Ok. So what if mixed populations of cooperators and defectors can segregate along a
risk gradient. Obviously, the cooperators can't survive mixed in with defectors like that,
so the initial state is implausible. A cooperator mutant would have to arise along the
periphery of the defector community, and migrate out into the boundary region. This is
exactly what happens in this simulation, and it always happens if you wait long enough. We
use a low mutation rate of 0.005 (half of one percent) on offspring. There is no bias, the
probability of mutation is equal in both directions, and as random as a random number
generator can make it.
If you compare the time series graph to the one in step 2 (no mutation, mixed initial
population) you will notice that the number of cooperators is lower on average and at its
peaks when there is mutation. Defectors are also maintained at a somewhat higher level.
Both phenomena are apparently due to the occurrence of defectors mutants within cooperator
patch-communities.
Nonetheless, the pattern is stable, and unlike the setup in Step 2, this always happens. Step 4 investigates the emergence and stabilization of a
discriminating strategy, TitForTat.
Robustness:
So how robust is this result? The parameter values used in the above simulation are not
"cooked", but were chosen for plausibility and convenience. We can explore the
question of robustness by considering at what parameter settings the stability of
the pattern (i.e. stable population of cooperators in the boundary region) breaks down, or
fails to form at all. The first group of parameters makes a difference, the second
doesn't.
- Movement Rate: benchmark value - 1%. Workable range: 0.001 -
0.03.
- High movement rates tend to compromise the spatial isolation pattern. Defectors find
cooperators faster, and Cooperators my not stay together long enough to reap the benefit
of reciprocity by their kin. Movement much faster than reproduction (once in about 50
cycles, or 2%) is bad for cooperators though stable communities may quickly form at rates
as high as 3%. Low movement rates aid the stability of the pattern but the pattern
itself may take longer to form. This is due to the original defector community being more
compact, allowing fewer avenues of escape for cooperator mutants, and to those
mutants being less likely to take advantage of those opportunities. Movement Rates
1/10th of the benchmark (i.e. 0.001 or about every 20th reproductive generation)
result in stable configurations with around 1800 cooperators.
Presumably lower rates would generate stable populations once formed. Movement rate
is possibly the most critical parameter, and further work needs to be done which allow the
mutation and evolution of movement rates for individual lineages.
- Mutation Rate: benchmark value - 0.5%. Workable range: 0.0001 -
0.1.
- Up to rates of 10% the rough pattern remains. At 10% the cooperator population is
sparse, fluctuating between about 125 and 250 individuals. As for lower values,
theoretically, any finite mutation rate will generate the pattern eventually. In practical
terms, values of 1/1000 reliably generate the pattern in less than 10,000 cycles (about
200 baseline generations). Lower rates simply don't result in very many mutants.
- Benefit from cooperation: benchmark value: 4 units. (Cost is 1 unit.)
Workable range (benefit:cost ratio): 2 - Infinity.
- Benefits as low as 2 may generate a somewhat more fragile population of around 100
cooperators. Such small populations are much more subject to drift, or "bad
luck". High benefits tend to create pattern quickly, but result in very large
populations which make the simulations unwieldy. Crowding costs need to be increased, as
well as risk levels in order to keep the population from exploding. What is important is
the benefit to cost ratio, however, and this can be investigated by reducing the cost.
Costs as low as 0.2 points (with benefit of 4) work just fine.
These parameters don't really make a difference to the emergence and stability of the
pattern.
- Carrying Capacity in the absence of play: benchmark value 5 agents.
- Carrying Capacities of 15 agents push the boundary region out into the risk = 2.5
region, increasing the size of the defector population to around 1000. Carrying Capacities
of 30, the approximate size of common human tribal groups (increasing crowding threshold
to 30 also) work just fine as well. Note that few patches actually end up with this
many individuals, due to mortality risks.
- Crowding
- Threshold: benchmark value 20 agents. This is mostly important for Step 4 when TitForTat
reinvades the benign region, keeping its population in check. Set to carrying capacity
(=5), for instance, TFT still reduces Defect to the level maintained by mutation.Set to
50, or 100, the same pattern results.
- Cost: benchmark value 3 units. In order for crowding costs to be effective, they need to
be at least as high as the joint cooperative payoff of Benefit - Cost. Higher values
simply enforce the threshold more severely. Lower ones allow unlimited growth of
cooperative communities in the benign region.
- Spatial Granularity: There is one sense in which the simulators do need
to be "tuned". As noted in the discussion of benefit:cost ratios, the simulation
can become unwieldy (on my computer) when there are many thousands of agents. More
importantly, in order for the pattern to be stable in the long term two things seem to be
required. First, the cooperator population must not "pack" up against the outer
edges. Such populations have no "firebreaks" and defector invasions can wipe out
the entire population. Second, cooperators must occupy enough patches to even out the
various kinds of "bad luck", invasion and mortality, that cooperators must deal
with. So first, mortality rates must be high enough to prevent stable communities
from forming at the outer edges. Second, mortality rates must be low enough so that the
boundary region is large enough to allow the required number of habitable patches. A grid,
say, 500x500 which approached mortality rates of 1 might be ideal from theoretical point
of view. Maybe next year's computer will be fast enough. But until then,
tractability of the simulation remains a consideration.
Notice that the real world does not have the kind of computational
limitations of our simulator. The question that arises there is how spatially broad
boundary regions need to be to support the phenomenon we see. As far as I know, ecologists
are not even looking for this kind of phenomenon yet.
Copyright © William Harms
1999. (Author, designer, and programmer.)
Evolving Artificial Moral Ecologies Project
Centre for Applied Ethics, UBC
