### Abstract

We consider the asymptotic behavior of the following model: balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = n^{p} for p > 1. In this case, one of the two bins eventually obtains a monopoly, in the sense that it obtains all balls thrown past some point. This model is motivated by the phenomenon of positive feedback, where the "rich get richer." We derive a simple asymptotic expression for the probability that bin 1 obtains a monopoly when bin 1 starts with x balls and bin 2 starts with y balls for the case f(n) = n^{p}. We then demonstrate the effectiveness of this approximation with some examples and demonstrate how it generalizes to a wide class of functions f.

Original language | English (US) |
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Journal | Electronic Journal of Combinatorics |

Volume | 11 |

Issue number | 1 R |

DOIs | |

State | Published - Apr 13 2004 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Electronic Journal of Combinatorics*,

*11*(1 R). https://doi.org/10.37236/1784