Convergence of the Quantile Admission Process with Veto Power

TL;DR

This paper proves that the opinion distribution in a stochastic quantile admission model with veto power converges to a limit, which can be deterministic or random, depending on initial conditions and parameters.

Contribution

It establishes convergence of the opinion distribution in the model and analyzes the nature of the limit, extending understanding of such stochastic processes.

Findings

Empirical opinion distribution converges for all initial distributions and parameters.

The limit distribution can be non-deterministic under certain conditions.

Conditions for the limit to be deterministic are characterized.

Abstract

The quantile admission process with veto power is a stochastic processes suggested by Alon, Feldman, Mansour, Oren and Tennenholtz as a model for the evolution of an exclusive social group. The model itself consists of a growing multiset of real numbers, representing the opinions of the members of the club. On each round two new candidates, holding i.i.d. $\mu$-distributed opinions, apply for admission to the club. The one whose opinion is minimal is then admitted if the percentage of current members closer in their opinion to his is at least $r$. Otherwise neither of the candidates is admitted. We show that for any $\mu$ and $r$, the empirical distribution of opinions in the club converges to a limit distribution. We further analyse this limit, show that it may be non-deterministic and provide conditions under which it is deterministic. The results rely on a recent work of the authors relating tail probabilities of mean and maximum of any pair of unbounded i.i.d. random variables, and on a coupling of the evolution of the empirical $r$-quantile of the club with a random walk in a changing environment.