Balls-in-bins models with asymmetric feedback and reflection

TL;DR

This paper investigates asymmetric feedback in balls-in-bins models, analyzing long-term behavior and probabilities of dominance, with results extending previous symmetric feedback models and including a classification of the process as a reflecting random walk.

Contribution

It introduces and analyzes asymmetric feedback functions in balls-in-bins models, providing new probabilistic results and a classification of the process as a reflecting random walk.

Findings

Single bin almost surely receives all but finitely many balls under superlinear feedback.

Normal approximation for the probability of a bin dominating, generalizing symmetric cases.

Complete classification of the long-term behavior of the asymmetric feedback model.

Abstract

Balls-in-bins models describe a random sequential allocation of infinitely many balls into a finite number of bins. In these models a ball is placed into a bin with probability proportional to a given function (feedback function), which depends on the number of existing balls in the bin. Typically, the feedback function is the same for all bins (symmetric feedback), and there are no constraints on the number of balls in the bins. In this paper we study versions of BB models with two bins, in which the above assumptions are violated. In the first model of interest the feedback functions can depend on a bin (BB model with asymmetric feedback). In the case when both feedback functions are power law and superlinear, a single bin receives all but finitely many balls almost surely, and we study the probability that this happens for a given bin. In particular, under certain initial conditions we derive the normal approximation for this probability, which generalizes the result in [5] obtained in the case of the symmetric feedback. The main part of the paper concerns the BB model with asymmetric feedback evolving subject to certain constraints on the numbers of allocated balls. The model can be interpreted as a transient reflecting random walk in a curvilinear wedge, and we obtain a complete classification of its long term behavior.