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Consider the following agent-based model:

  • There are $N$ agents
  • Every agent starts with $1
  • At each time interval (i.e. at each step), every agent gives \$1 to a randomly chosen agent.

I want to find how unequal the wealth distribution becomes over a long period of time.

After running a simulation for a large number of agents, I find that the wealth distribution becomes over a long period of time approaches (what I am "by eye" guessing to be) a Boltzmann distribution.

I am curious as to why this happens from a derivation standpoint. I have tried to find sample derivations online, but only find kinetic-model related Boltzmann distribution derivations. Can anyone point me to any resources or share a derivation that explain this result?

hu234
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    What if an agent has 0 dollars at some step? – Maximilian Janisch Nov 02 '22 at 16:03
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    Thanks for the question! If an agent has 0 dollars at some step, then the agent does not give $1 to any other agent in that step. Otherwise, the agent gives $1 to a randomly chosen agent in that step. – hu234 Nov 02 '22 at 19:01

1 Answers1

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Not a full answer (but too long for a comment).

Consider the set $E$ of all vectors in $\mathbb Z_{\ge 0}^N$ which sum to $N$. Your game can be seen as a homogeneous Markov chain with state space $E$.

It is irreducible (exercise) and therefore, since $E$ is finite, it is positively recurrent (see 1). The chain is also aperiodic since there is a probability $>0$ that you go from one state to itself (everyone who has money gives themselves money, this is defined to happen with probability $>0$).

Therefore, by the Markov chain convergence Theorem (cf. [2; Satz 17.52 and Satz 18.13]), starting from any probability distribution over $E$, the distribution over $E$ of state $n$ of the Markov chain (corresponding to iterating your game $n$ times) will converge, in the sense of the total variation distance, to the unique distribution on $E$ which is invariant with respect to one iteration of the Markov chain.

Therefore, you should compute this irreducible measure on $E$ and then check that the „typical vector in $E$“ according to this distribution corresponds approximately to your conjectured wealth distribution. (Not an easy task in my opinion.)

Footnote: If the agents cannot give money to themselves, the chain is not always irreducible: If you have $N=2$ then the two agents keep swapping one dollar so you can never leave your initial state in that case.

References

[2]: Achim Klenke, Wahrscheinlichkeitstheorie, 4. Auflage, 2020.

Maximilian Janisch
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