(I’m using the notation that a function applied to a set is the image of that set.)
So the big pi symbol stands for
∏i∈IAi=⋂i∈IUi(x)(Ai)
So it’s not a standalone operator: it’s context-dependent because it pops out an implicit x. The OP otherwise gives the impression of a more functional mindset, so I suspect the OP may mean something different from your guess.
Other problem with your interpretation: it yields the empty set unless all agents consider doing nothing an option. The only possible non-empty output is {x}. Reason: each set you are intersecting contains tuples with all elements equal to the ones in x, but for one. So the intersection will necessarily only contain tuples with all elements equal to those in x.
Edit: Cleo Nardo has confirmed that they intended ∏i∈I to mean the cartesian product of sets, the ordinary thing for that symbol to mean in that context. I misunderstood the semantics of what B(x) was intended to represent. I’ve updated my implementation to use the intended cartesian product when calculating the best response function, the rest of this comment is based on my initial (wrong) interpretation of B(x).
I write the original expression, and your expression rewritten using the OP’s notation:
(I’m using the notation that a function applied to a set is the image of that set.)
This is a totally clear and valid rewriting using that notation! My background is in programming and I spent a couple minutes trying to figure out how mathematicians write “apply this function to this set.”
I believe the way that B(x) is being used is to find Nash equilibria, using Cleo’s definition 6.5:
Like before, the Ψ-nash equilibria of g is the set of option-profiles x∈X such that x∈B(x).
These are going to be option-profiles where “not deviating” is considered optimal by every player simultaneously. I agree with your conclusion that this leads B(x) to take on values that are either {} or {x}. When B(x)={}, this indicates that x is not a Nash equilibrium. When B(x)={x}, we know that x is a Nash equilibrium.
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.
I’m weirded out by this. To look at everything together, I write the original expression, and your expression rewritten using the OP’s notation:
Original: B:x↦∏i∈Iψi(g∘Ui(x))
Yours: B(x)=⋂i∈I{x(i↦α):α∈ψi(g∘Ui(x))}=⋂i∈IUi(x)(ψi(g∘Ui(x)))
(I’m using the notation that a function applied to a set is the image of that set.)
So the big pi symbol stands for
∏i∈IAi=⋂i∈IUi(x)(Ai)
So it’s not a standalone operator: it’s context-dependent because it pops out an implicit x. The OP otherwise gives the impression of a more functional mindset, so I suspect the OP may mean something different from your guess.
Other problem with your interpretation: it yields the empty set unless all agents consider doing nothing an option. The only possible non-empty output is {x}. Reason: each set you are intersecting contains tuples with all elements equal to the ones in x, but for one. So the intersection will necessarily only contain tuples with all elements equal to those in x.
Edit: Cleo Nardo has confirmed that they intended ∏i∈I to mean the cartesian product of sets, the ordinary thing for that symbol to mean in that context. I misunderstood the semantics of what B(x) was intended to represent. I’ve updated my implementation to use the intended cartesian product when calculating the best response function, the rest of this comment is based on my initial (wrong) interpretation of B(x).
This is a totally clear and valid rewriting using that notation! My background is in programming and I spent a couple minutes trying to figure out how mathematicians write “apply this function to this set.”
I believe the way that B(x) is being used is to find Nash equilibria, using Cleo’s definition 6.5:
These are going to be option-profiles where “not deviating” is considered optimal by every player simultaneously. I agree with your conclusion that this leads B(x) to take on values that are either {} or {x}. When B(x)={}, this indicates that x is not a Nash equilibrium. When B(x)={x}, we know that x is a Nash equilibrium.
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.