Minority Game — Market Ecology & Volatility

Minority Game (Challet-Zhang 1997): N (odd) agents repeatedly choose between two rooms (0 or 1); those in the minority win. Each agent uses M bits of history to select from S strategies. The key control parameter is α = 2^M / N. At low α (many agents, short memory): herding dominates, high volatility σ² > N/4 (worse than random). At high α (few agents, long memory): random-like behavior, σ² ≈ N/4. A phase transition at α_c ≈ 0.34 separates these regimes — the minimum volatility point represents the most efficient market. The grand-canonical ensemble (with a fraction of random agents) smooths the transition.