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On the One Dimensional "Learning from Neighbours" Model

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	On the One Dimensional "Learning from Neighbours" Model
2.	Creator	Author's name, affiliation, country	Antar Bandyopadhyay; Indian Statistical Institute; India
2.	Creator	Author's name, affiliation, country	Rahul Roy; Indian Statistical Institute; India
2.	Creator	Author's name, affiliation, country	Anish Sarkar; Indian Statistical Institute; India
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Coexistence, Learning from neighbours, Markov chain, Random walk, Stationary distribution.
3.	Subject	Subject classification	Primary: 60J10, 60K35; Secondary: 60C05, 62E10, 90B15, 91D30.
4.	Description	Abstract	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".
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Work of the second author is supported by a grant from Department of Science and Technology, Government of India.
7.	Date	(YYYY-MM-DD)	2010-10-15
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/809
10.	Identifier	Digital Object Identifier	10.1214/EJP.v15-809
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 15
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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