Huh, that's interesting! I did "give away" the answer to it by asking in the same chat a number of questions about Iterated Prisoner's Dilemma. I wouldn't have thought that this could have been used for feedback so quickly, but who knows. Maybe so! Or maybe it sometimes it lucks out and gets the attention modules in the right place to successfully lock into the appropriate style of answer.

Edit: I probed it with a less obfuscated version of the story with all the details changed, and it analyzed the situation correctly (but failed to notice, or at least to reveal, that it is Prisoner's Dilemma). So it could just be that this is at the outer limits of its capabilities (I've mostly probed GPT-3.5, not -4) and sometimes it manages to converge on the right style of language usage to come up with the correct answer, but when I tried it the first time it made some bad random choices that led it to linguistic patterns that were not a good match to logical analysis.

Second edit: the reward-distribution logic was analyzed correctly by Bard in my somewhat de-obfuscated version, but it failed to give appropriate advice again because the mutual dependency wasn't appreciated when constructing the responses. When specifically asked what the problem is like, however, it does come up with Prisoner's Dilemma, and only then gives exactly the right answer for the strategy.