Restructuring the model...

[davidrpugh]

A few more thoughts from @markeschaffer (moved from issue #11)...

The structure of the model is in terms of proportions of adults rather than proportions of children, which makes sense. But it makes the analysis harder because the survivorship rates of children are different for males and females.

For any given family, the children consist of boys and girls. The distribution (in expectation) of genes A, a, G and g is uncorrelated with child gender, so in the population the distribution of genotypes AG, Ag, aG and ag will be the same for the population of male children and the population of female children.

One boy from each family survives to the next generation (become the male, take over the family farm, find a mate etc.), and that boy is randomly chosen. Genotype doesn't matter. But survivorship of girls to become females (join a male at the farm etc.) depends on whether they are carrying A or a. (More specifically, it depends on the interaction of Aa and Ga in the population blah blah.)

So unless we are at an edge solution, the distributions of genotypes AG, Ag, aG and ag wil be different for the population of adult males and females.

This complicates the analysis and the graphical presentations.

First question - since the distribution of genotypes across male children is the same as for female chldren, and is the same as for male adults, should we simplify the presentation by focusing on male proportions?

Second question - should we rewrite the model and recurrence relations in terms of children rather than adults? Since the distributions of genotypes is uncorrelated with child gender, we would have 4 (and only 3 linearly independent) recurrence relations.

Come to think of it, the male recurrence relations are already the recurrence relations for male children, which means they are already the recurrence relations for female children. So maybe we can do it by imposing some conditions on the existing set of 8 recurrence relations. Intuitively, we would be imposing initial conditions that require "consistency" in the sense that the initial distribution of genotypes across genders would have to have been generated by a previous generation (and not by a programmer doing a sweep!).

[davidrpugh]

More thoughts from @markeschaffer (moved from issue #11)...

Have looked into the above further and I think it can be done. It requires more notation (sigh) but it clarifies things. The notation follows on and expands pp. 16-22 of the paper, which uses C for a "family" C (1 male + 2 females ). CGA;ga,gA is a family where the male is GA, 1 female is ga and the other is gA.

Also distinguish between children and adults by lc/uc, i.e., Mga is a male carrying ga; mga is boy carrying ga.

Denote time by t, so Mga(t) are today's adult males carrying ga, and mga(t) are the male children carrying ga. Some of mga(t) have parents who are in Mga(t), but of course Mga(t) males can have other kinds of children (depending on wives F dated t) and mga(t) children can have other kinds of fathers (depending of mothers F dated t).

Male intergenerational dynamics are extremely simple: the genome shares of today's boys are the same as the genome shares of tomorrow's fathers: Mga(t+1) = mga(t), and the same for all the other males. This is a consequence of the randomness of male survival to adulthood (which boy in the family survives to take over the family farm - see p. 2).

Female survival from girl fga(t) to wife/mother Fga(t+1) is more complicated, because of the signalling/screening structure. The way to deal with this is to avoid working with adult female proportions F entirely, and also to make use of the fact that because inheritance of G and A is uncorrelated with gender, fga(t) = mga(t) and the same for all the other genome combos.

It's possible (I think) to write out the recurrence relations in terms of proportions of male children m only. We don't need to work with female children, since fxx(t)=mxx(t), and we don't need to work with adult males, Mxx(t+1)=mxx(t).

The reduction in dimensionality comes at very little cost, and indeed is probably sensible. In effect, the reduction in dimensionality is via initial conditions. We are imposing the constraint that comes from the uncorrelatedness of gender, i.e., that the distribution among children of AG, ag, etc. is the same for male children m and female children f. In the current model, this can happen for the initial period only, if we choose initial conditions that violate it. It is guaranteed to hold in periods 2 onwards. Simplifying in this way means it is guaranteed to hold in period 1 as well.

In the meantime, the implications for the analysis so far is that initial conditions that are pure subpopulations should give either exactly the same answers as in the reduced-dimensionality verison, or almost exactly the same answers (I have to check). The root-solver results are (probably) also unaffected. And for now (until I work out the reduced-dimen version) we should probably examine the results in terms of male proportions rather than female proportions. It won't make any difference but we should get used to it, I guess.

UPDATE by MS: the above is at least partly wrong, I think. Will come back to this.

[markeschaffer]

I'm thinking about notation changes that could help with the exposition and marketing of the paper. What do you think about the following? (David - can Github handle embedded latex?)

