The total amount of political capital can grow exponentially.
The total amount of political capital can NOT grow exponentially. The difference seems to be the number of validators and the strategies they use.
Plot of total PC for a simulation with 46 threshold spender validators (with threshold ~10). The y-axis is a log scale to illustrate the exponential growth. The red line is a plot of , which is a fit to the exponential part of the PC graph.
Plot of the PC for each validator (coloured lines) and the total (dashed line). At after some time there is not enough PC available for any validator to reach their threshold and so everything stops. This is the same simulation as above, but with 10 validators instead of 46.
I suspect that there is a dynamical bifurcation between the exponential growth behaviour and approximately constant behaviour. This bifurcation likely occurs as N (the number of validators) and f (the fraction earned from acknowledgements) are varied. The strategy also probably plays a role as that impacts how many blocks are acknowledgements vs. proposals. It would be nice to have a mathematical model which could predict this transition so that f could be varied dynamically with N to prevent the exponential growth of PC. This is desirable since otherwise it could become too easy to propose a block.