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Mapping Game Theory to Quantum Systems: Nash Equilibria via Neutral Atom Computing

Giovanni Ferrannini, Dario di Gregorio, Federico Fissore

TL;DR

Simulations show the effectiveness of the correspondence between Maximum Independent Sets and Nash equilibria on unit-disk graphs, and map these problems onto the ground state configurations of Rydberg atom arrays.

Abstract

Nash equilibria are crucial for understanding game behavior and systems in economics, physics, biology, and computer science. A significant application arises from the connection between Nash equilibria and optimization problems . However, finding Nash equilibria is challenging due to its NP-Hard complexity, specifically within the PPAD class. By exploiting the correspondence between Maximum Independent Sets (MIS) and Nash equilibria on unit-disk graphs, we map these problems onto the ground state configurations of Rydberg atom arrays. Simulations show the effectiveness of this quantum method, highlighting its potential for solving complex problems in game theory.

Mapping Game Theory to Quantum Systems: Nash Equilibria via Neutral Atom Computing

TL;DR

Simulations show the effectiveness of the correspondence between Maximum Independent Sets and Nash equilibria on unit-disk graphs, and map these problems onto the ground state configurations of Rydberg atom arrays.

Abstract

Nash equilibria are crucial for understanding game behavior and systems in economics, physics, biology, and computer science. A significant application arises from the connection between Nash equilibria and optimization problems . However, finding Nash equilibria is challenging due to its NP-Hard complexity, specifically within the PPAD class. By exploiting the correspondence between Maximum Independent Sets (MIS) and Nash equilibria on unit-disk graphs, we map these problems onto the ground state configurations of Rydberg atom arrays. Simulations show the effectiveness of this quantum method, highlighting its potential for solving complex problems in game theory.

Paper Structure

This paper contains 15 sections, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics definitions in Game Theory
  4. Public Goods in Networks Game
  5. Pay-off in the model
  6. Basics of Neutral atom quantum computing
  7. Hamiltonian and parameters
  8. Rydberg Blockade
  9. MIS and Ground State correspondence
  10. Analysis and Nash Equilibrium search
  11. Groundstates and Nash Equilibrium correspondence
  12. Simulations
  13. Mappable Network Games
  14. Computational time
  15. Conclusions

Figures (5)

  • Figure 1: Unit-disk graph embeddings for the two instances considered in the simulations. The disks of radius $R_b$ enforce the independence constraint corresponding to the Rydberg blockade.
  • Figure 2: Nash equilibria for Graph A (a) and B (b) obtained through classical simulation. Blue ones are free riders, red dots are contributing agents.
  • Figure 3: Linear profiles used in Bloqade simulation
  • Figure 4: Bloqade simulation for 1000 shots, zoomed in on the most frequently observed configurations of graph A (a) and graphs B (b, c, d, e). The full line in the histogram corresponds to the configuration shown on the right. Black dots represent excited atoms, which form an MIS. All the displayed configurations are ground states of their respective systems.
  • Figure : Pictorial abstract

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7