The limit point in the Jante's law process has an absolutely continuous distribution

Edward T Crane; Stanislav Volkov

doi:10.1016/j.spa.2023.104252

The limit point in the Jante's law process has an absolutely continuous distribution

Edward T Crane
, Stanislav Volkov^*

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

Abstract

We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante’s law process. We consider a version of the model where the space of possible opinions is a convex body B in R^d. N individuals in a population each hold a (multidimensional) opinion in B. Repeatedly, the individual whose opinion is furthest from the centre of mass of the N current opinions chooses a new opinion, sampled uniformly at random from B. Kennerberg and Volkov showed that the set of opinions that are not furthest from the centre of mass converges to a random limit point. We show that the distribution of the limit opinion is absolutely continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.

Original language	English
Article number	104252
Number of pages	26
Journal	Stochastic Processes and their Applications
Volume	168
Early online date	20 Nov 2023
DOIs	https://doi.org/10.1016/j.spa.2023.104252
Publication status	Published - 1 Feb 2024

Bibliographical note

Funding Information:
The research of S.V. is partially supported by the Swedish Science Foundation grant VR 2019-04173 and the Crafoord Foundation, Sweden grant no. 20190667 . S.V. would like to acknowledge the hospitality of the University of Bristol during his visits to Bristol. The research of E.C. is supported by the Heilbronn Institute for Mathematical Research, UK . We would like to thank John Mackay for pointing us to the notion of uniform geometry in [7] and Rami Atar for pointing out the relevant Brownian bees model. Finally, we would like to thank the anonymous referee for many useful comments and recommendations, in particular the suggestion to explain what can be done in the non-convex case.

Funding Information:
The research of S.V. is partially supported by the Swedish Science Foundation grant VR 2019-04173 and the Crafoord Foundation, Sweden grant no. 20190667. S.V. would like to acknowledge the hospitality of the University of Bristol during his visits to Bristol. The research of E.C. is supported by the Heilbronn Institute for Mathematical Research, UK . We would like to thank John Mackay for pointing us to the notion of uniform geometry in [7] and Rami Atar for pointing out the relevant Brownian bees model. Finally, we would like to thank the anonymous referee for many useful comments and recommendations, in particular the suggestion to explain what can be done in the non-convex case.

Publisher Copyright:
© 2023 The Author(s)

Keywords

Jante's law process
Consensus formation
Keynesian beauty contest
Rank-driven process
Interacting particle system

Access to Document

10.1016/j.spa.2023.104252Licence: CC BY

Handle.net

Persistent link

Cite this

@article{63a1917c01054189ba4ee501fd9a938f,

title = "The limit point in the Jante's law process has an absolutely continuous distribution",

abstract = "We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante{\textquoteright}s law process. We consider a version of the model where the space of possible opinions is a convex body B in Rd. N individuals in a population each hold a (multidimensional) opinion in B. Repeatedly, the individual whose opinion is furthest from the centre of mass of the N current opinions chooses a new opinion, sampled uniformly at random from B. Kennerberg and Volkov showed that the set of opinions that are not furthest from the centre of mass converges to a random limit point. We show that the distribution of the limit opinion is absolutely continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.",

keywords = "Jante's law process, Consensus formation, Keynesian beauty contest, Rank-driven process, Interacting particle system",

