Galton and Watson voluntarily skipping some generations

A riddle on a form of a Galton-Watson process, starting from a single unit, where no one dies but rather, at each of 100 generations, Dog either opts for a Uniform number υ of additional units or increments a counter γ by this number υ, its goal being to optimise γ. The solution proposed by the Riddler does not establish his solution’s is the optimal strategy and considers anyway average gains. Solution that consists in always producing more units until the antepenultimate hour (ie incrementing only at the 99th and 100th generations),  I tried instead various logical (?) rules and compared outputs by bRute foRce, resulting in higher maxima (over numerous repeated calls) for the alternative principle

s<-function(p=.66){ 
   G=0;K=1 for(t in 1:9){ 
      i=sample(1:K,1) 
      K=K+i*(i>=K*p)
      G=G+i*(i<K*p)}
  return(c(G+sample(1:K,1),K))}

interacting particle systems as… facebook

Among the many interesting arXived papers this Friday, I first read David Aldous’ “Interacting particle systems as stochastic social dynamics“. Being unfamiliar with those systems (despite having experts in offices down the hall in Paris-Dauphine!), I read this typology of potential models (published in Bernoulli) with a keen interest! The paper stemmed from a short course given in 2012 in Warwick and Cornell. I think the links exhibited there with (social) networks should be relevant for statisticians working on networks (!) and dynamic graphical models. Statistics is not mentioned in the paper, except for the (misleading) connection with statistical physics, but there is obviously a huge potential for statistical inference, from parameter estimation to model comparison. (As pointed out by David Aldous, there is usually “no data or evidence linking the model to the asserted real-world phenomena”.) The paper then introduces some basic models like the token, the pandemic and the averaging process, plus the voter model that relates to Kingman’s coalescent. A very nice read opening new vistas for sure (and a source of projects for graduate students most certainly!)