Intuitively, I would expect that we can collapse the result of doing this type of dollar scrambling over many iterations of the simulation into a single operation which takes a sequence of dollars and partitions it into N randomly-sized partitions for N people.
For instance, we can arrange the dollars into a sequence and then insert N-1 randomly-placed divisions to chop up the sequence. Then each of N people in the corresponding people sequence gets their corresponding (possibly empty) piece of the dollar sequence.
It is not intuitive at all to expect the pieces to be more or less identically long (which corresponds to the wrong intuition that everyone will have more or less their equal share of the dollars). That would mean that the chops are evenly spaced and not random.
For instance, we can arrange the dollars into a sequence and then insert N-1 randomly-placed divisions to chop up the sequence. Then each of N people in the corresponding people sequence gets their corresponding (possibly empty) piece of the dollar sequence.
If we distribute randomly placed chops into section of the real number line, we end up with string lengths that follow an exponential distribution: https://en.wikipedia.org/wiki/Exponential_distribution
It is not intuitive at all to expect the pieces to be more or less identically long (which corresponds to the wrong intuition that everyone will have more or less their equal share of the dollars). That would mean that the chops are evenly spaced and not random.
Randomly placing chops is a https://en.wikipedia.org/wiki/Poisson_process
However, part of the intuition here (possibly wrong) is that Poisson is applicable to a crudely discrete process like this, in some approximate way.