Finally, we consider the evolution of discriminating
mechanisms. In much of our experience as social animals, the stability of
cooperative behavior rests more on enforcement mechanisms than on spatial
segregation. It makes sense, however, that cooperation has to arise and succeed before
mechanisms of discrimination can evolve. This is why we don't allow discriminating
strategies to mutate directly from Defect, but only from Cooperate. The mutation scheme
looks like this:
| Defect <=> Cooperate <=> TitForTat |
The simplest (and most heavily studied) discriminating
cooperator strategy is TitForTat.1 Our
agents interact with their randomly assigned partners three times in a row. At each round
they have the opportunity to confer a benefit on their partener at small cost to
themselves. Cooperators always confer the benefit, defectors never do. TitForTat always
confers the benefit on the first move, and then copies the partner's previous moves
thereafter. For the three round game, with benefit worth 4 points and and a cost of 1,
this results in the payoff matrix to the right.
If you look at the submatrix that characterizes the interaction between Defect and
TitForTat (the lower right-hand four cells) you can see that TitForTat will grow faster if
there is lots of TitForTat, and Defect will grow faster (or at least survive better) if
there is mostly defect. But even if there are only a few defectors in a population of
TitForTat, they will do better than they would in a population of all defectors, and will
thus increase in numbers. (Defector communities stabilize at the patch resource
carrying capacity, 5 agents per patch in this case.) Which is to say, mixed communities of
TitForTat and Defect in the low risk region grow arbitrarily large. Not only is does this
bog down the simulator, but it seem rather unrealistic for a spatially explicit simulator.
There has to be some upward limit on how many agents can be crammed into one patch. Right? |
| Payoffs |
Cooperate |
TitForTat |
Defect |
| Cooperate |
9 |
9 |
-3 |
| TitForTat |
9 |
9 |
-1 |
| Defect |
12 |
4 |
0 |
|
So the simulator imposes crowding costs. If the number of agents in a
patch exceeds 20, each one is billed 3 units of resource per cycle. This keeps things
manageable and more realistic. It also has the effect of virtually eliminating defectors.
For TitForTat declines more slowly in crowded population dominated by TitForTat than
Defect does. It doesn't really matter how high the crowding threshold is set. TitForTat
will take it up to that level, and Defect will get frozen by the crowding cost. And
if the crowding cost is high enough to limit the growth of cooperative strategies then it
will be high enough to limit the growth of Defect. (See the discussion of Robustness.) The last defectors almost always
play against TitForTat, and get 4 points, with maybe an occasional point from the
patch's meager resource. 3 points for crowding cost, one for metabolism, they barely break
even. And eventually they wander out into the riskier regions.
Things are actually a little more complicated than that, since there may be cooperators
in the mixed patches as well. But defect tends to eliminate them rather quickly,
leaving the patch with only Defect and TitForTat. If you do the math, you will see that
the presence of cooperate does not result in the reduction of TitForTat. Cooperators are
simply replaced by Defectors. Defectors are driven down whenever
Frequency TFT > cost/[(repetitions -1)(benefit-cost)]
or in this case, when TitForTat is greater than 1/6 of the population.
The above series is not unusual. On the contrary, in running this simualation this
pattern always occurs. The only question is how long it takes.
Moreover, the phenomenon is robust over a wide variety of parameter settings.
Step 5 takes a preliminary look at what happens
when you let individual movement rates evolve via variation and selection.
Copyright © William Harms
1999. (Author, designer, and programmer.)
Evolving Artificial Moral Ecologies Project
Centre for Applied Ethics, UBC
