Exploring cooperation and defection in multi-agent systems via Ising model Hamiltonian

In this work, we model multi-agent decision-making scenarios by combining elements of game theory and statistical mechanics. This is done by utilizing the Ising model. We analyse three distinct cases which involves different interaction setups between players and an external arbiter. Our results show that as punishment increases, the system undergoes sharp phase transitions in magnetization. This transition reflects shifts from defection to cooperation. In case 1, where player A interacts with both the arbiter and player B, cooperation increases smoothly as punishment rises. In case 2, where player B interacts with both the arbiter and player A, a critical transition is observed at high punishment, especially when the sucker’s payoff and temptation are significant. In case 3, where both players are influenced by the arbiter but do not directly interact, cooperation only emerges at high punishment values. These findings provide insight into the role of external influence and punishment in promoting cooperative behaviour in large-scale systems.