# Normal-Form Game

A normal-form game is a game where each player only makes a single choice, like RPS.

We define a normal-form game as a tuple $(N,A,u)$ where:

- $N={1,…,n}$ is a finite set of $n$ players
- $A=S_{1}×S_{2}×⋯×S_{n}$ is the set of all possible combinations of simultaneous actions, where $S_{i}$ is the finite set of actions for player $i$
- $u$ is a vector of utilities/payoffs/rewards that we define

Two types of strategies:

**Pure strategy**(deterministic policy): Chooses a single action with probability 1.**Mixed strategy**($σ$): At least two actions played with positive probability- $σ_{i}(s)$ is the probability that player $i$ chooses action $s∈S_{i}$

Other convention: $−i$ refers to player $i$’s opponents

To find the optimal strategy to win a normal-form game, we use Regret Matching to find the Regret Matching to find the Nash Equilibrium.