> "I don't know"

Which is exactly the same as “no” in this situation. If you were capable of proving the opposite I assume he would have been willing to hear you out.

> not be able to give you the answer to this on the spot

Realizing the most obvious strategy would be suboptimal if the game is adversarial is the first step of the correct answer though. If you didn’t know what to do next obviously the correct answer is “no, because I don’t know”.

Someone who is trying to absolve himself from making any decision ls is presumably not the sort of a person they were looking for.

Also would Balmer have been hiring for engineers to begin with?

No, it's about understanding what an interview is for. They're not trying to get the answer. They're trying to find out what you know, how you think and how you communicate.

Do you spot that it's different with one Vs many plays? Do you spot the binary search? Do you spot that an adversarial opponent can push things? Can you clearly communicate these?

If you just say "I don't know" and that's it you are showing you don't know how to communicate important information and miss soft skills about understanding the context of an interview.

If you say "I don't know" and talk through your thoughts then great. The point is talking things through, even if you have gotten the wrong answer.

Maybe you'd be able to say "and here's how I'd code a simulation to check"

In fairness, if someone proposes a complex and contrived game which cannot be easily analysed but is potentially adversarial... you shouldn't play them.