  We show that, for a continuous set of entangled four-partite states, the task
of maximizing the payoff in the symmetric-strategy four-player quantum Minority
game is equivalent to maximizing the violation of a four-particle Bell
inequality with each observer choosing the same set of two dichotomic
observables. We conclude the existence of direct correspondences between (i)
the payoff rule and Bell inequalities, and (ii) the strategy and the choice of
measured observables in evaluating these Bell inequalities. We also show that
such a correspondence between Bell polynomials (in a single plane) and
four-player, symmetric, binary-choice quantum games is unique to the
four-player quantum Minority game and its "anti-Minority" version. This
indicates that the four-player Minority game not only plays a special role
among quantum games but also in studies of Bell-type quantum nonlocality.
