It was an interesting game to think about, thanks for the exercise!

For P2, the dominant strategy if he is B, is A (100% win). But if he is C or D, I think it is to randomize between A or B (50% win), because due to the dominant strategy of P1, he cannot update his beliefs based on the fact that P1 is playing B.

But honestly, I'm not quite sure about this.

As in the second case it does not change his chance of winning, I would argue that the dominant strategy for P1 is to play B because he is both minimizing losing chance and revealing less information.

But if P1 has B, due to the first part, playing C or D would reveal he has B, so he would have 50% chance of winning in first guess and would be sure to lose if he missed.

If he has B and guesses B, then he has 0% of winning by first guess and 50% winning with P2's guess.

I would say that the dominant strategy for P1 is to guess B if he is A because he has 1/3 of winning in first guess while if the other was really B and P1 guessed something else, he has 100% chance of losing in second guess, so what minimizes losing chance is guessing B.

Good question, I don't know much about this type of games. I would intuitively say no and that there would be Bayesian eqs.

If we assume players are fully rational, then :

Sometimes the simplest reasoning are the most effective!

Do you have another reasoning process in mind? @adrien-fabre.bsky.social

Probably it would be interesting to also think about the fact that he did not choose C or D so, if he is not level 0, he preferred to give me information to try to trick me than not and forcing me to randomly play A or B.

So I think I would answer B. Both because I believe they are the largest part of the distribution and they are the only sophistication level that can be tricked.

And finally, I would simplify by assuming that anyone being able to think about the "double trick" (belief about belief and so on) is trapped as me in this binary loop.

For instance I suppose there are not so many level 0, people not understanding or who are not caring about the outcome (let's quite randomly say 20%), 50% thinking about what I could do so trying to trick me (here they would be B)

Clearly not trivial I would say. That's difficult to assume a level (k-1) to which I could find the best response because there is no reason to believe the other is a odd or even level of thinking. So I would think about the best response to the distribution of levels in the population.

he could have anticipated my belief that I see this level0+ thinking and then be Player B. And so on with beliefs about beliefs about beliefs...
So it's a level k thinking issue.

If I unterstood correctly the rules, either he chose B because he is a random level thinker, or because he anticipated that I believe that if he chose B, I would think the only remaining possibility for him is to be A.
But if he is more sophisticated...

Thanks for the discovery! It's all I've listened to for the two last days while working

Been waiting on this one since your keynote in Lyon like two (three?) years ago!

I did not know about the fact that registrations were made public after a 4-year embargo on OSF. That's great regarding the drawer effect issue.