Electronic Journal of Probability

On the One Dimensional "Learning from Neighbours" Model

Antar Bandyopadhyay, Rahul Roy, and Anish Sarkar

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We consider a model of a discrete time "interacting particle system" on the integer line where infinitely many changes are allowed at each instance of time. We describe the model using chameleons of two different colours, viz., red (R) and blue (B). At each instance of time each chameleon performs an independent but identical coin toss experiment with probability α to decide whether to change its colour or not. If the coin lands head then the creature retains its colour (this is to be interpreted as a "success"), otherwise it observes the colours and coin tosses of its two nearest neighbours and changes its colour only if, among its neighbours and including itself, the proportion of successes of the other colour is larger than the proportion of successes of its own colour. This produces a Markov chain with infinite state space. This model was studied by Chatterjee and Xu (2004) in the context of diffusion of technologies in a set-up of myopic, memoryless agents. In their work they assume different success probabilities of coin tosses according to the colour of the chameleon. In this work we consider the symmetric case where the success probability, $\alpha$, is the same irrespective of the colour of the chameleon. We show that starting from any initial translation invariant distribution of colours the Markov chain converges to a limit of a single colour, i.e., even at the symmetric case there is no "coexistence" of the two colours at the limit. As a corollary we also characterize the set of all translation invariant stationary laws of this Markov chain. Moreover we show that starting with an i.i.d. colour distribution with density $p\in[0,1]$ of one colour (say red), the limiting distribution is all red with probability $\Pi(\alpha,p)$ which is continuous in $p$ and for $p$ "small" $\Pi(p)>p$. The last result can be interpreted as the model favours the "underdog".

Article information

Electron. J. Probab. Volume 15 (2010), paper no. 51, 1574-1593.

Accepted: 15 October 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability 62E10: Characterization and structure theory 90B15: Network models, stochastic 91D30: Social networks

Coexistence Learning from neighbours Markov chain Random walk Stationary distribution

This work is licensed under a Creative Commons Attribution 3.0 License.


Bandyopadhyay, Antar; Roy, Rahul; Sarkar, Anish. On the One Dimensional "Learning from Neighbours" Model. Electron. J. Probab. 15 (2010), paper no. 51, 1574--1593. doi:10.1214/EJP.v15-809. https://projecteuclid.org/euclid.ejp/1464819836.

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  • Bala, Venkatesh; Goyal, Sanjeed. Learning from Neighbors. Rev. Econ. Stud. 65 (1998), 595–621.
  • Bala, Venkatesh; Goyal, Sanjeev. A noncooperative model of network formation. Econometrica 68 (2000), no. 5, 1181–1229.
  • Banerjee, Abhijit; Fudenberg, Drew. Word-of-mouth learning. Games Econom. Behav. 46 (2004), no. 1, 1–22.
  • Billingsley, Patrick. Probability and measure. Third edition. Wiley Series in Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. xiv+593 pp. ISBN: 0-471-00710-2.
  • Chatterjee, Kalyan; Xu, Susan H. Technology diffusion by learning from neighbours. Adv. in Appl. Probab. 36 (2004), no. 2, 355–376.
  • Glenn Ellison; Drew Fudenberg. Rules of Thumb for Social Learning. J. Political Economy 101 (1993) no. 4, 612–643.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • Georgii, Hans-Otto. Gibbs measures and phase transitions. de Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 1988. xiv+525 pp. ISBN: 0-89925-462-4.
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6.
  • Liggett, Thomas M. Interacting particle systems. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005. xvi+496 pp. ISBN: 3-540-22617-6.
  • Rudin, Walter. Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1.