Normal form and extensive form

There are two commonly-found kinds of games in game theory: games in normal form and games in extensive form.

Games in normal form were formalized in the 1920s by Borel and Von Neumann. A game in normal form, also called strategic game, is basically a matrix (or a tensor if more than two players). Each cell in the matrix corresponds to an outcome -- and an outcome has a payoff for each player, as explained previously.

In the case of two players, each row corresponds to a possible choice (called "strategy") of the row player (here, Alice), and each column corresponds to a possible choice of the column player (here, Bob). So, imagine that Alice and Bob are sitting in different rooms with no means of communication. They each make their choice separately. Then, based on their two decisions, the corresponding outcome (at the crossing of the row chosen by Alice and the column chosen by Bob) is taken and the payoffs distributed to them. For example, in the game shown below (which is the Prisoner's Dilemma), if Alice Cooperates and Bob Defects, then Alice gets $0 and Bob gets $3.

A game in normal form can also have three players, in which case the outcomes would be organized as a cube, or more, which is more difficult to visualize, but formally possible.

A game in normal form: the prisoner's dilemma

A second kind of game is a game in extensive form. This form dates back to Kuhn in the 1950s. A game of extensive form is basically a tree, which each leaf being an outcome. Thus, each leaf contains payoffs for each player. In a game in extensive form, the players take turn. The game starts at the root (here, Bob) who can go left or right. Then, the next player decides, and so on. The tree indicates which decisions are available after other decisions have been made. Like games in normal form, there can be any number of players (and as many payoffs labelling each outcome).

A game in extensive form with two players

Each outcome is associated with the path leading to it. For example, outcome (2,7) is reached with Bob going left, then Alice going left, then Bob going right. Instead of just "left" and "right", one can also use labels, and there can be more than two possibilities (called "actions") that an agent can select from at each node.