Normal form of a game

The normal form of a game , just normal form , referred to in the game theory a representation form of games , which mainly refers to the a priori to strategy amounts of each player and a payoff function is limited as a function of the chosen strategy combinations. This form of representation is best suited to games in which all players define their strategies at the same time and without knowing the choice of the other players.

An alternative is the extensive form of a game , the strength of which lies in the clear presentation of chronological or logical sequences.

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

definition

The normal form of a game is a tuple with the following elements: ${\displaystyle \Gamma =(N,\Sigma ,u)}$

Crowd of players

${\displaystyle N=\{1,2,\ldots ,n\}}$

Strategy room

${\displaystyle \Sigma =\Sigma _{1}\times \Sigma _{2}\times \dotsb \times \Sigma _{n}}$

${\displaystyle \Sigma _{i}}$describes the amount of strategy the player can choose from. ${\displaystyle i}$

Utility function

${\displaystyle u\colon \Sigma \to \mathbb {R} ^{n}}$

Here is the utility function of the player . Depending on their own strategy and the strategy of the other players, the player has a benefit or a payout of . ${\displaystyle u_{i}\colon \Sigma \to \mathbb {R} }$ ${\displaystyle i}$ ${\displaystyle u_{i}(\sigma _{1},\ldots ,\sigma _{i},\ldots ,\sigma _{n})}$

Mixed and pure strategies

In the so-called pure strategies , the players choose precisely . However, for some games, it is necessary to additionally give the possibility to the players randomly select the strategies and previously only the probability distribution over specify with which the individual will be selected. It denotes the parameters of this probability distribution and the set of possible parameter combinations. ${\displaystyle \sigma _{i}\in \Sigma _{i}}$ ${\displaystyle \Sigma _{i}}$ ${\displaystyle \sigma _{i,j}\in \Sigma _{i}}$ ${\displaystyle s_{i}}$ ${\displaystyle S_{i}}$

If finite or countable , then is a vector , where the probability indicates that the strategy is chosen. One speaks of a mixed strategy . ${\displaystyle \Sigma _{i}}$ ${\displaystyle s_{i}}$ ${\displaystyle s_{i,j}}$ ${\displaystyle \sigma _{i,j}}$ ${\displaystyle s_{i}}$

The tuple is the normal form of such a mixed strategy game. The following applies and is the expected benefit . ${\displaystyle S(\Gamma )=(N,S,u)}$ ${\displaystyle S=S_{1}\times S_{2}\times \dotsb \times S_{n}}$ ${\displaystyle u\colon S\to \mathbb {R} ^{n}}$

Presentation in tabular form

Bimatrix ( battle of the sexes )

Only games with two players considered and the strategy sets and finite and manageable, it can be a game in normal form as a table payoff matrix ( Bimatrix ) representing: ${\displaystyle N=\{1,2\}}$ ${\displaystyle \Sigma _{1}}$ ${\displaystyle \Sigma _{2}}$

Player 1 \ Player 2	${\displaystyle \sigma _{2,1}}$	${\displaystyle \sigma _{2,2}}$
${\displaystyle \sigma _{1,1}}$	(3.3)	(1.2)
${\displaystyle \sigma _{1,2}}$	(2.1)	(1.1)

In this case, the first number in brackets denotes the payout of player 1 and the second number the payout of player 2 for the corresponding strategy combination. For example , if player 1 chooses strategy and player 2 , player 1 receives a payout of 2 and player 2 a payout of 1. ${\displaystyle \sigma _{1,2}}$ ${\displaystyle \sigma _{2,1}}$

Individual evidence

1. ↑ Wolfgang Leininger and Erwin Amann: Introduction to game theory. , P. 14 ff.