The Annals of Applied Probability

Resource dependent branching processes and the envelope of societies

F. Thomas Bruss and Mitia Duerinckx

Full-text: Open access

Abstract

Since its early beginnings, mankind has put to test many different society forms, and this fact raises a complex of interesting questions. The objective of this paper is to present a general population model which takes essential features of any society into account and which gives interesting answers on the basis of only two natural hypotheses. One is that societies want to survive, the second, that individuals in a society would, in general, like to increase their standard of living. We start by presenting a mathematical model, which may be seen as a particular type of a controlled branching process. All conditions of the model are justified and interpreted. After several preliminary results about societies in general we can show that two society forms should attract particular attention, both from a qualitative and a quantitative point of view. These are the so-called weakest-first society and the strongest-first society. In particular we prove then that these two societies stand out since they form an envelope of all possible societies in a sense we will make precise. This result (the envelopment theorem) is seen as significant because it is paralleled with precise survival criteria for the enveloping societies. Moreover, given that one of the “limiting” societies can be seen as an extreme form of communism, and the other one as being close to an extreme version of capitalism, we conclude that, remarkably, humanity is close to having already tested the limits.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 1 (2015), 324-372.

Dates
First available in Project Euclid: 16 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1418740188

Digital Object Identifier
doi:10.1214/13-AAP998

Mathematical Reviews number (MathSciNet)
MR3297775

Zentralblatt MATH identifier
1308.60103

Subjects
Primary: 69J85 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
Controlled branching processes extinction criteria Borel–Cantelli lemma almost-sure convergence complete convergence order statistics stopping times Galton–Watson processes Lorenz curve society structures laissez-faire society mercantilism communism capitalism

Citation

Bruss, F. Thomas; Duerinckx, Mitia. Resource dependent branching processes and the envelope of societies. Ann. Appl. Probab. 25 (2015), no. 1, 324--372. doi:10.1214/13-AAP998. https://projecteuclid.org/euclid.aoap/1418740188


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