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Perfect information game – action function

i can’t uderstand the activation function in the perfect-information game definition given by the book i’m studying on.

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A (finite) perfect-information game (in extensive form) is a tuple G=(N, A, H, Z, χ, ρ, σ, u) where:

  • N is a set of n players
  • A is a (single) set of actions
  • Z is a set of terminal nodes, disjoint from H
  • χ : H → 2^A is the action function, wich assigns to each choice node a set of possible actions
  • ρ : H → N is the player function, wich assigns to each non terminal node a player i ∈ N who chooses an action at that node
  • σ : H x A is the successor function, which maps a choice node and an action
    to a new choice node or terminal node such that for all h1, h2 ∈ H and a1, a2 ∈ A, if
    σ(h1, a1) = σ(h2, a2) then h1 = h2 and a1 = a2
  • u = (u1,…, un), where ui : Z → R is a real-valued utility function for player i on the
    terminal nodes Z

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The problem that i have is that i do not understand, first, how the set of actions A is defined. In a normal form game it is the product A = (A1 x A2 x …. x An) where each Ai is the set of all possible actions to player i and this is pretty straight forward. But i cannot understand what "A is a (single) set of actions" means.
Second the action function χ. I do not understand why it goes form H to 2^A. What is the codomain 2^A made up of?

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