Deterministic dynamics in the minority game

The minority game (MG) behaves as a stochastically disturbed deterministic system due to the coin toss invoked to resolve tied strategies. Averaging over this stochasticity yields a description of the MG's deterministic dynamics via mapping equations for the strategy score and global information. The strategy-score map contains both restoring-force and bias terms, whose magnitudes depend on the game's quenched disorder. Approximate analytical expressions are obtained and the effect of "market impact" is discussed. The global-information map represents a trajectory on a de Bruijn graph. For small quenched disorder, a Eulerian trail represents a stable attractor. It is shown analytically how antipersistence arises. The response to perturbations and different initial conditions is also discussed.