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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.

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great question. I hope one or all answers address group selection. Which I don't think was widely accepted at the time that the selfish gene was published. –  David Apr 29 '12 at 2:00
    
@David It still isn’t widely accepted. There are some attempts to put it on a new footing but they certainly don’t represent the majority view of the evolutionary biology crowd. –  Konrad Rudolph May 1 '12 at 10:55

2 Answers 2

up vote 10 down vote accepted

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.

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Thanks for the detailed answer :) After reading on, I found an explanation two chapters later, essentially what you said - Cheaters may exist in a small number and so may Suckers, but they would be decimated by the Cheaters again once both became significant. Dawkins admits this is not truly an ESS and instead offers what Axelrod called it, a 'collectively stable strategy'. –  Armatus Apr 29 '12 at 8:15
    
“… as the product of” is missing a few crucial words at the end. As an aside, your gibe at Dawkins’ opposition to group selection is really mis-placed. Last I looked his is still the majority position among evolutionary biologists, and this isn’t the right place for a detailed discussion of the subject, and your current formulation is putting Dawkins’ intelligence in question (“does not believe in”) and accuses him of using a sleazy argument (“[which is why] he likes this discussion so much”) which is quite inacceptable in polite conversation. –  Konrad Rudolph Apr 29 '12 at 21:57
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@Konrad Rudolph yes, but EO Wilson et al's arguments (to cite one example) are serious ones. Fortunately Science is not a democracy and alternative investigation is still allowed; There is still substantial reason for question and the best journal editorial boards still see this as a reasonable question to raise. –  shigeta Apr 30 '12 at 20:47
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@shigeta It is very clear in his book that 'Selfish genes' are not a kind of genes. He explains how all genes are 'selfish' - please read the correct section of the Wikipedia article (en.wikipedia.org/wiki/The_Selfish_Gene#.22Selfish.22_genes). I personally find his choice of word an unlucky one because it asks for misunderstandings like this. –  Armatus May 1 '12 at 18:22
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Yes, they may use the term "Selfish Gene" this way, but the conversation here was about Dawkins and as it seems, he uses the term differently (which is his good right). His theme is not that "selfish genes" exist, but that all genes are selfish, and nothing else - since all altruism/cooperation is merely a product of lower-level selfishness. I think I remember the "selfish gene" example you mentioned, but he used it only to illustrate that genes' interest does not have to be the organism's and he most definitely did not say that only this specifically was a selfish gene. –  Armatus May 10 '12 at 0:14

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.

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+1 for reducing my marriage to a for loop :) –  DVK Feb 21 '13 at 21:05
    
BTW, if you have even a single cheater (which due to random mutations you usually will end up with), "Grudgers and Suckers both have the same payoff" becomes false. –  DVK Feb 21 '13 at 21:07

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