Convergence in a multidimensional randomized Keynesian beauty contest

Michael Grinfeld, Stanislav Volkov, Andrew R. Wade

Advances in Applied Probability, 47, no. 1, March 2015, 57–82. DOI: 10.1239/aap/1427814581

Abstract

We study the asymptotics of a Markovian system of $N \geq 3$ 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 $\xi_N$ in $[0,1]^d$. 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 $\xi_N$, 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, $N=3$.