Oh I see now, B just needs to work to pinpoint Nash equilibria, I did not make that connection.
But anyway, the reason I’m suspicious of your interpretation is not that your math is not correct, but that it makes the OP notation so unnatural. The unnatural things are:
∏ being context-dependent.
∏ not having its standard meaning.
Ui used implicitly instead of explicitly, when later it takes on a more important role to change decision theory.
Using x∈B(x) as condition without mentioning that already B(x)≠∅⟺x is Nash if |I|≥2.
So I guess I will stay in doubt until the OP confirms “yep I meant that”.
Suppose Alice and Bob are playing prisoner’s dilemma. Then the best-response function of every option-profile is nonempty. But only one option-profile is nash.
Oh I see now, B just needs to work to pinpoint Nash equilibria, I did not make that connection.
But anyway, the reason I’m suspicious of your interpretation is not that your math is not correct, but that it makes the OP notation so unnatural. The unnatural things are:
∏ being context-dependent.
∏ not having its standard meaning.
Ui used implicitly instead of explicitly, when later it takes on a more important role to change decision theory.
Using x∈B(x) as condition without mentioning that already B(x)≠∅⟺x is Nash if |I|≥2.
So I guess I will stay in doubt until the OP confirms “yep I meant that”.
B(x)≠∅ isn’t equivalent to x being Nash.
Suppose Alice and Bob are playing prisoner’s dilemma. Then the best-response function of every option-profile is nonempty. But only one option-profile is nash.
x∈B(x) is equivalent to x being Nash.