3,141 research outputs found
Wealth distribution and collective knowledge. A Boltzmann approach
We introduce and discuss a nonlinear kinetic equation of Boltzmann type which
describes the influence of knowledge in the evolution of wealth in a system of
agents which interact through the binary trades introduced in Cordier,
Pareschi, Toscani, J. Stat. Phys. 2005. The trades, which include both saving
propensity and the risks of the market, are here modified in the risk and
saving parameters, which now are assumed to depend on the personal degree of
knowledge. The numerical simulations show that the presence of knowledge has
the potential to produce a class of wealthy agents and to account for a larger
proportion of wealth inequality.Comment: 21 pages, 10 figures. arXiv admin note: text overlap with
arXiv:q-bio/0312018 by other author
Portfolio Optimization and Model Predictive Control: A Kinetic Approach
In this paper, we introduce a large system of interacting financial agents in
which each agent is faced with the decision of how to allocate his capital
between a risky stock or a risk-less bond. The investment decision of
investors, derived through an optimization, drives the stock price. The model
has been inspired by the econophysical Levy-Levy-Solomon model (Economics
Letters, 45). The goal of this work is to gain insights into the stock price
and wealth distribution. We especially want to discover the causes for the
appearance of power-laws in financial data. We follow a kinetic approach
similar to (D. Maldarella, L. Pareschi, Physica A, 391) and derive the mean
field limit of our microscopic agent dynamics. The novelty in our approach is
that the financial agents apply model predictive control (MPC) to approximate
and solve the optimization of their utility function. Interestingly, the MPC
approach gives a mathematical connection between the two opponent economic
concepts of modeling financial agents to be rational or boundedly rational.
Furthermore, this is to our knowledge the first kinetic portfolio model which
considers a wealth and stock price distribution simultaneously. Due to our
kinetic approach, we can study the wealth and price distribution on a
mesoscopic level. The wealth distribution is characterized by a lognormal law.
For the stock price distribution, we can either observe a lognormal behavior in
the case of long-term investors or a power-law in the case of high-frequency
trader. Furthermore, the stock return data exhibits a fat-tail, which is a well
known characteristic of real financial data
Kinetic models for goods exchange in a multi-agent market
We introduce a system of kinetic equations describing an exchange market
consisting of two populations of agents (dealers and speculators) expressing
the same preferences for two goods, but applying different strategies in their
exchanges. We describe the trading of the goods by means of some fundamental
rules in price theory, in particular by using Cobb-Douglas utility functions
for the exchange. The strategy of the speculators is to recover maximal utility
from the trade by suitably acting on the percentage of goods which are
exchanged. This microscopic description leads to a system of linear
Boltzmann-type equations for the probability distributions of the goods on the
two populations, in which the post-interaction variables depend from the
pre-interaction ones in terms of the mean quantities of the goods present in
the market. In this case, it is shown analytically that the strategy of the
speculators can drive the price of the two goods towards a zone in which there
is a marked utility for their group. Also, the general system of nonlinear
kinetic equations of Boltzmann type for the probability distributions of the
goods on the two populations is described in details. Numerical experiments
then show how the policy of speculators can modify the final price of goods in
this nonlinear setting
Social climbing and Amoroso distribution
We introduce a class of one-dimensional linear kinetic equations of Boltzmann
and Fokker--Planck type, describing the dynamics of individuals of a
multi-agent society questing for high status in the social hierarchy. At the
Boltzmann level, the microscopic variation of the status of agents around a
universal desired target, is built up introducing as main criterion for the
change of status a suitable value function in the spirit of the prospect theory
of Kahneman and Twersky. In the asymptotics of grazing interactions, the
solution density of the Boltzmann type kinetic equation is shown to converge
towards the solution of a Fokker--Planck type equation with variable
coefficients of diffusion and drift, characterized by the mathematical
properties of the value function. The steady states of the statistical
distribution of the social status predicted by the Fokker--Planck equations
belong to the class of Amoroso distributions with Pareto tails, which
correspond to the emergence of a \emph{social elite}. The details of the
microscopic kinetic interaction allow to clarify the meaning of the various
parameters characterizing the resulting equilibrium. Numerical results then
show that the steady state of the underlying kinetic equation is close to
Amoroso distribution even in an intermediate regime in which interactions are
not grazing
Refining self-propelled particle models for collective behaviour
Swarming, schooling, flocking and herding are all names given to the wide variety of collective behaviours exhibited by groups of animals, bacteria and even individual cells. More generally, the term swarming describes the behaviour of an aggregate of agents (not necessarily biological) of similar size and shape which exhibit some emergent property such as directed migration or group cohesion. In this paper we review various individual-based models of collective behaviour and discuss their merits and drawbacks. We further analyse some one-dimensional models in the context of locust swarming. In specific models, in both one and two dimensions, we demonstrate how varying the parameters relating to how much attention individuals pay to their neighbours can dramatically change the behaviour of the group. We also introduce leader individuals to these models with the ability to guide the swarm to a greater or lesser degree as we vary the parameters of the model. We consider evolutionary scenarios for models with leaders in which individuals are allowed to evolve the degree of influence neighbouring individuals have on their subsequent motion
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