Mimicry as a mechanism for supporting cooperation?

[davidrpugh]

@markeschaffer, @PaulSeabright

Does it make sense to consider cases where dA=da=0? In this extreme case "altruistic" females mimic "selfish" females and vice versa. From the paper, the matching probabilities reduce to...

SGA = Sga = 0
SGa = SgA = 1

...and are independent of male screening ability. My simulations are indicating that, in this extreme case, the steady state equilibrium is unique with mgA=fgA = 1.0 irrespective of initial conditions.

Does this result conform with your intuitions? Is it an interesting or unexpected result?

[markeschaffer]

It's not surprising.

                SGa = SgA = 1

means that little g males always gather together 2 big A females. So you get 100% cooperation, perversely via the mechanism that little g males are promoting it. I think it's an artefact of the functional form that male screening ability is irrelevant.

...and which is a good reason why we should think a bit about the functional form before exploring mimicry. Definitely worth looking into, but I suspect the functional form will need to change

[davidrpugh]

Is it interesting? If not, would it be interesting if dA=da=0.1 so that there was imperfect mimicry by both female types? My simulation results are indicating that cooperation/altruism is the dominant outcome in this case as well.

[markeschaffer]

Prob need a different functional form, I think.

[davidrpugh]

Prob need a different functional form, I think.

What do you mean by this? Which functional form are you referring to?

[markeschaffer]

The functional forms of the S functions on page 5. But I think maybe I'm wrong, in that perfect signalling (d=1) and perfect mimicry (d=0) are symmetric.

Perfect signalling: dA=da=1, so SGA=Sga=1, G-men always get 2 A-wives, and g-men always get 2 a-wives. The larger payoff to cooperative AA families means their female children dominate the adoption pool, the A gene spreads, and the steady-state equilibrium should be 100% A, with mostly (entirely?) G men.

Perfect mimicry: dA=da=0, so SGa=SgA=1, g-men always get 2 A-wives, and G-men always get 2 a-wives. The larger payoff to cooperative AA families means their female children dominate the adoption pool, the A gene spreads, and the steady-state equilibrium should be 100% A, with mostly (entirely?) g men.

[markeschaffer]

And for the record...

Mixed perfect signalling/mimicry (1): dA=1, da=0, so SgA=1, SGA=fA/(fA+(1-eA)fa) (<1) and SGa=1-SGA. A-females signal perfectly, a-females mimic As perfectly, but whereas g-men always get 2 A-wives (mimicry is 100% effective), G-men are hampered by their imperfect screening (eA) and sometimes get a-wives.

Mixed perfect signalling/mimicry (2): da=1, dA=0, so SGa=1, Sga=fa/(fa+(1-ea)fA) (<1) and SgA=1-Sga. a-females signal perfectly, A-females mimic as perfectly, but whereas G-men always get 2 a-wives (mimicry is 100% effective), g-men are hampered by their imperfect screening (ea) and sometimes get A-wives.

[davidrpugh]

The functional forms of the S functions on page 5. But I think maybe I'm wrong, in that perfect signalling (d=1) and perfect mimicry (d=0) are symmetric.

I claim that perfect signaling and perfect mimicry are not symmetric. Specifically when dA=da=1.0, any initial condition of the form mGA = 1 - mga so that mGa=maG=0 will self replicate and therefore constitute an equilibrium. Equilibrium shares for females can be derived from recurrence relations as follows.

fGA = mGA * (PiAA / (mG * PiAA + mg * Piaa)) < 1
fga = 1 - fGA > 0

Such a setup would constitute an interior equilibrium! Put differently, starting from "pure" sub-populations, perfect female signaling can support the coexistence of altruism and selfishness.

So far as I can tell, out numerical simulations are consistent with my derivation (which means that we now have our first unit test...and our code passes!).

[markeschaffer]

Hmm... something not quite right here (?). If the initial condition is of the form mGA=1-mga but there are all 4 kinds of females (fGA, fGa, fgA and fga), then some of the mGA males will get fgA wives, and therefore will have a mix of mGA and mgA sons. Some of these mgA sons will make it to the next generation. Are you assuming the same initial conditions for men and women, so that mGA=1-mga and fGA=1-fga? Then it is indeed self-replicating, because mGA males will always get fGA wives and the subpopulations will breed true.

[davidrpugh]

@markeschaffer

Are you assuming the same initial conditions for men and women, so that mGA=1-mga and fGA=1-fga?

I am indeed assuming the the initial condition for female shares is the same as the initial condition for male shares. This has been a maintained assumption in all simulations that I run. I was under the impression, from our previous discussions, that there are good biological reasons for imposing this restriction. Is that not the case? Should we consider relaxing this requirement?

Regardless, I am pleased that we now have some analytic results that can be used as unit tests for the code.

[markeschaffer]

And here's the symmetry. If we have perfect mimicry, so dA=da=0, then any initial condition of the form mGa=1-mgA and fGa=1-fgA so that mGA=mga=fGA=fga=0 should be self-replicating. The mGa males will always get fGa females, and the mgA males will always get fgA females, and again the subpopulations breed true.

[markeschaffer]

For the moment, let's stick with initially pure subpopulations and ignore mimicry. Let's first see if I can rewrite the model sensibly so that initial conditions provide additional constraints.

[davidrpugh]

OK. Sounds good.