The Minority Game (MG) behaves as a stochastically perturbed 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' discussed. The
global-information map represents a trajectory on a De Bruijn graph. For small
quenched disorder, an Eulerian trail represents a stable attractor. It is shown
analytically how anti-persistence arises. The response to perturbations and
different initial conditions are also discussed.Comment: 16 pages, 5 figure
