Editing Glossary of game theory (section)

==Definitions of a game==
=== Notational conventions ===

; [[Real numbers]] : <math> \mathbb{R} </math>.
; The set of '''players''' : <math> \mathrm{N} </math>.
; Strategy space : <math> \Sigma\ = \prod_{i \in \mathrm{N}} \Sigma\ ^i </math>, where
; Player i's strategy space : <math> \Sigma\ ^i </math> is the space of all possible ways in which player '''i''' can play the game.
; A strategy  for player '''i''' : 
<math> \sigma\ _i </math> 
is an element of 
<math> \Sigma\ ^i </math>.
; Complements : 
<math> \sigma\ _{-i} </math> 
an element of <math> \Sigma\ ^{-i} = \prod_{ j \in \mathrm{N}, j \ne i} \Sigma\ ^j </math>, is a tuple of strategies for all players other than '''i'''.
; Outcome space : <math> \Gamma</math> is in most textbooks identical to - 
; Payoffs : <math> \mathbb{R} ^ \mathrm{N} </math>, describing how much [[utility function|gain]] (money, pleasure, etc.) the players are allocated by the end of the game.

===Normal form game===
A game in normal form is a function:
:<math> \pi\ : \prod_{i\in \mathrm{N}} \Sigma\ ^ i \to \mathbb{R}^\mathrm{N}</math>

Given the ''tuple'' of ''strategies'' chosen by the players, one is given an allocation of ''payments'' (given as real numbers).

A further generalization can be achieved by splitting the '''game''' into a composition of two functions: 
:<math> \pi\ : \prod_{i \in \mathrm{N}} \Sigma\ ^i \to \Gamma</math>
the '''outcome function''' of the game (some authors call this function "the game form"), and:
:<math> \nu\ : \Gamma\ \to \mathbb{R}^\mathrm{N} </math>
the allocation of '''payoffs''' (or '''preferences''') to players, for each outcome of the game.

===Extensive form game===
This is given by a [[tree (graph theory)|tree]], where at each [[vertex (graph theory)|vertex]] of the ''tree'' a different player has the choice of choosing an [[graph theory|edge]]. The ''outcome'' set of an extensive form game is usually the set of tree leaves.

===Cooperative game===
A game in which players are allowed to form coalitions (and to enforce coalitionary discipline). A cooperative game is given by stating a ''value'' for every coalition:
:<math> \nu\ : 2^{\mathbb{P}(N)} \to \mathbb{R}</math>
It is always assumed that the empty coalition gains nil. ''Solution concepts'' for cooperative games 
usually assume that the players are forming the ''grand coalition'' <math> N </math>, whose value <math> \nu(N) </math> is then divided among the players to give an allocation.

===Simple game===
 
A Simple game is a simplified form of a cooperative game, where the possible gain is assumed to be either '0' or '1'. A simple game is couple ('''N''', '''W'''), where '''W''' is the list of "winning" '''coalitions''', capable of gaining the loot ('1'), and '''N''' is the set of players.