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 = {x1, . . . , xm},
Y = {y1, . . . , yn} and payoff function A is given by
a11 . . . a1n
. . .
. . .
. . .
am1 . . . amn
where aij = A(xi, yj), 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 xi ∈ X and Player 2 chooses strategy yj ∈ 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 = {x1, . . . , xm}
and Player 2 has strategy set Y = {y1, . . . , yn} 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
a11 . . . a1n b11 . . . b1n
. . . . . .
. . . . . .
. . . . . .
am1 . . . amn bm1 . . . bmn
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 X1, . . . ,Xn consists of strategy sets
X1, . . . ,Xn corresponding to the players; and real valued functions a1 . . . an : X1 × . . . × Xn → R such that the payoff to player i when the strategies chosen by each player j is xj ∈ Xj is given by ai(x1, . . . , xn).
Of course, bimatrix games then become the special case n=2 (identifying X with X1, Y with X2 and the functions A,B with a1, b1. 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.