Convergence of the quantile admission process with veto power

The quantile admission process with veto power is a stochastic process suggested by Alon, Feldman, Mansour, Oren and Tennenholtz as a model for the evolution of an exclusive social club. Each member is represented by a real number (his opinion). On every round two new candidates, holding i.i.d. μ-distributed opinions, apply for admission. 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 is admitted. We show that for any μ and r, the empirical distribution of opinions in the club converges a.s. 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 coupling of the evolution of the empirical r-quantile of the club with a random walk in a changing environment.