Normal form of play

In game theory , a game in a normal or strategic form ( English normal form ) consists of three elements: the set of players, the set of pure strategies for each player, the set of payment functions of each player. Thus, the game in normal form can be represented as an n-dimensional matrix (table), the elements of which are n-dimensional payment vectors. This table is called the payoff matrix .

Content

Formal Definition

A game in normal form is called a troika $G=\left\langle P,\mathbf {S} ,\mathbf {F} \right\rangle$ ${\ displaystyle G = \ left \ langle P, \ mathbf {S}, \ mathbf {F} \ right \ rangle}$ where

P=\{1,2,\ldots ,m\}

{\ displaystyle P = \ {1,2, \ ldots, m \}}

- many players

\mathbf {S} =\{S_{1},S_{2},\ldots ,S_{m}\}

{\ displaystyle \ mathbf {S} = \ {S_ {1}, S_ {2}, \ ldots, S_ {m} \}}

- many sets of clean strategies for each player,

\mathbf {F} =\{F_{1},F_{2},\ldots ,F_{m}\}

{\ displaystyle \ mathbf {F} = \ {F_ {1}, F_ {2}, \ ldots, F_ {m} \}}

- Many payment functions for each player.

Each player $i\in P$ ${\ displaystyle i \ in P}$ there is a finite set of clean strategies $S_{i}=\{1,2,\ldots ,n_{i}\}$ ${\ displaystyle S_ {i} = \ {1,2, \ ldots, n_ {i} \}}$ and utility function (payment function) $F_{i}:S_{1}\times S_{2}\times \ldots \times S_{m}\rightarrow \mathbb {R}$ ${\ displaystyle F_ {i}: S_ {1} \ times S_ {2} \ times \ ldots \ times S_ {m} \ rightarrow \ mathbb {R}}$ .

The outcome of the game is a combination of each player’s pure strategies:

{\vec {s}}=(s_{1},s_{2},\ldots ,s_{m})

{\ displaystyle {\ vec {s}} = (s_ {1}, s_ {2}, \ ldots, s_ {m})}

Where $s_{1}\in S_{1},s_{2}\in S_{2},\ldots ,s_{m}\in S_{m}$ ${\ displaystyle s_ {1} \ in S_ {1}, s_ {2} \ in S_ {2}, \ ldots, s_ {m} \ in S_ {m}}$ .

Two Players / Two Strategies

	Player 2 L	Player 2 R
Player 1 U	4 , 3	–1 , –1
Player 1 D	0 , 0	3 , 4
Normal form for a game with 2 players, each of which has 2 strategies.

The case of two players - two pure strategies is shown on the table. Pure strategies of the first player: U and D. Pure strategies of the second player: L and R. If the first player chooses U and the second player (at a time) chooses L, then the corresponding payments are 4 and 3 (the first element of the vector (4, 3) means payment of the first player, and the second - payment of the second player if strategies U and L were chosen). That is, to find the distribution of payments corresponding to each set of strategies played, you just need to find the vector located at the intersection of the corresponding rows and columns of the table (the rows correspond to the strategies of the first player, and the columns correspond to the strategies of the second player). The combination of strategies played is called the outcome of the game. In this example, the outcome of the game is (U, L). All possible outcomes for this game: {(U, L), (U, R), (D, L), (D, R)}. Obviously, each cell in the table corresponds to one of the possible outcomes.

Utility Function

In the general case, it is assumed that the player has preferences on a variety of outcomes. That is, for each player, binary relations between the elements of this set are given. This means that the player can compare any two outcomes: the player either prefers one of the two outcomes or remain indifferent between both outcomes. Under certain additional assumptions regarding the preferences of the player, it can be shown that there is a Neumann-Mongenshtern utility function representing the utility of each outcome as a real number u (s), moreover, if u (s) ≥u (s') <=> the player prefers (or is indifferent ) outcome s to outcome s'. In our example, the first player prefers the outcome (U, L) to the outcome (D, R) since 4> 3.

Games with full / incomplete information

In games with full information, the description of the game is known to all players (all players know the pure strategies and utility functions of all other players). In games with incomplete information, some players may not know the utility functions of other players (that is, do not know some specific values for the table cells from our example).

Any game in extensive form can be represented by a game in normal form (not necessarily equivalent). Normal representation of the game can be used to find dominant strategies.

Literature

Petrosyan L.A. , Zenkevich N.A., Semina E.A. Game Theory: Study Guide for Universities. - M .: Higher. school, Book House "University", 1998. - S. 304. - ISBN 5-06-001005-8 , 5-8013-0007-4.