Glossary of game theory

Real numbers

${\displaystyle \mathbb {R} }$.

The set of players

${\displaystyle \mathrm {N} }$.

Strategy space

${\displaystyle \Sigma \ =\prod _{i\in \mathrm {N} }\Sigma \ ^{i}}$, where

Player i's strategy space

${\displaystyle \Sigma \ ^{i}}$ is the space of all possible ways in which player i can play the game.

A strategy for player i

${\displaystyle \sigma \ _{i}}$ is an element of ${\displaystyle \Sigma \ ^{i}}$.

Complements

${\displaystyle \sigma \ _{-i}}$ an element of ${\displaystyle \Sigma \ ^{-i}=\prod _{j\in \mathrm {N} ,j\neq i}\Sigma \ ^{j}}$, is a tuple of strategies for all players other than i.

Outcome space

${\displaystyle \Gamma }$ is in most textbooks identical to -

Payoffs

${\displaystyle \mathbb {R} ^{\mathrm {N} }}$, describing how much gain (money, pleasure, etc.) the players are allocated by the end of the game.

A game in normal form is a function:

${\displaystyle \pi \$ :\prod _{i\in \mathrm {N} }\Sigma \ ^{i}\to \mathbb {R} ^{\mathrm {N} }}

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:

${\displaystyle \pi \$ :\prod _{i\in \mathrm {N} }\Sigma \ ^{i}\to \Gamma }

the outcome function of the game (some authors call this function "the game form"), and:

${\displaystyle \nu \$ :\Gamma \ \to \mathbb {R} ^{\mathrm {N} }}

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:

${\displaystyle \nu \ :2^{\mathbb {P} (N)}\to \mathbb {R} }$

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 ${\displaystyle N}$, whose value ${\displaystyle \nu (N)}$ 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.