Group Selection

This was inspired by last week’s Less Wrong discussion.

First of all, taboo “group selection” – it seems to be responsible for too many cached thoughts.

The game is iterated prisoner’s dilemma. There is a large population, and random individuals from the population have to play IPD against each other. Over time they will accumulate a “utility” (just the sum of all the scores from the different games they played) and this utility is exponentiated to determine the number of asexual offspring that individual will have in the next generation.

We would expect successful strategies to dominate in the population, but also that whether a strategy is “successful” depends on the mix of strategies present in the rest of the population.

The payoffs are as follows:

  D C
D 1,1 3,0
C 0,3 2,2

So on each turn, if both players defect then they both score 1. If they both cooperate they both score 2. If one cooperates and the other defects then the defector scores 3 and the cooperator scores 0. Each game lasts for 10 turns.

If this was real life then we could imagine the creatures evolving all kinds of strategy. But I’m only modelling three possible strategies:

  • Always defect
  • Always cooperate
  • Tit-for-tat. Always cooperate on the first turn and then on subsequent turns, do whatever the other player did on the previous turn.

I’ve made some modelling simplifications here:

  • There is no mutation
  • Individual interactions are not modelled. Instead I compute a matrix of how well each strategy scores against each other, then work out how the populations of different strategies will change based on the mix of other strategies.
  • The total population is held constant. (This would correspond to there being some resource which is both tightly constrained and fully utilised by the population).
  • There are discrete time steps.

So what happens?

As we know, tit-for-tat is quite a good strategy in IPD so we wouldn’t be surprised to see it dominate. But sometimes we will end up with a few “always cooperates” in the mix as well – the reason being that if everybody’s cooperating all the time, natural selection has no way of distinguishing the “always cooperate” strategy from “tit-for-tat”.

"Cooperate" starts as the largest group but quickly plummets. "Defect" rises and then falls once it runs out of suckets. "Tit-for-tat" emerges as the most successful, although a few "cooperates" remain in the mix.

But depending on the starting configuration, it doesn’t always go this way. “Always defect” has a slight advantage in a game against tit-for-tat because tit-for-tat gets clobbered on the opening move. This means that if there are too few “tit-for-tats” then their mutual cooperation isn’t enough to help them survive the mild bruising they get from the “always defects”:

The initial population of "cooperates" is quickly replaced by "defects". "Tit-for-tat" steadily declines over time.

Digital information

The universe isn’t really digital, so whenever we want to store some digital information we first need to create a system with multiple equilibria. We can store information by nudging it from one equilibrium point to another, or maybe the equilibrium isn’t perfectly stable and the system nudges itself (which we might call a “mutation”).

So the above system appears to store one bit of digital information – it can be in one of two stable equilibrium zones. I’ll develop this idea further down, although I may be reading too much into this.


Two groups

In this new model, games only take place between members of the same group. But individuals play the same number of games regardless of the size of the group. This means each group essentially operates independently – the relative sizes of the populations following different strategies will be the same as before (within each group), but the size of the group now changes depending on the success of the group as a whole. (This information was invisible in the previous examples – the group size was always limited by the fixed resource constraint and there was no information showing how hard we were bumping against that constraint).

In the first example, I start off with a small group of mostly tit-for-tat and a larger group of mostly always-defect. As expected, the cooperating group takes over completely:

The group consisting of mostly "tit-for-tat" and "always-cooperate" takes over. The group consisting of mostly "defect" dies out completely.

But this only works if the cooperating group represents an evolutionarily stable strategy. If the cooperating group is mostly “always cooperate” then it will win against the other group initially, but this advantage is unsustainable and it will end up reverting to the “always defect” state. At this point, both groups are made up of defectors and so neither will gain any more ground against each other.

The group consisting of mostly "always cooperate" does well initially but gets taken over by "always defect". At that point it doesn't lose any ground, but doesn't gain any either (since it's now essentially identical to the other group).

So all that I’ve really shown is that if two groups have settled on different evolutionarily stable strategies, the group with the “better” strategy (from the group’s point of view) will tend to dominate. This should be entirely uncontroversial – it’s right there on Wikipedia. But I’m still curious about what the implications might be, and when it might be an oversimplification.

(While I remember, there’s another thing that I’m leaving unexplained here, which is why individuals form themselves into groups, and only interact within that group, in the first place. I think that will require a more sophisticated kind of model though).

What if there’s a whole bunch of different ESSs

In the case where there’s only two possible ESSs, it’s not very exciting. While we might see “selection” of groups, we don’t see any meaningful “evolution” of groups. But if the games played between individuals were more complicated, and we allowed more sophisticated behaviours in our models, then we might see a much richer ESS search space. To the extent that you might see something like evolution happening at the group level? Possibly.

So this kind of process can’t produce groups whose individuals aren’t following an evolutionarily stable strategy. Just as individual organisms can’t evolve into something which breaks the laws of physics. Evoking this kind of “group selection” doesn’t remove the need to explain what’s happening at the individual level. In fact in general, each layer of abstraction we add means an extra thing that we need to explain. We now need to explain why this behaviour is good for individuals and why this behaviour is good for groups.

Budding

But are we so certain that we’ll never find groups that don’t exhibit an ESS? E.g. what if there were no tit-for-tats, might we find groups consisting of “always cooperate”?

It seems possible that we might do. We assume that from time to time, groups will split into two. It can do this by splitting right down the middle, or by a small group splitting off, or (if individuals are asexual) by an individual wandering off. If the group that splits off always contains the same mix of strategies as the original group, then it seems that the “always defects” will always take over.

But suppose the strategies are not uniformly mixed and the group that splits off contains a non-representative sample (which seems most likely if it’s a single individual or a very small group). Then while the proportion of “always defects” in the main group will always increase, in the little groups that split off it might get reset back to zero. (Or to 100% in some of them, but those groups won’t do so well).

Now, it should be obvious that this actually happens. Our “individuals” now are cells and our “groups” are multicellular organisms. The equivalent of “always defect” here would be cancer or (arguably) some kind of infectious agent. “Always cooperate” are healthy cells.

Sometimes organisms reproduce by large pieces falling off and turning into new organisms. But as far as I know, it’s rare for a species to only reproduce this way. (Anyone know of any examples?) Generally they go down to the single cell in order to reproduce, either sexually or asexually (I remember reading something along these lines inThe Selfish Gene so I assume the science backs this up somewhat).

So can the same thing happen with groups? If the individuals in a group are all genetically identical then the answer is a qualified “definitely yes”. The qualification is that the level of cooperation between individuals can become so great that it’s not really obvious what we should label as “group” and what we should label as “organism” (e.g. Corals(wp), Portuguese Man o’ War(wp)).

If they aren’t genetically identical (i.e. there’s sex going on and it doesn’t result in a new group each time) then it’s less obvious. Can something like this explain eusociality(wp)? I’m not sure. I’d want to model it before putting it forward as an idea, and I’m not sure how to do that yet.

Purity and running away

The other way we might get groups of “always cooperates” is if the groups keep themselves completely clean of defectors. If the number of defectors is 0 then it will stay at 0 until there is a mutation or an individual wanders in from another group. If we suppose both of those events are rare then would this be sustainable?

The problem is that when a group splits off it needs to get “far enough away” that the groups won’t accidentally share individuals. But they also need to be close enough that they can steal resources from each other (otherwise there’s no group competition at all). This could be plausible if the groups have a clear enough idea of who’s “in” and who’s “out”. A group splitting event would require a new sense of group identity.

So this one may just work but seems somehow precarious.

Conclusion

None really until I’m able to model more interesting things.

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