In game theory, strategic or normal form games are used to describe the effect of a single, simultaneous decision by each participant. Thus a strategic form game is more akin to a single turn in chess than the entire chess match; despite this simplicity many interesting (and famous) problems can be formulated in terms of strategic form.

## Strategic form for two player zero-sum games

**Definition 1**: A two player zero-sum game in strategic (or normal) form X,Y,A consists of two strategy sets X and Y, corresponding to the players, and a function A : X × Y → **R** representing the pay-off. The game is finite if both X and Y are finite sets.

**Definition 2**: A play of the game consists of Player 1 choosing a strategy x ∈ X and Player 2 simultaneously choosing a strategy y ∈ Y . Player 1 is then awarded A(x, y) in winnings, and Player
2 awarded -A(x, y) (i.e, Player 2 loses whatever Player 1 wins; this is the zero-sum condition).

**Definition 3**: The payoff matrix or game matrix for the game with X = {x_{1}, . . . , x_{m}},
Y = {y_{1}, . . . , y_{n}} and payoff function A is given by
a_{11} . . . a_{1n}
. . .
. . .
. . .
a_{m1} . . . a_{mn}

where a_{ij} = A(x_{i}, y_{j}), that is, the (i, j)th entry of A determines the winnings for Player 1 and losses for Player 2 when Player 1 chooses strategy x_{i} ∈ X and Player 2 chooses strategy y_{j} ∈ Y . As a shorthand, we may describe Player 1 as choosing the row and Player 2 as choosing the column.

## Strategic form for two player general sum games

The zero-sum condition is not vital to the strategic form, but its relaxation necessitates an increase in complexity of the notation, since the payoff to Player 2 is no longer immediate from knowledge of the payoff to Player 1.

**Definition 4**: The 2 player general sum game where Player 1 has strategy set X = {x_{1}, . . . , x_{m}}
and Player 2 has strategy set Y = {y_{1}, . . . , y_{n}} such that we can represent Player 1’s payoff A(x, y)
by a matrix A and Player 2’s B(x, y) by a matrix B (both m × n ) is described as being a game in
strategic form X, Y,A,B:
Player 1 Player 2
a_{11} . . . a_{1n} b_{11} . . . b_{1n}
. . . . . .
. . . . . .
. . . . . .
a_{m1} . . . a_{mn} b_{m1} . . . b_{mn}

Thus any zero-sum game in strategic form X,Y,A is a game of strategic form X,Y,A,B where B=-A i.e., (b

_{ij})=(-a

_{ij}) . The more compact

Bimatrix notation is often employed, and this class of games referred to as

Bimatrix games.

## Strategic form for n player general sum games

With some further adjustment of notation, strategic form can accomodate any number of participants.

**Definition 5:** A finite n-player game in strategic form X_{1}, . . . ,X_{n} consists of strategy sets
X_{1}, . . . ,X_{n} corresponding to the players; and real valued functions a_{1} . . . a_{n} : X_{1} × . . . × X_{n} → **R** such that the payoff to player *i* when the strategies chosen by each player *j* is x_{j} ∈ X_{j} is given by a_{i}(x_{1}, . . . , x_{n}).

Of course, bimatrix games then become the special case n=2 (identifying X with X_{1}, Y with X_{2} and the functions A,B with a_{1}, b_{1}. There is no convenient notation for an n player game; further, for large numbers of players individual return may be less interesting than group dynamics, in which case the coalitional form may be a more suitable model.

Part of *A survey of game theory*- see project homenode for details and links to the print version.