Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T16:50:20.993Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence in a Multidimensional Randomized Keynesian Beauty Contest

Part of: Special processes Limit theorems Game theory Markov processes Time-dependent statistical mechanics (dynamic and nonequilibrium) Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

Michael Grinfeld ,
Stanislav Volkov  and
Andrew R. Wade
Michael Grinfeld*
Affiliation:
University of Strathclyde
Stanislav Volkov*
Affiliation:
Lund University and University of Bristol
Andrew R. Wade*
Affiliation:
Durham University
*
∗ Postal address: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK.
∗∗ Postal address: Centre for Mathematical Sciences, Lund University, Box 118, Lund, SE-22100, Sweden.
∗∗∗ Postal address: Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Keynesian beauty contestradius of gyrationrank-driven processsum of squared distances

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60D05: Geometric probability and stochastic geometry 60F15: Strong theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C22: Interacting particle systems 91A15: Stochastic games
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 57 - 82
Copyright
© Applied Probability Trust 

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