Evolving communities with individual preferences

Journal article

The goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals. We model the population by a measure space (𝛺,,𝜈) where 𝜈 determines the abundance of individual preferences. The preferences of an individual 𝜔∈𝛺 are described by a measurable choice 𝑋(𝜔) of a rough path. We aim to identify, for each individual, a choice for the forward evolution 𝑌𝑡(𝜔) for an individual in the community. These choices 𝑌𝑡(𝜔) must be consistent so that 𝑌𝑡(𝜔) correctly accounts for the individual's preference and correctly models their interaction with the aggregate behaviour of the community. In general, solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid. The evolution of the population need not be governed by any over‐arching partial differential equation (PDE). Although one can match the standard non‐linear parabolic PDEs of McKean–Vlasov type with specific examples of communities in this case. The bulk behaviour of the evolving population provides a solution to the PDE. We focus on the case of weakly interacting systems, where we are able to exhibit the existence and uniqueness of consistent solutions. An important technical result is continuity of the behaviour of the system with respect to changes in the measure 𝜈 assigning weight to individuals. Replacing the deterministic 𝜈 with the empirical distribution of an independent and identically distributed sample from 𝜈 leads to many standard models, and applying the continuity result allows easy proofs for propagation of chaos. The rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences. We make no presumption that the macroscopic dynamics are modelled by a PDE. This work builds on the fine probability literature considering the limit behaviour for systems where a large number of particles are interacting with independent preferences; there is also work on continuum models with preferences described by a semi‐martingale measure. We mention some of the key papers.

Authors: Cass, T; Lyons, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: London Mathematical Society
• Journal: Proceedings of the London Mathematical Society
• Publication date: 2014-08-19
• DOI: 10.1112/plms/pdu040
• EISSN: 1460-244X
• ISSN: 0024-6115
• Keywords: SBTMR