Actually, just to move this forward, how about I'll propose a formal model for the game that I am discussing, and we'll see what parts of it you disagree. We'll also limit ourselves to assurance contracts to simplify things; if we agree on the economics of the AC then we can move on to the DAC.

1. There exist N players, each of which receive $V utility from the production of a hypothetical public good.

2. An assurance contract is set up where people can contribute either $0 or $C.

3. If more than N/k people contribute, the funds are sent to the entrepreneur, otherwise they are sent back to the donors. k is set by the entrepreneur, because the entrepreneur knows from prior experience that each person has a probability of 1/k of contributing (the reason why the entrepreneur wants to set the threshold to N/k is so that the threshold is right at the top of the bell curve for the probability distribution of total contributions, maximizing the probability that someone is pivotal and therefore maximizing the incentive to contribute). As another consequence of this optimization on the part of the entrepreneur, the probability of success is 0.5.

4. The game lasts for R rounds, round 1 ... round R. People who have not yet contributed can become contributors during any round.

Note that there are plenty of simplifications here. If you think that given these simplifications my analysis is correct, but under your preferred simplifications my analysis is not correct, then we can focus on the simplifications. If you think that my analysis is not correct even given the simplifications, then we move on.

1. There is no incentive to contribute to rounds 1 ... R-1 (this is because you have more information in round R, and because contributing earlier means that you are pushing the probability of success toward the right side of the gaussian, where the derivative of the probability of success is lower, so fewer people will contribute)

2. Let p be the probability of being pivotal.

3. The utility of contributing is pV - C * 0.5. Hence, someone will contribute if 2pV > C.

The stable equilibrium is the one where 2pV = C, so some people contribute and some do not, and the equilibrium probability of contributing is 1/k. If more than 1/k people contribute, then the Gaussian will move to the right, so the threshold will no longer be at the top of the Gaussian, so p will be lower and thus 2pV < C so others will be less likely to contribute to compensate (so it would be the same result except you are expected to pay more); less than 1/k people contributing has the same result, except instead of compensating it drives the success probability to zero (which nobody wants).

This equilibrium does not exist if there are no values of C and k such that 2pV = C.