summary.txt

A threshold Public Goods Game (PGG) in which success requires overall cooperative collective action, and decisions
must be made knowing that failure to cooperate implies a risk of overall collapse

This is a first summary text from a first read of the article

Population (Z):
 -> Zr = rich individuals
 -> Zp = poor individuals
 -> Z = Zr + Zp
 -> N = size of the groups formed from individuals randomly sampled from the population

Endowment (b):
 -> br = money start for rich individuals
 -> bp = money start for poor individuals
 -> br > bp
 -> c = a fraction of the endowment (b) to help solve the group task

Behaviors:
 -> cooperators (Cs) = contribute c to help solve the group task
 -> defector (Ds) = do not contribute anything to solve the group task

If the next intermediate target is not met:
 -> Cs will lose: br/p - c
 -> Ds will lose: br/p
_
b is the average endowment of the population
  _
Mcb is the threshold to be met with the contributions (c)

r = perception of risk of collective disaster, probability of individuals to lose everything if the threshold is not met

Homophily (h):
 -> 0 ≤ h ≤ 1
 -> h = 1, individuals are restricted to influence (and be influenced) by those of the same wealth status
 -> h = 0, no wealth discrimination takes place

ηG = fraction of time a group succeeds in achieving Mcb as a function of risk

Payoff function (Π):
    -> jr = nbr of rich Cs in the group of size N
    -> jp = nbr of poor Cs in the group of size N
    -> nbr of Ds = N−jr−jp   _
    -> Δ = cr*jr + cp*jp − Mcb
    -> ΠDs(payoff for defectors) = b * {Θ(Δ) + (1-r) * [1− Θ(Δ)]}
    -> ΠCs(payoff for cooperators) = ΠDs - c
    -> Θ(Δ)=1 whenever Δ>1 being 0 otherwise

fitness (f) of an individual adopting a given strategy (X) in a population of wealth class (k) (fXk):
    -> supplementary material 4 different fitness functions

The number of individuals adopting a given strategy will evolve in time according to a stochastic
birth–death process combined with the pairwise comparison rule = each individual of strategy X adopts
the strategy Y of a randomly selected member of the population, with probability  given by the Fermi function


A mutation probability μ, individuals adopt a randomly chosen different strategy, in such a way that when
μ=1, the individual does change strategy

The transition probabilities T can all be written in terms of the following expression, which
gives the probability that an individual with strategy X ∈{R,P}; Dg in the subpopulation k ∈{C,D};
Pg changes to a different strategy Y ∈{C,D}; Dg, both from the same subpopulation k and from the
other population l (that is, l = P if k = R, and l = R if k = P)
    -> supplementary material to check the function

There is also a gradient of selection ∇i

Markov process -> eigen search problem -> ng

Graphs:
 -> Average group achievement (ηG) as a function of risk (r)
                                    _
    1) initial endowment of all b = b = 1 (no wealth inequality)
    2) h = 0
    3) h = 1
    Other parameters :
    br = 2.5; bp = 0.625 (for 2 and 3)
    cost of cooperation = 0.1b
    cr = 0.1br; cp = 0.1bp
    Z = 200; Zr = 40; Zp = 160
    N = 6; M = 3

 -> Stationary distribution and gradient of selection for different values of risk r and of the homophily parameter h
    2 different risks
    for each risk homophily of 0 - 0.7 - 1
    A(r=0.2;h=0):B(r=0.2;h=0.7),C(r=0.2;h=1),D(r=0.3;h=0),E(r=0.3;h=0.7),F(r=0.3;h=1.0)
    Z = 200; Zr = 40; Zp = 160;
    c = 0.1;     _  _
    N = 6; M = 3cb (b = 1);
    bp = 0.625; br = 2.5; pk
    pkmax = {pAmax, ..., pFmax} = {2,40,75,3,2,20} * 10e-3
    ∇kmax = {∇Amax, ..., ∇Fmax} = {16,6,2,16,6,3} * 10e-2

 -> Stationary distribution and gradient of selection for populations comprising 10% of individuals exhibiting an obstinate cooperative behavior
    h = 1
    Z = 200; Zr = 40; Zp = 160;
    c = 0.1;     _  _
    N = 6; M = 3cb (b = 1);
    bP = 0.625; bR = 2.5;
    r = 0.2;
    β = 5.0; pk
    pkmax = {pAmax, ..., pCmax} = {76,4,2} * 10e-3
    ∇kmax = {∇Amax, ..., ∇Cmax} = {3,3,4} * 10e-2

 -> others in SM