Index genotypes by i=1..4 (AG, Ag, aG, ag).

Indices below are i, j, k, and x

U(t) is the population share of a (geno)type of family "unit" at time t (better than "C", which makes me think of "children")

M_i(t) and F_i(t) are the population share of adult males and females of genotype i, respectively

m_i(t) and f_i(t) are the population shares of children of genotype i, before any selection effects, dating games, whatever

Subscripts i, j, k, on U_ijk(t) indicate the genotype of the male, female 1 and female 2, respectively. NB: this is wasteful because it distinguishes unnecessarily between the two females. We have to track combinations, not permutations. It would become useful if we want to introduce such distinctions, e.g., we want family hierarchies (senior female, junior female, and similarly for males if we ever move to a 2M2F setup.

p(i,j,k,x) = probability that a child in a family unit with male of genotype i, female 1 of genotype j and female 2 of genotype k has genotype x. For example,

p(1,1,1,x) = 1 for x=1, p(1,1,1,x) = 0 otherwise

because p(1,1,1,1) = p(AG, AG, AG, AG) and if all possible parents are AG, then all the children are AG.

Payoffs Pi are indexed by genotypes j,k=1..4: (NB: this is also wasteful notation, but let's keep it for now.) Following notation in the paper,

Pi_jk is the payoff to female #1 of genotype carrying genotype j when paired with a female #2 carrying genotype k.

The above should mean that we can write out large parts of the recurrence relations using summation notation and single equations instead of equations for all the genotypes. For example

Sum_i Sum_j Sum_k U_ijk(t) * p(i,j,k,x)

is M_x(t+1), the share of adult males of genotype x in generation t+1. (I think....)

[markeschaffer]

Made some progress on rewriting the model. If you read the first draft of this comment, ignore the bit about the model being wrong - the model is fine, it's just the writeup in the paper that's misleading.

Modification of notation above:

p(x|i,j) = probability of genotype x resulting from the mating of male of genotype i and female of genotype j. (Of course, p(x|i,j) = p(x|j,i). It doesn't matter whether genotype i is male and j is female or visa-versa.)

Finally, write the mating probabilities in terms of genotypes instead of specific genes:

S(i,j, . ) = probability that a male of genotype i gets a female of genotype j as a mate. This is exactly what we have now, but with repetition, e.g., S(1,1)=S(2,1)=S(1,3)=S(2,3) by definition. (Because 1=GA and 2=Ga, males of type=1 have the same S as males of type 2, since males express only G and not a.) The 3rd argument in S(i,j,.) is shorthand for all the shares of female girls in the adoption pool at time t.

Here is the expression for M_x(t+1), the share of males of genotype x in generation t+1:

M_x(t+1) = Sum_i Sum_j Sum_k U_ijk(t) * { p(x|i,j)_P_jk/(P_jk + Pi_kj) + p(x|i,k)_P_kj/(P_jk + Pi_kj) }

Note that m_x(t+1)=M_x(t+1), i.e., the shares of boys and men are identical. Total payoffs for an individual family = P_jk + P_kj. We weight p(x|i,j) by the share that female j has in the family's total payoffs, and we weight p(x|i,k) by the share that female k has in the family's total payoffs.

Denote by C(t+1) the number of girls produced by the females in generation t. This is just half the total payoffs, since half turn into boys (m_x(t+1)) and half turn into girls who enter the t+1 adoption pool.

C(t+1) = (1/2) * Sum_i Sum_j Sum_k U_ijk(t) * [Pi_jk + Pi_kj]

The shares of girls that go into the t+1 adoption pool are:

f_x(t+1) = Sum_i Sum_j Sum_k U_ijk(t) * { p(x|i,j)_P_jk + p(x|i,k)_P_kj } / C(t+1)

NB: we could alternatively use f_x(t+1) for the total NUMBER of girls in the adoption pool and remove C(t+1).

And the family shares for period t+1 are:

U_ijk(t+1) = Sum_i Sum_j Sum_k M_i(t_1) * S(i,j, .) * S(i,k, .)

where the shares (or numbers) of girls in the adoption pool appear in the S() functions.

That's the entire model.

...at least, I think that's what the model is. Will review tomorrow.

[PaulSeabright]

sounds good to me!

