Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.
is an element of .
an element of , is a tuple of strategies for all players other than i.
A game in normal form is a function:
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:
the outcome function of the game (some authors call this function "the game form"), and:
the allocation of payoffs (or preferences) to players, for each outcome of the game.
This is given by a tree, where at each vertex of the tree a different player has the choice of choosing an edge. The outcome set of an extensive form game is usually the set of tree leaves.
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:
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 , whose value is then divided among the players to give an allocation.
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.