  We introduce a mean field model for optimal holding of a representative agent
of her peers as a natural expected scaling limit from the corresponding
$N-$agent model. The induced mean field dynamics appear naturally in a form
which is not covered by standard McKean-Vlasov stochastic differential
equations. We study the corresponding mean field game of mutual holding in the
absence of common noise. Our first main result provides an explicit equilibrium
of this mean field game, defined by a bang--bang control consisting in holding
those competitors with positive drift coefficient of their dynamic value. We
next use this mean field game equilibrium to construct (approximate) Nash
equilibria for the corresponding $N$--player game. We also provide some
numerical illustrations of our mean field game equilibrium which highlight some
unexpected effects induced by our results.
