fri Dec 12 1997 - 11:50:28 PST

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

*
This archive was generated by hypermail 2.1.8
: Tue Mar 07 2006 - 14:45:29 PST
*