normal-form game

Definition

Lists what payoffs players get as function of their actions. Suitable representation of a zero-sum game when each player's information is static in nature, as it suppresses all dynamic aspects of the decision problem.

finite $n$-player normal form game $\langle N, A, u \rangle$

• Player (index) set: $N = \{1,2,...,n\}$
• Action set $A_i$ for player $i$: $a = (a_1, a_2, ..., a_n) \in A = A_1 \times A_2 \times ... \times A_n$ is an action profile
• Utility function (or payoff function) for player $i$: $u_i : A \mapsto \mathbb{R}$, $u = (u_1, u_2, ..., u_n)$ is a profile of utility

$N$-person finite static game in normal form

• players make decisions independently and each one unilaterally seeks the minimum possible loss, taking into account possible rational choices of the other players

Matrix description

Games in strategic form are sometimes are sometimes called matrix games due to being described by matrices. This is an $n$-dimensional matrix when there are $n \geq 2$ players. Each cell contains a vector with $n$ coordinates, containing payoffs to the $n$ players. Named bimatrix game when $n=2$, as there are two payoff matrices, one for each player.

Compare

• extensive form game (tree)

References

1. Quanyuan Zhu, Lecture 2 of ECE-GY 6263 Game Theory, Fall 2019. https://bpb-us-e1.wpmucdn.com/wp.nyu.edu/dist/5/2123/files/2019/12/Lecture-2-Scribe.pdf
2. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 77-78.
3. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999, p. 88.