Normal-form game

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

A partial topology of two-player, two-strategy games, including such games as Prisoner's dilemma, Stag hunt, and Chicken

Often, symmetric games (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.

Both players Player 2 Player 1 Stag Hare Stag 3, 3 0, 2 Hare 2, 0 2, 2

Just row Player 2 Player 1 Stag Hare Stag 3 0 Hare 2 2

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.

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

${\displaystyle S_{i}=\{1,2,\ldots ,k\}.}$

A pure strategy profile is an association of strategies to players, that is an I-tuple

${\displaystyle {\vec {s}}=(s_{1},s_{2},\ldots ,s_{I})}$

such that

${\displaystyle s_{1}\in S_{1},s_{2}\in S_{2},\ldots ,s_{I}\in S_{I}}$

A payoff function is a function

${\displaystyle u_{i}:S_{1}\times S_{2}\times \ldots \times S_{I}\rightarrow \mathbb {R} .}$

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= {1, 2, ..., I}.

Definition: A game in normal form is a structure

${\displaystyle \mathrm {T} =\langle I,\mathbf {S} ,\mathbf {u} \rangle }$

where:

${\displaystyle I=\{1,2,\ldots ,I\}}$

is a set of players,

${\displaystyle \mathbf {S} =\{S_{1},S_{2},\ldots ,S_{I}\}}$

is an I-tuple of pure strategy sets, one for each player, and

${\displaystyle \mathbf {u} =\{u_{1},u_{2},\ldots ,u_{I}\}}$

is an I-tuple of payoff functions.

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