  Quantum games represent the really 21st century branch of game theory,
tightly linked to the modern development of quantum computing and quantum
technologies. The main accent in these developments so far was made on
stationary or repeated games. In the previous paper of the author the truly
dynamic quantum game theory was initiated with strategies chosen by players in
real time. Since direct continuous observations are known to destroy quantum
evolutions (so-called quantum Zeno paradox) the necessary new ingredient for
quantum dynamic games represented the theory of non-direct observations and the
corresponding quantum filtering. Another remarkable 21st century branch of game
theory represent the so-called mean-field games (MFG), with impressive and ever
growing development.
  In this paper we are merging these two exciting new branches of game theory.
Building a quantum analog of MFGs requires the full reconstruction of its
foundations and methodology, because in $N$-particle quantum evolution
particles are not separated in individual dynamics and the key concept of the
classical MFG theory, the empirical measure defined as the sum of Dirac masses
of the positions of the players, is not applicable in quantum setting.
  As a preliminary result we derive the new nonlinear stochastic Schr\"odinger
equation, as the limit of continuously observed and controlled system of large
number of interacting quantum particles, the result that may have an
independent value. We then show that to a control quantum system of interacting
particles there corresponds a special system of classical interacting particles
with the identical limiting MFG system, defined on an appropriate Riemanian
manifold. Solutions of this system are shown to specify approximate Nash
equilibria for $N$-agent quantum games.
