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In
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
, normal form is a description of a ''game''. Unlike
extensive form An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, th ...
, normal-form representations are not graphical ''per se'', but rather represent the game by way of a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. While this approach can be of greater use in identifying strictly dominated strategies and
Nash equilibria In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equi ...
, some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable
strategies Strategy (from Greek στρατηγία ''stratēgia'', "art of troop leader; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the " ar ...
, and their corresponding payoffs, for each player. In static games of complete,
perfect information In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market p ...
, 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, whereas 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 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 Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **'' Cardinalis'', genus of cardinal in the family Cardinalidae **'' Cardinalis cardinalis'', or northern cardinal, t ...
or
ordinal utility In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to as ...
—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.


An example

The matrix provided 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 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. If one can change the identities of the players without changing the payoff to ...
s (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. The topological space of games with related payoff matrices can also be mapped, with adjacent games having the most similar matrices. This shows how incremental incentive changes can change the game.


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 The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
, we can see that each prisoner can either "cooperate" or "defect". If exactly one prisoner defects, he gets off easily and the other prisoner is locked up for a long time. However, if they both defect, they will both be locked up for a shorter time. One can determine that ''Cooperate'' is strictly dominated by ''Defect''. One must compare the first numbers in each column, in this case 0 > −1 and −2 > −5. 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 0 > −1 and −2 > −5. This shows that no matter what row does, column does better by choosing ''Defect''. This demonstrates the unique
Nash equilibrium In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equ ...
of this game is (''Defect'', ''Defect'').


Sequential games in normal form

These matrices only represent games in which moves are simultaneous (or, more generally, information is
imperfect The imperfect ( abbreviated ) is a verb form that combines past tense (reference to a past time) and imperfective aspect (reference to a continuing or repeated event or state). It can have meanings similar to the English "was walking" or "used to ...
). 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 In game theory, a sequential game is a game where one player chooses their action before the others choose theirs. The other players must have information on the first player's choice so that the difference in time has no strategic effect. Seque ...
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: # ''Left if player 1 plays Top and Left otherwise'' # ''Left if player 1 plays Top and Right otherwise'' # ''Right if player 1 plays Top and Left otherwise'' # ''Right if player 1 plays Top 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 with the following data: There is a finite set ''I'' of players, each player is denoted by ''i''. Each player ''i'' has a finite ''k'' number of pure strategies :: S_i = \. A is an association of strategies to players, that is an ''I''-
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
: \vec = (s_1, s_2, \ldots,s_I) such that : s_1 \in S_1, s_2 \in S_2, \ldots, s_I \in S_I A is a function : u_i: S_1 \times S_2 \times \ldots \times S_I \rightarrow \mathbb. 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 ''I''= . Definition: A ''game in normal form'' is a structure : \Tau=\langle I, \mathbf, \mathbf\rangle where: :I=\ is a set of players, :\mathbf= \ is an ''I''-tuple of pure strategy sets, one for each player, and : \mathbf = \ is an ''I''-tuple of payoff functions.


References

* * . An 88-page mathematical introduction
free online
at many universities. * * . A comprehensive reference from a computational perspective; see Chapter 3

* * J. von Neumann and O. Morgenstern, ''Theory of games and Economic Behavior'', John Wiley Science Editions, 1964. Which was originally published in 1944 by Princeton University Press. {{Matrix classes Game theory game classes