The Annals of Applied Probability

Resource dependent branching processes and the envelope of societies

F. Thomas Bruss and Mitia Duerinckx

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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

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

First available in Project Euclid: 16 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • Asmussen, S. and Kurtz, T. G. (1980). Necessary and sufficient conditions for complete convergence in the law of large numbers. Ann. Probab. 8 176–182.
  • Bingham, N. H. and Doney, R. A. (1974). Asymptotic properties of supercritical branching processes. I. The Galton–Watson process. Adv. in Appl. Probab. 6 711–731.
  • Bruss, F. T. (1978). Branching processes with random absorbing processes. J. Appl. Probab. 15 54–64.
  • Bruss, F. T. (1980). A counterpart of the Borel–Cantelli lemma. J. Appl. Probab. 17 1094–1101.
  • Bruss, F. T. (1984). A note on extinction criteria for bisexual Galton–Watson processes. J. Appl. Probab. 21 915–919.
  • Bruss, F. T. and Robertson, J. B. (1991). “Wald’s lemma” for sums of order statistics of i.i.d. random variables. Adv. in Appl. Probab. 23 612–623.
  • Coffman, E. G. Jr., Flatto, L. and Weber, R. R. (1987). Optimal selection of stochastic intervals under a sum constraint. Adv. in Appl. Probab. 19 454–473.
  • Cohn, H. (1996). On the asymptotic patterns of supercritical branching processes in varying environments. Ann. Appl. Probab. 6 896–902.
  • Cohn, H. and Klebaner, F. (1986). Geometric rate of growth in Markov chains with applications to population-size-dependent models with dependent offspring. Stoch. Anal. Appl. 4 283–307.
  • Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996). On the Lambert $W$ function. Adv. Comput. Math. 5 329–359.
  • Daley, D. J., Hull, D. M. and Taylor, J. M. (1986). Bisexual Galton–Watson branching processes with superadditive mating functions. J. Appl. Probab. 23 585–600.
  • González, M., Molina, M. and del Puerto, I. (2005). On $L^{2}$-convergence of controlled branching processes with random control function. Bernoulli 11 37–46.
  • González, M., Molina, M. and Del Puerto, I. (2002). On the class of controlled branching processes with random control functions. J. Appl. Probab. 39 804–815.
  • Haccou, P., Jagers, P. and Vatutin, V. A. (2007). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge Univ. Press, Cambridge.
  • Hautphenne, S. (2012). Extinction probabilities of supercritical decomposable branching processes. J. Appl. Probab. 49 639–651.
  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • Hsu, P. L. and Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33 25–31.
  • Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, New York.
  • Klebaner, F. C. (1985). A limit theorem for population-size-dependent branching processes. J. Appl. Probab. 22 48–57.
  • Molina, M. (2010). Two-sex branching process literature. In Workshop on Branching Processes and Their Applications. Lect. Notes Stat. Proc. 197 279–293. Springer, Berlin.
  • Pearson, K. (1968). Tables of the Incomplete Beta-Function, 2nd ed. Cambridge Univ. Press, Cambridge.
  • Schuh, H.-J. (1976). A condition for the extinction of a branching process with an absorbing lower barrier. J. Math. Biol. 3 271–287.
  • Sevast’janov, B. A. and Zubkov, A. M. (1974). Controlled branching processes. Teor. Verojatnost. i Primenen. 19 15–25.
  • Xu, K. and Mannor, S. (2012). Rate-optimal control for resource-constrained branching processes. Available at arxiv:1203.1072v1.
  • Yakovlev, A. Y. and Yanev, N. M. (2009). Relative frequencies in multitype branching processes. Ann. Appl. Probab. 19 1–14.
  • Yanev, N. M. (1976). Conditions for degeneracy of $\varphi$-branching processes with random $\varphi$. Theory Probab. Appl. 20 421–428.