author = "Crane, \{Edward T\} and Stanislav Volkov",

note = "Funding Information: The research of S.V. is partially supported by the Swedish Science Foundation grant VR 2019-04173 and the Crafoord Foundation, Sweden grant no. 20190667 . S.V. would like to acknowledge the hospitality of the University of Bristol during his visits to Bristol. The research of E.C. is supported by the Heilbronn Institute for Mathematical Research, UK . We would like to thank John Mackay for pointing us to the notion of uniform geometry in [7] and Rami Atar for pointing out the relevant Brownian bees model. Finally, we would like to thank the anonymous referee for many useful comments and recommendations, in particular the suggestion to explain what can be done in the non-convex case. Funding Information: The research of S.V. is partially supported by the Swedish Science Foundation grant VR 2019-04173 and the Crafoord Foundation, Sweden grant no. 20190667. S.V. would like to acknowledge the hospitality of the University of Bristol during his visits to Bristol. The research of E.C. is supported by the Heilbronn Institute for Mathematical Research, UK . We would like to thank John Mackay for pointing us to the notion of uniform geometry in [7] and Rami Atar for pointing out the relevant Brownian bees model. Finally, we would like to thank the anonymous referee for many useful comments and recommendations, in particular the suggestion to explain what can be done in the non-convex case. Publisher Copyright: {\textcopyright} 2023 The Author(s)",

year = "2024",

month = feb,

day = "1",

doi = "10.1016/j.spa.2023.104252",

language = "English",

volume = "168",

journal = "Stochastic Processes and their Applications",

issn = "0304-4149",

publisher = "Elsevier B.V.",

}

TY - JOUR

T1 - The limit point in the Jante's law process has an absolutely continuous distribution

AU - Crane, Edward T

AU - Volkov, Stanislav

N1 - Funding Information: The research of S.V. is partially supported by the Swedish Science Foundation grant VR 2019-04173 and the Crafoord Foundation, Sweden grant no. 20190667 . S.V. would like to acknowledge the hospitality of the University of Bristol during his visits to Bristol. The research of E.C. is supported by the Heilbronn Institute for Mathematical Research, UK . We would like to thank John Mackay for pointing us to the notion of uniform geometry in [7] and Rami Atar for pointing out the relevant Brownian bees model. Finally, we would like to thank the anonymous referee for many useful comments and recommendations, in particular the suggestion to explain what can be done in the non-convex case. Funding Information: The research of S.V. is partially supported by the Swedish Science Foundation grant VR 2019-04173 and the Crafoord Foundation, Sweden grant no. 20190667. S.V. would like to acknowledge the hospitality of the University of Bristol during his visits to Bristol. The research of E.C. is supported by the Heilbronn Institute for Mathematical Research, UK . We would like to thank John Mackay for pointing us to the notion of uniform geometry in [7] and Rami Atar for pointing out the relevant Brownian bees model. Finally, we would like to thank the anonymous referee for many useful comments and recommendations, in particular the suggestion to explain what can be done in the non-convex case. Publisher Copyright: © 2023 The Author(s)

PY - 2024/2/1

Y1 - 2024/2/1

N2 - We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante’s law process. We consider a version of the model where the space of possible opinions is a convex body B in Rd. N individuals in a population each hold a (multidimensional) opinion in B. Repeatedly, the individual whose opinion is furthest from the centre of mass of the N current opinions chooses a new opinion, sampled uniformly at random from B. Kennerberg and Volkov showed that the set of opinions that are not furthest from the centre of mass converges to a random limit point. We show that the distribution of the limit opinion is absolutely continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.

AB - We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante’s law process. We consider a version of the model where the space of possible opinions is a convex body B in Rd. N individuals in a population each hold a (multidimensional) opinion in B. Repeatedly, the individual whose opinion is furthest from the centre of mass of the N current opinions chooses a new opinion, sampled uniformly at random from B. Kennerberg and Volkov showed that the set of opinions that are not furthest from the centre of mass converges to a random limit point. We show that the distribution of the limit opinion is absolutely continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.

KW - Jante's law process

KW - Consensus formation

KW - Keynesian beauty contest

KW - Rank-driven process

KW - Interacting particle system

U2 - 10.1016/j.spa.2023.104252

DO - 10.1016/j.spa.2023.104252

M3 - Article (Academic Journal)

SN - 0304-4149

VL - 168

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

M1 - 104252

ER -

The limit point in the Jante's law process has an absolutely continuous distribution

Abstract

Bibliographical note

Keywords

Access to Document

Handle.net

Fingerprint

Cite this