Avoiding defeat in a balls-in-bins process with feedback
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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 α<1/2 of the balls; in fact, this probability is \exp(c_p(α)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.
30 pages; to be submitted. V.2 has some minor corrections
30 pages; to be submitted. V.2 has some minor corrections