Unlike the classical kinetic theory of rarefied gases, where microscopic interactions among gas molecules are described as binary collisions, the modelling of socio-economic phenomena in a multi-agent system naturally requires to consider, in various situations, multiple interactions among the individuals. In this paper, we collect and discuss some examples related to economic and gambling activities. In particular, we focus on a linearisation strategy of the multiple interactions, which greatly simplifies the kinetic description of such systems while maintaining all their essential aggregate features, including the equilibrium distributions.
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Figure 1. The asymptotic wealth variance $ \Sigma^\infty $ vs. the taxation rate $ \alpha $ for both constant and $ \alpha $-dependent mean wealth $ m $ of the background
Figure 2. Evolution at times $ t = 1,\,5,\,25 $ of the multiple-interaction model (8), (9) with $ N = 5 $, $ N = 100 $ and of its linearised version (12), (13)
Figure 3. Comparison between the large time solution ($ t = 25 $) of the linearised model (12), (13) and the equilibrium distribution $ f^\infty $ (17) computed from the Fokker-Planck equation (16) in the quasi-invariant regime. Top row: $ \kappa = 0.1 $, bottom row: $ \kappa = 0.01 $. The right column displays the log-log plots of the graphs in the left column
Figure 4. Comparison between the equilibrium distribution $ f^\infty $ (17) and the large time solution of the multiple-interaction model with $ N = 10^2 $ and $ N = 10^3 $ for $ \kappa = 0.1 $. The right panel is the log-log plot of the graph in the left panel, which gives a closer insight into the tails of the compared distributions
Figure 5. Evolution at times $ t = 5,\,10,\,25 $ of the multiple-interaction model (18), (20) with $ N = 2 $, $ N = 5 $, $ N = 100 $ and of its linearised version (22), (23). The bottom row displays the log-log plots of the graphs in the top row to better appreciate the approximation of the tail of the distribution
Figure 6. Left: comparison of the equilibrium distribution 30 and the large time distribution of the linearised model (22), (24) in the quasi-invariant limit. Right: comparison of the evolution of the energy of the multiple-interaction model (20) with $ N = 10 $ for decreasing $ \lambda $ and the solution to (28) obtained in the quasi-invariant limit
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Figures(6)
The asymptotic wealth variance
Evolution at times
Comparison between the large time solution (
Comparison between the equilibrium distribution
Evolution at times
Left: comparison of the equilibrium distribution 30 and the large time distribution of the linearised model (22), (24) in the quasi-invariant limit. Right: comparison of the evolution of the energy of the multiple-interaction model (20) with