Electronic Journal of Probability

The compulsive gambler process

David Aldous, Daniel Lanoue, and Justin Salez

Full-text: Open access

Abstract

In the  compulsive gambler process there is a finite set of agents who meet pairwise at random times ($i$ and $j$ meet at times of a rate-$\nu_{ij}$ Poisson process) and, upon meeting, play an instantaneous fair game in which one wins the other's money.  We introduce this process and describe some of its basic properties. Some properties are rather obvious (martingale structure;  comparison with Kingman coalescent) while others are more subtle (an "exchangeable over the money elements" property, and a construction reminiscent of the Donnelly-Kurtz look-down construction).  Several directions for possible future research are described. One - where agents meet neighbors in a  sparse graph - is studied here, and another - a continuous-space extension called the metric coalescent - is studied in Lanoue (2014).

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 35, 18 pp.

Dates
Accepted: 1 April 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067141

Digital Object Identifier
doi:10.1214/EJP.v20-3582

Mathematical Reviews number (MathSciNet)
MR3335826

Zentralblatt MATH identifier
1335.60141

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
coalescent exchangeable interacting particle system martingale social dynamics

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Aldous, David; Lanoue, Daniel; Salez, Justin. The compulsive gambler process. Electron. J. Probab. 20 (2015), paper no. 35, 18 pp. doi:10.1214/EJP.v20-3582. https://projecteuclid.org/euclid.ejp/1465067141


Export citation

References

  • Aldous, David. Interacting particle systems as stochastic social dynamics. Bernoulli 19 (2013), no. 4, 1122–1149.
  • Aldous, David; Lanoue, Daniel. A lecture on the averaging process. Probab. Surv. 9 (2012), 90–102.
  • Aldous, David J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), no. 1, 3–48.
  • Berestycki, Nathanael. Recent progress in coalescent theory. Ensaios Matematicos [Mathematical Surveys], 16. Sociedade Brasileira de Matematica, Rio de Janeiro, 2009. 193 pp. ISBN: 978-85-85818-40-1
  • Bertoin, Jean. Random fragmentation and coagulation processes. Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006. viii+280 pp. ISBN: 978-0-521-86728-3; 0-521-86728-2
  • Bertoin, Jean. Exchangeable coalescents, Nachdiplom Lectures, ETH Zurich. www.fim.math.ethz.ch/lectures/Lectures_Bertoin.pdf, 2010.
  • S. Chatterjee and P. S. Dey, phMultiple phase transitions in long-range first-passage percolation on square lattices, 2013, arXiv 1309.5757.
  • Dembo, Amir; Montanari, Andrea. Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 (2010), no. 2, 565–592.
  • Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166–205.
  • Gnedin, Alexander V. On convergence and extensions of size-biased permutations. J. Appl. Probab. 35 (1998), no. 3, 642–650.
  • Kreer, Markus; Penrose, Oliver. Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel. J. Statist. Phys. 75 (1994), no. 3-4, 389–407.
  • Daniel Lanoue, The metric coalescent, 2014, arXiv:1406.1131.
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4
  • Justin Salez, The compulsive gambler process, 2012, Early draft of this work, available at http://www.stat.berkeley.edu/simaldous/Papers/salez-cg.pdf.