Whether the threshold is met in each case depends on the number of other signatories. Let’s call the number of other signatories XX, where X∼Binomial(N−1,p)X∼Binomial(N−1,p). Then:
There’s not a general closed form for pp, so we used numerical methods to find values for pp and the probability that the threshold number of signatories is reached given different values for NN and mm.
You solved for the solution of your equation numerically, but I think a decent analytical proxy would probably be that the good gets funded if m*sqrt(N)>1.
The analytical intuition is:
The standard deviation of the sum of N independent events grows as sqrt(N).
The probability of causing the good to go from unfunded to funded is proportional to the standard deviation (if the distribution is centered around the threshold, which it will be when the probability of getting funded is 50%, ie when the good goes from probably unfunded to probably funded).
The gains are proportional to N and m, so the expected gains are proportional to m*N/sqrt(N)=m*sqrt(N)
The value of funding your selfish good is constant at 1.
Re: The appendix on assurance contract.
You solved for the solution of your equation numerically, but I think a decent analytical proxy would probably be that the good gets funded if m*sqrt(N)>1.
The analytical intuition is:
The standard deviation of the sum of N independent events grows as sqrt(N).
The probability of causing the good to go from unfunded to funded is proportional to the standard deviation (if the distribution is centered around the threshold, which it will be when the probability of getting funded is 50%, ie when the good goes from probably unfunded to probably funded).
The gains are proportional to N and m, so the expected gains are proportional to m*N/sqrt(N)=m*sqrt(N)
The value of funding your selfish good is constant at 1.
So the good gets funded if m*sqrt(N)>1.
And it seems to roughly match the graph.