Strategy (game theory)

Under a strategy of a player is understood in game theory a complete plan on how the player will behave in every possible game situation. Through the strategy so the game behavior of a player is fully described.

Examples

Set of strategies, gameplay, normal form

As the strategy the player's engagement fully describes ( eg without dice in a game without random external factors ) are the gameplay and thus the payoffs of each player set, if you know which player plays which strategy. ( In the above example, we know that Player A "stone" and player B plays "paper", so we know that B will win, play the two to one euro, so A and B will lose a euro a payoff of a receive EUR Formal:. 's strategy combination ( "stone", "paper" ) leads to the payoff vector (-1, 1)).

The amount a player of all strategies is called strategy set ( with often abbreviated, whereby the player called ). In the above example, " rock, paper, scissors ", the strategy sets of all players are the same, namely

A game like "scissors, paper, stone " in which all the players again and pulling at the same time, can be formally described by the specification of the strategy sets for each player and the payoff function that determines the payoffs for each strategy combination. If a game is defined in this way, it is called a game in normal form.

Pull the player (such as playing chess ) and not simultaneously such a simple description is often insufficient; then you have to rely on the extensive form. Since all possible reactions of the players must be taken into account, the strategies can be very complicated in such games.

Pure and mixed strategies

Strictly speaking, was previously only pure strategies of the speech, ie of strategies in which each player always clearly opts for a specific action. Frequently, however, games have no equilibria in pure strategies. "Scissors, paper, stone " for example has a ( Nash ) equilibrium in pure strategies: Lay down a player clearly to set an icon (such as " paper " ), the other player would better choose ( here " scissors " ), what the first anticipated and will therefore not set up.

A solution is offered here mixed strategies, in which the player does not set on a pure strategy, but mixes several pure strategies according to a probability distribution. Mixed strategies in the game " scissors, paper, stone " would be (besides, of course, many others) as " choose, stone 'and' scissors ' with probability 1/2" or "select, scissors ', ' stone ' and ' paper ' each with probability 1/ 3 ". If you play " rock, paper, scissors " to a fixed amount of money and the players want to maximize their expected payoff, so an equilibrium is produced in that both players this " third " strategy play. Once one of the players plays the third strategy, it is for the expected payoff no matter what strategy the other player chooses. In contrast, any other strategy the opponent can choose a strategy that provides a better for him than the expected value of third strategy. Conversely, this means for the player, that the exception to the third strategy represents a handicap for him, if it is known to the adversary.

Continuous strategy

If the ( infinite) set of actions ( and thus the strategies) of a player in a game is not countable, it is called continuous strategies. An example could be a game in which two players must choose a number from the real numbers between 0 and 1, with the wins with the greater number. ( 1 To exclude the obvious choice here is one banned in the game. )

In games with continuous strategies, the game is often characterized by so-called reaction functions. The Nash equilibrium (ie the tuple that consists of the best responses of all players ) is determined from the intersection of the reaction curves of the players.

Strategies of nature

Games with non- deterministic elements, so-called games with random trains ( about dice games ), can be understood as strategic games without chance moves, to which chance (nature) participates and in which this plays a mixed strategy itself ( a cube would thus cover the strategy "select any eye number with probability 1/ 6" play ). The "real" players anticipate this strategy of nature in their decisions. In contrast to a "real" players obviously can not be assumed that nature is "strategic", ie behaves rationally.