2014-08-13 2:39 GMT+02:00 markeschaffer notifications@github.com:

Made some progress on rewriting the model. The good news is that the parsimony is there. The bad news is that this has brought out what I think is a strange feature (alt: a mistake) in the existing model. The good news is that fixing it isn't hard, and the new notation makes this soooo much clearer.

Modification of notation above:

p(x|i,j) = probability of genotype x resulting from the mating of male of genotype i and female of genotype j. (Of course, p(x|i,j) = p(x|j,i). It doesn't matter whether genotype i is male and j is female or visa-versa.)

Finally, write the mating probabilities in terms of genotypes instead of specific genes:

S(i,j) = probability that a male of genotype i gets a female of genotype j as a mate. This is exactly what we have now, but with repetition, e.g., S(1,1)=S(2,1)=S(1,3)=S(2,3) by definition. (Because 1=GA and 2=Ga, males of type=1 have the same S as males of type 2, since males express only G and not a.)

Then here is what the current model has as the expression for M_x(t+1), the share of males of genotype x in generation t+1:

M_x(t+1) = Sum_i Sum_j Sum_k U_ijk(t) * { p(x|i,j) + p(x|i,k) }

And here is what I think is the correct (or at least more sensible) version:

M_x(t+1) = Sum_i Sum_j Sum_k U_ijk)t) * { p(x|i,j)_P_jk/(P_jk + Pi_kj) + p(x|i,k)_P_kj/(P_jk + Pi_kj) }

The old version uses the sum of the ps; the new version uses a weighted sum of the p's.

The mistake (or at least peculiarity) in the current model is described on p. 6. Here is what we say now:

"Each "family" has a single surviving male child that grows to adulthood who is randomly chosen from all the male children. ... This means that the fecundity of the mother is irrelevant. The probability that the surviving male child will be drawn from one of the possible 4 genotypes depends only on the distribution of these genotypes among the mother’'s children, and not on the total number of children she produces."

This is wrong, or at least weird. The number of male children of female 1 should depend on her fecundity, and similarly for female 2. If female 1 is an altruist, and female 2 is selfish, then it is more likely that one of female 2's sons will inherit the farm. In the extreme case that Pi_Aa=0 (old notation) then it is impossible for one of the altruist's sons to inherit the farm, because her payoff is zero and she won't have any kids!

That's why the 2nd version makes more sense. Total family payoffs = P_jk + P_kj. We weight p(x|i,j) by the share that female j has in the family's total payoffs, and we weight p(x|i,k) by the share that female k has in the family's total payoffs.

And the good news is that the notation is so much more transparent that it's easy to spot things like this. Programming it will be very easy. Maybe there's even a closed-form solution lurking in there.

— Reply to this email directly or view it on GitHub https://github.com/davidrpugh/population-ecology-approach/issues/13#issuecomment-51996778 .

Toulouse School of Economics, Manufacture des Tabacs, 21 allée de Brienne, 31015 Toulouse Cedex 6, France www.tse-fr.eu

Institute for Advanced Study in Toulouse www.iast.fr

Personal website:

http://paulseabright.com/

http://press.princeton.edu/titles/9169.html

[markeschaffer]

Some thoughts on a possible version of the model/paper for a biology outlet:

We should look at a 1M1F pure sexual selection version. The 1M2F altruism version is a natural extension of the sexual selection structure and biologists would want to see both. Economists would be interested only in the altruism version.

Note that in the wild, sexual selection is usually practised by females, i.e., females carry the Gamma gene and males express the theta gene. But let's keep it as it is for now.

The 1M1F version is a simplified version of the existing model. In the new format, it's very easy to write out. Female payoffs are now a simply function of whether they are carrying A or a. Let's say Pi(A) >= Pi(a). Pi(A)>Pi(a) is the case where the behaviour has an impact on fitness (so a connection with natural selection). Pi(A)=Pi(a) is the case where the behaviour is completely neutral, e.g., different appearance/colouring/mating sound.

The 1M1F model should show that multiple SS edge equilibria are possible even in the strict inequality Pi(A)>Pi(a) case. A trait that is harmful to fitness (a) and that natural selection would normally drive out can persist via support from sexual selection. (Not sure about interior equilibria.)

In the 1M1F model we could possibly relate A vs. a to screening. A big difference between A and a is visible to males. A peacock with a big tail gets payoff small a (growing the tail is costly); a peacock with a big tail gets payoff big A.

