Convergence in a multidimensional randomized Keynesian beauty contest

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

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.