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Experiments in longevity vs resource distribution

Experiments in longevity vs resource distribution

Idea

This page contains a silly model.

Model

QuantityVariable
Timesteptt
Current populationP tP_t
Knowledgeκ\kappa
Resourcesρ\rho
QuantityParameterConstraints
Initial populationP 0P_0P 0>1P_0\gt 1
No timestepsTTT>1T\gt 1
Variability of interactionsΔ\Delta0<Δ<10\lt \Delta\lt 1
Learning probabilityp λp_{\lambda}0<p λ<10\lt p_{\lambda}\lt 1
Learning incrementλ\lambda0<λ<10\lt \lambda\lt 1
Population growthγ\gamma0<γ<10\lt \gamma\lt 1
Fortune split rateϕ\phi0<ϕ<10\lt \phi\lt 1

Changing share of resources

For each index ii in 11 to P tP_t:

  • Uniformly randomly pick another index jj to engage in interaction. If iji\ne j compute

    (1)w i=ρ i(1λ κ i)U(1Δ,1+Δ)andw j=ρ j(1λ κ j)U(1Δ,1+Δ) w_i=\rho_i (1-\lambda^{\kappa_i}) U(1-\Delta,1+\Delta) \quad and \quad w_j=\rho_j (1-\lambda^{\kappa_j}) U(1-\Delta,1+\Delta)

    where U(a,b)U(a,b) is a uniformly distributed random variable.

  • Compute

    (2)R=ρ i+ρ jw i+w j R=\frac{\rho_i+\rho_j}{w_i+w_j}

    and set the updated ρ\rho values

    (3)ρ i=w iRandρ j=w jR \rho_i'=w_i R \quad and \quad \rho_j'=w_j R
  • Possibly increase the participants’ knowledge through learning

    (4)κ i=κ i+Bernoulli(p λ)andκ j=κ j+Bernoulli(p λ) \kappa_i'=\kappa_i + Bernoulli(p_{\lambda}) \quad and \quad \kappa_j'=\kappa_j +Bernoulli(p_{\lambda})

    Note that the possible knowledge increment is taken to be independent of the change in share of resources as it is generally possible to learn, and equally importantly not to learn, from either success or failure. (There is an argument about whether the amount learned, in the event something is learned, ought to scale in some way with the amount already known. For simplicity we start with the additive term here.)

Generation of offspring

At each time tt, after the changing of resources and gaining of experience calculation, randomly select γP t\lfloor\gamma P_t\rfloor individuals as “parents”. From each parent

  • Create a new descendant (with index kk) with knowledge of κ k=1\kappa_k=1.

  • Split the resources of each parent into ρ i=(1ϕ)ρ i\rho_i'=(1-\phi)\rho_i kept by the parent and a resource value of ρ k=ϕρ i\rho_k=\phi\rho_i is used to initialise the child.

Experimental observations

Summary