Robin Hood model versus Sheriff of Nottingham model: transfers in population dynamics

TL;DR

The study formulates a rigorous, measure-valued framework for transfers in a population where the exchanged quantity is wealth-like, using a transfer kernel $K$ to drive a continuous-time dynamics on the space of measures. It proves well-posedness via a continuous semiflow on $\mathcal{M}_+(I)$, establishes $L^1$-invariance under suitable kernel conditions, and analyzes long-time behavior across RH, SN, and distributed variants. The RH rule promotes consensus, driving the distribution to a Dirac mass at the initial mean, while SN induces unbounded inequality; distributed RH yields a unique stationary distribution with exponential convergence. Numerical simulations corroborate the analytical results and reveal that even a small SN fraction leads to rapid segregation, highlighting the sensitivity of wealth distributions to transfer rules and offering a framework applicable to biological and economic transfer phenomena.

Abstract

We study the problem of transfers in a population structured by a continuous variable corresponding to the quantity being transferred. The model takes the form of an integro-differential equations with kernels corresponding to the specific rules of the transfer process. We focus our interest on the well-posedness of the Cauchy problem in the space of measures. We characterize transfer kernels that give a continuous semiflow in the space of measures and derive a necessary and sufficient condition for the stability of the space $L^1$ of integrable functions. We construct some examples of kernels that may be particularly interesting in economic applications. Our model considers blind transfers of economic value (e.g. money) between individuals. The two models are the ``Robin Hood model'', where the richest individual unconditionally gives a fraction of their wealth to the poorest when a transfer occurs, and the other extreme, the ``Sheriff of Nottingham model'', where the richest unconditionally takes a fraction of the poorest's wealth. Between these two extreme cases is a continuum of intermediate models obtained by interpolating the kernels. We illustrate those models with numerical simulations and show that any small fraction of the ``Sheriff of Nottingham'' in the transfer rules leads to a segregated population with extremely poor and extremely rich individuals after some time. Although our study is motivated by economic applications, we believe that this study is a first step towards a better understanding of many transfer phenomena occurring in the life sciences.

Figures (8)

• Figure 1: In the figure, a transfer between two individuals, and consider $x_1$ and $x_2$ (respectively $y_1$ and $y_2$) the values of the transferable quantities before transfer (respectively after transfer). We plot a transfer whenever $x_2>x_1$, and $f\in [0,1/2]$ (on the left hand side), and $(1-f)\in [1/2,1]$ (on the right hand side). We observe that the values $y_1$ and $y_2$ are the same on both sides.
• Figure 2: In the figure, the two values before transfers are $x_1$ and $x_2$, and the two values after transfer are $y_1$ and $y_2$. We plot a transfer whenever $x_2>x_1$, and $f\in [0,1]$ (on the left hand side), and $f$ is replaced by $1+f$ (on the right hand side). We observe that the values $y_1$ and $y_2$ are the same on both sides.
• Figure 3: In this figure, we use $p=1$ (i.e. $100 \%$ RH model), $f_1=f_2=0.1$, $1/\tau=1$ years. We start the simulations with $100 \, 000$ individuals. The figures (a) (b) (c) (d) are respectively the initial distribution at time $t=0$, and the distribution $10$ years, $50$ years and $100$ years.
• Figure 4: In this figure, we zoom on the distribution for $t=100$ in Figure \ref{['Fig3']} (d). The figure on the right-hand side corresponds to the yellow region in the left figure.
• Figure 5: In this figure, we use $p=0.5$ (i.e. $50 \%$ RH model and $50\%$ SN model), $f_1=f_2=0.1$, $1/\tau=1$ years. We start the simulations with $100 \, 000$ individuals. The figures (a) (b) (c) (d) are respectively the initial distribution at time $t=0$, and the distribution $10$ years, $50$ years and $100$ years.

Theorems & Definitions (27)

• Remark 3.1
• Theorem 4.1
• proof
• Proposition 4.2
• Theorem 4.3
• Remark 4.4
• Remark 4.5
• Proposition 4.6: Improved regularity of the semiflow
• Lemma 4.7
• proof