Game Theory: Basic Concepts
High-rationality solution concepts in game theory can
emerge in a world populated by low-rationality agents.
The philosophers kick up the dust and then complain that
they cannot see.
An extensive form game G consists of a number οι players, a game tree, and a set οf payoffs. A game tree consists of a number of nodes connected by branches. Each branch connects a head node to a distinct tail node. If b is a branch of the game tree, we denote the head node of b by bh, and the tail node of b by bt.
A path from node a to node a' in the game tree is a connected sequence of branches starting at a and ending at a'.1 If there is a path from node a to node a', we say a is an ancestor of a', and a' is a successor to a. We call the number of branches between a and a' the length of the path. If a path from a to a' has length 1, we call a the parent of a', and a' is a child of a.
We require that the game tree have a unique node r, called the root node, that has no parent, and a set Τ of nodes, called terminal nodes or leaf nodes, that have no children. We associate with each terminal node t e Τ (e means “is an element of”), and each player i, a payoff πi (t) ∈ R (R is the set of real numbers). We say the game is finite if it has a finite number of nodes. We assume all games are finite unless otherwise stated.
We also require that the graph of G have the following tree property. There must be exactly one path from the root node to any given terminal
1Technically, a path is a sequence b1, …, bk of branches such that
for i = 1,…, k − 1, and ; i.e., the path starts at a, the tail of each branch is the
head of the next branch, and the path ends at a'. The length of the path is k.