The normal-form solver computes Nash equilibria for strategic interactions where players choose actions simultaneously. You describe the players, their available actions, and the payoffs for every combination of actions — the solver finds the strategies where no player can improve by changing their choice alone.

## Normal-Form Games

A normal-form game (also called a matrix game) is defined by three things:

1. **Players** — the decision-makers (currently two players are supported)
2. **Actions** — the choices available to each player
3. **Payoffs** — a number for each player for every combination of actions

For example, the classic Prisoner's Dilemma:

|  | Cooperate | Defect |
|---|---|---|
| **Cooperate** | -1, -1 | -3, 0 |
| **Defect** | 0, -3 | -2, -2 |

Each cell shows (Row player's payoff, Column player's payoff). The solver finds the equilibrium: both players Defect, earning -2 each — because neither can improve by switching unilaterally.

## What the Solver Returns

For any game you provide, the solver computes:

- **Nash equilibria** — one or more strategy profiles where no player wants to deviate
- **Mixed strategies** — probability distributions over actions when pure strategies aren't stable (as in [Rock-Paper-Scissors](/docs/what-is-an-equilibrium))
- **Expected payoffs** — what each player can expect to earn at equilibrium

## When to Use It

The normal-form solver is the right tool when:

- You have a one-shot strategic decision with simultaneous moves
- You want to find unexploitable strategies for competitive situations
- You need to analyze pricing, bidding, resource allocation, or any scenario where outcomes depend on others' choices

For sequential games with multiple decision points, see [What is a Strategic Context?](/docs/what-is-a-strategic-context) and the game analysis tools.

## See also
- [What is an Equilibrium?](/docs/what-is-an-equilibrium) — the concept behind Nash equilibrium
- [Solving Your First Game](/docs/solving-your-first-game) — hands-on tutorial
- [What are Repeated-Game Policies?](/docs/what-are-repeated-game-policies) — when the same game is played multiple times