Let me propose a positive, generative theory that generates multiple agents/firms/etc.
Any agent (let’s say nation) when it starts has a growth rate and a location. As it grows, it gains more power in its location, and power nearby at levels declining exponentially with distance. If a new nation starts at some minimum level of power, not too late and somewhat far from a competing nation, then it can establish itself and will end up having a border with the other nation where their power levels are the same.
Certainly there are growth rates for different nations that would give us fluctuating or expanding borders for new nations or that would wipe out new nations. Major changes in history could be modeled like different coefficients on the rates of growth or the exponent of decay (or maybe more clearly by different measures of distance, especially with inventions like horses, boats, etc.)
In particular if the growth rates of all nations are linear, then as long as a nation can come into existence it will be able to expand as it ages from size zero to some stable size. The linear rates could be 5000 and .0001 an the smaller growth nation would still be able to persist—just with control of much less space.
In practice, in history, some nations have become hundreds of times as large as others, but only a certain size has been achieved. There are a few ways this model could account for a different single-nation scenario.
Non-linear growth rates; in particular growing at a rate that equals the rate of distance decay (no matter what those rates are) would overwhelm any nation growing at a linear rate. Probably it’s easier to overwhelm those nations than that.
Massive changes to the distance decay. This has been attested historically at least in small part—European countries expanding into the Americas and Africa with boats, Mongols with horses, even back to vikings. This is also analogous to principal-agent problems (and maybe Roman republicanism is another example?) Mathematically, if the exponents are similar it won’t matter too much (or at least the ‘strongest’ state will take ‘some time’ to overpower others) but an immediate, large change on the part of the strongest nation would cause it to overwhelm every other nation
Not having enough space for equilibrium. A new nation can only start in this model if it’s far enough away from established nations to escape their influence. We don’t see much in the way of new nations starting these days, for example.
This is a toy model that I literally came up with off the top of my head, so I don’t mean to claim that it has any really thrilling analogies to reality that I haven’t listed above.
I do think it’s robustly useful in that if you massively change all the parameters numerically it should still usually predict that there will be many nations but if you tweak the parameters at a higher level it could collapse into a singleton.
Let me propose a positive, generative theory that generates multiple agents/firms/etc.
Any agent (let’s say nation) when it starts has a growth rate and a location. As it grows, it gains more power in its location, and power nearby at levels declining exponentially with distance. If a new nation starts at some minimum level of power, not too late and somewhat far from a competing nation, then it can establish itself and will end up having a border with the other nation where their power levels are the same.
Certainly there are growth rates for different nations that would give us fluctuating or expanding borders for new nations or that would wipe out new nations. Major changes in history could be modeled like different coefficients on the rates of growth or the exponent of decay (or maybe more clearly by different measures of distance, especially with inventions like horses, boats, etc.)
In particular if the growth rates of all nations are linear, then as long as a nation can come into existence it will be able to expand as it ages from size zero to some stable size. The linear rates could be 5000 and .0001 an the smaller growth nation would still be able to persist—just with control of much less space.
In practice, in history, some nations have become hundreds of times as large as others, but only a certain size has been achieved. There are a few ways this model could account for a different single-nation scenario.
Non-linear growth rates; in particular growing at a rate that equals the rate of distance decay (no matter what those rates are) would overwhelm any nation growing at a linear rate. Probably it’s easier to overwhelm those nations than that.
Massive changes to the distance decay. This has been attested historically at least in small part—European countries expanding into the Americas and Africa with boats, Mongols with horses, even back to vikings. This is also analogous to principal-agent problems (and maybe Roman republicanism is another example?) Mathematically, if the exponents are similar it won’t matter too much (or at least the ‘strongest’ state will take ‘some time’ to overpower others) but an immediate, large change on the part of the strongest nation would cause it to overwhelm every other nation
Not having enough space for equilibrium. A new nation can only start in this model if it’s far enough away from established nations to escape their influence. We don’t see much in the way of new nations starting these days, for example.
This is a toy model that I literally came up with off the top of my head, so I don’t mean to claim that it has any really thrilling analogies to reality that I haven’t listed above.
I do think it’s robustly useful in that if you massively change all the parameters numerically it should still usually predict that there will be many nations but if you tweak the parameters at a higher level it could collapse into a singleton.