strategic form

A strategic form of a game in game theory is generally a presentation in a table like the following:

                                          Coop          Defect
                                    +--------------+--------------+
                                    |              |              |
                              Coop  |   5  ,  5    |   -10 , 10   |
                                    |              |              |
                                    +--------------+--------------+
                                    |              |              |
                             Defect |   10 , -10   |   -5 ,  -5   |
                                    |              |              |
                                    +--------------+--------------+

That’s one version of the classic Prisoner’s Dilemma game in strategic form, also called normal form, where “Coop” means cooperate. The row player chooses either the top row (cooperate) or the bottom row (defect). The column player chooses either the left or right column.

The outcome of the game is the cell of the table selected by both the row and column chosen. The payoff in utility to the row player is the first number in the outcome cell, and the payoff to the column player is the second number.

The strategic form is used for games which are:

• static, in that each player makes their move simultaneously, and only makes one move; and
• finite, in that there's a finite number of moves per player, and a finite number of players.

For dynamic games, the extensive form is used.

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, . . . , x_m}, Y = {y₁, . . . , y_n} and payoff function A is given by

a11 . . . a1n
.  .      .
.     .   .
.       . .
am1 . . . amn

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 = {x₁, . . . , x_m} and Player 2 has strategy set Y = {y₁, . . . , 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
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, . . . ,X_n consists of strategy sets X₁, . . . ,X_n corresponding to the players; and real valued functions a₁ . . . a_n : X₁ × . . . × 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(x₁, . . . , x_n).

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.

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