Edit: Cleo Nardo has confirmed that they intended to mean the cartesian product of sets, the ordinary thing for that symbol to mean in that context. I misunderstood the semantics of what 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 .
I write the original expression, and your expression rewritten using the OP’s notation:
Original:
Yours:
(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 is being used is to find Nash equilibria, using Cleo’s definition 6.5:
Like before, the -nash equilibria of is the set of option-profiles such that .
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 to take on values that are either or . When , this indicates that is not a Nash equilibrium. When , we know that is a Nash equilibrium.
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 my initial (wrong) interpretation of B(x).
I needed to go back to one of the papers cited in Part 1 to understand what that ∏i∈I was doing in that expression. I found the answer in A Generalization of Nash’s Theorem with Higher-Order Functionals. I’m going to do my best to paraphrase Hedges’ notation into Cleo’s notation, to avoid confusion.
TLDR: B(x) is picking out the set of option-profiles P(X) that are simultaneously best-responses by all players to that option-profile x. It does this by considering all of the option-profiles that can result by each player best-responding, then takes the intersection of those sets.
On page 6, Hedges defines the best response correspondence B∈X→P(X)
B(x)=⋂i∈IBi(x)
Where
Bi∈X→P(X)
Hedges builds up the idea of Nash Equilibria using quantifiers rather than optimizers, (like max rather than argmax), but I believe the approaches are equivalent. Unpacking B:x↦∏i∈Iψi(g∘Ui(x)) from the inside out:
Ui(x)∈Xi→X
g∘Ui(x)∈Xi→R
That makes g∘Ui(x) a ψi-task. Since ψi∈(Xi→R)→P(Xi), we know that ψi(g∘Ui(x))∈P(Xi).
This is where I had to go looking through papers. What sort of product takes a set of best-responses from each player, relative to a given option-profile, and returns a set of option-profiles that are simultaneously regarded by each player as a best-response? I thought about just taking the Cartesian product of the sets, but that wouldn’t get us only the mutual best-responses.
Let’s call the way that each player maps option-profiles to best-responses bi∈X→P(Xi). This is exactly the sets we want to take the product of:
bi(x)=ψi(g∘Ui(x))
Hedges introduces notation on page 3 to handle the operation of taking an option-profile, varying one player’s option, and leaving the rest the same. Paraphrasing, Hedges defines x(i↦α)∈∏j∈IXj by
x(i↦α)j={αif i=jxjotherwise
You can read x(i↦α) as “give me a new copy of x, where the ith entry has been set to the value α.” Hedges uses this to define the deviation maps equivalently to the way Cleo did. Ui:X→(Xi→X)
Ui(x)(α)=x(i↦α)
The correspondences Bi∈X→P(X) take as input an option profile, and returns the set of option-profiles which are player i’s optimal unilateral deviations from that option profile. To construct Bi from bi, we want to map bi(x)∈P(Xi) to the option-profiles which deviate from x in those exact ways.
Bi(x)={x(i↦α):α∈bi(x)}
We can then use Hedges’ B(x)=⋂i∈IBi(x) to get the best-response correspondence! We can unpack this to get a definition of B using objects that Cleo defined, using that deviation notation from Hedges:
B(x)=⋂i∈I{x(i↦α):α∈ψi(g∘Ui(x))}
Thank you Cleo for writing this article! This was my first introduction to Higher-Order Game Theory, and I wrote up an implementation in TypeScript to help me understand how all of the pieces fit together!