Balls and bins models with feedback
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
The onset of dominance in balls-in-bins processes with feedback
Random Structures & Algorithms
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In a balls-in-bins process with feedback, balls are sequentiallythrown into bins so that the probability that a bin with nballs obtains the next ball is proportional to f(n)for some function f. A commonly studied case where there aretwo bins and f(n) = np for p 0, and our goal is to study the fine behaviour of this processwith two bins and a large initial number t of balls. Perhapssurprisingly, Brownian Motions are an essential part of both ourproofs.For p 1/2, it was known that with probability 1 oneof the bins will lead the process at all large enough times. Weshow that if the first bin starts witht+λ√t balls (for constantλεR, the probability that it always oreventually leads has a non-trivial limit depending on λ.For p ≤ 1/2, it was known that with probability 1 thebins will alternate in leadership. We show, however, that if theinitial fraction of balls in one of the bins is 1/2, the timeuntil it is overtaken by the remaining bin scales likeθ(t1+1/(1-2p)) for p t)) for p = 1/2. In fact, theovertaking time has a non-trivial distribution around the scalingfactor, which we determine explicitly.Our proofs use a continuous-time embedding of the balls-in-binsprocess (due to Rubin) and a non-standard approximation of theprocess by Brownian Motion. The techniques presented also extend tomore general functions f.