Nash Equilibria in Static Games: Agent-Based Simulation on Complex Networks

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Paper: Influence of complex network topologies on nash equilibria in static games under dynamic simulations

Volume 82, article number 397, (2026) Published: 05 May 2026

Cite this article :

Shafaati, S., Azimi, M., Mohammadzadeh, J. et al. Influence of complex network topologies on nash equilibria in static games under dynamic simulations. J Supercomput 82, 397 (2026). https://doi.org/10.1007/s11227-026-08421-9

DOI:https://doi.org/10.1007/s11227-026-08421-9

Full version: https://rdcu.be/fg6oG

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This project simulates the evolution of strategies in 2-player static games using agent-based modeling on various types of complex networks. It explores how network topology and different strategy update rules influence the convergence toward Nash equilibria.

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📌 Objectives

• Model strategic behavior of agents playing classic games (e.g., Prisoner's Dilemma) on networks.
• Examine how different update mechanisms (e.g., replicator dynamics, imitation) affect strategy evolution.
• Evaluate how network types (e.g., scale-free, small-world, community-based) influence the emergence and stability of Nash equilibria.

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🧠 Core Concepts

• Nash Equilibrium: A state where no agent can improve its payoff by unilaterally changing strategy.
• Agent-Based Modeling: Each node in the network is an agent with an evolving strategy.
• Evolutionary Game Theory: Agents adapt strategies based on payoff-based learning or imitation.
• Complex Networks: Topologies used include:
  • Erdos-Renyi (random)
  • Barabasi-Albert (scale-free)
  • Watts-Strogatz (small-world)
  • Complete graph
  • Homogeneous (regular)
  • LFR benchmark (community-rich)

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📂 Structure

🔧 Game Definitions

Classic 2-player games are defined using payoff matrices:

• Prisoner's Dilemma
• Hawk-Dove
• Stag Hunt
• Public Goods Game (function-based)

🌐 Network Generation

Creates networks using networkx, supporting:

• Random graphs
• Scale-free networks
• Small-world networks
• Modular/community-based graphs via LFR

🔁 Strategy Update Mechanisms

Implements various strategy dynamics:

• Static: No change in strategy
• Replicator: Strategies reproduced in proportion to payoffs
• Unconditional Imitation: Agents copy the highest earner
• Moran: Reproducer and replacer selected probabilistically
• Fermi Rule: Random comparisons with payoff-based imitation

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📊 Metrics & Analysis (Expected / To Be Expanded)

• Proportion of strategies over time
• Average payoff and system-wide fitness
• Community cohesion and internal/external cooperation
• Nash Equilibrium detection (by verifying strategy stability)
• Convergence patterns (e.g., consensus, coexistence, oscillation)

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🖼️ Visualization

Plots include:

• Evolution of strategy proportions
• Network diagrams showing agent strategies
• Community-based heatmaps
• Stability over time

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📦 Requirements

Install required Python packages:

pip install numpy pandas networkx matplotlib seaborn