  Imagine that there are two bins to which balls are added sequentially, and
each incoming ball joins a bin with probability proportional to the p-th power
of the number of balls already there. A general result says that if p>1/2,
there almost surely is some bin that will have more balls than the other at all
large enough times, a property that we call eventual leadership.
  In this paper, we compute the asymptotics of the probability that bin 1
eventually leads when the total initial number of balls $t$ is large and bin 1
has a fraction \alpha<1/2 of the balls; in fact, this probability is
\exp(c_p(\alpha)t + O{t^{2/3}}) for some smooth, strictly negative function
c_p. Moreover, we show that conditioned on this unlikely event, the fraction of
balls in the first bin can be well-approximated by the solution to a certain
ordinary differential equation.
