Normal-form game

The normal form called in game theory a representation of the form of games, which relate primarily to the a priori strategy amounts of the individual players and a payoff function is limited as a function of the chosen strategy combinations. Accessible, this form of representation most such games in which all players for their strategies simultaneously and without knowing the choice of the other players.

An alternative is the extensive form, whose strength lies in the ideological representation of temporal or logical sequences.

The normal form of games was first described by Émile Borel (1921) and John von Neumann ( 1928) described, who realized that in principle any strategy game can be transformed into such a form.

Definition

The normal form of the game is a tuple with the following elements:

Mixed and pure strategies

In the so-called pure strategies, players choose one carefully. However, for some games it is necessary, in addition to the players should be allowed to randomly select the strategies and previously only specify the probability distribution over, with which the individual will be selected. This means the parameters of the probability distribution and the set of possible parameter combinations.

Is finite or countable, so is a vector, which indicates the probability that the strategy is chosen. One speaks in a mixed strategy.

The tuple is the normal form of such a game in mixed strategies. Where, and the expected benefit.

Representation in tabular form

If only games with two players, and are considered the strategy amounts of finite and manageable, you can represent a game in normal form as a table, the payoff matrix:

In this case, the first number in the parenthesis denotes the payoff of player 1 and the second number is the payoff of player 2 at the corresponding strategy combination. Selects player 1, for example, strategy and player 2, so both will each receive a disbursement 3

• Game theory