Advances in Applied Probability

Convergence in a multidimensional randomized Keynesian beauty contest

Michael Grinfeld, Stanislav Volkov, and Andrew R. Wade

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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.

Article information

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

Dates
First available in Project Euclid: 31 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1427814581

Digital Object Identifier
doi:10.1239/aap/1427814581

Mathematical Reviews number (MathSciNet)
MR3327315

Zentralblatt MATH identifier
1318.60100

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aap/1427814581


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References

  • Benassi, C. and Malagoli, F. (2008). The sum of squared distances under a diameter constraint, in arbitrary dimension. Arch. Math. (Basel) 90, 471–480.
  • De Giorgi, E. and Reimann, S. (2008). The $\alpha$-beauty contest: choosing numbers, thinking intervals. Games Econom. Behav. 64, 470–486.
  • Erdős, P. (1940). On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math. 62, 180–186.
  • Grinfeld, M., Knight, P. A. and Wade, A. R. (2012). Rank-driven Markov processes. J. Statist. Phys. 146, 378–407.
  • Hughes, B. D. (1995). Random Walks and Random Environments; Vol. 1, Random Walks. Clarendon Press, New York.
  • Johnson, N. L. and Kotz, S. (1995). Use of moments in studies of limit distributions arising from iterated random subdivisions of an interval. Statist. Prob. Lett. 24, 111–119.
  • Keynes, J. M. (1957). The General Theory of Employment, Interest and Money. Macmillan, London.
  • Krapivsky, P. L. and Redner, S. (2004). Random walk with shrinking steps. Amer. J. Phys. 72, 591–598.
  • Moran, P. A. P. (1968). An Introduction to Probability Theory. Clarendon Press, Oxford.
  • Moulin, H. (1982). Game Theory for the Social Sciences. New York University Press.
  • Muratov, A. and Zuyev, S. (2013). LISA: locally interacting sequential adsorption. Stoch. Models 29, 475–496.
  • Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surv. 4, 1–79.
  • Penrose, M. D. and Wade, A. R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. Appl. Prob. 36, 691–714.
  • Penrose, M. D. and Wade, A. R. (2008). Limit theory for the random on-line nearest-neighbor graph. Random Structures Algorithms 32, 125–156.
  • Pillichshammer, F. (2000). On the sum of squared distances in the Euclidean plane. Arch. Math. (Basel) 74, 472–480.
  • Witsenhausen, H. S. (1974). On the maximum of the sum of squared distances under a diameter constraint. Amer. Math. Monthly 81, 1100–1101.