  Consider a balls-in-bins process in which each new ball goes into a given bin
with probability proportional to f(n), where n is the number of balls currently
in the bin and f is a fixed positive function. It is known that these so-called
balls-in-bins processes with feedback have a monopolistic regime: if f(x)=x^p
for p>1, then there is a finite time after which one of the bins will receive
all incoming balls.
  Our goal in this paper is to quantify the onset of monopoly. We show that the
initial number of balls is large and bin 1 starts with a fraction \alpha>1/2 of
the balls, then with very high probability its share of the total number of
balls never decreases significantly below \alpha. Thus a bin that obtains more
than half of the balls at a "large time" will most likely preserve its position
of leadership. However, the probability that the winning bin has a
non-negligible advantage after n balls are in the system is ~ C n^{1-p}, and
the number of balls in the losing bin has a power-law tail. Similar results
also hold for more general functions f.
