Wikipedia:Reference desk/Archives/Mathematics/2021 June 19

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June 19

Steal from the rich/poor, give to the poor/rich

I have wondered about how the scenarios "steal from the rich, give to the poor" and "steal from the poor, give to the rich" would eventually end up, if allowed to proceed to the end in idealistic conditions.

What I have come up is:

• Steal from the rich, give to the poor: Everyone has the exact same amount of money.
• Steal from the poor, give to the rich: One person has all the money in the world. No one else has anything.

Is this correct, and is there way to formally prove this? JIP | Talk 01:46, 19 June 2021 (UTC)

Assuming there are $N$ individuals, we can set up a system of $N$ first-order differential equations for the flow of wealth to individual $i$:

${\frac {dw_{i}}{dt}}=\sum _{j=1}^{N}f(w_{i},w_{j}).$

Here, function $f$ regulates the rate of transfer. People can have negative wealth, but for the sake of simplicity we exclude this possibility. By postulating that

$f(v,u)=-f(u,v),$

we have that $\textstyle {\sum _{i}w_{i}}$ is constant.

For the scenario of social redistribution of wealth, we can use:

$f(u,v)=-\lambda (u-v),$

where $\lambda >0$. Equilibrium is reached when all wealth values $w_{i}$ are equal.

A negative value for $\lambda$ means that the rich are getting richer and the poor are getting poorer, but will lead, with this function definition, to explosive negative wealth. So for this scenario we use:

$f(u,0)=f(0,v)=0,$

$f(u,v)=\lambda \operatorname {sign} (u-v)~\mathrm {if} ~u,v>0,$

again with $\lambda >0$. Here, equilibrium is reached when all non-zero wealth values are equal. If at any time two individuals have equal wealth, their wealth values will from then on remain equal. More generally, if at some time we have $w_{i}>w_{j}$, then at any later time still $w_{i}>w_{j}$, or $w_{i}=w_{j}=0.$ If, initially, there was one individual who was the unique richest, they will remain the richest and emerge in the limit as the sole possessor of any wealth. If several initially possess exactly equal and maximal wealth, they will end up sharing the totality equally. This is, however, an unstable equilibrium. If we add quantum fluctuations to the model, there will again be a unique Emperor Rat survivor of the rat race. In the end you can prove anything with some model; this simplistic model supports the conjecture, and by looking at the proof it should be clear that this is also the case for a much wider class of transfer functions.  --Lambiam 09:31, 19 June 2021 (UTC)