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Interacting nonlinear reinforced stochastic processes: Synchronization or non-synchronization

Part of: Mathematical sociology Limit theorems Special processes Applications Sequential methods

Published online by Cambridge University Press:  01 August 2022

Irene Crimaldi ,
Pierre-Yves Louis Open the ORCID record for Pierre-Yves Louis [Opens in a new window]  and
Ida G. Minelli
Irene Crimaldi*
Affiliation:
IMT School for Advanced Studies Lucca
Pierre-Yves Louis*
Affiliation:
PAM UMR 02.102 and Institut de Mathématiques de Bourgogne
Ida G. Minelli*
Affiliation:
Università degli Studi dell’Aquila
*
*Postal address: Piazza San Ponziano 6, 55100 Lucca, Italy. Email address: irene.crimaldi@imtlucca.it
**Postal address: Université Bourgogne Franche-Comté, Institut Agro Dijon, 1 esplanade Erasme, 21000, Dijon, France.
****Postal address: Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio (Coppito 1), 67100 L’Aquila, Italy. Email address: idagermana.minelli@univaq.it
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Abstract

The rich-get-richer rule reinforces actions that have been frequently chosen in the past. What happens to the evolution of individuals’ inclinations to choose an action when agents interact? Interaction tends to homogenize, while each individual dynamics tends to reinforce its own position. Interacting stochastic systems of reinforced processes have recently been considered in many papers, in which the asymptotic behavior is proven to exhibit almost sure synchronization. In this paper we consider models where, even if interaction among agents is present, absence of synchronization may happen because of the choice of an individual nonlinear reinforcement. We show how these systems can naturally be considered as models for coordination games or technological or opinion dynamics.

Keywords

Interacting agentsinteracting stochastic processesreinforced stochastic processurn modelnonlinear Pólya urngeneralized Pólya urnreinforcement learningstochastic approximationgame theorytechnological dynamicsCODA models

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 62L20: Stochastic approximation 91D30: Social networks
Secondary: 60F15: Strong theorems 62P20: Applications to economics 62P25: Applications to social sciences
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 275 - 320
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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