Extensive-form game

The extensive form called in game theory a representation of the form of games, in contrast to the normal form takes into account the temporal sequence of decisions and this a game tree called tree representation used.

Definition

In the extensive form of a game is a formal mathematical description of a game, with which the possible according to the rules of the game play histories are fully characterized. Specifically, it is the following:

• The number of players.
• Each Score ( position called ) the information about who is on the train,
• Which possible moves for the player in question exist and
• Based on what information (eg, the knowledge of their own and already played cards ), he has to make his decision.

The formalization of the extensive form is based on a mathematical graph, where the node positions and the edges correspond to possible moves. Specifically, this includes formalization

• A tree (which is a connected graph without loops),
• A node that is the root of the tree and the initial position of the game symbol ( and the tree to a directed graph does )
• A lot of players ( including if a fictitious player who "decides" the chance moves ),
• A mapping which assigns to each node a player ( which pulls in this position, that is a permitted train selects )
• For each player a partition of the nodes in which he draws, in amounts of information,
• A mapping which assigns to each terminal node a payoff for each player.

The amount of information each include those positions that are indistinguishable to the pulling player based on the information it currently available - for example, because the previous branching is based on a non- recognizable by the withdrawing player decision of another player within the game tree. All positions an amount of information must therefore contain the same number of possible moves. Within the extensive form the possible moves of all the items an amount of information must be identified in each case consistent ( for example, by numbering). Within a graphical representation of the game tree, the positions of the individual amounts of information are usually clustered together as shown above. Based on this representation, we also speak of information districts.

A game in which all information sets contain only one element is called a game with perfect information. Some authors also speak of perfect information. A withdrawing player knows then, as with most board games usual, always the entire history of the current game. Counter-examples are card games, where players only know each have their own cards. Such games are examples of games with imperfect (or incomplete ) information.

Even a game with imperfect information can have complete information, which means that there is certainty about the rules of the game with the players.

Properties of games and their representation

The difference between the display in the form of extensive and that in normal form is that the extensive form a game is modeled as a series of decisions of the players, while all the decisions are considered as the same time taking place in the normal form.

Sequential structures of games make solutions required beyond the Nash equilibrium. In particular, Nash equilibria may contain threats that are not credible when one takes into account the sequential structure of the game. One way to exclude such equilibria consists in the application of the concept of subgame perfect equilibria.