Abstract
We consider a finite number of particles characterised by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of another particle, the leader. The follower chooses its leader according to the proximity rank of the latter with respect to the former. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit equation is akin to the Boltzmann equation. However, it exhibits a spatial non-locality instead of the classical non-locality in velocity space. This result relies on the approximation properties of Bernstein polynomials.
Highlights
We explore collective dynamics driven by rank-based interactions, i.e. that’s to say interactions determined by the rank of the agents with respect to certain criterion
In economics for instance, it was extensively analysed in [14] that agents are more sensitive to their rank compared to others than their own independent cardinal level
Under a propagation of chaos assumption, we have shown that the large system size limit is described by a spatially nonlocal kinetic model of Boltzmann type
Summary
We explore collective dynamics driven by rank-based interactions, i.e. that’s to say interactions determined by the rank of the agents with respect to certain criterion. There are many examples where such interactions take place. In economics for instance, it was extensively analysed in [14] that agents are more sensitive to their rank compared to others (salary or wealth for example) than their own independent cardinal level. To go further, [17] studies, in an organisation, compensation schemes which pay according to an individual’s ordinal rank rather than their output level. Such payoff based on the rank approach . Degond appears very naturally in a variety of economics applications such as bids, the labour market, portfolio management, the oil market, academic production, reputation, etc
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