Nash Equilibrium in Quantum Games

This is a brute force method similar to gradient descent method inorder to find Nash Equilibrium points in random quantum games.

Prerequisite

This is a MATLAB program. This program requires QETLAB (Quantum Entanglement Theory LABoratory) which is a MATLAB toolbox for exploring quantum entanglement theory.

To install QETLAB, vist this page.

Running the code

• First run the file PartialTraceModified.m: This is a modified version of the inbuilt PartialTrace function included in QETLAB. The modification allows us to calculate the partial trace of symbolic matrices.

• Next run the file generate_random_game.m: This file is use to generate a random quantum game. This file will have two inputs:

  • Number of strategies available for player A
  • Number of strategies available for player B

• Run the file find_equilibrium.m: This file will run the brute force algorithm to find the equilibrium for the random quantum game generated in the previous step. The important parameters in this file are:

  • Set linear_update_method = true to use the Linear update method and set linear_update_method = false to use the matrix exponential update method
  • Set total_iterations to a desired value. Current value is total_iterations = 1500
  • Set weight to desired value. The recommended weight in linear update method is in the range 0.1 to 10. The recommended weight in matrix exponential update method is in the range 1 to 15.
  • Set the error tolerance by changing parameter epsilon.

• Run the file brute_force_check_equilibrium: This is a brute force method which checks if player A's response is indeed the best response to Player B's strategy and vice-versa, by using random density matrices.