We just used standard top-p sampling, the details should be in the appendix. We just sample one explanation. I think I did not follow your suggestion.
Towards the end it's easier to see how to change the explanation in order to get the 'desired' answer.
Some relevant discussion here: https://twitter.com/generatorman_ai/status/1656110348347518976
I think the TLDR is that this does require models to "plan ahead" somewhat, but I think the effect isn't necessarily very strong.
I don't think "planning" in language models needs to be very mysterious. Because we bias towards answer choices, models can just use these CoT explanations as post-hoc rationalizations. They may internally represent a guess at the answer before doing CoT, and this internal representation can have downstream effects on the explanations (in our case, this biases the reasoning). I think models probably do this often -- models are trained to learn representations that will help for all future tokens, not just the next token. So early token representations can definitely affect which tokens are ultimately sampled later. E.g. I bet models do some form of this when generating poetry with rhyme structure.
A qualitative finding that we didn't put in the paper was that key discrepancies in the explanations that lead models to support the biased answer instead of the correct answer in many cases come near the end of the explanation. Sometimes the model does normal reasoning, and then gives some caveat or makes a mistake at the end, that leads it to ultimately give the biased prediction. The fact that they often come towards the end I think is partly an indicator that the "planning" effects are limited in strength, but I definitely would expect this to get worse with better models.
Thanks! Glad you like it. A few thoughts: