After reading the post on the dominant assurance contract implementation, I read through the original DAC paper, and I'm trying to understand what it is about DACs that makes them effective in funding public goods where traditional ACs are not. Tabarrok claims in http://mason.gmu.edu/~atabarro/PrivateProvision.pdf
that they solve the public goods provision problem via a solution with one equilibrium, and argues that even under imperfect information they can be used to provide public goods with 1/2 probability.
I tried to do the math behind the contracts in a simplified form that makes things clearer, but I'm arriving at a completely different conclusion: DACs are not a single bit better than traditional ACs.
First, the definition. The way an AC works is that there are N participants, a public good with per-person cost C, per-person reward ~V (that's shorthand for "a distribution centered around V") with V > C, and a contract is set up where people can pledge R, and if at least C/R people contribute then the contract spends the funds and produces the public good, and if fewer people contribute then everyone is refunded. From the point of view of a participant, there are three possible scenarios:
1. The number of people who will pledge not including them is less than C/R - 1. In this case, if they pledge or don't pledge their return is 0.
2. The number of people who will pledge not including them is exactly C/R - 1. In this case, the return is V - R for pledging and 0 for not pledging.
3. The number of people who will pledge including them exceeds C/R. In this case, the return is V-R for pledging and V for not pledging.
Presuppose that the probability for (1) is 1/2-p/2, for (2) is p, and for (3) is 1/2-p/2. By central limit theorem p ~= 1 / sqrt(N). Then, the expected return from pledging is:
p * (V - R) - R*(1/2-p/2)
People will contribute if that value is greater than zero. Then, we have:
p * (V - R) - R/2+pR/2 > 0
2pV - pR - R+pR > 0
2pV > R
Since R = 2C, that's:
pV > C
Hence, people have the incentive to contribute if the probability that they are "pivotal", ie. the chance the goal will be reached with them and fail without them, is greater than the inverse of the social return coefficient (V/C). Thus, a public good with 100x social return will succeed up to N = 10000 people, 10x social return with up to N = 100 people, etc.
Now, let's try a dominant assurance contract. Now, there is an entrepreneur who makes the following deal: if at least kN people (let k=1/2) sign up, then they will be required to provide a payment of S to the entrepreneur, and the entrepreneur will pay C per person for the public good, which has value ~V per person. Otherwise, the entrepreneur pays everyone F. The idea is that if the contract always fails then everyone thinks the contract will fail, then everyone has the incentive to contribute in order to receive F, but then if that happens then the contract will eventually end up succeeding at least part of the time. Now, let's look at the incentives of the entrepreneur (who controls F and S), with success probability 1/2 as before.
1/2 * S - 1/2 * F - C > 0
S > F + 2C
S - F > 2C
Now, given that, let's look at the individual participants:
p * (V - S) + F * (1/2 - p/2) - S * (1/2 - p/2) > 0
2p * (V - S) + (F - S) - p * (F - S) > 0
2p * (V - S) + p * (S - F) > S - F
2p(V-S) > (1-p)(S-F)
2p(V-S) > (1-p)2C
p(V-S) > (1-p)C
p(V-C) > C - pC (since obviously S > C for the entrepreneur to be profitable)
pV - pC > C - pC
pV > C
Exactly the same inequality. What in my above analysis is wrong?