Chapter summary

In this chapter we introduce a graphic way of describing a game, the description in extensive form, which depicts the rules of the game, the order in which the players make their moves, the information available to players when they are called to take an action, the termination rules, and the outcome at any terminal point. A game in extensive form is given by a game tree, which consists of a directed graph in which the set of vertices represents positions in the game, and a distinguished vertex, called the root, represents the starting position of the game. A vertex with no outgoing edges represents a terminal position in which play ends. To each terminal vertex corresponds an outcome that is realized when the play terminates at that vertex. Any nonterminal vertex represents either a chance move (e.g., a toss of a die or a shuffle of a deck of cards) or a move of one of the players. To any chance-move vertex corresponds a probability distribution over the edges emanating from that vertex, which correspond to the possible outcomes of the chance move.

To describe games with imperfect information, in which players do not necessarily know the full board position (like poker), we introduce the notion of information sets. An information set of a player is a set of decision vertices of the player that are indistinguishable by him given his information at that stage of the game.