Another angle to explore in the 1M1F model could be speciation. If an interior equilibrium exists with high assortativity (sp?), i.e., GA males and females mostly mate with each other, ga males and females mostly mate with each other, then we have the conditions for speciation caused by sexual selection.

The current model is rigged to show how altruism can appear via exogamy of females. In the current setup, competition between male children is entirely intra-family. We can loosen this so that males, like females, compete against the rest of the male population ("playing the field"). Using the family farm metaphor, we change competition to inherit a farm from competition between brothers for a single farm vs. competition among all males for all the farms. The effect is that a high female payoff not only sends out lots of female children looking for mates, but also sends out lots of males looking for territory.

[PaulSeabright]

thanks, will try and give some thought to this before our skype on Tuesday.

2014-08-17 23:18 GMT+02:00 markeschaffer notifications@github.com:

Some thoughts on a possible version of the model/paper for a biology outlet:

We should look at a 1M1F pure sexual selection version. The 1M2F altruism version is a natural extension of the sexual selection structure and biologists would want to see both. Economists would be interested only in the altruism version.

Note that in the wild, sexual selection is usually practised by females, i.e., females carry the Gamma gene and males express the theta gene. But let's keep it as it is for now.

The 1M1F version is a simplified version of the existing model. In the new format, it's very easy to write out. Female payoffs are now a simply function of whether they are carrying A or a. Let's say Pi(A) >= Pi(a). Pi(A)>Pi(a) is the case where the behaviour has an impact on fitness (so a connection with natural selection). Pi(A)=Pi(a) is the case where the behaviour is completely neutral, e.g., different appearance/colouring/mating sound.

The 1M1F model should show that multiple SS edge equilibria are possible even in the strict inequality Pi(A)>Pi(a) case. A trait that is harmful to fitness (a) and that natural selection would normally drive out can persist via support from sexual selection. (Not sure about interior equilibria.)

In the 1M1F model we could possibly relate A vs. a to screening. A big difference between A and a is visible to males. A peacock with a big tail gets payoff small a (growing the tail is costly); a peacock with a big tail gets payoff big A.

Another angle to explore in the 1M1F model could be speciation. If an interior equilibrium exists with high assortativity (sp?), i.e., GA males and females mostly mate with each other, ga males and females mostly mate with each other, then we have the conditions for speciation caused by sexual selection.

The current model is rigged to show how altruism can appear via exogamy of females. In the current setup, competition between male children is entirely intra-family. We can loosen this so that males, like females, compete against the rest of the male population ("playing the field"). Using the family farm metaphor, we change competition to inherit a farm from competition between brothers for a single farm vs. competition among all males for all the farms. The effect is that a high female payoff not only sends out lots of female children looking for mates, but also sends out lots of males looking for territory.

— Reply to this email directly or view it on GitHub https://github.com/davidrpugh/population-ecology-approach/issues/13#issuecomment-52435406 .

Toulouse School of Economics, Manufacture des Tabacs, 21 allée de Brienne, 31015 Toulouse Cedex 6, France www.tse-fr.eu

Institute for Advanced Study in Toulouse www.iast.fr

Personal website:

http://paulseabright.com/

http://press.princeton.edu/titles/9169.html

[markeschaffer]

Something important to think about: the additional condition used in the iterated PD that

2*Pi_AA > (Pi_aA + Pi_Aa)

as well as the standard PD requirement that

Pi_aA > Pi_AA > Pi_aa > Pi_Aa

The additional condition above is used in the iterated PD to rule out strategies where the players maximize their payoff by taking turns exploiting each other.

Our new payoff structure (10>3>2>0) violates this, because 2*3 < (10+0).

This is important, because these payoffs mean that from the male perspective, he will have the most children if he has one selfish wife and one cooperative wife. So good signalling and good screening can backfire for the males, because assortativity is bad. If, say, the adoption pool is 50% A and 50% a, then male is best off - will maximize his number of children - with random screening.

Better is probably if our benchmark case is satisfies the additional condition, e.g.,

5 > 3 > 2 > 0 and 2*3 > (5+0)

so that the male payoff in terms of expected offspring is maximized with two cooperative wives.

[PaulSeabright]

5>3>2>0 won't work because it's back to the same problem, 5-3 = 2-0, thus the relative payoff to selfishness versus altruism is independent of the number of selfish people in the population, so no returns to scarcity.

