Advances in Applied Probability

Convergence in a multidimensional randomized Keynesian beauty contest

Michael Grinfeld, Stanislav Volkov, and Andrew R. Wade

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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.

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Adv. in Appl. Probab., Volume 47, Number 1 (2015), 57-82.

First available in Project Euclid: 31 March 2015

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Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F15: Strong theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35] 91A15: Stochastic games

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


Grinfeld, Michael; Volkov, Stanislav; Wade, Andrew R. Convergence in a multidimensional randomized Keynesian beauty contest. Adv. in Appl. Probab. 47 (2015), no. 1, 57--82. doi:10.1239/aap/1427814581.

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