# Perfect-information game in extensive form definition

I'm writing a paper and I need to define what a perfect-information game in extensive form is. My paper includes material from Game Theory and Reinforcement Learning. Since the notation of both fields is different, I changed that definition a bit, but I don't know if this is correct.

A (finite) perfect-information game (in extensive form) is a tuple $G = (\mathcal{N}, \mathcal{A}, \mathcal{H}, \mathcal{Z}, \chi, \rho, \sigma, u)$, where:

• $\mathcal{N}$ is a set of $n$ players;

• $\mathcal{A}$ is a (single) set of actions;

• $\mathcal{H}$ is a set of nonterminal choice nodes;

• $\mathcal{Z}$ is a set of terminal nodes, disjoint from $\mathcal{H}$;

• $\chi : \mathcal{H} \mapsto 2^A$ is the action function, which assigns to each choice node a set of possible actions;

• $\rho : \mathcal{H} \mapsto \mathcal{N}$ is the player function, which assigns to each nonterminal node a player $i \in \mathcal{N}$ who chooses an action at that node;

• $\sigma : \mathcal{H} \times \mathcal{A} \mapsto \mathcal{H} \cup \mathcal{Z}$ is the successor function, which maps a choice node and an action to a new choice node or terminal node such that for all $h_1, h_2 \in H$ and $a_1, a_2 \in A$, if $\sigma(h_1, a_1) = \sigma(h_2, a_2)$ then $h_1 = h_2$ and $a_1 = a_2$; and

• $u = (u_1,\dots,u_n)$, where $u_i : Z \mapsto \mathbb{R}$ is a real-valued utility function for player $i$ on the terminal nodes $Z$.

In Reinforcement Learning the set of all states is $\mathcal{S}$, the set of actions $\mathcal{A}$ and the successor function $\mathcal{A(s)}$.

I wanted to know if it makes sense to change the Game Theory definition to:

• $\mathcal{N(\cdot)}$ for the player function.
• $\mathcal{A(\cdot)}$ for the successor function.
• $\mathcal{S^T}$ for the set of terminal nodes/states.
• $\mathcal{S^N}$ for the set of nonterminal nodes/states.

Any ideas? Suggestions?