🎮 Game Theory

Always Cooperate (green) and Always Defect (red) ignore the opponent. Tit-for-Tat (blue) repeats the neighbour's last move towards it. Pavlov, or Win-Stay Lose-Shift (purple), keeps its move if the last payoff was good and switches if it was poor. Random (amber) cooperates or defects with a 50/50 coin flip each interaction.

In Robert Axelrod's 1980 tournaments Tit-for-Tat beat 62 rival strategies because it is nice (never defects first), retaliatory (punishes defection at once), forgiving (returns to cooperation immediately) and clear. On a grid, clusters of TFT agents shield one another from exploitation, letting cooperation hold a foothold even amid defectors.

In a one-shot game mutual defection is the Nash equilibrium: neither player can gain by switching alone. Mutual cooperation is the Pareto optimum: you cannot improve one agent without harming the other. The tragedy of the dilemma is that rational self-interest drives players to the worse, Nash outcome.

Noise sets a mutation rate, from 0 to 20 per cent. After the imitation step, each agent has that probability of instead adopting a randomly chosen strategy. A little noise keeps the system from freezing into a single absorbing state and lets new strategies be re-seeded so the dynamics stay lively.

Yes. Because agents only interact with eight immediate neighbours rather than the whole population, cooperators can form protective clusters whose interior members all cooperate and score well. This local structure is exactly why cooperation survives spatially when it would be wiped out in a well-mixed population.

All Defectors seeds a sea of defectors with a tiny TFT core to test invasion; TFT Invasion uses a larger cooperative cluster; Pavlov World fills the grid mostly with Pavlov agents; and Mixed starts from a weighted blend of all five strategies. They are quick ways to explore which configurations let cooperation take hold.

The payoff matrix, strategy definitions and Nash and Pareto analysis are faithful to standard theory, and the imitate-best dynamic is a recognised model used by Nowak and May. It is a simplified teaching tool, though: real ecologies involve continuous strategies, memory of many past rounds and reproduction rather than pure copying.

It models any situation where private incentives undermine shared gains: nations emitting carbon despite a collective interest in restraint, rivals locked in arms races, bacteria producing public goods that cheaters exploit, and firms tempted to undercut a mutually profitable price. The simulation hints at how repetition and structure can rescue cooperation.