In Game theory, a play in normal form is specification of the Space of the strategies and the functions of payment of each player to all the stages possible of the play.

Mathmetic definition

Preliminary definitions

The normal form is employed to describe plays with many blows, of players and of finished strategies.

That is to say thus a unit finished P players indexed by {1, 2,…, m }. Each player K can employ a finished number of strategies pure selected as a whole $S\_k\; =\; \backslash \; \{1,\; 2,\; \backslash \; ldots,\; n\_k\; \backslash \}$.

A profile of strategies is then the m - uplet:

$\backslash \; vec\; \{\backslash \; sigma\}\; =\; (\backslash \; sigma\_1,\; \backslash \; sigma\_2,\; \backslash \; ldots,\; \backslash \; sigma\_m)\; \backslash \; in\; (S\_1\; \backslash \; times$

S_2 \ times \ ldots \ times S_n)

One notes $\backslash \; Sigma$ the whole of the profiles of strategies.

A function of payments is a function F :

$F:\; \backslash \; Sigma\; \backslash \; rightarrow\; \backslash \; mathbb\; \{R\}.$

It interprête like the profit, possibly measured in terms of Utility, for each player and any result of the play.

Play in normal form

The normal form of a play is then the data of

$(P,\; \backslash \; mathbf\; \{S\},\; \backslash \; mathbf\; \{F\})$

with the definitions of the preceding paragraph.

Infinite plays

The definitions given above are also valid for the plays comprising an infinite number of player or strategies possible. However, their study asks for tools of analyzes functional which are not necessary in game theory finished.

Mixed strategies in normal form

It is possible to integrate the possibility of mixed strategies in a play in normal form. It is supposed whereas each player associates with a Probabilité $Pr\_k$ with each element of $S\_k$:

$\backslash \; operatorname\; \{Pr\}\; \_k=\; \{\backslash \; operatorname\; \{Pr\}\; \_k\; (1),\; \backslash \; operatorname\; \{Pr\}\; \_k\; (2),\; \backslash \; ldots,\; \backslash \; operatorname\; \{Pr\}\; \_k\; (n\_k)\}.$

A profile of mixed strategies is then the data of the $\backslash \; operatorname\; \{Pr\}\; \_k,\; K\; \backslash \; in\; \{1,2,\; \backslash \; ldots,\; m\}$.

Space $\backslash \; sigma$ of the profiles of strategies is then one Space probabilized such as:

$\backslash \; operatorname\; \{Pr\}\; (\backslash \; vec\; \{\backslash \; sigma\}\; =\; (\backslash \; sigma\_1,\; \backslash \; sigma\_2,\; \backslash \; ldots,\; \backslash \; sigma\_m))=\; \backslash \; operatorname\; \{Pr\}\; \_1\; (\backslash \; sigma\_1)\; \backslash \; times\; \backslash \; operatorname\; \{Pr\}\; \_2\; (\backslash \; sigma\_2)\; \backslash \; times\; \backslash \; cdots\; \backslash \; times\; \backslash \; operatorname\; \{Pr\}\; \_m\; (\backslash \; sigma\_m)$.

The function of payments is then a Random variable on $(Sigma,\; \backslash \; operatorname\; \{Pr\})$. One then considers the hope of it according to $\backslash \; operatorname\; \{Pr\}$.

Stamp profits

Definition

When there are only two players and a sufficiently restricted number of strategies, it is possible to give the normal form of a play under form of a table with m lines and N columns, where m and N are the number of strategies at the disposal of the player represented respectively in line and column. The boxes of the table are then filled with a doublet giving the payment for each player if it result of the play is the pair of strategies corresponding to the line and to column of the box considered.

Example

Let us consider the play known under the name of Dilemme of the prisoner. Both players are two criminals, heard at the same time, separately one of the other and without possibility of communicating in connection with a crime committed in commun run. Each prisoner can is to deny the crime ( C , to cooperate), that is to say to plead guilty and be used as witness for the prosecution against his accomplice ( D , to deviate). The result of each strategy of many years of prison is as follows:

The first prisoner (Line) can thus choose to cooperate or deviate. In the same way, the second prisoner (Column) can choose between cooperating and deviating. If both cooperate, they bail out one year of prison each one. If they deviate both, they bail out ten years each one. If Line cooperates and that Colonne deviates, Colonne is released, and Line takes twenty years of prison. Conversely, if Line deviates and Column cooperates, Ligne is free and Column takes some for twenty years.

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