In game theory, normal form is a way of describing a game. Unlike extensive form, normal form representations are not graphical per se, but rather represents the game with a matrix. This can be of greater use in identifying strictly dominated strategies and Nash equilibria, on the other hand some information is lost as compared to extensive form representations. The normal form representation of game includes all strategies of each player and payoffs for each strategy profile are represented. Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ... It has been suggested that Game tree be merged into this article or section. ... Look up matrix in Wiktionary, the free dictionary. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ... In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ...

In static games of complete, perfect information, a normal form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of all strategies available to that player, where a strategy is a complete plan of action for every stage of the game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from that the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility - often cardinal in the normal form representation) of a player, i.e. the payoff function of a player takes as its input a strategy profile (that is a specification of strategies for every player) and yields a representation of payoff as its output. Complete information is a term used in economics and game theory to describe a economic situation or game in which knowledge about other market participants or players is available to all participants and is instantaneously updated as new information arises. ... Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ...

An example

The matrix to the right is a normal form representation of a game in which players move simultaneously (or at least do not observe the other player's move before making their own) and receive the payoffs as specified for the combinations of actions played. For example, if player 1 plays top and player 2 plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player (in this case player 2).

Other representations

Often symmetric games (where the payoffs do not depend on which player chooses each action) are represented with only one payoff. This is the payoff for the row player. For example, the payoff matrices on the right and left below represent the same game. In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. ...

Uses of normal form

Dominated strategies

The payoff matrix facilitates elimination of dominated strategies, and it is usually used to illustrate this concept. For example, in the prisoner's dilemma (to the right), one can determine that Cooperate is strictly dominated by Defect. One must compare the first numbers in each column, in this case 3>2 and 1>0. This shows that no matter what the column player chooses, the row player does better by choosing Defect. Similarly, one compares the second payoff in each row; again 3>2 and 1>0. This shows that no matter what row does, column does better by choosing Defect. This demonstrates the unique Nash equilibrium of this game is (Defect, Defect). In game theory, dominance (also called strategic dominance) occurs when one strategy is better than another strategy for one player, no matter how that players opponents may play. ... Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? Many points in this article may be difficult to understand without a background in the elementary concepts of game theory. ... In game theory, the Nash equilibrium (named after John Nash, who proposed it) is a kind of solution concepts of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. ...

Sequential games in normal form

These matrices only represent games in which moves are simultaneous (or, more generally, information is imperfect). The above matrix does not represent the game in which player 1 moves first, observed by player 2 and then player 2 moves because it does not specify each of player 2's strategies in this case. In order to represent this sequential game we must specify all of player 2's actions, even in contingencies that can never arise in the course of the game. In this game, player 2 has actions, as before, Left and Right. Unlike before he has four strategies, contingent on player 1's actions. The strategies are: Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ... In game theory, a sequential game is a game where one player chooses his action before the other chooses hers. ...

1. Left if player 1 plays Top and Left otherwise
2. Left if player plays Up and Right otherwise
3. Right if player plays Up and Left otherwise
4. Right if player plays Up and Right otherwise

On the right is the normal form representation of this game.

General Formulation

In order for a game to be in normal form, we are provided the following data:

• There is a finite set P of players, which we label {1, 2, ..., m}

• Each player k in P has a finite number of pure strategies

A pure strategy profile is an association of strategies to players, that is an m-tuple A pure strategy is a term used to refer to strategies in Game theory. ... In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. ...

such that

We will denote the set of strategy profiles by Σ

A payoff function is a function

whose intended interpretation is the award given to a single player at the outcome of the game. Accordingly, to completely specify a game, the payoff function has to be specified for each player in the player set P= {1, 2, ..., m}.

Definition. A game in normal form is a structure

where P = {1,2, ...,m} is a set of players,

is an m-tuple of pure strategy sets, one for each player, and

is an m-tuple of payoff functions.

There is no reason in the previous discussion to exclude games which have an infinite number of players or an infinite number of strategies per player. The study of infinite games is more difficult however, since it requires use of functional analytic techniques. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

References

• R. D. Luce and H. Raiffa, Games and Decisions, Dover Publications, 1989.

• J. Weibull, Evolutionary Game Theory, MIT Press, 1996

• J. von Neumann and O. Morgenstern, Theory of games and Economic Behavior, John Wiley Science Editions, 1964. This book was initially published by Princeton University Press in 1944.

External Links

• http://www.whalens.org/Sofia/choice/matrix.htm