Game (game theory)
A game in the sense of game theory is a mathematical model for describing processes in which several actors mutually influence the results of their decision. In contrast to the common meaning of the word game, z. B. Including processes of coordinating radio frequencies in badly arranged rescue missions, but excluding all one-person games. This is related to the fact that game theory has developed from the consideration of certain games into a very general language for strategic conflicts.
Formalization
According to John von Neumann and Oskar Morgenstern , the founders of mathematical game theory, a game is “simply the entirety of all the rules that describe it”. This characterization is universal, once you get the concept of rule of the game from the original context of a board game dissolves and interpreted as a summary of the following:
- The number of players.
- For each score ( called position ) the information about
- whose turn it is
- which moves are available for the player concerned and
- On the basis of which information (e.g. knowledge of his own cards and those that have already been played) he has to make his decision.
- For final positions, who won how much (a player's profit is called the payout ).
- In the case of random moves, how likely the possible outcomes are.
Mathematical models
The rules of a game can be described in terms of a purely mathematical model using mathematical objects ( numbers , quantities and functions ). In addition to the actual parlor games, any number of interactive decision-making processes of an economic nature can be modeled. In contrast to situations that are examined in classical decision theory, at least two decision-makers ( players ) are involved in games.
Émile Borel (1921) and John von Neumann (1928) recognized that all possible decisions that a player may have to make during a game can be combined into a complete plan of action, a so-called ( pure ) strategy . Without restricting a player's options, he can theoretically even be required to determine his strategy secretly at the beginning of the game. In addition, it can make perfect sense for a player not to choose his strategy in a fixed manner, but to “throw it out” at random according to a probability distribution that he has determined - such a procedure is called a mixed strategy .
Game theory essentially knows two forms of mathematical modeling of a game:
- The so-called normal form corresponds to the imaginable organization of a game, in which all players have to select their strategies simultaneously at the beginning, as is usual in the simple game of scissors, rock, paper . A two-person game in normal form can be displayed as a table, although this is only possible realistically for very simple games. Your entries are the payouts ( winnings ) to the players. In a game with random influence, the normal form contains expected values .
- In contrast, in an extensive game, the chronological course of the game is explicitly modeled. The players make their moves at different times and some of them know the moves they made before. Every single move decision and the knowledge of the previous game that the moving player has is modeled by mathematical objects. Random influences are taken into account in the model through probability distributions, according to which a fictitious player makes the relevant move decisions.
Another game model is the subject of combinatorial game theory . This can only be used for special games.
Properties of games
When examining games, the properties described below are decisive:
Zero-sum property
In a zero-sum game , the sum of all payouts is always 0. In the special case of a two-person zero-sum game , one also speaks of matrix games , since the normal form can be represented as a bimatrix : the rows correspond to the (pure) strategies of the first player, the columns to the (pure) strategies of his opponent and the matrix coefficients of the payouts to the first player or the deposits of the second player. The first player's win is equal to the second player's loss.
Perfect information
In games with perfect information , each player always has the previous game event at the time of a decision, i.e. H. the previously made decisions of his teammates as well as the previously made random decisions, fully known.
Examples of games with perfect information are board games such as chess , mill, and backgammon . Counterexamples are card games like skat and poker as well as games with simultaneous moves like rock-paper-scissors .
A special class of random two-person zero-sum games with perfect information are the subject of combinatorial game theory.
Perfect memory
In games with perfect memory, the information that was known to him at the time of a previous decision made by him is still known to each player, also for later decisions.
Games like Skat show that this condition is not inevitable: There the declarer plays against a team that behaves like a single "player" in terms of interests, but whose members only know their own cards and therefore do not have all the information that they need Team partners were known to their previous decisions.
In games with a perfect memory, every player can find a behavior strategy that is equivalent to the expected payouts for every mixed strategy , in which the random influence of the strategy selection is implemented "locally": For this purpose, the player chooses for each move decision that he may make in a game has to meet a probability distribution.
outlook
Game theory tries in particular to characterize rational behavior in games. How these are designed, for example with regard to existence and uniqueness, depends on the properties of the game in question.
Known applications include a. Conception of auctions, e.g. B. of broadcasting and mobile communications licenses .
An example of a concept of an all-round rationality is the so-called Nash equilibrium . In the case of a two-person zero-sum game with perfect information, the associated payout is clearly determined and, in extensive games, can be calculated using the Minimax algorithm .
Web links
Individual evidence
- ↑ John von Neumann, Oskar Morgenstern: Game Theory and Economic Behavior , Würzburg 1961, p. 48.
- ↑ Jörg Bewersdorff : Luck, Logic and Bluff: Mathematics in Play - Methods, Results and Limits , Springer Spectrum , 6th edition 2012, ISBN 978-3-8348-1923-9 , doi : 10.1007 / 978-3-8348-2319 -9 , p. IX.
- ↑ Émile Borel: La théorie du jeu et les équations intégrales à noyau symétrique gauche In: Comptes rendus hebdomadaires des séances de l'Académie des sciences , 173, 1921, pp. 1304-1308 ( online version ).
- ↑ J. v. Neumann: On the theory of parlor games , Mathematische Annalen, 100, 1928, pp. 295-320, doi: 10.1007 / BF01448847 , online (freely accessible) .