I think 5>3>1>0 would work

2014-08-18 14:31 GMT+02:00 markeschaffer notifications@github.com:

Something important to think about: the additional condition used in the iterated PD that

2*Pi_AA > (Pi_aA + Pi_Aa)

as well as the standard PD requirement that

Pi_aA > Pi_AA > Pi_aa > Pi_Aa

The additional condition above is used in the interated PD to rule out strategies where the players maximize their payoff by taking turns exploiting each other.

Our new payoff structure (10>3>2>0) violates this, because 2*3 < (10+0).

This is important, because these payoffs mean that from the male perspective, he will have the most children if he has one selfish wife and one cooperative wife. So good signalling and good screening can backfire for the males, because assortativity is bad. If, say, the adoption pool is 50% A and 50% a, then male is best off - will maximize his number of children - with random screening.

Better is probably if our benchmark case is satisfies the additional condition, e.g.,

5 > 3 > 2 > 0 and 2*3 > (5+0)

so that the male payoff in terms of expected offspring is maximized with two cooperative wives.

— Reply to this email directly or view it on GitHub https://github.com/davidrpugh/population-ecology-approach/issues/13#issuecomment-52485490 .

Toulouse School of Economics, Manufacture des Tabacs, 21 allée de Brienne, 31015 Toulouse Cedex 6, France www.tse-fr.eu

Institute for Advanced Study in Toulouse www.iast.fr

Personal website:

http://paulseabright.com/

http://press.princeton.edu/titles/9169.html

[markeschaffer]

Another idea: M-F cooperation in the 1M1F model. The male chooses his mate according to his Gamma, and then he and his mate forage with payoffs based on their thetas.

This relates to the point above about the extra condition in the iterated PD. What will be important now is the size of the joint payoff, since the number of offspring will be based on the joint payoff.

A feature here is that, compared to the 1M2F model, the G genes have a harder time getting coordination between two A players, because the G gene can reside on a Ga male.

Whether the G gene is unambiguously better off with an A mate (will have a bigger combined payoff => more offspring) depends on the size of the joint payoff.

If the iterated PD condition is satisfied, then a GA male is best off finding an A female (follows directly from the iterated PD condition).

But for a Ga male to be unambiguously better off finding an A female requires another condition:

(Pi_aA + Pi_Aa) > 2*Pi_aa

So for G males to always maximize their fitness by choosing an A female requires this:

2_Pi_AA > (Pi_aA + Pi_Aa) > 2_Pi_aa

And the same applies to g males - a g male will always maximize fitness by choosing an A female if the above condition is satisfied. So if G males prefer As, and g males prefer as, then the above condition will support the evolution of M-F cooperation.

If the condition isn't satisfied, maybe it gets interesting...

[markeschaffer]

Something else related is the 2M-2F case. If you have an internal hierarchy so that only one male gets to choose who enters the group, it's similar to the 1M-2F setup. If it's a democratic or devolved setup so that the males choose their mates separately, then you run into coordination issues. If it's an internal hierarchy, the senior male can get coordination by always aiming to get two A females if he's G (or 2 a females if he's g). In the democratic setup, you will get coordination failures if one male is G and one male is g.

[markeschaffer]

Some more thoughts on payoffs for the 1M2F case:

I think we should look at 2 cases, one where the iterated PD (IPD?) condition,

2*Pi_AA > (Pi_aA + Pi_Aa)

is NOT satisfied (as with the current payoffs, 10>3>2>0 (?)), AND one where it is satisfied (as with 5>3>1>0).

The reason is that the case where it isn't satisfied is still interesting. Taking the perspective of the gatekeeper gamma gene, the number of children of the male depends on the total (joint) payoff of the females. If the condition is not satisfied, the number of children of the male is maximised when he has a mix of A-a wives - one to be exploited and one to do the exploiting. So we have a novel mechanism for supporting cooperation - a polymorphism that can generate an internal female hierarchy with a large joint payoff. Whereas in the IPD this is supported by reciprocity (the females take turns exploiting each other), here it is supported by males who can maximise the number of their offspring by setting up an exploitative hierarchy. (Reminds me a bit of social insects, actually.)

