I get \sigma < 4 \gamma for 3.c, did I understand this correctly?
The model doesn't make much sense to me. Why would the per person coordination cost be quadratic. You'd expect something maybe linear.
The model doesn't make much sense to me. Why would the per person coordination cost be quadratic. You'd expect something maybe linear.
In a group of N people, there are N x (N-1) potential direct 1-to-1 interactions, and (N-1) x (N-2) interactions with one predetermined intermediary. Those are both N^2 terms. Per capita, it would be an N^1 term. At the human scale of around 150 personal relationships, everybody knows everybody, so the burden for anyone is knowing everyone else.
Beyond that size, you need at least one intermediary. There are N x (N-1) x (N-2) interactions between two people with an unspecified intermediary. That is an N^3 term, so N^2 per capita. The burden there is knowing at least one person that knows the person you want to deal with. At another critical size, again dependent on human social limits, you can no longer effectively use friend-of-a-friend connections.
Then you create sociological systems to take the place of personal intermediaries. These tend to scale relatively proportionally with the size of the population. A road system for a town with twice the population will be about twice as big. Miles of road per person will be similar near the median town size. The same goes for cops per person and sewer pipes per person and cash available for withdrawal from bank branches per person, and so on. For simplicity, we can just assume that all the systems of civil society scale according to N. So now all trades with two counterparties have complexity N x N x (N-1), which is again an N^3 term with an N^2 per-capita component. But the scale of the system is not necessarily N^1. We just know it's a positive number. So the per-capita coordination burden is more rightfully N^(1+something).
That "something" can be found by performing a statistical analysis of city budgets by city population. That seems like a task for someone more interested in the topic than I am.