Normal-Form Game¶

Definition¶

A normal-form game (strategic-form game) is a triple

G=\big\langle N,\ \{S_i\}_{i\in N},\ \{v_i\}_{i\in N}\big\rangle,

where $N=\{1,\ldots,n\}$ is a finite set of players, $S_i$ is the nonempty set of pure strategies of player $i$ , and $v_i\colon S\to\mathbb{R}$ is player $i$ 's payoff function on the set of profiles $S=S_1\times\cdots\times S_n$ $S_{n}$ , a Cartesian product.

Payoff matrix. When $n=2$ and $S_1$ , $S_2$ are finite, the game is displayed as a bimatrix: player 1's strategies label the rows, player 2's strategies label the columns, and the cell in row $s_1$ and column $s_2$ contains the pair $v_1(s_1,s_2),\,v_2(s_1,s_2)$ . The first number always belongs to the row player.

Interpretation¶

The normal form lists who plays, what each player can do and what each profile pays. It is the natural description of a static game of complete information; dynamic games, written as extensive-form games, can also be converted to normal form (Week 3), but the timing of moves is then no longer visible.

Prisoner's Dilemma bimatrix with player 1's rows in blue and player 2's columns in red; a blue arrow joins the first entries down column M, a red one the second entries along row F.

The figure labels the parts of the triple in the Prisoner's Dilemma example: the rows are $S_1=\{M,F\}$ ${M, F}$ , the columns are $S_2=\{M,F\}$ ${M, F}$ , and the cell in row $s_1$ and column $s_2$ holds $v_1(s_1,s_2)$ (blue, first) and $v_2(s_1,s_2)$ (red, second); for instance, row $F$ , column $M$ holds $-1,-5$ . The arrows show which numbers each player compares: player 1 fixes the column and compares first entries down it ( $-2$ against $-1$ in column $M$ ), while player 2 fixes the row and compares second entries along it ( $-5$ against $-4$ in row $F$ ).

To read a matrix correctly: player 1 compares first entries down a column (the column fixes $s_2$ ), and player 2 compares second entries along a row (the row fixes $s_1$ ). Comparing the wrong entries is a frequent error when looking for best responses.

For pure-strategy analysis (dominance, best responses, pure Nash equilibrium) only each player's ranking of profiles matters. For mixed strategies (Week 2) the payoff values themselves matter, because players then take expectations.

Key Results¶

A finite two-player game with $m$ rows and $k$ columns is described by $2mk$ numbers; in general a finite game needs $n\cdot|S_1|\cdots|S_n|$ numbers.
Applying any strictly increasing transformation to one player's payoffs leaves strict dominance between pure strategies, best responses to pure strategies and pure Nash equilibria unchanged; a positive affine transformation $\alpha v_i+\beta$ $β$ , $\alpha>0$ $0$ , also leaves the mixed-strategy analysis unchanged.
A two-player game is zero-sum if $v_1(s)+v_2(s)=0$ for every profile $s$ .
A two-player game is symmetric if $S_1=S_2$ and $v_1(a,b)=v_2(b,a)$ for all $a,b$ .
Strategy sets may be infinite, in which case the game is given by formulas rather than a matrix.

Examples¶

Prisoner's Dilemma (one common parametrization; payoffs are minus years in prison):

	$M$	$F$
$M$	$-2,-2$	$-5,-1$
$F$	$-1,-5$	$-4,-4$

Here $v_1(F,M)=-1$ and $v_2(F,M)=-5$ $- 5$ . The game is symmetric.

Battle of the Sexes:

	$O$	$F$
$O$	$2,1$	$0,0$
$F$	$0,0$	$1,2$

Matching Pennies (zero-sum; player 1 wins if the coins match):

	$H$	$T$
$H$	$1,-1$	$-1,1$
$T$	$-1,1$	$1,-1$

Infinite strategy sets. In a Cournot duopoly, $S_i=[0,\infty)$ and $v_i(q_1,q_2)=\big(a-b(q_1+q_2)-c\big)q_i$ as long as the price is nonnegative; there is no matrix.

Prerequisites¶

Sources¶

Game Theory · Week 1

Aliases: Payoff Matrix, strategic-form game, normal form, strategic form, bimatrix game, game in normal form, game in strategic form Keywords: bimatrix, row player, column player, (N, S_i, v_i), Prisoner's Dilemma, Battle of the Sexes, Matching Pennies, zero-sum game Symbols: alpha, beta, element of, infinity, real numbers

Last updated: 17 October 2026 21:53