Why is 'Grudger' an evolutionary stable strategy?

Question

I am currently reading 'The Selfish Gene' by Richard Dawkins, which I am sure many here have read. The topic are evolutionary stable strategies (ESS) regarding cooperation.

I apologise for the long question. If you are already familiar with the topic and Dawkins' model of Cheat, Sucker and Grudger: my question is, how can Grudger be an ESS if it could be invaded both by Suckers (because they have no disadvantage against Grudger) and Cheats (because a Cheat minority is unlikely to meet the same Grudger twice, turning Grudger into Sucker effectively)?

More detailed:

The model

Near the end of chapter 10 (p 185 in my version), Dawkins uses a model of birds who clean each other of parasites, therefore helping in survival (as cleaning themselves they cannot reach every spot of their body). He defines three different behaviours for the model:

• Sucker - birds who indiscriminately help and clean other birds
• Cheat - birds who let others help them but never do so themselves
• Grudger - birds who help others and remember who they helped. If the same bird does not help them later (reciprocate), they will not help that bird again.

Claim: Cheat and Grudger are ESS

He claims that both Cheat and Grudger in themselves are ESS - that is, if all birds behave this way, none of the other behaviours can develop because they will be immediately penalised by lower chances of reproducing.

The part that makes sense: Suckers is not an ESS, Cheat is

Sucker is of course not an ESS. If all birds were Suckers, any Cheat that developed would have a huge reproductive advantage and Cheat genes would overtake the population.

Being an ESS makes sense for Cheat. If all birds cheat, nobody will ever be helping each other. A minority of Suckers would be spending all their time helping and not getting anything in return, Cheats have the advantage and Suckers die out again. Grudger would be unlikely to meet a Cheat who they helped before again, so they too will spend all their time helping and die out again.

The part that confuses: Grudger is an ESS?

But Dawkins also claims that Grudger is an ESS, and he seems very confident in that. Now I don't consider myself enough of a smartypants to claim that he's wrong, but I don't understand how Grudger can be an ESS. If all birds behave in this way, and for any reason some Sucker developed - the Sucker would have no disadvantage. All birds would still always be helping each other, so nothing would stop the Suckers from propagating equally well as the Grudgers, invading the gene pool. That's already the ESS broken, but even further, the presence of Suckers would mean that if Cheats came up, they would have a realistic chance of surviving - Grudgers would shun them after having helped once, but if the number of Suckers is large enough, Cheats will have an advantage.

Moreover, back to the initial setting of Grudgers only - if a Cheat developed, he would be unlikely to meet the same Grudger twice, receiving the benefit all the time but never paying the cost. He would have an advantage and spread Cheat genes.

The problem

I'm not familiar enough with how these kinds of models are calculated in order to state chances that Cheats will take over completely, but however I think of it Grudger does not seem to be an ESS to me.

Does anyone have an explanation why Dawkins is so sure that it is? Seeing as in nature we do see patterns like Sucker and Grudger all the time, I must be missing something important here.

Answer 1

Unfortunately it is not necessary to invoke group selection to answer this question. This is one of the reasons that Dawkins likes this discussion so much - he does not believe in group selection and so the discussion in SG does not invoke group selection. ESSs are described in the book as the product of direct competition or interaction between genes.

ESSs in this case, can be described in terms of game theory. In the famous Prisoner's Dilemma experiment, Grudger is similar to tit for tat, which 'won' the competition in the original Axelrod contest.

To see how this works you make a simple win/loss game matrix:

     G        NG
G    win 1    win 3
NG   lose 3   lose 1

if you are groomed you win 1, if you are groomed, but you don't have to groom - even better win 3 (say) If you groom but are not groomed, lose 3 If neither of you are groomed, you both lose 1

one might argue the exact proportions, but the point is that getting some thing for nothing is better than reciprocating, and getting nothing for your efforts and time are a loss, because you could have been getting groomed by someone else. As you can see cheaters end up in the top row all the time. grudgers end up along the diagonal, and once in a while in the lower left, Suckers get stuck in the lower left a lot whenever there is a cheater around.

now run this encounter over and over. A behavior which scores negative the more times you run is not stable - they are going to disappear from the population, at least if this disadvantage is real

It has more than one stable outcomes in populations, a population that is full of Grudgers will all groom each other as before you know everyone, you assume they will reciprocate. Everyone wins!

Any invading Cheaters will quickly be at a disadvantage, in that they will not be groomed more than N times where N is the number of grudgers in the community. Note that there is an equilibrium here - the Cheaters may exist in a small number - when N is large enough for a cheater to get enough grooming to make a 'living'.

Suckers can also exist within a population of grudgers, but a population of Suckers where Cheaters show up are quickly sucked dry by the cheaters over several generations where you tally up 'points' and give more, healthier offspring to high scorers. They are not ESS stable.

Cheaters are also stable - nobody ever wins, but they don't lose big either and any invading grudgers can't get groomed.

Answer 2

In an infinite, well mixed population with single pairwise encounters, Grudger is indeed not an ESS. In fact, as you correctly note, in such a model the Grudger and Sucker strategies are indistiguishable, as the probability of anyone encountering the same individual twice is zero.

To make it possible for the Grudger strategy to survive against invasion by Cheaters, we must somehow extend the model to allow pairs of individuals to meet more than once. Some ways to achieve this include:

• Finite population size: if there are n individuals and they each participate on average in m encounters during their lifetime (or during the average time for which their memory persists), then each of them will encounter every other individual m / (n−1) times on average.

• Viscous population: this is a general term for populations that are not well mixed. For example, if individuals live on a spatially extensive landscape, have limited movement rates and interact only with nearby individuals, then two individuals that meet once have a higher probability of meeting again due to spatial proximity.

• Iterated encounters: in these types of models, pairs of individuals are assumed to interact with each others some (fixed or random) number of times before parting and finding new partners to interact with. In this way, repeat encounters can be included even in infinite, well mixed population models. While this may be a reasonable approximation in some cases (e.g. for models of spousal cooperation in serially monogamous species), frankly the main reason for studying such models seems to be that they're mathematically simpler than finite or viscous populations.

Not entirely coincidentally, many of these mechanisms can also permit the survival of pure Sucker or altruist strategies against invasion by Cheaters through group and/or kin selection (or more general forms of assortment).

Ps. Even with these mechanisms, Grudger will never be a strict ESS anyway, since in any population consisting of only Grudgers and Suckers both have the same payoff.