🔍 Nash Equilibrium Finder

This repository contains a Python script and a Jupyter Notebook for calculating the Nash Equilibrium in two-player games based on their payoff matrix. The Nash Equilibrium is a key concept in game theory that describes a stable state where no player can benefit by unilaterally changing their strategy.

What is a Nash Equilibrium?

In game theory, the Nash Equilibrium is a central concept that describes a combination of strategies in which no player can improve their outcome by changing their strategy, provided the other players keep their strategies unchanged. It represents a stable state in which the strategies of all players are optimal given the strategies of the others.

Example: Payoff Matrix

Consider the following payoff matrix for two players:

Player 1 / Player 2	Strategy A	Strategy B
Strategy X	(3, 2)	(1, 4)
Strategy Y	(2, 1)	(4, 3)

Here:

• The first value in each cell represents Player 1's payoff.
• The second value in each cell represents Player 2's payoff.

Nash Equilibrium: In this example, the pair of strategies (Player 1: Strategy Y, Player 2: Strategy B) forms a Nash Equilibrium because neither player can improve their payoff by unilaterally changing their strategy.

Code Overview

The Python script includes:

• best_response_player1(matrix): Computes the best responses for Player 1.
• best_response_player2(matrix): Computes the best responses for Player 2.
• strict_equilibrium(matrix): Identifies all Nash Equilibria in pure strategies.

Usage

1. Clone this repository:

   git clone https://github.com/alexanderk001/nash-equilibrium.git

2. Modify the matrix variable in the script to test different scenarios.

3. Run the script with your own payoff matrix:

   python nash.py

See the Jupyter Notebook for further explanations

Example Input

The following is an example payoff matrix defined in the script:

matrix = [
    [(3, 3), (1, 4)],
    [(4, 1), (2, 2)]
]

The combination of strategy 2 (Player 1) and strategy 2 (Player 2) is a pure Nash equilibrium with payoffs (2, 2).

This example is a so-called Prisoner's dilemma.

License

This project is licensed under the MIT License.