Convergence in a multidimensional randomized Keynesian beauty contest

Grinfeld, Michael and Volkov, Stanislav and Wade, Andrew R. (2015) Convergence in a multidimensional randomized Keynesian beauty contest. Advances in Applied Probability, 47 (1). pp. 57-82. ISSN 0001-8678

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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.