Is there a way of analytically determining the minimum length of time when cooperation becomes evolutionarily stable? Put another way… what does “long-term” mean?
Your analyses of robustness of intrinsic growth differences… Am I understanding this right that an optimum value of 𝛼 can be determined that for entities growing at different rates and that each entity would have a different optimum value of 𝛼 as a result?
We don’t have an analytical solution for the value of T critical (that is the minimum time horizon you should have in was for cooperation to be stable). We have only numerical solutions that, as one could have guessed, show a dependence on sigma (the noise amplitude). This implies that the higher the noise the shorter will T critical be.
For the second: As long as the difference in intrinsic growth is smaller than the noise amplitude (sigma^2), the evolutionary stable alphas tend to infinity, for both individuals. They tend to infinity but at the same time they keep a precise ratio in order to be stable (like two non-parallel lines with positive coefficients do).
Always remember this is a model, probably in real life they wouldn’t really reach infinity. They would probably reach some finite values that satisfy that ratio (so both positive but different).
Thank you Lorenzo! I would like to build an intuition for these principles. For example – the idea that higher noise leads to a shorter T-critical is counter-intuitive to me at first, but makes sense if I think about it.
Great talk, Lorenzo, and quite understated. I like it!
I wanted to ask not just you but maybe also Athena and others if the following characterization makes sense to you. The Nowak paper is sort of a standard reference of perhaps the standard view on cooperation, and I think at least one way of seeing it is that we’re doing something totally different from Nowak. Here is why I think that.
Nowak starts essentially with the following setup. Two people can choose to cooperate or not. If they cooperate, we give them a “cooperation benefit.”
He then goes on to list different ways in which this cooperation benefit we magicked up can find its way back to the donor so that both donor and recipient benefit.
My view of this has always been that it’s too ad-hoc. I mean: it’s valid, you can set things up this way, and you can ask the questions Nowak asks. But what I wanted to point out in the paper with Alex was just how ubiquitous our specific cooperation benefit is, the generic way of producing this benefit, and what its properties are.
In the Nowak setup, I can make the cooperation benefit as big as I want, and I don’t have to explain where it came from.
Our setup specifies where the benefit comes from (it arises in any multiplicative setup, and all evolutionary (all living) setups are multiplicative). There can be other benefits but the one we study is very fundamental. Because we know where it comes from and what it means, physically, we can study its properties and make predictions based on it.
I think that’s the difference I wanted emphasize more: yes, Nowak-type benefits by fiat are one way of modelling but I think they make any result quite ad-hoc, and it’s very much “you get out what you put in.” Our approach is not arbitrary, not benefit by fiat, and you get results out which were not at all obvious at the outset.
From an analogous point of view, you can look at organisations like cooperatives and examine the rules that govern exchange of benefits between members.
These rules are maintained by committees and can become quite complex and ad-hoc in the same way that you describe the Nowak-style benefits.
All the while, there is a deeper principle operating that is benefitting all members. But with all the attention on the written down rules and ad-hoc benefits, the deeper principle goes unnoticed.
My interest is whether it would be possible to maximise the “deeper” growth benefit while keeping the organisation overheads and ad-hoc arrangements to a bare minimum.
Question prompted by James King’s talk but maybe more answerable by Lorenzo: For a fixed number of agents, is there any cooperative advantage to hierarchical cooperation. Non-hierarchical would be just pool equally between everyone in the population, hierarchical would be group into say cells of 10, then supercells of 10 cells, and so on for any level of depth, with aggregation of resources at higher levels in the hierarchy happening at different time scale to the lower levels.
Great talk Lorenzo, thank you!
A couple of questions…
Is there a way of analytically determining the minimum length of time when cooperation becomes evolutionarily stable? Put another way… what does “long-term” mean?
Your analyses of robustness of intrinsic growth differences… Am I understanding this right that an optimum value of 𝛼 can be determined that for entities growing at different rates and that each entity would have a different optimum value of 𝛼 as a result?
Dear James, Thanks 🙂
We don’t have an analytical solution for the value of T critical (that is the minimum time horizon you should have in was for cooperation to be stable). We have only numerical solutions that, as one could have guessed, show a dependence on sigma (the noise amplitude). This implies that the higher the noise the shorter will T critical be.
For the second: As long as the difference in intrinsic growth is smaller than the noise amplitude (sigma^2), the evolutionary stable alphas tend to infinity, for both individuals. They tend to infinity but at the same time they keep a precise ratio in order to be stable (like two non-parallel lines with positive coefficients do).
Always remember this is a model, probably in real life they wouldn’t really reach infinity. They would probably reach some finite values that satisfy that ratio (so both positive but different).
Thank you Lorenzo! I would like to build an intuition for these principles. For example – the idea that higher noise leads to a shorter T-critical is counter-intuitive to me at first, but makes sense if I think about it.
Great talk, Lorenzo, and quite understated. I like it!
I wanted to ask not just you but maybe also Athena and others if the following characterization makes sense to you. The Nowak paper is sort of a standard reference of perhaps the standard view on cooperation, and I think at least one way of seeing it is that we’re doing something totally different from Nowak. Here is why I think that.
Nowak starts essentially with the following setup. Two people can choose to cooperate or not. If they cooperate, we give them a “cooperation benefit.”
He then goes on to list different ways in which this cooperation benefit we magicked up can find its way back to the donor so that both donor and recipient benefit.
My view of this has always been that it’s too ad-hoc. I mean: it’s valid, you can set things up this way, and you can ask the questions Nowak asks. But what I wanted to point out in the paper with Alex was just how ubiquitous our specific cooperation benefit is, the generic way of producing this benefit, and what its properties are.
In the Nowak setup, I can make the cooperation benefit as big as I want, and I don’t have to explain where it came from.
Our setup specifies where the benefit comes from (it arises in any multiplicative setup, and all evolutionary (all living) setups are multiplicative). There can be other benefits but the one we study is very fundamental. Because we know where it comes from and what it means, physically, we can study its properties and make predictions based on it.
I think that’s the difference I wanted emphasize more: yes, Nowak-type benefits by fiat are one way of modelling but I think they make any result quite ad-hoc, and it’s very much “you get out what you put in.” Our approach is not arbitrary, not benefit by fiat, and you get results out which were not at all obvious at the outset.
I have been wondering this too!
From an analogous point of view, you can look at organisations like cooperatives and examine the rules that govern exchange of benefits between members.
These rules are maintained by committees and can become quite complex and ad-hoc in the same way that you describe the Nowak-style benefits.
All the while, there is a deeper principle operating that is benefitting all members. But with all the attention on the written down rules and ad-hoc benefits, the deeper principle goes unnoticed.
My interest is whether it would be possible to maximise the “deeper” growth benefit while keeping the organisation overheads and ad-hoc arrangements to a bare minimum.
Question prompted by James King’s talk but maybe more answerable by Lorenzo: For a fixed number of agents, is there any cooperative advantage to hierarchical cooperation. Non-hierarchical would be just pool equally between everyone in the population, hierarchical would be group into say cells of 10, then supercells of 10 cells, and so on for any level of depth, with aggregation of resources at higher levels in the hierarchy happening at different time scale to the lower levels.