Normal-Form Game

A simultaneous-move representation via strategy sets and payoff functions.

Ask the Game theory assistant1 min read · Updated September 9, 2026

Definition

A tuple $\langle N,\{S_i\}_{i\in N},\{u_i:\prod_j S_j\to\mathbb{R}\}\rangle$ ; two-player finite games are often displayed as a payoff matrix.

Every extensive-form game induces a normal form over strategies (complete contingent plans).

Intuition

All players choose once and simultaneously, without knowing others' choices.

It abstracts away timing and information flow to focus on strategy-level reasoning.

Worked example

The $2\times 2$ Prisoner's Dilemma matrix is the canonical normal form.

Any $n$ -player finite game can be listed as an $n$ -dimensional payoff tensor.

The math

Nash equilibria in the normal form always exist in mixed strategies (Nash, 1950).

The normal form captures strategic equivalence classes coarser than extensive-form equivalence.

Where it is used

The normal form is the default representation for computing Nash equilibria algorithmically: the Lemke-Howson algorithm operates directly on bimatrix payoff tables. Most introductory game theory textbooks use normal-form analysis as their starting point.

In mechanism design, the designer first specifies the normal form — strategy spaces and payoff functions — before analyzing what equilibrium emerges. This abstraction underlies auction design, voting rule analysis, and contract theory.

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Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.