  We present a dynamical theory of a multi-agent market game, the so-called
Minority Game (MG), based on crowds and anticrowds. The time-averaged version
of the dynamical equations provides a quantitatively accurate, yet intuitively
simple, explanation for the variation of the standard deviation (`volatility')
in MG-like games. We demonstrate this for the basic MG, and the MG with
stochastic strategies. The time-dependent equations themselves reproduce the
essential dynamics of the MG.
