I don't have a solution for you, but a related probably-unsolvable problem is what some friends of mine call “cashing in your reputation capital”: having done the work to build up a reputation (for trustworthiness, in particular), you betray it in a profitable way and run.

… otherwise trustworthy people are forced to act against their will. … But if we make everything secret, is there a way to verify whether the system is really working as described?

This is a problem in elections. In the US (I believe depending on state) there are rules which are intended to prevent someone from being able to provide proof that they have voted a particular way (to make coercion futile), and the question then is whether the vote counting is accurate. I would suggest that the topic of designing fair elections contains the answer to your question insofar as an answer exists.