Yes Requires the Possibility of No
1. A group wants to try an activity that really requires a lot of group buy in. The activity will not work as well if there is doubt that everyone really wants to do it. They establish common knowledge of the need for buy in. They then have a group conversation in which several people make comments about how great the activity is and how much they want to do it. Everyone wants to do the activity, but is aware that if they did not want to do the activity, it would be awkward to admit. They do the activity. It goes poorly. 2. Alice strongly wants to believe A. She searches for evidence of A. She implements a biased search, ignoring evidence against A. She finds justifications for her conclusion. She can then point to the justifications, and tell herself that A is true. However, there is always this nagging thought in the back of her mind that maybe A is false. She never fully believes A as strongly as she would have believed it if she just implemented an an unbiased search, and found out that A was, in fact, true. 3. Bob wants Charlie to do a task for him. Bob phrases the request in a way that makes Charlie afraid to refuse. Charlie agrees to do the task. Charlie would have been happy to do the task otherwise, but now Charlie does the task while feeling resentful towards Bob for violating his consent. 4. Derek has an accomplishment. Others often talk about how great the accomplishment is. Derek has imposter syndrome and is unable to fully believe that the accomplishment is good. Part of this is due to a desire to appear humble, but part of it stems from Derek's lack of self trust. Derek can see lots of pressures to believe that the accomplishment is good. Derek does not understand exactly how he thinks, and so is concerned that there might be a significant bias that could cause him to falsely conclude that the accomplishment is better than it is. Because of this he does not fully trust his inside view which says the accomplishment is good. 5. Eve is has an aversio
I proposed this same voting system here: https://www.lesswrong.com/s/gnAaZtdwjDBBRpDmw
It is not strategy proof. If it were, that would violate https://en.wikipedia.org/wiki/Gibbard–Satterthwaite_theorem [Edit: I think, for some version of the theorem. It might not literally violate it, but I also believe you can make a small example that demonstrates it is not strategy proof. This is because the equilibrium sometimes extracts all the value from a voter until they are indifferent, and if they lie about their preferences less value can be extracted.]
Further, it is not obviously well defined. Because of the discontinuities around ties, you cannot take advantage of the compactness of the space of distributions, so it is not clear that Nash equilibria exist. (It is also not clear that they don't exist. My best guess is that they do.)