Convergence in a multidimensional randomized Keynesian beauty contest

Michael Grinfeld, Stanislav Volkov, Andrew R. Wade

Research output: Contribution to journalArticle

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Abstract

We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]d in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U [0,1]d random particle. We show that the limiting configuration contains N - 1 coincident particles at a random location ξN ∈ [0, 1]d. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξN, showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.
LanguageEnglish
Pages57-82
Number of pages26
JournalAdvances in Applied Probability
Volume47
Issue number1
DOIs
StatePublished - 31 Mar 2015

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Lyapunov functions
Set theory
Coincident
Barycentre
Lyapunov Function
Thing
Discrete-time
Limiting
Radius
Configuration
Interval
Subset

Keywords

  • Keynesian beauty contest
  • radius of gyration
  • rank-driven process
  • sum of squared distances

Cite this

Grinfeld, Michael ; Volkov, Stanislav ; Wade, Andrew R./ Convergence in a multidimensional randomized Keynesian beauty contest. In: Advances in Applied Probability. 2015 ; Vol. 47, No. 1. pp. 57-82
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Convergence in a multidimensional randomized Keynesian beauty contest. / Grinfeld, Michael; Volkov, Stanislav; Wade, Andrew R.

In: Advances in Applied Probability, Vol. 47, No. 1, 31.03.2015, p. 57-82.

Research output: Contribution to journalArticle

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