Avoiding defeat in a balls-in-bins process with feedback

Imagine that there are two bins to which balls are added sequentially, and each incoming ball joins a bin with probability proportional to the pth 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 α < 1/2 of the balls; in fact, this probability is exp(cp(α)t + O ( t ) ) for some smooth, strictly negative function cp. 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.