  We investigate the percolation properties of a planar reinforced network
model. In this model, at every time step, every vertex chooses $k \ge 1$
incident edges, whose weight is then increased by 1. The choice of this
$k$-tuple occurs proportionally to the product of the corresponding edge
weights raised to some power $\alpha > 0$. Our investigations are guided by the
conjecture that the set of infinitely reinforced edges percolates for $k = 2$
and $\alpha \gg 1$. First, we study the case $\alpha = \infty$, where we show
the percolation for $k = 2$ after adding arbitrarily sparse independent
sprinkling and also allowing dual connectivities. We also derive a finite-size
criterion for percolation without sprinkling. Then, we extend this finite-size
criterion to the $\alpha < \infty$ case. Finally, we verify these conditions
numerically.