[PaulSeabright]

Yes, this is indeed an interesting case to look at - with a natural social insect interpretation. More generally cooperative breeding species are appropriate here - meerkats for example

2014-08-20 14:41 GMT+02:00 markeschaffer notifications@github.com:

Some more thoughts on payoffs for the 1M2F case:

I think we should look at 2 cases, one where the iterated PD (IPD?) condition,

2*Pi_AA > (Pi_aA + Pi_Aa)

is NOT satisfied (as with the current payoffs, 10>3>2>0 (?)), AND one where it is satisfied (as with 5>3>1>0).

The reason is that the case where it isn't satisfied is still interesting. Taking the perspective of the gatekeeper gamma gene, the number of children of the male depends on the total (joint) payoff of the females. If the condition is not satisfied, the number of children of the male is maximised when he has a mix of A-a wives - one to be exploited and one to do the exploiting. So we have a novel mechanism for supporting cooperation - a polymorphism that can generate an internal female hierarchy with a large joint payoff. Whereas in the IPD this is supported by reciprocity (the females take turns exploiting each other), here it is supported by males who can maximise the number of their offspring by setting up an exploitative hierarchy. (Reminds me a bit of social insects, actually.)

— Reply to this email directly or view it on GitHub https://github.com/davidrpugh/population-ecology-approach/issues/13#issuecomment-52771815 .

Toulouse School of Economics, Manufacture des Tabacs, 21 allée de Brienne, 31015 Toulouse Cedex 6, France www.tse-fr.eu

Institute for Advanced Study in Toulouse www.iast.fr

Personal website:

http://paulseabright.com/

http://press.princeton.edu/titles/9169.html

[markeschaffer]

Something else we need to look at is the functional form of the S(.) matching function. The reason is the "depletion" issue that is raised on p. 4 of the old draft.

"Signalling" means females separate into 2 adoption pools; "screening" means males then choose from their preferred adoption pool.

The "depletion" problem is that there may not be enough females in a pool for the males who prefer that pool to get mates.

I think this is connected to an odd feature of the current functional form. We can generate the depletion problem by setting dA=1 and da=0, so that all females choose to be in the A pool, and the a pool is empty.

Intuitively, this should make it harder for the g males to find their preferred mates than the G males. Indeed it does. But bizarrely, the current S(.) function implies that S(g,A)=1 > S(G,A).

Put another way: when dA=1 and da=0, g males are 100% (in)accurate at getting A females as mates, regardless of their screening ability, whereas G males get matched with their preferred A mates less than 100% of the time.

[edited to remove typo about mimicry]

[markeschaffer]

Just had a nice thought on how to characterise our model in terms of traditional Darwinian models etc.

1. Is there a term for sexual selection (=mate selection) generalized to group membership as we do?
2. It can't be called "group selection", since that term is already used.
3. "Group selection" is, roughly speaking, selection pressure that operates at the group level. It's a messy debate, and my personal view is that the best way to think about it is that it's like saying the same model can be written in structural or reduced form. Structural form is group selection; reduced form is individual selection. It's just packaging.
4. And the nice thought: we rule out "group selection", or - in our model - "family selection". Competition between males to take over the family farm is entirely INTRA-family competition, (competition between brothers).

[markeschaffer]

And here is a link to a new article in Nature that analyzes the the INTER-family competition in chimps and bonobos that we are ruling out:

http://www.nature.com/nature/journal/v513/n7518/full/nature13727.html

Abstract:

Observations of chimpanzees (Pan troglodytes) and bonobos (Pan paniscus) provide valuable comparative data for understanding the significance of conspecific killing. Two kinds of hypothesis have been proposed. Lethal violence is sometimes concluded to be the result of adaptive strategies, such that killers ultimately gain fitness benefits by increasing their access to resources such as food or mates1, 2, 3, 4, 5. Alternatively, it could be a non-adaptive result of human impacts, such as habitat change or food provisioning6, 7, 8, 9. To discriminate between these hypotheses we compiled information from 18 chimpanzee communities and 4 bonobo communities studied over five decades. Our data include 152 killings (n = 58 observed, 41 inferred, and 53 suspected killings) by chimpanzees in 15 communities and one suspected killing by bonobos. We found that males were the most frequent attackers (92% of participants) and victims (73%); most killings (66%) involved intercommunity attacks; and attackers greatly outnumbered their victims (median 8:1 ratio). Variation in killing rates was unrelated to measures of human impacts. Our results are compatible with previously proposed adaptive explanations for killing by chimpanzees, whereas the human impact hypothesis is not supported.

[davidrpugh]

Done results merged to master.