I hope so. Here is my attempt:
Imagine that I have 1.0 ETH and a project that only I care about: let us call it "OctopusWealth". I would be foolish to donate my 1.0 ETH directly to "OctopusWealth", even though this represents my real feeling. It would be much better for me to find a friend (whether a real person or a bot/sockpuppet account), give this other entity 0.5 of my ETH, and have two donations of 0.5 ETH. The problem is that there's no limit to this: OctopusWealth will get more matching funds with 10 donations of 0.1ETH, even more matching funds with 100 donations of 0.01ETH, and even matching funds for 10000 donations of 0.0001ETH. I can drain more and more of the pool towards my nuisance project, without spending any more money - the only challenge is recruiting more users (whether real people or bots). I repeat because this idea is so crucial: any time someone can recruit a new user (whether or a real person or not)
Notice that it's possible for this strategy to be so profitable in terms of matching funds that I could even bribe users to implement it: e.g. I'll give you 0.1 ETH to keep for yourself if you donate 0.0001 ETH to my project. Additionally, due to the Matthew Effect, my artificially popular grant is likely to get authentic donations from users who are new and just want to try out giving.
The key idea is that unless creating more cooperators becomes too expensive, there is literally no limit to how much of the matching funds I can accrue. In theory, if I can get enough people to give enough small donations, I could take 99.99% of the matching funds. One practical way to stop this is to put a minimum donation amount in place. If I have $10000 to spend, but the minimm donation is $1, then this limits to me 10000 donations of $1 each and stops the "infinite spiral".
In addition to placing a definitive ceiling on how much matching funds I can direct towards "OctopusWealth", placing a floor on the donation amount probably makes strategic play clearer If my account gets 10000 donations of $1, that is highly suspicious. I will be forced to vary the donation amounts to avoid detection, and this creates more work for m.
Q: Could we find an Optimum Minimum Bound? (prevents sybil the best while disenfranchising the least number of users)
My A: From a pure mathematical optimization standpoint, probably not. However, if we made assumptions (reasonable or based on analysis of data) about the distribution of what users would "naturally" give, then yes - we could model what % of users are disenfranchised vs. how much in strategic exploitation the system is likely to use.
For instance, if you say something like "the pool of GitCoin user donations follows an exponential distribution with a mean of lambda; how much would setting a minimum donation of 1.0 save in sybil attacks vs. how many users would it disenfranchise?" -- this can be answered fairly easily. It's just that the community will need to make on additional assumption.
Q: Maybe we could model the effect of different methods of decentralizing the power to select what is valid as a grant in a round?
My A: Yes, potentially. This is probably more of a Game Theory model, which is less in my immediate wheelhouse -- but I"m always happy to do research, recruiting, and collaboration to solve a problem. I don't have any ideas at the moment about how to decentralize this, but I'm happy to give precise qualitative analysis to any ideas that y'all have.
Q: What other questions do you think are worth answering?
My A:
MyQ1: I think one possible way forward would be for you to propose specific scenarios, either from your previous rounds data or hypotheticals. We can answer almost any specific question, and that often makes a better starting point for "big ideas" than starting with abstraction.
MyQ2: Are you still interested in doing a wargame (red vs. blue) simulation? Now that my students are on break, I could put together a strong red team on my own, and of course you have many other avenues for players.
MyQ3: It's not a question, but an idea that I want to emphasize that came out of the Math Working Group's research: you now have a precise formula at your disposal for calculating what "optimal play" would have brought to a particular grant.
The process for determining the max amount a grant could have received in QF:
Calculate x = "number of users who gave to the grant"
Calculate B = "total amount that those users gave to all grants" (summing)
3 Plug x and B into the formula 0.5x(x-1)*(B/(x+B)).