Convergence in a Multidimensional Randomized Keynesian Beauty Contest

Grinfeld, Michael; Volkov, Stanislav; Wade, Andrew R.

doi:10.1017/s0001867800007709

Cited by 5 publications

(39 citation statements)

References 14 publications

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“…We study a generalization of the model presented in Grinfeld et al [2]. Fix an integer N ≥ 3 and some d-dimensional random variable ζ .…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

“…Now arbitrarily choose N distinct points on R d , d ≥ 1. The process in [2], called the Keynesian beauty contest process, is a discrete-time process with the following dynamics: given the configuration of N points we compute its centre of mass μ and discard the point most distant from μ; if there is more than one, we choose each one with equal probability. Then this point is replaced with a new point drawn independently each time from the distribution of ζ .…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

“…Then this point is replaced with a new point drawn independently each time from the distribution of ζ . In [2] it was shown that when ζ has a uniform distribution on a unit cube, then the configuration converges to some random point on R d , with the exception of the most distant point.…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

“…Here and throughout, x denotes the Euclidean norm in R d , x • y is a dot product of two vectors x, y ∈ R d , and B r (x) = {y ∈ R d : y − x < r} is an open ball of radius r centred at x. As in [2], we also define, for X n = (x 1 , x 2 , . .…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

“…In [2], where K = 1, the authors called x (n) the extreme point of X n , that is, a point of x 1 , . .…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

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Jante's law process

Kennerberg¹,

Volkov²

2018

Adv. Appl. Probab.

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Consider the process which starts with N ≥ 3 distinct points on ℝd, and fix a positive integer K < N. Of the total N points keep those N - K which minimize the energy amongst all the possible subsets of size N - K, and then replace the removed points by K independent and identically distributed points sampled according to some fixed distribution ζ. Repeat this process ad infinitum. We obtain various quite nonrestrictive conditions under which the set of points converges to a certain limit. This is a very substantial generalization of the `Keynesian beauty contest process' introduced in Grinfeld et al. (2015), where K = 1 and the distribution ζ was uniform on the unit cube.

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“…We study a generalization of the model presented in Grinfeld et al [2]. Fix an integer N ≥ 3 and some d-dimensional random variable ζ .…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

“…In [2], where K = 1, the authors called x (n) the extreme point of X n , that is, a point of x 1 , . .…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 99%

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Jante's law process

Kennerberg¹,

Volkov²

2018

Adv. Appl. Probab.

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Convergence in the p-Contest

Kennerberg

Volkov

2020

J Stat Phys

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We study asymptotic properties of the following Markov system of $$N \ge 3$$N≥3 points in [0, 1]. At each time step, the point farthest from the current centre of mass, multiplied by a constant $$p>0$$p>0, is removed and replaced by an independent $$\zeta $$ζ-distributed point; the problem, inspired by variants of the Bak–Sneppen model of evolution and called a p-contest, was posed in Grinfeld et al. (J Stat Phys 146, 378–407, 2012). We obtain various criteria for the convergences of the system, both for $$p<1$$p<1 and $$p>1$$p>1. In particular, when $$p<1$$p<1 and $$\zeta \sim U[0,1]$$ζ∼U[0,1], we show that the limiting configuration converges to zero. When $$p>1$$p>1, we show that the configuration must converge to either zero or one, and we present an example where both outcomes are possible. Finally, when $$p>1$$p>1, $$N=3$$N=3 and $$\zeta $$ζ satisfies certain mild conditions (e.g. $$\zeta \sim U[0,1]$$ζ∼U[0,1]), we prove that the configuration converges to one a.s. Our paper substantially extends the results of Grinfeld et al. (Adv Appl Probab 47:57–82, 2015) and Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018) where it was assumed that $$p=1$$p=1. Unlike the previous models, one can no longer use the Lyapunov function based just on the radius of gyration; when $$0<p<1$$0<p<1 one has to find a more finely tuned function which turns out to be a supermartingale; the proof of this fact constitutes an unwieldy, albeit necessary, part of the paper.

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Jante's law process

Kennerberg,

Volkov

2017

Preprint

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Consider the process which starts with N ≥ 3 distinct points on R d , and fix a positive integer K < N . Of the total N points keep those N −K which minimize the energy (defined as the sum of all pairwise distances squared) amongst all the possible subsets of size N − K, and then replace the removed points by K i.i.d. points sampled according to some fixed distribution ζ. Repeat this process ad infinitum. We obtain various quite non-restrictive conditions under which the set of points converges to a certain limit. This is a very substantial generalization of the "Keynesian beauty contest process" introduced in [3], where K = 1 and the distribution ζ was uniform on the unit cube.

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Convergence in a Multidimensional Randomized Keynesian Beauty Contest

Cited by 5 publications

References 14 publications

Jante's law process

Jante's law process

Convergence in the p-Contest

Jante's law process

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