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 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.

中文翻译：

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社会攀登和阿莫罗索分布

我们引入了一类玻尔兹曼和福克-普朗克类型的一维线性动力学方程，描述了多主体社会中个体在社会等级制度中寻求高地位的动态。在玻尔兹曼水平上，根据 Kahneman 和 Twersky 的前景理论的精神，建立了围绕普遍期望目标的代理状态的微观变化，引入了一个合适的价值函数作为状态变化的主要标准。在放牧相互作用的渐近线中，玻尔兹曼型动力学方程的解密度显示为收敛于具有可变扩散和漂移系数的 Fokker-Planck 型方程的解，其特征在于值函数的数学性质。福克-普朗克方程预测的社会地位统计分布的稳态属于具有帕累托尾的阿莫罗索分布类，对应于社会精英的出现。微观动力学相互作用的细节可以阐明表征所得平衡的各种参数的含义。然后数值结果表明，即使在相互作用不是放牧的中间状态下，基本动力学方程的稳态也接近阿莫罗索分布。