Convergence in a multidimensional randomized Keynesian beauty contest

Grinfeld, Michael; Volkov, Stanislav; Wade, Andrew R.

doi:10.1239/aap/1427814581

Convergence in a multidimensional randomized Keynesian beauty contest

Grinfeld, Michael; Volkov, Stanislav; Wade, Andrew R.

Authors

Michael Grinfeld

Stanislav Volkov

Professor Andrew Wade andrew.wade@durham.ac.uk
Professor

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.

Citation

Grinfeld, M., Volkov, S., & Wade, A. R. (2015). Convergence in a multidimensional randomized Keynesian beauty contest. Advances in Applied Probability, 47(1), 57-82. https://doi.org/10.1239/aap/1427814581

Journal Article Type	Article
Acceptance Date	Mar 13, 2014
Online Publication Date	Mar 31, 2015
Publication Date	Mar 1, 2015
Deposit Date	May 7, 2014
Publicly Available Date	May 26, 2014
Journal	Advances in Applied Probability
Print ISSN	0001-8678
Electronic ISSN	1475-6064
Publisher	Applied Probability Trust
Peer Reviewed	Peer Reviewed
Volume	47
Issue	1
Pages	57-82
DOI	https://doi.org/10.1239/aap/1427814581
Keywords	Keynesian beauty contest, Radius of gyration, Rank-driven process, Sum of squared distances.