extensive form game

Definition (two person zero-sum finite game)

An extensive form of a two-person zero-sum finite game without chance moves is a finite tree structure with

a specific vertex indicating the starting point of the game
a payoff function assigning a real number to each terminal vertex of the tree, determining the payoff (or respectively, loss) to P2 (respectively, P1)
partition of the nodes of the tree into two player sets, $\overline{N}^1$ and $\overline{N}^2$ for P1 and P2 respectively
subpartition of each player set into information sets $\{\eta_j^i\}$ , such that the same number of intermediate branches emanates from every node belonging to the same information set, and no node follows another node in the same information set

(convention: P1 minimizer, P2 maximizer)

an equivalent normal form exists

A game in extensive form (or extensive-form game) is an ordered vector

\Gamma = (N, V, E, x^0, (V_i)_{i \in N}, O, u)

where

A game in extensive form (with chance moves and with imperfect information) is a vector

\Gamma = (N, V, E, x^0, (V_i)_{i \in N}, (p_x)_{x \in V_0}, (U_i^j)_{i \in N}^{j = 1,...,k_i}, O, u)

where, in addition to the variables above,

$p_x$ is a probability distribution over the set of edges from $x$ ,
$(U_i^j)^{j = 1,...,k_i}$ is a partition of $V_i$ for a player $i \in N$
the pair $(U_i^j, A(U_i^j))$ is an information set of player $i$ for each player $i \in N$ and $j \in \{1,2,...,k_i\}$
Here, each element $U_i^j$ in the partition $(U_i^j)^{j = 1,...,k_i}$ of $V_i$ is an information set of player $i$ .

may be described by directed graph $G = (V,E)$
finite games may be described by finite graphs, infinite games by infinite graphs
measure theory may be used to describe infinite games