  The aim of this paper is to study the asymptotic behavior of strongly
reinforced interacting urns with partial memory sharing. The reinforcement
mechanism considered is as follows: draw at each step and for each urn a white
or black ball from either all the urns combined (with probability $p$) or the
urn alone (with probability $1-p$) and add a new ball of the same color to this
urn. The probability of drawing a ball of a certain color is proportional to
$w_k$ where $k$ is the number of balls of this color. The higher the $p$, the
more memory is shared between the urns. The main results can be informally
stated as follows: in the exponential case $w_k=\rho^k$, if $p\geq 1/2$ then
all the urns draw the same color after a finite time, and if $p<1/2$ then some
urns fixate on a unique color and others keep drawing both black and white
balls.
